Numerical mathematics and advanced applications ENUMATH 2017 9783319964140, 9783319964157

Citation preview

126

Florin Adrian Radu · Kundan Kumar Inga Berre · Jan Martin Nordbotten Iuliu Sorin Pop Editors

Numerical Mathematics and Advanced Applications ENUMATH 2017 Editorial Board T. J.Barth M.Griebel D.E.Keyes R.M.Nieminen D.Roose T.Schlick

Lecture Notes in Computational Science and Engineering Editors: Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick

126

More information about this series at http://www.springer.com/series/3527

Florin Adrian Radu • Kundan Kumar • Inga Berre • Jan Martin Nordbotten • Iuliu Sorin Pop Editors

Numerical Mathematics and Advanced Applications ENUMATH 2017

123

Editors Florin Adrian Radu University of Bergen Bergen, Norway

Kundan Kumar University of Bergen Bergen, Norway

Inga Berre University of Bergen Bergen, Norway

Jan Martin Nordbotten University of Bergen Bergen, Norway

Iuliu Sorin Pop Hasselt University Diepenbeek, Belgium

ISSN 1439-7358 ISSN 2197-7100 (electronic) Lecture Notes in Computational Science and Engineering ISBN 978-3-319-96414-0 ISBN 978-3-319-96415-7 (eBook) https://doi.org/10.1007/978-3-319-96415-7 Library of Congress Control Number: 2018964409 Mathematics Subject Classification (2010): 90-08, 35-XX, 65-XX, 68U20, 65M60, 68Uxx, 65M08 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: Image reprinted with kind permission from Lene Sælen EuropeMap image: Image reprinted with kind permission from © zentilia/Fotolia This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) is a series of conferences held every two years, providing a forum for discussion of basic aspects and new trends in numerical mathematics and challenging scientific and industrial applications. The first ENUMATH was held in Paris (1995). Subsequent ENUMATH conferences were held at the universities of Heidelberg (1997), Jyvaskyla (1999), Ischia Porto (2001), Prague (2003), Santiago de Compostela (2005), Graz (2007), Uppsala (2009), Leicester (2011), Lausanne (2013), and Ankara (2015). Since 1995, ENUMATH has become one of the most established and recognised series of conferences on numerical mathematics and scientific computing. This volume contains 98 proceedings papers from ENUMATH 2017, which was held in Voss, Norway, 25–29 September, 2017. The contributions are based on a selection of invited plenary lectures, mini-symposia and contributed talks. The conference attracted 334 scientists from 128 institutions worldwide. There were 11 plenary talks, 230 mini-symposium talks and 73 contributed talks, all of excellent scientific standard. The importance of computational algorithms and their development is wellrecognised in science and engineering. Computational mathematics is now regarded as a technology in its own right and plays a crucial role in industrial and research activities. This book excellently reflects that. It contains recent results and new trends in the designing, implementation and analysis of numerical algorithms as well as their applications to pressing scientific and industrial problems. Along with the theoretical developments, the book presents numerical methods for applications of high societal relevance within, for example, water, climate, energy, life sciences, technology and finance. ENUMATH 2017 was a successful conference, which was made possible by a number of people. First of all, we thank all the participants for attending the conference, for having given interesting presentations and for the enthusiastic discussions during the conference. A special thanks goes to the mini-symposia organisers, who attracted numerous presentations of high quality. Furthermore, we would like to thank the scientific committee of the conference and the 11 invited v

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Preface

plenary speakers: V. Girault (Paris, France), B. Kaltenbacher (Klagenfuert, Austria), K-A. Mardal (Oslo, Norway), F. Nobile (Austin, USA), R. Nochetto (Maryland, USA), A. Patera (MIT, USA), I. S. Pop (Hasselt, Belgium), M. Rognes (Oslo, Norway), R. Scheichl (Bath, UK), L. da Veiga (Milan, Italy) and A. Zanna (Bergen, Norway). They all made ENUMATH 2017 a great event, a spring for new ideas and further collaborations. Additionally, we would like to mention our sincere gratitude to the programme committee members: F. Brezzi, M. Feistauer, R. Glowinski, G. Kreiss, Y. Kuznetsov, P. Neittaanmaki, J. Periaux, A. Quarteroni, R. Rannacher, E. Süli and B. Wohlmuth (chair). The organisers are grateful for the generous financial support from the Norwegian Research Council (NFR, grant no. 26321) and from the University of Bergen (UiB). Finally, we would like to thank the Department of Mathematics of UiB and especially the Porous Media Group for their help in organising the conference. We are especially grateful to J. Both for providing nearly all the technical support, from the design of the conference poster to the webpage and the final programme of the conference. Special thanks to I. Gjerde and S. Haugsbø for taking care of most of the organisational problems. Additionally, we thank H. Dahle, H. Munthe-Kass and I. Bjørstad for being part of the local organisation committee of the ENUMATH 2017. Last but not least, we would like to thank all the authors for their scientific contributions, the reviewers for their valuable work and Springer for publishing this book. Bergen, Norway Bergen, Norway Bergen, Norway Bergen, Norway Diepenbeek, Belgium May 2018

Florin Adrian Radu Kundan Kumar Inga Berre Jan Martin Nordbotten Iuliu Sorin Pop

Organisation

Organisation Committee of ENUMATH 2017

Florin Adrian Radu Kundan Kumar Inga Berre Helge Dahle Hans Munthe-Kaas Peter Bjorstad

University of Bergen, Norway University of Bergen, Norway University of Bergen, Norway University of Bergen, Norway University of Bergen, Norway University of Bergen, Norway

Programme Committee

Franco Brezzi Miloslav Feistauer Roland Glowinski Gunilla Kreiss Yuri Kuznetsov Pekka Neittaanmaki Jacques Periaux Alfio Quarteroni Rolf Rannacher Endre Süli Barbara Wohlmuth

Universitè˘a degli studi di Pavia, Italy Universitas Carolina Pragensis, Czech Republic University of Houston, USA Uppsala Universitet, Sweden University of Houston, USA University of Jyvaskula, Finland Dassault Aviation St. Cloud, France EPFL, Lausanne, Switzerland Ruprecht-Karls-Universität Heidelberg, Germany University of Oxford, UK Technische Universität Munich, Germany

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Contents

Part I

Plenary Lectures

PDE Apps for Acoustic Ducts: A Parametrized Component-to-System Model-Order-Reduction Approach . . . . . . . . . . . . . . . Jonas Ballani, Phuong Huynh, David Knezevic, Loi Nguyen, and Anthony T. Patera Sub-voxel Perfusion Modeling in Terms of Coupled 3d-1d Problem.. . . . Karl Erik Holter, Miroslav Kuchta, and Kent-André Mardal Iterative Linearisation Schemes for Doubly Degenerate Parabolic Equations . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Jakub Wiktor Both, Kundan Kumar, Jan Martin Nordbotten, Iuliu Sorin Pop, and Florin Adrian Radu Mathematics and Medicine: How Mathematics, Modelling and Simulations Can Lead to Better Diagnosis and Treatments .. . . . . . . . . . . . . . Erik A. Hanson, Erlend Hodneland, Rolf J. Lorentzen, Geir Nævdal, Jan M. Nordbotten, Ove Sævareid, and Antonella Zanna Part II

3

35

49

65

Kernel Methods for Large Scale Problems: Algorithms and Applications

Convergence of Multilevel Stationary Gaussian Convolution .. . . . . . . . . . . . Simon Hubbert and Jeremy Levesley Anisotropic Weights for RBF-PU Interpolation with Subdomains of Variable Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . R. Cavoretto, A. De Rossi, G. E. Fasshauer, M. J. McCourt, and E. Perracchione Radial Basis Function Approximation Method for Pricing of Basket Options Under Jump Diffusion Model. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Ali Safdari-Vaighani

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103

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Contents

Greedy Algorithms for Matrix-Valued Kernels. . . . . . . . . . .. . . . . . . . . . . . . . . . . . Dominik Wittwar and Bernard Haasdonk

113

GPU Optimization of Large-Scale Eigenvalue Solver .. . .. . . . . . . . . . . . . . . . . . Pavel K˚us, Hermann Lederer, and Andreas Marek

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Part III

Advanced Discretization Methods for Computational Wave Propagation

On the Efficiency of the Peaceman–Rachford ADI-dG Method for Wave-Type Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Marlis Hochbruck and Jonas Köhler

135

Trefftz-Discontinuous Galerkin Approach for Solving Elastodynamic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Hélène Barucq, Henri Calandra, Julien Diaz, and Elvira Shishenina

145

Part IV

Unfitted Finite Element Methods: Analysis and Applications

FETI-DP Preconditioners for the Virtual Element Method on General 2D Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Daniele Prada, Silvia Bertoluzza, Micol Pennacchio, and Marco Livesu

157

Modeling Flow and Transport in Fractured Media by a Hybrid Finite Volume: Finite Element Method . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Alexey Chernyshenko and Maxim Olshanskii

165

A Cut Cell Hybrid High-Order Method for Elliptic Problems with Curved Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Erik Burman and Alexandre Ern

173

A Cut Finite Element Method with Boundary Value Correction for the Incompressible Stokes Equations . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Erik Burman, Peter Hansbo, and Mats G. Larson

183

Numerical Integration on Hyperrectangles in Isoparametric Unfitted Finite Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Fabian Heimann and Christoph Lehrenfeld

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Part V

Advances in Numerical Linear Algebra Methods and Applications to Partial Differential Equations

On a Generalization of Neumann Series of Bessel Functions Using Hessenberg Matrices and Matrix Exponentials . . . . . . . . . .. . . . . . . . . . . . . . . . . . A. Koskela and E. Jarlebring

205

Influence of the SIPG Penalisation on the Numerical Properties of Linear Systems for Elastic Wave Propagation . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Uwe Köcher

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Function-Based Algebraic Multigrid Method for the 3D Poisson Problem on Structured Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Ali Dorostkar Part VI

Numerical Methods in Biophysics

Mathematical Modelling of Phenotypic Selection Within Solid Tumours . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Mark A. J. Chaplain, Tommaso Lorenzi, Alexander Lorz, and Chandrasekhar Venkataraman Uncertainty Assessment of a Hybrid Cell-Continuum Based Model for Wound Contraction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Fred Vermolen Part VII

225

237

247

Structure Preserving Discretizations and High Order Finite Elements for Differential Forms

The Discrete Relations Between Fields and Potentials with High Order Whitney Forms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Ana M. Alonso Rodríguez and Francesca Rapetti

259

Model Order Reduction of an Elastic Body Under Large Rigid Motion .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Ashish Bhatt, Jörg Fehr, and Bernard Haasdonk

269

On Surface Area and Length Preserving Flows of Closed Curves on a Given Surface .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Miroslav Koláˇr, Michal Beneš, and Daniel Ševˇcoviˇc

279

Derivation of Higher-Order Terms in FFT-Based Numerical Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Felix Dietrich, Dennis Merkert, and Bernd Simeon

289

Part VIII

Monge-Ampère Solvers with Applications to Illumination Optics

A Least-Squares Method for a Monge-Ampère Equation with Non-quadratic Cost Function Applied to Optical Design.. . . . . . . . . . . . . . . . . N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman Solving Inverse Illumination Problems with Liouville’s Equation . . . . . . . Bart S. van Lith, Jan H. M. ten Thije Boonkamp, and Wilbert L. IJzerman Part IX

301 311

Mixed and Nonsmooth Methods in Numerical Solid Mechanics

Strong vs. Weak Symmetry in Stress-Based Mixed Finite Element Methods for Linear Elasticity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Bernhard Kober and Gerhard Starke

323

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Contents

Evolution of Load-Bearing Structures with Phase Field Modeling. . . . . . . Ingo Muench An Accelerated Newton Method for Nonlinear Materials in Structure Mechanics and Fluid Mechanics . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Thomas Richter and Carolin Mehlmann Part X

335

345

A Posteriori Error Estimation, Adaptivity and Approximation

Best Error Localizations for Piecewise Polynomial Approximation of Gradients, Functions and Functionals . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Andreas Veeser

357

Adaptive Discontinuous Galerkin Methods for Flow in Porous Media . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Birane Kane, Robert Klöfkorn, and Andreas Dedner

367

An Adaptive E-Scheme for Conservation Laws . . . . . . . . . .. . . . . . . . . . . . . . . . . . Ebise A. Abdi, Christian V. Hansen, and H. Joachim Schroll

379

Adaptive Filtered Schemes for First Order Hamilton-Jacobi Equations . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Maurizio Falcone, Giulio Paolucci, and Silvia Tozza

389

Goal-Oriented a Posteriori Error Estimates in Nearly Incompressible Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Dustin Kumor and Andreas Rademacher

399

Nitsche’s Method for the Obstacle Problem of Clamped Kirchhoff Plates . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Tom Gustafsson, Rolf Stenberg, and Juha Videman

407

Part XI

Noncommutative Stochastic Differential Equations: Analysis and Simulation

Stochastic B-Series and Order Conditions for Exponential Integrators . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Alemayehu Adugna Arara, Kristian Debrabant, and Anne Kværnø

419

What Is a Post-Lie Algebra and Why Is It Useful in Geometric Integration . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Charles Curry, Kurusch Ebrahimi-Fard, and Hans Munthe-Kaas

429

On Non-commutative Stochastic Exponentials . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Charles Curry, Kurusch Ebrahimi-Fard, and Frédéric Patras

439

Contents

Part XII

xiii

Innovative Numerical Methods and Their Analysis for Elliptic and Parabolic PDEs

An Introduction to the Gradient Discretisation Method.. . . . . . . . . . . . . . . . . . Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, and Raphaèle Herbin

451

Quasi-Optimal Nonconforming Methods for Second-Order Problems on Domains with Non-Lipschitz Boundary .. . .. . . . . . . . . . . . . . . . . . Andreas Veeser and Pietro Zanotti

461

Convergence of Adaptive Finite Element Methods with Error-Dominated Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Christian Kreuzer and Andreas Veeser

471

Finite Element Methods for Parabolic Problems with Time-Dependent Domains: Application to a Milling Simulation . . . . . . . . . Carsten Niebuhr and Alfred Schmidt

481

Part XIII

Polyhedral Methods and Applications

Numerical Investigation of the Conditioning for Plane Wave Discontinuous Galerkin Methods . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Scott Congreve, Joscha Gedicke, and Ilaria Perugia

493

The Virtual Element Method for the Transport of Passive Scalars in Discrete Fracture Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . S. Berrone, M. F. Benedetto, Andrea Borio, S. Pieraccini, and S. Scialò

501

On the Implementation of a Multiscale Hybrid High-Order Method .. . . Matteo Cicuttin, Alexandre Ern, and Simon Lemaire VEM for the Reissner-Mindlin Plate Based on the MITC Approach: The Element of Degree 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Claudia Chinosi Part XIV

509

519

Recent Advances in Space-Time Galerkin Methods

DGM for the Solution of Nonlinear Dynamic Elasticity .. . . . . . . . . . . . . . . . . . Miloslav Feistauer, Martin Hadrava, Jaromír Horáˇcek, and Adam Kosík

531

Higher Order Space-Time Elements for a Non-linear Biot Model . . . . . . . Manuel Borregales and Florin Adrian Radu

541

Iterative Coupling of Mixed and Discontinuous Galerkin Methods for Poroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Markus Bause

551

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Contents

Stability of Higher-Order ALE-STDGM for Nonlinear Problems in Time-Dependent Domains.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Monika Balázsová and Miloslav Vlasák Part XV

561

PDE Software Frameworks

Implementation of Mixed-Dimensional Models for Flow in Fractured Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Eirik Keilegavlen, Alessio Fumagalli, Runar Berge, and Ivar Stefansson

573

Fast Matrix-Free Evaluation of Hybridizable Discontinuous Galerkin Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Martin Kronbichler, Katharina Kormann, and Wolfgang A. Wall

581

Part XVI

Numerical Methods for Simulating Processes in Porous Media

Numerical Benchmarking for 3D Multiphase Flow: New Results for a Rising Bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Stefan Turek, Otto Mierka, and Kathrin Bäumler

593

A Linear Domain Decomposition Method for Two-Phase Flow in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . David Seus, Florin A. Radu, and Christian Rohde

603

A Numerical Method for an Inverse Problem Arising in Two-Phase Fluid Flow Transport Through a Homogeneous Porous Medium . . . . . . . . Aníbal Coronel, Richard Lagos, Pep Mulet, and Mauricio Sepúlveda

615

A Fully-Implicit, Iterative Scheme for the Simulation of Two-Phase Flow in Porous Media.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Anna Kvashchuk and Florin Adrian Radu

625

Mass Lumping for MHFEM in Two Phase Flow Problems in Porous Media . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Jakub Solovský and Radek Fuˇcík

635

Uncertainty Quantification in Injection and Soil Characteristics for Biot’s Poroelasticity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Menel Rahrah and Fred Vermolen

645

Reactivation of Fractures in Subsurface Reservoirs—A Numerical Approach Using a Static-Dynamic Friction Model . . . . . . .. . . . . . . . . . . . . . . . . . Runar L. Berge, Inga Berre, and Eirik Keilegavlen

653

Part XVII

Model Reduction Methods for Simulation and (Optimal)Control

POD-Based Economic Model Predictive Control for Heat-Convection Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Luca Mechelli and Stefan Volkwein

663

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xv

Real-Time Optimization of Thermal Ablation Cancer Treatments . . . . . . Zoi Tokoutsi, Martin Grepl, Karen Veroy, Marco Baragona, and Ralph Maessen

673

Parametric Model Reduction via Interpolating Orthonormal Bases . . . . . Ralf Zimmermann and Kristian Debrabant

683

A Spectral Element Reduced Basis Method in Parametric CFD . . . . . . . . . Martin W. Hess and Gianluigi Rozza

693

POD for Optimal Control of the Cahn-Hilliard System Using Spatially Adapted Snapshots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Carmen Gräßle, Michael Hinze, and Nicolas Scharmacher Part XVIII

703

Recent Advances on Polyhedral Discretizations

New Strategies for the Simulation of the Flow in Three Dimensional Poro-Fractured Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Stefano Berrone, Andrea Borio, Sandra Pieraccini, and Stefano Scialò

715

The Virtual Element Method on Anisotropic Polygonal Discretizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Paola F. Antonietti, Stefano Berrone, Marco Verani, and Steffen Weißer

725

Part XIX

FEM Meshes with Guaranteed Geometric Properties

On Zlámal Minimum Angle Condition for the Longest-Edge n-Section Algorithm with n ≥ 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Sergey Korotov, Ángel Plaza, José P. Suárez, and Tania Moreno

737

Adaptive Solution of a Singularly-Perturbed Convection-Diffusion Problem Using a Stabilized Mixed Finite Element Method . . . . . . . . . . . . . . . María González and Magdalena Strugaru

743

Spaces of Simplicial Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Jon Eivind Vatne Simplicial Vertex-Normal Duality with Applications to Well-Centered Simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Jan Brandts and Michal Kˇrížek Maximum Angle Condition for n-Dimensional Simplicial Elements . . . . . Antti Hannukainen, Sergey Korotov, and Michal Kˇrížek Part XX

753

761 769

Discretizations and Solvers for Multi-Physics Problems

An Oscillation-Free Finite Volume Method for Poroelasticity . . . . . . . . . . . . Massimiliano Ferronato, Herminio T. Honorio, Carlo Janna, and Clovis R. Maliska

779

xvi

Contents

Numerical Investigation on the Fixed-Stress Splitting Scheme for Biot’s Equations: Optimality of the Tuning Parameter ... . . . . . . . . . . . . . . . . . Jakub W. Both and Uwe Köcher Numerical Simulation of Biofilm Formation in a Microchannel .. . . . . . . . . David Landa-Marbán, Iuliu Sorin Pop, Kundan Kumar, and Florin A. Radu Numerical Methods for Biomembranes Based on Piecewise Linear Surfaces . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . John P. Brogan, Yilin Yang, and Thomas P.-Y. Yu Heavy Metals Phytoremediation: First Mathematical Modelling Results . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Aurea Martínez, Lino J. Alvarez-Vázquez, Carmen Rodríguez, Miguel E. Vázquez-Méndez, and Miguel A. Vilar Urban Heat Island Effect in Metropolitan Areas: An Optimal Control Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Lino J. Alvarez-Vázquez, Francisco J. Fernández, Aurea Martínez, and Miguel E. Vázquez-Méndez

789 799

809

819

829

Nitsche-Based Finite Element Method for Contact with Coulomb Friction . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Franz Chouly, Patrick Hild, Vanessa Lleras, and Yves Renard

839

Electrically Driven MHD Flow Between Two Parallel Slipping and Partly Conducting Infinite Plates.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Munevver Tezer-Sezgin and Pelin Senel

849

Two Methods for the Numerical Modelling of the PM Transport and Deposition on the Vegetation.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . ˇ Ludˇek Beneš and Hynek Rezníˇ cek

859

On a New Mixed Formulation of Kirchhoff Plates on Curvilinear Polygonal Domains .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Katharina Rafetseder and Walter Zulehner

869

Part XXI

Reduced Order Models for Time-Dependent Problems

POD-Based Multiobjective Optimal Control of Time-Variant Heat Phenomena ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Stefan Banholzer, Eugen Makarov, and Stefan Volkwein

881

Greedy Kernel Methods for Accelerating Implicit Integrators for Parametric ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Tim Brünnette, Gabriele Santin, and Bernard Haasdonk

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Contents

Part XXII

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Limiter Techniques for Flow Problems

Third-Order Limiter Functions on Non-equidistant Grids . . . . . . . . . . . . . . . Birte Schmidtmann and Manuel Torrilhon A Linearity Preserving Algebraic Flux Correction Scheme of Upwind Type Satisfying the Discrete Maximum Principle on Arbitrary Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Petr Knobloch Part XXIII

899

909

New Frontiers in Domain Decomposition Methods: Optimal Control, Model Reduction, and Heterogeneous Problems

Optimized Schwarz Methods for Advection Diffusion Equations in Bounded Domains .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Martin J. Gander and Tommaso Vanzan Optimal Coarse Spaces for FETI and Their Approximation . . . . . . . . . . . . . Faycal Chaouqui, Martin J. Gander, and Kévin Santugini-Repiquet

921 931

Coupling MPC and HJB for the Computation of POD-Based Feedback Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Giulia Fabrini, Maurizio Falcone, and Stefan Volkwein

941

Adaptive Multiple Shooting for Nonlinear Boundary Value Problems . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Thomas Carraro and Michael Ernst Geiger

951

Part XXIV

Error Analysis for Finite Element Methods for PDEs

Exponential Scaling and the Time Growth of the Error of DG for Advection-Reaction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Václav Kuˇcera and Chi-Wang Shu

963

Maximum Norm Estimates for Energy-Corrected Finite Element Method . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Piotr Swierczynski and Barbara Wohlmuth

973

Digital Operators, Discrete Equations and Error Estimates . . . . . . . . . . . . . . Alexander Vasilyev and Vladimir Vasilyev A Simple Boundary Approximation for the Non-symmetric Coupling of the Finite Element Method and the Boundary Element Method for Parabolic-Elliptic Interface Problems . . . . . . .. . . . . . . . . . . . . . . . . . Christoph Erath and Robert Schorr

983

993

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Part XXV

Contents

Fluid Dynamics

Galerkin Projection and Numerical Integration for a Stochastic Investigation of the Viscous Burgers’ Equation: An Initial Attempt .. . . . Markus Wahlsten and Jan Nordström Nonlinear Flux Approximation Scheme for Burgers Equation Derived from a Local BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . J. H. M. ten Thije Boonkkamp, N. Kumar, B. Koren, D. A. M. van der Woude, and A. Linke

1005

1015

A Spectral Solenoidal-Galerkin Method for Flow Past a Circular Cylinder . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Hakan I. Tarman

1025

Conservative Mimetic Cut-Cell Method for Incompressible Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . René Beltman, Martijn Anthonissen, and Barry Koren

1035

Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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Part I

Plenary Lectures

PDE Apps for Acoustic Ducts: A Parametrized Component-to-System Model-Order-Reduction Approach Jonas Ballani, Phuong Huynh, David Knezevic, Loi Nguyen, and Anthony T. Patera

Abstract We present an SCRBE PDE App framework for accurate and interactive calculation and visualization of the parametric dependence of the pressure field and associated Quantities of Interest (QoI)—such as impedance and transmission loss— for an extensive family of acoustic duct models. The Static Condensation Reduced Basis Element (SCRBE) partial differential equation (PDE) numerical approach incorporates several principal ingredients: component-to-system model construction, underlying “truth” finite element PDE discretization, (Petrov)-Galerkin projection, static condensation at the component level, parametrized model-order reduction for both the inter-component (port) and intra-component (bubble) degrees of freedom, and offline-online computational decompositions; we emphasize in this paper reduced port spaces and QoI evaluation techniques, especially frequency sweeps, particularly germane to the acoustics context. A PDE App constitutes a Web User Interface (WUI) implementation of the online, or deployed, stage of the SCRBE approximation for a particular parametrized model: User model parameter inputs to the WUI are interpreted by a PDE App Server which then invokes a parallel cloud-based SCRBE Online Computation Server for calculation of the pressure and associated QoI; the Online Computation Server then downloads the spatial field and scalar outputs (as a function of frequency) to the PDE App Server for interrogation and visualization in the WUI by the User. We present several examples of acousticduct PDE Apps: the exponential horn, the expansion chamber, and the toroidal bend;

J. Ballani Swiss Federal Institute of Technology in Lausanne, Lausanne, Switzerland e-mail: [email protected] P. Huynh · D. Knezevic · L. Nguyen Akselos S.A., Lausanne, Switzerland e-mail: [email protected]; [email protected]; [email protected] A. T. Patera () Massachusetts Institute of Technology, Cambridge, MA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_1

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in each case we verify accuracy, demonstrate capabilities, and assess computational performance.

1 Introduction Acoustic ducts [20] appear in a broad range of contexts. We cite here a few important applications: mufflers; audio systems and musical instruments; the auditory and vocal functions; sound measurement devices; and Non-Destructive Testing systems. We formulate any particular acoustic duct problem as a parametrized model: a partial differential equation (PDE) mapping from a parameter P -tuple μ in parameter domain P ⊂ RP to pressure field p(μ) and associated scalar Quantities of Interest, QoI(μ). The parameter μ prescribes the topology and geometry of the spatial domain, Ω(μ), as well as constitutive constants, boundary conditions, and in particular frequency. The QoI, or outputs, are functionals of the pressure field p(μ): the QoI are important for assessment and design of engineering systems; we consider in this paper inlet impedance and transmission loss. We also emphasize the full pressure field, and in particular visualization of the full pressure field, p(x; μ), x ∈ Ω(μ): the full pressure field is important for verification, but also for insight related to (say) resonance spatial structure and radiation directivity. In this paper we shall focus on PDE Apps. A PDE App is an approximation to a particular parametrized model (and hence associated parametrized PDE) which we contend should furthermore satisfy four “PDE App Requirements”: Req 1: Req 2: Req 3: Req 4:

Rapid problem specification: ≤5 s to input μ. Rapid output computation: ≤5 s to evaluate the QoI(μ). Rapid field visualization: ≤5 s to render the pressure p(μ) over the duct domain. Sufficient accuracy: ≤5% error in p(μ), in a suitable norm, and the QoI(μ).

The choice of 5 s is not arbitrary, but rather imposed by the human attention span: PDE Apps are intended to be interactive; we are also interested in the real-time and many-query contexts. In actual practice, in the acoustics context, the PDE App performance requirements are even more demanding: we wish to evaluate the QoI as a function of frequency f (in Hz), or equivalently nondimensional wavenumber ka0 ; here k is dimensional wavenumber and a0 is inlet radius. Towards that end, we first express our parameter μ as (μ , ka0 ), where the (P − 1)-tuple μ ∈ P  represents all parameters except wavenumber. We then define a QoI wavenumber sweep: we ask for the QoI for given μ and nsweep values of ka0 in the interval [0, (ka0)max ]. A PDE App which delivers 5-s response for a single query may not deliver 5-s response for a wavenumber sweep; however, sweeps are “embarrassingly parallel,” and thus through concurrency we can in principle satisfy Req 2 even for QoI wavenumber sweeps for nsweep large.

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There are a variety of techniques for the estimation of the pressure field and associated QoI: analytical or semi-analytical approaches such as (patched) planewave [7, 20] and in particular (patched) multimodal duct expansions [10, 25, 26]; and computational approaches such as Finite Difference (FD) [10], Finite Element (FE) [16, 19], and Boundary Element (BE) [23, 25, 26] methods. The semi-analytical multimodal approaches are typically quite fast and also highly accurate—but can not readily treat complex duct geometries. Conversely, the computational approaches can treat general geometries—but are typically quite expensive, in particular in three space dimensions. We note more generally that many techniques, including response-surface approaches and even the boundary element method, are well-suited to QoI evaluation but less effective for full pressure recovery and visualization. The PDE App criteria, in particular the interaction imperative combined with the visualization requirement, suggest a projection-based model order reduction approach. In this paper we pursue the SCRBE (Static Condensation Reduced Basis Element) model order reduction procedure. The SCRBE numerical approach incorporates several principal ingredients: component-to-system model construction; underlying “truth” finite element PDE discretization; (Petrov)-Galerkin projection; static condensation at the component level; parametrized model order reduction for both the inter-component and intra-component degrees of freedom, termed “port reduction” and “bubble reduction” respectively; and finally, offline-online computational decompositions. The offline-online decomposition transforms the rapid response requirement from a challenge into an opportunity: an expensive offline stage, performed once, “reduces” the PDE approximation; a very inexpensive online stage, performed many times, invokes the reduced PDE to provide interactive response. The SCRBE method [14, 15] incorporates aspects of the Component Mode Synthesis (CMS) technique [8, 11, 12] and the Reduced Basis (RB) method [1, 21, 24]; the former provides the concepts of components and ports and associated approximations, whereas the latter provides the framework for parametric analysis. The Reduced Basis Element (RBE) method of [18] represents the first combination of the CMS and RB approaches. The SCRBE method may thus be viewed as an RBE method for a particular (Static Condensation [29]) choice for the interface treatment and particular strategies for port [3, 9, 22, 27] and bubble [6, 28] approximation spaces. The port approximation of the SCRBE formulation can also be interpreted, in the acoustics context, as a generalization of more classical separation-of-variables multi-modal approaches. For the purposes of this paper we shall narrow the definition of a PDE App: a web implementation of the online (or deployed) stage of the SCRBE approximation for a particular parametrized model. In a PDE App, User model parameter (μ) inputs to a Web User Interface (WUI) are interpreted by a PDE App Server which then invokes a parallel cloud-based SCRBE Online Computation Server for calculation of the approximate pressure field and associated QoI; the Online Computation Server subsequently downloads the spatial field and scalar outputs (as a function of frequency) to the PDE App Server for interrogation and visualization in the WUI

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by the User.1 We note that this web implementation requires attention not only to the computation of the pressure field and associated QoIs but also to the the download of these quantities from the Online Computation Server to the PDE App Server. We shall, however, impose our four PDE App Requirements only on the former and exclusive of the latter, in particular since download times will depend sensitively on many factors outside our control, most notably Internet bandwidth and service, geographic location of the servers, and local network configuration and contention. In this paper we describe and demonstrate several enhancements to the SCRBE method with particular attention to the context of acoustics and acoustic ducts: extension of the transfer eigenproblem port-mode training procedure as originally formulated for equilibrium elasticity [27] to the Helmholtz PDE of (frequencydomain) acoustics—designed to obtain an economical port space relevant to a large family of acoustic duct models; port space refinement strategies (within the WUI)— designed to optimize performance for any particular acoustic duct model-cum-PDE App; parallel treatment of the frequency (or wavenumber) parameter—designed to preserve interactive response even for QoI sweeps. We shall also emphasize the description and demonstration of the PDE App framework. We summarize the overall PDE App web architecture and the respective tasks of the PDE App Server and Online Computation Server. We present several examples of acoustic-duct PDE Apps, in particular the exponential horn, the expansion chamber, and the toroidal duct bend. The reader may also directly access the PDE Apps described—albeit a small sample of the many acousticduct PDE Apps which can be readily constructed from our library of archetype components—through the WUI available at https://atpwui.akselos.com/acoustics/ pdeappsdata.html. We close with an outline of the paper. In Sect. 2 we describe the mathematical formulation: governing equations and boundary conditions; Quantities of Interest; formulation of parametrized models and associated PDE Apps. In Sect. 3 we introduce the SCRBE method and associated PDE App web implementation. In Sect. 4 we present our small set of illustrative PDE Apps.

2 Mathematical Formulation 2.1 Governing Equations We begin with the governing equations. We denote the domain of the acoustic duct by Ω ⊂ Rd≡3 , a point in Ω by x ≡ (x1 , x2 , x3 ), and the boundary of the domain

1 A first realization of the PDE App concept is provided in [13], however the latter invokes a much simpler and less general model order reduction approach—the Reduced Basis method but without components—that is furthermore restricted by an “onboard” implementation which is much less powerful than the web cloud formulation described in the current paper.

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by ∂Ω. We identify two special portions of the boundary: the inlet, Γinlet, a disk of radius a0 , and the outlet, Γoutlet. In most cases the outlet, Γoutlet, is a disk of radius aoutlet; unless otherwise indicated, we shall assume aoutlet = a0 . In the case of a flanged radiation termination we append to our duct a half-sphere Srad of radius rrad such that our “outlet” Γoutlet is now the hemispherical surface of Srad . We shall assume a harmonic pressure of the form (p(x; ω) exp(iωt)) where √  denotes real part, i = −1, ω is angular frequency, and t is time; we shall henceforth only consider the (complex) frequency-domain pressure, p(x; ω). The complex particle velocity is then related to the pressure as v(x; ω) ≡ iρ0 ω−1 (1 + ik)∇p(x; ω). Here ρ0 is the density, k ≡ ω/c0 is the dimensional wavenumber, c0 is the speed of sound, and  is a dissipation factor with dimensions of length (we elaborate upon the dissipation treatment below). Our working medium shall be air at room temperature: unless otherwise indicated, we take c0 = 343 m/s and ρ0 = 1.2 kg/m3; in fact, most of our results are presented in a format which is independent of these dimensional values. For future reference we also introduce the frequency f = ω/(2π) measured in Hz, the nondimensional wavenumber ka0 , and the characteristic specific impedance Z0 ≡ ρ0 c0 . We now present the governing equations and boundary conditions associated with any given problem. We look for pressure p(x; ω) the solution of the Helmholtz equation (1 + ik)∇ 2 p + k 2 p = 0 for x ∈ Ω ,

(1)

subject to a specified incoming wave (IW) condition at the inlet and a radiation condition at the outlet, ∂p + ik(p − 2pinc ) = 0 on Γinlet (IW), ∂n ∂p 1 )p = 0 on Γoutlet (R), (1 + ik) + (ik + ∂n reff

(1 + ik)

(2)

and zero-velocity conditions on the remainder of the boundary (walls), ∂p ∂n = 0 on ∂Ωwall. Here pinc is the amplitude of the incoming wave, n denotes the outward normal, and reff is an effective radiation radius. The choice reff = ∞ yields a Sommerfeld condition, corresponding to an anechoic termination.2 The choice reff = rrad , for rrad the radius of our radiation hemisphere Srad , yields a first-order radiation condition, corresponding to a flanged termination into an infinite medium. (Recall that we impose zero-velocity conditions on all walls, and hence on the wall of the flange.)

2 We note that Eq. (2) (IW), and also Eq. (2) (R) in the Sommerfeld case, can be slightly improved: for Eq. (2) (IW) √ in the term ik(p − 2pinc ) and for Eq. (2) (R) in the term ikp we might replace k with k d = k/ 1 + ik. However, the dissipation term is extremely small, and hence the effect of this improvement correspondingly unimportant.

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In one example, the toroidal bend, we shall also consider velocity boundary conditions at inlet and outlet. In this case we replace the inlet and outlet conditions, Eq. (2), with (1 + ik)

∂p = −ikZ0 vinlet on Γinlet (VI), ∂n

∂p = 0 on Γoutlet , (VO) ∂n

(3)

where vinlet is the imposed inward-normal uniform velocity at the inlet to the duct. The remainder of the formulation remains as before. Note that in the variational context all boundary conditions considered in this paper are natural. We make a few comments on the treatment of dissipation. Our interest is in the inviscid limit,  → 0. Consideration of the Navier-Stokes equations with slip velocity boundary conditions [7] would suggest  = ξ d ν0 /c0 for ν0 the kinematic viscosity and ξ d an order-unity dimensionless constant (for air thermophysical properties at room temperature). We shall interpret  as primarily a numerical dissipation term intended to regularize the inviscid Helmholtz equations: we choose  = 1 × 10−5 alib where alib is a reference radius for our library which is always of the same order as a0 , the duct inlet radius. For this choice of —in fact on the order of ν0 /c0 for a0 on the order of 1 cm—the numerical dissipation affects our results very little except for (1) ducts which are very long relative to a0 , or (2) frequencies which are very close to resonances (and in the absence of other forms of resistive impedance). We confirm the small effect of dissipation in our examples on a caseby-case basis.

2.2 Quantities of Interest (QoI) We now turn to outputs. We shall consider several problem outputs, notably the nondimensional inlet (or input) specific impedance and the transmission loss:  Z inlet ≡

pinlet vinlet



 /Z0 (II),

TL = −20 log10

|poutlet| |pinc |

 (dB) (TL) .

(4)

Note the transmission loss is defined for the particular case of boundary conditions Eq. (2) and furthermore inlet radius equal to outlet radius. Here pinlet and vinlet are defined as the spatial averages over Γinlet of respectively the pressure and the inward normal velocity, and poutlet is defined as the spatial average over Γoutlet of the pressure. In actual practice, for the ducts—with sufficient inlet and outlet straightduct segments—and frequencies that we consider, the pressure is sensibly uniform (planewave) over inlet and outlet. (It then also follows that we may evaluate, for boundary condition Eq. (2) (IW), vinlet as (2pinc −pinlet)/Z0 .) We note that a variety of other outputs, such as inlet reflection coefficient and power, may be readily expressed in terms of Z inlet and TL; for example, the inlet reflection coefficient, R inlet , is given by (Z inlet − 1)/(Z inlet + 1).

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We next consider “subproblem” outputs. Here subproblem refers to a building block to be incorporated into an acoustic duct problem; the corresponding subproblem outputs—impedances and transfer matrices—supply the requisite pressure and velocity interface information. The procedure is standard: we consider an appropriate duct “test problem” which includes the subproblem of interest; we exploit planewave theory to extract from the test problem outputs the desired subproblem outputs. We shall consider here only single-terminal subproblems. However, the methodology readily extends to two-terminal subproblems and in particular transfer matrices; several transfer matrix examples are available at https:// atpwui.akselos.com/acoustics/pdeappsdata.html. As an example of a subproblem output we consider a horn impedance. We assume that ka0 is smaller than the first (azimuthal-mode) circular duct cut-off wavenumber ≈ 1.8412. Our test problem comprises a straight segment of length L connected to a subproblem which terminates in flanged radiation to an infinite quiescent medium. We may view the subproblem in this case as a horn; the single terminal of the subproblem—the junction between the straight segment and the subproblem—is thus the throat of the horn. We define Z throat as the specific impedance at the horn throat normalized by Z0 ≡ ρ0 c0 ; we calculate Z throat in standard fashion as Z throat ≡ (e−ikL + R inlet eikL )/(e−ikL − R inlet eikL ), where R inlet here refers to the reflection coefficient at the inlet to the test problem (and not at the throat of the horn). The impedance Z throat is independent of L as L → ∞ (for  = 0); in practice, the pressure relaxes to plane wave over a distance on the order of a0 . We may apply Z throat as a termination impedance in any acoustic duct problem. Finally, we emphasize that the QoI, or outputs, in this section are scalar. In addition, we must consider the full pressure field—or various slices of the pressure—as a high-dimensional output.

2.3 Models and Apps We next formally introduce the concept of a parametrized model. A parametrized model is a problem (or test problem, henceforth test model) as introduced in the previous sections but now formalized as regards parametric dependence. In particular, a parametrized model is defined by a model parameter μ (≡ (μ , ka0 )) in a model parameter domain P ≡ P  × [0, (ka0)max ] ⊂ RP . The (P − 1)-tuple parameter μ describes the topology and geometry of the spatial domain, Ω(μ ) (which we will write more simply as Ω(μ)), as well as the medium, sources, and boundary conditions. Our interest is in the map from model parameter μ ≡ (μ , ka0 ) ∈ P to (1) pressure field p(x; μ)—solution p of Eq. (1) over the domain Ω ≡ Ω(μ), and to (2) associated QoI(μ) such as (model outputs) Z inlet(μ) and TL(μ) and (submodel output) Z throat (μ). We also consider wavenumber sweeps: for given μ ∈ P  , we wish to evaluate (say) TL(μ , ka0 ) for nsweep values of ka0 uniformly distributed over the interval [0, (ka0 )max ]; for future reference we define the sweep wavenumber set K ≡ {(ka0)j = j (ka0 )max /nsweep , 1 ≤ j ≤ nsweep }.

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Fig. 1 Spatial domain Ω(μ) of the flanged exponential horn parametrized model

We present briefly an example of a parametrized model. We consider a circular duct connected to a flanged exponential horn, the spatial domain for which is depicted in Fig. 1. The model parameters are given by μ ≡ (L/a0 , mhorn a0 , amouth /a0 , ka0 ) in parameter domain P≡[2, 20]×[0.0334, 0.1666] ×[4, 12]×[0, 1], where L is the (pre-horn) length of the circular duct, a0 is the (prehorn) circular duct radius and hence also the horn throat radius, mhorn is the exponent such that horn radius a(x3) satisfies a(x3) = a0 exp(mhornx3 ), and ka0 is nondimensional wavenumber; here x3 is axial distance along the duct measured from the horn throat. We recall that we impose the incoming wave condition Eq. (2) (IW) on the inlet (green in Fig. 1) and the first-order radiation condition Eq. (2) (R) for reff = rrad over the hemispherical surface Srad (cyan in Fig. 1). In this example we shall interpret our model as a test model such that our submodel QoI shall be Z throat (μ). In the current paper a PDE App is an implementation of the online stage of the SCRBE approximation for a particular parametrized model; the PDE App is accessed through a WUI which accepts parameter μ ∈ P and provides pressure pApp (x; μ) ≈ p(x; μ) and scalar outputs QoIApp (μ) ≈ QoI(μ). We shall denote a PDE App which provides model outputs such as Z inlet and TL a “PDE App” without qualification, and a PDE App (for a test model) which provides submodel outputs such as Z throat a “PDE App-SUB.” In this paper, in the interest of brevity, we do not present visualizations, but instead refer the interested reader to the interactive PDE Apps; we shall, however, discuss the computational cost of full pressure reconstruction.

3 Numerical Approach 3.1 Library A library L is a defined by a set of nfid fiducial ports, LFP , and narch archetype components, LAC . For our particular acoustic duct library, nfid = 2 and narch = 22.

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Fig. 2 The two fiducial ports in LFP : geometry; FE mesh; control point

We now define fiducial port and archetype component attributes, the ensemble of which shall be denoted the Library Dataset. The two fiducial ports in LFP are denoted FP DISK and FP ANNULUS, respectively. Any given fiducial port, FP β (for example, β ≡ DISK), is defined by a (fixed) spatial β domain γfid and an associated P2 finite element (FE) space W h;β of dimension β NP,fid . We present in Fig. 2 the spatial domains associated with our two fiducial ports: FP DISK is simply a disk of radius 1/2; FP ANNULUS is an annulus with inner radius 1/2 and outer radius 1. (In general, we may consider any desired geometry for the ports.) We also depict in Fig. 2 the nodes of the P2 (triangle) FE meshes DISK ANNULUS associated with W h;β : NP,fid = 977 and NP,fid = 1184. Note for fiducial ports with nontrivial rotation groups we also identify a control point to fix orientation; the latter are indicated in Fig. 2 as small open circles at “noon.” Any given archetype component, AC α (for example, α ≡ DUCT), is defined by several attributes: α

1. A local parameter ν α in parameter domain V α ⊂ RP . The P α -tuple includes geometry, constitutive, boundary condition, and operation parameters; the latter shall always include a nondimensional wavenumber, kalib, where alib is a fiducial radius common to all archetype components in the library. 2. A reference domain Dˆ α with boundary ∂ Dˆ α on which we identify I α mutually disjoint reference ports γˆ α,i , 1 ≤ i ≤ I α , as well as an inhomogeneous Neumann α . Each reference port domain γˆ α,i is defined in terms of a parameterportion γˆNeu β independent (typically affine) transformation applied to the domain γfid of a α,i “parent” fiducial port β = βpar . (In practice, we may include in LAC port-rotation variants of AC α each associated with a different map from the parent fiducial port.) 3. A parametrized mapping Tνα such that the component physical domain is deduced from the component reference domain Dˆ α as Dνα = Tνα (Dˆ α ); the component port physical and reference domains are related in a similar fashion. Note Tνα will depend only on the geometry parameters. (For quantities in which

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the archetype component α already appears as superscript we suppress the superscript for the local parameter ν α .) In general, Dˆ α = Dναˆ for some νˆ α ∈ V α . 4. The parametrized Helmholtz sesquilinear form ν α ∈ V α → aˆ να : H 1 (Dˆ α ; C) × H 1 (Dˆ α ; C) → C given by (summation over repeated indices)  aˆ να (v, ˆ w) ˆ ≡

 Dˆ α

1 + i(kalib)(

∂ wˆ   ∂ vˆ ∗ α −1 α ) (Jν )ij |Jν |(Jνα )−1 j i alib ∂xi ∂xi 

− (kalib)2 vˆ ∗ w|J ˆ να | dV ;

(5)

here H 1 (Dˆ α ; C) is the space of complex fields over Dˆ α with square integrable first derivative, Jνα is the Jacobian of Tνα , and ∗ denotes complex conjugate. We also introduce an anti-linear form associated with the Neuman data ν α ∈ V α → fˆνα : H 1 (Dˆ α ; C) → C. 5. A P2 (tetrahedron) FE approximation space Xˆ h;α (Dˆ α ) ⊂ H 1 (Dˆ α ; C) of dimension N α ; recall that all boundary conditions are natural. We must construct Xˆ h;α such that the component mesh conforms to the (mapped) parent fiducial port mesh. We decompose our FE approximation space as the direct sum of a port space and a bubble space: Xh;α = XPh;α ⊕ XBh;α , where the port space is of dimension NPα and the bubble space is of dimension NBα such that N α = NPα + NBα . We further introduce a standard nodal basis in which we number the NPα nodes associated with the port degrees of freedom first and the NBα nodes associated with the bubble degrees of freedom subsequently. α α 6. The archetype component FE stiffness matrix Aνh;α ∈ CN ×N (respectively, α archetype component FE load vector F h;α ∈ CN ) associated with the ν α Helmholtz sesquilinear form aˆ ν (·, ·) (respectively, the inhomogeneous Neumann anti-linear form fˆνα (·)) and our nodal basis. We partition Aνh;α in terms of port and bubble degrees of freedom ⎛

NP ×NP ⎜ APP

Ah;α ν

⎜ ⎜ ≡ ⎜ N ×N ⎜ B P ⎝ ABP

NP ×NB APB NB ×NB ABB

⎞α ⎟ ⎟ ⎟ ⎟ . ⎟ ⎠

(6)

ν

the first NPα rows of Aνh;α (and F h;α ν ) correspond to port test functions and the remaining NBα rows of Aνh;α (and F h;α ν ) correspond to bubble test functions; the first NPα columns of Aνh;α correspond to port trial functions and the remaining NBα columns of Aνh;α correspond to bubble trial functions. 7. An affine representation (or approximate representation) of the sesquilinear form, α

aˆ να (w, v) =

Q

q=1

Θqα (ν α )aˆ qα (w, v) ,

(7)

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Fig. 3 Several parametrized archetype components in LAC : reference domain; FE mesh; ports

typically developed by the EIM [5]; note that the aˆ qα are independent of ν α . A similar expansion may be developed for fˆνα . As point of reference: for an archetype component α in which (the geometry is fixed and) the only parameter is kalib we obtain an exact affine representation for Qα = 2. The archetype component information and fiducial port information together constitute a Library Dataset. We present in Fig. 3 the reference domains and P2 (tetrahedron) FE meshes associated with a few of the archetype components in LAC : (from left to right) AC ANNULAR_ DUCT , AC DUCT _ TO _ COAXDUCT , AC SIDEBRANCH , AC SIDEHOLE , AC EXPANSION , AC BEND , AC DUCT , and AC INCOMING _ WAVE ; note that, absent any qualifier, “duct” shall refer to circular duct. We present two views of each component so as to expose all ports: the ports with parent fiducial port β ≡ DISK are coded red, and the ports with parent fiducial port β ≡ ANNULUS are coded yellow; note that, in general, green shading indicates an incoming wave inlet boundary condition (AC INCOMING_WAVE), and cyan shading indicates a radiation boundary condition (AC SIDEHOLE). The selected components are equipped with either two ports or ( AC DUCT _ TO _ COAXDUCT and AC SIDEBRANCH ) three ports. All ports are disjoint; we can also consider non-disjoint ports, albeit less simply, and not in this paper.

3.2 Model Synthesis We now relate any given parametrized model to our library of fiducial ports and archetype components. We create, for any μ ∈ P, a SYSTEM(μ): ninst (μ) instantiated components from our library LAC of narch archetype components. Each instantiated component IC , 1 ≤  ≤ ninst(μ), is defined as (1) a parent archetype par component, α = α (μ), for (2) a particular value of the local parameter ν α , ν (μ) ∈ V α . Two (or more) instantiated components IC  and IC  with common par par parent archetype component α (μ) = α (μ) = α (say α = DUCT) may, and typically will, be assigned different local component parameter values, ν (μ) =

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ν (μ); there may, however, be global variables—in our case kalib—shared by all instantiated components of all parents. In general, the local parameter value ν (μ) assigned to any particular instantiated component IC  will depend on the value of the model parameter, μ. In fact, as indicated by the argument μ for ninst(μ) and par α (μ), even the number and parentage of instantiated components will depend on parameter; the associated flexibility is ultimately important for numerical accuracy even in fixed topology but also permits parametrized models with variable topology. For any μ ∈ P, our SYSTEM(μ) must satisfy two assembly rules. (Union Requirement) The union of the instantiated component (physical) domains reproduces the model spatial domain, Ω(μ). (Intersection Requirement) The intersection of the instantiated component (physical) domains, is either (1) empty, or (2) an entire port (physical domain) of both components; in the latter case the participating ports in IC  and IC  must share the same parent fiducial port and align at respective control points. Thanks to the intersection requirement we may identify nglob (μ) global ports the (physical) domains of which we shall denote {Γj }j =1,...,nglob (μ) ; each global port of the system is the coalescence of two (connected, hence coincident) instantiated component ports. In actual practice we do not rely on geometry to ensure connection. Rather we specify a list of connections E: each connection is a pair of (instantiated component, instantiated port) couples from which we then deduce docking parameters; the latter control rigid body motions under which the equations are invariant and which can thus serve to mate instantiated components in the desired fashion. Typically E is deduced from a script associated to a particular parametrized model, denoted a PDE App Script, which instantiates components and then establishes pairwise connections: an explicit map for each μ ∈ P to a corresponding SYSTEM(μ). To close this section we present briefly an example of a parametrized model and associated μ → SYSTEM(μ) synthesis. We consider a circular duct with flanged exponential horn, as described in Sect. 2. In order to construct our system we require the four parametrized archetype components AC α in LAC shown in Fig. 4: (from left to right) α ≡ INCOMING_WAVE, a duct segment with incoming wave inlet boundary condition—one DISK port, and local component parameter ν α ≡ {segment radius/alib, segment length/alib, kalib }; α ≡ DUCT, a circular duct—two DISK ports, and local component parameter ν α = {segment radius/alib , segment Fig. 4 The parametrized archetype components (reference domain) in LAC needed to form the flanged exponential horn SYSTEM(μ) for any μ ∈ P

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length/alib, kalib }; α ≡ EXP_HORN, an exponential horn segment—two DISK ports, and local component parameter ν α ≡ {segment smaller port radius/alib , segment exponent· alib, segment length/alib, kalib }; α ≡ FLANGED_RADIATION, an infinite flange with radiation hemisphere—one DISK port, and local component parameter ν α ≡ {port radius/alib (which also dictates rrad as 2 · port radius), kalib }. All these components are also provided with docking parameters: rigid-body translation and rotation. Note that unless otherwise indicated, zero velocity conditions are imposed on all parts of the component boundary except the ports. We construct the SYSTEM(μ) associated with parameter value μ = (12, 0.1076, 10.67, ·) which corresponds to the exponential horn in PostHixson (henceforth abbreviated “PH”) [23]. We depict the SYSTEM(μ) in Fig. 1: the black lines delineate the (physical domains of the) instantiated components and in particular identify the respective perimeters of the global ports {Γj }j =1,...,nglob (μ)≡19 ; we do not depict the (very dense) FE mesh associated with Xh (Ω(μ)) of dimension NSYS (μ) = 725,594. The instantiated components par are derived from the archetype components of Fig. 4 as follows: α=1 = par par par INCOMING_ WAVE , {α }=2,...,5 = DISK , {α }=6,...,19 = EXP _ HORN , and α=20 = FLANGED_RADIATION. For the particular μ considered, the instantiated DUCT components all share the same value of local parameter, however the instantiated EXP _ HORN components will each have different local parameter values and in particular different values of segment smaller port radius/alib; all instantiated components (derived from any archetype component) share the same value of the nondimensional wavenumber, kalib. We emphasize that we can only address models which can be represented, for all μ ∈ P, as a SYSTEM(μ) which (satisfies our two assembly requirements and) can be synthesized from archetype components available in our library. In the remainder of this section, unless otherwise indicated, we shall make certain expositional assumptions on the fiducial ports and archetype components in our library. These assumptions are not actual limitations of the computational approach (or the actual PDE Apps presented later in the paper) and are introduced solely to simplify the presentation. In particular, we shall suppose that there is a single fiducial port, nfid = 1, say β = DISK, such that all component ports share this common parent. We further suppose that all archetype components have exactly two ports, hence I α = 2 for all α in LAC . We further suppose that all archetype components share common FE approximation dimensions such that N α = N , NPα = NP , and NBα = NB . Note it follows from our assumptions on fiducial and component ports that NP = 2NP,fid .

3.3 Truth Finite Element Approximation We may now identify, thanks to the fiducial ports and our intersection requirement, a global conforming “truth” FE approximation space Xh (Ω(μ)) and corresponding Galerkin pressure approximation ph (μ) ∈ Xh (Ω(μ)). Here Xh (Ω(μ))

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“stitches together” the FE spaces of each instantiated component; the latter, in turn, are mapped from the FE spaces of the respective parent archetype component. The dimension of Xh (Ω(μ)) is given, under our expositional assumptions, by NSYS (μ) = nglob(μ)NP,fid + ninst(μ)NB . We emphasize that the truth FE approximation space depends on the FE spaces associated with the archetype components but also on the particular instantiated SYSTEM(μ)—for example, a straight duct constructed with more and shorter DUCT components induces a finer truth FE approximation space than a straight duct constructed with fewer and longer DUCT components. The basis coefficients ph (μ) ∈ CNSYS (μ) for ph (μ) ∈ Xh (Ω(μ)) satisfy the linear system AhSYS (μ)ph (μ) = F hSYS (μ)—shorthand [AhSYS (μ), F hSYS (μ)]—for AhSYS (μ) ∈ CNSYS (μ)×NSYS (μ) and F hSYS (μ) ∈ CNSYS (μ) given by AhSYS (μ) =

ninst (μ)

h;α

par

 A Aν (μ)

=1

(μ)

,

F hSYS (μ) =

ninst (μ)

h;α

par

 A F ν (μ)

=1

(μ)

,

(8)

respectively. The symbol A refers to standard direct stiffness assembly but executed here at the level of instantiated components-cum-substructures to enforce weak continuity of velocity on global ports; direct stiffness assembly at nodes interior to the component is already reflected in the component stiffness matrix. Note we assemble over instantiated components but refer to the corresponding parent par archetype component α (μ) for appropriate local parameter value ν = ν (μ). This truth FE approximation shall serve as the foundation on which we will build the SCRBE model reduction approximation and hence also the PDE App. We must thus confirm not only that the SCRBE approximation pApp (μ) is sufficiently close to ph (μ), but also that ph (μ) is sufficiently close to p(μ), the exact solution.

3.4 Static Condensation: Finite Element We describe here standard static condensation [29], also known as substructuring, as preparation for subsequent model order reduction. In particular, we can eliminate the bubble degrees of freedom in Eq. (6) to obtain the archetype component FE Schur (complement) matrix and Schur vector, Ah;α ∈ CNP ×NP and Fνh;α ∈ CNP , ν respectively, associated with each archetype component α ∈ LAC . In this paper, for brevity, we focus on the Schur matrix (in particular since the computational cost associated with the Schur vector is typically negligible by comparison): −1 α Ah;α ν ≡ ( APP − APB ABB ABP )ν ;

(9)

Equation (9) is readily derived from Eq. (6) by application of block Gaussian elimination. We note that all matrices in Eq. (9) depend on local parameter ν.

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Next, we proceed to model synthesis. We presume that for some parametrized model and given μ ∈ P we form a SYSTEM(μ) which satisfies our assembly requirements. We then perform direct stiffness assembly, now on the instantiated component FE Schur matrices and Schur vectors, to arrive at the SYSTEM(μ) FE Schur matrix AhSYS (μ) ∈ Cnglob (μ)NP,fid ×nglob (μ)NP,fid and Schur vector F h (μ) ∈ Cnglob (μ)NP,fid , AhSYS (μ) =

ninst (μ)

h;α

par

 A Aν (μ)

=1

(μ)

,

h FSYS (μ) =

ninst (μ)

h;α

par

A Fν (μ)

=1

(μ)

,

(10)

respectively. In general we must anticipate that the SYSTEM(μ) FE Schur matrix will be of large dimension—in particular since NP,fid will be large—and thus h (μ)] will be solution of the SYSTEM(μ) FE Schur linear system [AhSYS (μ), FSYS costly. Furthermore, and even more crucially, the SYSTEM(μ) FE Schur matrix is very expensive to form: for 1 ≤  ≤ ninst (μ), each instantiated component FE par Schur matrix Ah;α for α = α and ν = ν (μ) requires solution of NP (sparse) ν FE linear systems of size NB —the inverse in Eq. (9); hence AhSYS (μ) requires in total solution of ninst (μ)NP large FE linear systems. Note that for each model parameter μ (respectively, local parameters ν (μ), 1 ≤  ≤ ninst(μ)) we must form the SYSTEM(μ) FE Schur complement AhSYS (μ) anew.

3.5 SCRBE Method Reduced Schur Complement We now proceed to the reduced Schur complement. β We first introduce, for each fiducial port β ∈ LFP , a set of port modes ψm , 1 ≤ h;β m ≤ NP,fid , which constitute a basis for W (we shall henceforth eliminate the superscript β given our expositional assumption of a single fiducial port). We then construct, for each archetype component, a matrix Ψ αPP ∈ RNP ×NP : column j of Ψ αPP comprises the FE nodal coefficients of fiducial port mode ψk mapped to port i such that j = 2(k − 1) + i. We next define Ψ αPP (M) (typically abbreviated to Ψ αPP for some prescribed M) as the first 2M columns of Ψ αPP : we assign M (mapped) fiducial port modes to each of the two ports of the archetype component. We similarly introduce, for each archetype component α ∈ LAC , and each port mode m ∈ {1, . . . , 2M}, a set of bubble modes ϕnα,m , 1 ≤ n ≤ NB , which constitute a basis for the reduced basis (RB) space XBh;α,m . We then construct, for NB ×NB : column j of Φ α,m comprises each archetype component, a matrix Φ α,m BB ∈ R BB the FE nodal coefficients of bubble mode ϕjα,m . Finally, we define Φ α,m BB (N α,m ) α,m ) as the first N α,m columns (typically abbreviated to Φ α,m for some prescribed N  BB α,m = N is of Φ α,m BB ; for simplicity of exposition, we shall henceforth assume that N independent of α and m.

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We then introduce, for given α ∈ LAC , two spaces: the space Υ trial,α given by  α,m −1 span{ Ψ αPP w | LαBP w − Φ α,m BB (AB B ν )

 H α α α α , w ∈ R2M } (Φ α,m ) (A L + A Ψ ) w    ·,m m BB ν BP BP ν PP BB

(with summation over m); the space Υ test,α ≡ span{( Ψ αPP w | LαBP w ), w ∈ R2M }. (For simplicity we define the spaces implicitly in terms of the FE nodal coefficients α α,m H α,m α,m of the pressure, (·P | ·B ) ∈ RN .) Here Aα,m B B ν = (ΦBB ) ABB ν ΦBB is the N × α N RB projection matrix, and LBP is a prescribed lifting function. Standard static condensation would consider a Galerkin approximation based on common trial and α,m −1 test space Υ trial,α , however this choice leads to the product of (AB terms. We  B ν ) thus choose a Petrov-Galerkin approach, trial space Υ trial,α and test space Υ test,α , which avoids the bubble-bubble terms and greatly reduces the computational cost. For simplicity we present here only the case in which Φ α,m BB is independent of α m and furthermore LBP = 0: the archetype component SCRBE Schur matrix  h,N,M;α 2M×2M Aν ∈R is then given by Ah,N,M;α ≡ (Ψ αPP )H AαPP ν Ψ αPP − (Ψ αPP )H AαPB ν Φ αBB (AαB B ν )−1 (Φ αBB )H AαBP ν Ψ αPP , ν

(11) where H refers to Hermitian transpose; a similar expression, but less compact, is α,m obtained for the case in which Φ α,m BB depends on m. (The choice Φ BB independent of m, for which the Galerkin and Petrov-Galerkin approximations in fact coincide, leads to RB spaces of prohibitively high dimension; thus, in practice, we prefer to construct a tailored RB space for each port mode.) A similar expression may be obtained for the component SCRBE Schur vector Fνh,N,M;α ∈ R2M . We then proceed to form the SYSTEM(μ) SCRBE Schur matrix and vector, h,N,M Ah,N,M ∈ Rnglob (μ)M×nglob (μ)M and FSYS ∈ Rnglob (μ)M , respectively, by direct SYS stiffness assembly: Ah,N,M (μ) = SYS

ninst (μ) =1

par

h,N,M;α (μ)

A Aν (μ)

,

h,N,M FSYS (μ) =

ninst (μ) =1

par

hN,M;α (μ)

A Fν (μ)

. (12)

h,N,M Solution of the SCRBE Schur linear system [Ah,N,M (μ), FSYS (μ)] yields the SYS SCRBE coefficients for the reduced port modes on each of the global ports of SYSTEM(μ). Evaluation of the Quantities of Interest, QoIApp (μ), typically requires only these port coefficients. Evaluation of the pressure pApp (x; μ) over the domain Ω(μ) (more precisely, at the nodes of the FE mesh) requires further calculation: we first obtain, within each instantiated component, the RB bubble coefficients and then, through Φ αBB , we evaluate the bubble modes at the FE nodes.

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Reduction Procedures In this section we discuss the choice of port modes and bubble modes, and thus through the respective truncation to M port modes and N bubble modes, the SCRBE approximation. In fact, the SCRBE bubble reduction [14] is a standard extension of single-component reduced basis (RB) best practices [6, 28] and furthermore requires very little specialization within the acoustics context of this paper. We emphasize here only that the training of the reduced basis (bubble) spaces may proceed independently on each archetype component in the library and furthermore without reference to any particular system or parametrized model. We now turn to port reduction, and in particular to the adaptations effected within the current acoustics context. We require that the retained port modes ψm , 1 ≤ m ≤ M, will provide with increasing M an increasingly accurate approximation to the pressure over the global ports of any SYSTEM(μ) associated with any parameter value μ ∈ P of any model in the family of acoustic ducts associated with our library L. We furthermore intend that we can achieve sufficient accuracy with relatively few port modes, M  NP,fid . We first argue why we can expect that only a relatively few port modes will suffice. For acoustic waveguide problems it is well-known and easily demonstrated that only relatively few (lowest, or propagating) “duct” modes travel down a waveguide [20]: the vast majority of the (higher, or evanescent) duct modes decay exponentially fast in the axial direction. It is thus plausible from this separation-ofvariables argument that the pressure can be well represented on the global ports of a system to high accuracy with relatively few degrees of freedom: even if within instantiated components many higher modes may be (locally) excited by geometric variations, these evanescent modes will largely decay before reaching the corresponding instantiated ports; we thus need only retain the propagating modes and the more slowly decaying evanescent modes. We thereby also identify an archetype component design best practice: to the extent possible, we should separate ports from regions of rapid or discontinuous geometric or property variation; on the ports of such components we can reasonably expect to observe only propagating modes. It remains to determine an appropriate hierarchy of port modes for any given library. Towards that end, we first define a training-pair set: all distinct subsystems of two instantiated components which may be formed from our library such that our rules of assembly are honored; note two subsystems are distinct if any local parameters are different or if the pair of ports which define the connection between the two components is different. In each member (a two-component subsystem) of the training-pair set we identify non-shared global ports, which we denote “sending ports,” and a single shared global port, which we denote the “receiving port.” We next form a training-pair sample by consideration of some suitably rich but finite subset of the training-pair set. We then solve, for each member of our training-pair sample, the transfer eigenproblem developed in [27] and here extended to the Helmholtz equation of acoustics. The transfer eigenproblem derives from a Rayleigh quotient which measures the energy of the pressure on the receiving port relative to the energy of the pressure on the sending ports; here the pressure on the receiving port is the trace

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of the solution to the Helmholtz Dirichlet problem defined by the sending ports—a Helmholtz “map.” The transfer eigenproblem yields a hierarchy of transfer eigenfunctions which capture the most persistent sending-port behavior. To conclude the process we transform, for any given member of the training-pair sample, the retained transfer eigenfunctions—the transfer eigenfunctions corresponding to the largest eigenvalues—to the receiving port parent fiducial port. Finally, we perform a POD on the port data on each fiducial port and truncate the resulting POD expansion as informed by our accuracy requirements. We emphasize that this “port training” procedure is independent of any particular model, and requires consideration only of small—two-component—subsystems. In this work we apply the transfer eigenproblem in a somewhat special fashion, in particular for the FP DISK port space: we first apply the transfer eigenproblem to just the subset of archetype components required for the exponential horn model— which in particular includes representative axisymmetric archetype components; we then apply the transfer eigenproblem to the (full) library of archetype components; we next concatenate the axisymmetric modes from the horn series and all the modes from the full-library series; finally, we orthonormalize and truncate at Mmax modes. The explicit placement of the axisymmetric modes at the front of the deck, which we term “axisymmetric promotion,” will permit us to choose M small for (axisymmetrically loaded) axisymmetric models. We apply the transfer eigenproblem to the FP ANNULUS port space in more standard fashion—a single library series—except that once again we promote the axisymmetric modes to the front of the deck. The treatment of the annular port demonstrates the generality of the approach to effectively any library and in particular to any fiducial port geometry. We show in Fig. 5 several modes for both the DISK and ANNULUS fiducial ports. For FP DISK we obtain Mmax = 40: port modes m = 1, . . . , 6 are axisymmetric, and port modes m = 7, . . . , 40 are non-axisymmetric (hence with azimuthal dependence). For FP ANNULUS we obtain Mmax = 40: port modes m = 1, 2 are axisymmetric, and port modes m = 3, . . . , 40 are non-axisymmetric. We note that the modes will often resemble the propagating and least rapidly decaying evanescent classical duct modes associated with a homogeneous waveguide (in this case, of circular and annular cross section, respectively), however we also typically observe some “empirical modes” particularly suited to the components of the library. We note for the FP DISK some noise in mode m = 2, an unwanted artifact most likely caused by the late appearance of some axisymmetric modes (associated with area variation) in the horn series; in fact, mode m = 2 is the noisiest port mode present in our port space, hence selected for Fig. 5 to disclose worst-case behavior. We shall observe that for (axisymmetrically loaded) axisymmetric models M = 6 shall suffice, and that for non-axisymmetric models M = 10 shall typically suffice; we do thus realize substantial dimension reduction with respect to the dimensions of DISK ANNULUS the FE port spaces, NP,fid = 977 and NP,fid = 1184.

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β

Fig. 5 Port modes χm for m = 1 (planewave), 2, 5, and 10 for β ≡ DISK (left) and β ≡ ANNULUS (right)

3.6 Computational Procedure: Offline-Online Decomposition We can now summarize the computational procedure. There are two stages, an offline stage, and an online stage. In the offline stage—prior to deployment—we may afford ourselves ample computational resources—though in actual practice control of offline costs is very important. In the online stage we deploy the PDE App: we must then consider our PDE App Requirements Req 1, Req 2, Req 3, and Req 4 introduced earlier. We emphasize that the offline stage is associated to a library of archetype components but with no reference to any particular model; the connection between the library of components and any particular model is effected in the online stage. The offline-online approach is particularly well-suited to our PDE Apps as we place a premium on interactive response—the online stage—within a many-query context which amortizes the offline stage. We now summarize the full offline-online decomposition. In the offline stage, we first define our library L and in particular all the attributes associated with each fiducial port in LFP and each archetype component in LAC : the Library Dataset. We next apply the transfer eigenproblem technology to identify β the port modes {Ψm , 1 ≤ m ≤ Mmax , β ∈ LFP }. (In our exposition, we assume for bookkeeping convenience that these modes are known at the outset; in actual fact, we first work with a standard FE nodal port basis in order to identify the optimal port modes.) We subsequently identify the RB modes {ϕnα,m , 1 ≤ n ≤ N, 1 ≤ m ≤ 2M,α ∈ LAC }, for N ≤ NB (and, in actual fact, N  NB ). Finally we calculate, and store in an Online Dataset, parameter-independent FE inner products—faciliated by the affine expansion Eq. (7)—required to form the entries of the RB projection matrices and also the archetype component SCRBE Schur matrix; the actual size of our particular Online Dataset is roughly 3.9 GB. We also store a Visualization Dataset which contains notably the truth FE nodal basis coefficients of the RB modes as required for reconstruction of the pressure field over Ω(μ).

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In the online stage, we consider some particular parametrized model. For any given model parameter μ ∈ P, we first specify the port mode truncation M ≤ Mmax (typically, and in this paper, the bubble reduction truncation, N, is fixed in the Offline stage). Next, we invoke the Online Dataset to form the archetype component Schur matrix (and similarly, Schur vector); we subsequently apply direct stiffness summation to assemble the SYSTEM(μ) SCRBE Schur matrix and vector, h,N,M Eq. (12). We then solve the SCRBE Schur linear system [Ah,N,M (μ), FSYS (μ)] SYS to obtain the SCRBE coefficients. Finally, we evaluate the QoI, and we invoke the Visualization Dataset to reconstruct the pressure for visualization. The operation count for the Online stage, except for pressure reconstruction, is independent of the dimension of the underlying truth (stitched together) FE space Xh (Ω(μ)), NSYS (μ). (The reconstruction of the pressure will of course depend on NSYS (μ), but only linearly.) In particular, the SCRBE approach addresses the two difficulties associated with FE static condensation. First, thanks to port reduction, the SCRBE Schur matrix is very small, of size nglob(μ)M × nglob (μ)M, and furthermore block sparse, since only global ports which originate in a common instantiated component interact. (In practice, we may thus invoke robust direct methods to solve the block-sparse SCRBE Schur system.) Second, the SCRBE Schur matrix is much less expensive to form: for 1 ≤  ≤ ninst(μ), each instantiated par component SCRBE Schur matrix Ah,N,M;α for α = α and ν = ν (μ) requires ν α,m only solution of 2M (dense) RB linear systems of size N ×N—the inverse of AB  B ν that appears in Eq. (11); both port reduction and bubble reduction play a role, the former to reduce the number of linear systems from NP to 2M, the latter to reduce the size of each linear system from NB to N. For our acoustics duct library and PDE Apps we shall typically take 2M ≤ 20 and N ≈ 40, and hence the savings relative to NP = 2NP,fid ≈ 2000 and NB ≈ 30000 will be substantial. We comment on the important role of components. First, as regards parametrization, we reduce a single large problem with many model parameters—the full model—to many small problems with just a few local parameters—instantiated components; we thus address, by divide and conquer, the curse of dimensionality which often plagues reduced basis techniques. Second, components also permit the development of local approximation spaces (over each instantiated component) which offer flexibility and ultimately computational efficiency. In particular, components facilitate models in which the topology may vary as a function of the parameter. Third, in the offline stage, we solve FE problems only over pairs of components (port training) or single components (bubble training): we never solve truth FE problems for entire systems. We may thus consider large models, and in particular models which may be too large to admit FE solution. Fourth, we may amortize our offline effort not over just a particular model, but over all models which we can and may synthesize from the library of archetype components; we can thus tolerate higher offline costs. A common impediment to model order reduction without components is the relative inflexibility imposed by a priori specification of a particular problem and model parameter domain. Fifth, and finally, we facilitate— through the PDE App Script—the development of new models.

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3.7 PDE App Architecture The PDE App is a web implementation of the online stage of the SCRBE approximation for a particular parametrized model from a “Repertoire” of available models (and hence PDE Apps). The web architecture has five elements: the User; the User’s interface device (laptop or tablet or phone), denoted the Client; the Web User Interface (WUI) on the Client browser; a PDE App Server; and an Online Computation Server. The PDE App Server, available at https://atpwui.akselos.com/acoustics/ pdeappsdata.html, hosts the data and codes required to communicate between the User and the Online Computation Server: the Repertoire of PDE App Scripts associated with the different parametrized models available; the PDE App Postprocessor required to interrogate any particular model; the Library Dataset which includes the component specifications and meshes needed for rendering the pressure field. The Online Computation Server hosts the Online Dataset and also Visualization Dataset associated with the entire library of ports and components. The Online Computation Server—which performs the majority of the calculations— will typically comprise many cores or certainly several cores: in the implementation exercised in this paper, we exploit an 8-core instance of Google Compute Engine (GCE). We now present the flow of information, from User and back to User, for any given (model and hence) PDE App from the Repertoire (note that in the below “WUI” should be interpreted as “PDE App Server through WUI”): 1. User invokes in WUI a particular PDE App from PDE App Repertoire. 2. PDE App Server launches associated PDE App Script and PDE App Postprocessor and loads PDE App Minimal Component Dataset—a subset of the Library Dataset which includes the FE reference meshes only for those archetype components required for the particular model.3 3. User inputs to WUI a request—a desired model parameter value μ ∈ P and any discretization choices, in particular M and nsweep ; recall that nsweep determines the sweep wavenumber set K ≡ {(ka0 )j = j (ka0 )max /nsweep , 1 ≤ j ≤ nsweep }. 4. PDE App Script translates μ = (μ , kvis a0 ) ∈ P into a SYSTEM(μ) specification expressed as instantiated components and associated port connections. 5. PDE App Script sends SYSTEM(μ) specification to Online Computation Server. 6. Online Computation Server, for given μ and the single “visualization” wavenumber kvisa0 , loads Online Dataset and Visualization Dataset (an initialization step), forms and solves the reduced Schur complement linear system, and reconstructs pressure field pApp (μ).

3 The Minimal Component Dataset is not small, but it is loaded by the WUI only upon launch of a particular parametrized model; the per-query response is thus not affected.

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7. Online Computation Server, for given μ and sweep wavenumber set K, forms and solves the reduced Schur complement linear system and evaluates QoIApp (μ , ka0 ) for all ka0 in K. In practice, the wavenumber set K is divided into subsets which are then processed in parallel, with a corresponding improvement in response time; it is through parallelism that we scale performance to accommodate wavenumber sweeps within the tight PDE App Requirements. 8. Online Computation Server sends reconstructed pressure field pApp (μ) and wavenumber sweep QoIApp (μ ; ·) to WUI. 9. User invokes Postprocessor and Minimal Component Dataset in WUI to select and view results: three-dimensional pressure field pApp (μ , kvisa0 ) over lines or surfaces or volumes for parameter value (μ , kvis a0 ); QoI(μ , ka0 ) wavenumber sweep for the nsweep values of ka0 in K. 10. User downloads, as needed, particular QoI sweeps to Client storage for permanent archiving. 11. User inputs to WUI the next request: we return to Step 3 and repeat the cycle. Each User request results in a pressure visualization at a single wavenumber and a QoI sweep associated with many wavenumbers. The framework is many-query in the sense of many requests to the Online Computation server but also many QoI evaluations (sweep) within each request.

4 PDE Apps: Examples (Infinite) Flanged Exponential Horn We consider here a PDE App for the test model introduced in Sect. 2 and further elaborated upon, as regards model synthesis, in Sect. 3; we recall that our submodel QoI is Z throat(μ). We make one further point concerning the mathematical model. The radius of our radiation hemisphere, Srad , is given by rrad = 2amouth, where amouth is the radius of the horn mouth. It is known that a finite flange for rrad /amouth = 2 is quite far from infinite as regards pressure distribution and QoI. However, it is crucial to note that our outlet boundary condition does not model a finite flange—which would require the radiation condition to be imposed at a radius considerably larger than the radius of the flange—but a truncated infinite flange—with radiation condition imposed at radius rrad . In this context, rrad /amouth = 2 in fact already provides a rather good approximation for an infinite flanged horn, as we confirm below. We have performed self-consistent verification studies which confirm that the FE mesh (h), port space for truncation M = 6, and RB bubble spaces (characterized by N) are adequate to provide the throat impedance to within an accuracy of several percent, hence consistent with PDE App Req 4; we present some results for convergence in M shortly. The verification procedure is typical for any given model-cum-App: we perform stitched-together truth FE calculations at several “difficult” parameter values on several meshes to determine adequate convergence in h such that plausibly ph (μ) is sufficiently close to p(μ) for all μ ∈ P; we then

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compare stitched-together truth FE model predictions to SCRBE predictions, again for several parameter values, and as needed refine Mmax and N (or the tolerances which determine Mmax and N) such that plausibly ph,M,N (μ)(≡ pApp (μ)) is sufficiently close to ph (μ) for all μ ∈ P; for radiation problems we must in addition confirm insensitivity to the hemisphere radius. Note that we do not refine the model per se but rather the relevant archetype components in the library—and indeed, all the archetype components in the library if the FE resolution on the fiducial ports changes. In this sense, verification of each model is a continuation of the library training process, and inasmuch all models benefit from verification of each model: the library rapidly matures to a steady state. We recall also that the port training procedure treats the components of the horn in a special fashion to ensure (more generally) efficient approximation of axisymmetric problems. We also extensively exploit third-party results for both verification and also validation; in this case the Post-Hixson exponential horn study (henceforth PH) [23] provides complete coverage for a particular parameter value. In our language, the PH horn corresponds to the model of Sect. 2 for parameter values mhorn a0 = 0.1076 and amouth/a0 = 10.67. We further select L/a0 = 4: L/a0 refers to the geometry of the test model from which we deduce the submodel output Z throat (μ); the choice L/a0 = 4 is sufficiently large to recover the L/a0 → ∞ limit and sufficiently small to effectively eliminate any inlet dissipative effects. PH presents results for the Boundary Element (BE) method and Experiment. Agreement between PH BE predictions and PH Experiment predictions is for the most part very good, and we will thus in our comparisons take PH BE as our reference. We present in Fig. 6a, derived from figure 4.24 of PH, our PDE App-SUB results (M = 6) and PH BE results for the real part and imaginary part of Z throat (μ) as a function of kamouth. We observe very good agreement, certainly to within our 5% tolerance, between the PDE App-SUB and PH BE. We present in Fig. 6b the PDE App-SUB predictions for the throat impedance as a function of wavenumber for several values of M, the number of port modes retained: we observe that M = 1, effectively planewave, provides a very poor approximation, that M = 5 provides a good approximation but only for lower wavenumbers, and that M = 6—all the axisymmetric modes in our port space—provides a good approximation over the entire range of wavenumbers [0, (ka0)max ≡ 1] (as demonstrated by comparison with PH BE in Fig. 6a). Further increases in M will have no effect since modes m > 6 are no longer axisymmetric and hence not excited in our axisymmetric model. The remainder of the results in this section, including timings, correspond to the particular choice M = 6. We now assess the computational performance for M = 6. In all timings in this paper, we consider an 8-core GCE (Google Compute Engine) instantiation; to ensure representative timings, the Online Computation Server is dedicated to a single request. The sweep of Fig. 6a—nsweep = 200—is effected in less than 4.7 s (all times reported are wall-clock time): the time is roughly evenly split between reduced Schur complement linear system formation and reduced Schur complement linear system solution; we conclude that both port reduction and bubble reduction

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Fig. 6 Normalized horn throat specific impedance, Z throat , as a function of kamouth : (a) comparison of PDE App and PH BE. (b) PDE App-SUB prediction (imaginary part) for different values of port truncation, M

(RB approximation) are important. The pressure reconstruction, for any single wavenumber kvisa0 , is completed in less than 0.5 s. Expansion Chamber We consider here the expansion chamber shown in Fig. 7. We also indicate in Fig. 7 the global ports which in turn demarcate the instantiated components which comprise the system: an inlet, an outlet, two expansion (or contraction) components, and several smaller and larger circular ducts; in all, four archetype components are required. The model parameters are given by μ ≡ (Lpre /a0 , Lec /a0 , Lpost /a0 , aec /a0 , ka0 ) in parameter domain P ≡ [4, 12] × [1.2, 25] × [4, 12] × [1.5, 6.5] × [0, 1.5], where a0 is the inlet and outlet radius, Lpre and Lpost are the respective lengths of the straight ducts before and after the

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Fig. 7 Expansion chamber geometry and system synthesis

expansion chamber, Lec is the length of the expansion chamber, and aec is the radius of the expansion chamber. Our (model) QoI is in this case the transmission loss, TL, as defined in Eq. (4) (TL). The self-consistent verification proceeds following the template described in the context of the flanged horn. We thus turn to verification and validation with results reported in the literature, in particular Selemat and Radavich (henceforth SR) [26]. In our language, the configurations studied in SR correspond to several different values of Lec /a0 —all in our parameter domain—but always aec /a0 = 3.1525. The (dissipation-free) transmission loss will not depend on Lpre /a0 or Lpost/a0 as Lpre /a0 → ∞ and Lpost/a0 → ∞; we choose Lpre /a0 = Lpost/a0 = 8 to ensure relaxation to planewave form yet also control inlet and outlet dissipation effects. (In fact, Lpre /a0 = Lpost /a0 = 8 also appears to closely match the corresponding choice of SR.) All results reported here, unless otherwise indicated, are for Lpre /a0 = Lpost/a0 = 8, Lec /a0 = 22.23, and aec /a0 = 3.1525. Furthermore, given that our model is axisymmetric, we choose M = 6 for our port truncation. SR develops results for a multimodal expansion (hereafter “Analysis”), a Boundary Element (BE) method, and also Experiment. The translation ka0 → f (Hz) is based on c0 = 343 m/s and, from SR, inlet and outlet duct radius a0 = 0.0243 m. We present in Fig. 8a, derived from figure 18 of SR, our PDE App results for M = 6, SR BE, and SR Experiment for the transmission loss as a function of frequency f for Lec /a0 = 22.23; note SR Analysis and SR BE effectively coincide, and hence we include only the former in our plot. In contrast to the planewave approximation, SR Analysis and the PDE App (for M = 6) capture well the higher propagating modes excited as f approaches and exceeds the cut-off frequency: the SR multimodal expansion explicitly includes the higher duct modes; the SCRBE method deduces the necessary modes from the port training procedure, as described in Sect. 3. We do note, however, for the case considered in Fig. 8a, that the resonance at ≈2.86 kHz—the PDE App second-to-last resonance—is not clearly evident in SR BE or SR Experiment. We present in Fig. 8b the TL as predicted by the PDE App in the immediate vicinity of the second-to-last resonance. We also include in Fig. 8b the predictions of our underlying truth FE approximation. We observe that our

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Fig. 8 Expansion chamber transmission loss for Lec /a0 = 22.23: Comparison of PDE App (M = 6) and SR BE, SR Experiment (a) for the full frequency range, and (b) in the immediate neighborhood of the second-to-last resonance

second-to-last resonance is quite narrow and furthermore falls in between the SR BE and SR Experiment points—hence not inconsistent with SR. We also note the good agreement between the SCRBE (PDE App) and truth FE approximations: we can conclude that the PDE App resonance is not an artifact of model order reduction. (In fact, we have also confirmed the resonance at ≈2.86 kHz on a refinement of our truth FE approximation, and we are thus quite certain that the resonance is in general not an artifact of discretization but rather a feature of the exact solution to the Helmholtz equation.) This second-to-last resonance demonstrates the importance of very fast computations, especially for acoustics in which resolution in parameter— in particular, wavenumber—is crucial. Finally, we discuss computational performance (for M = 6, and the core allocations described in Sect. 3.7). The sweep of Fig. 8a—nsweep = 200—is effected

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Fig. 9 Toroidal bend geometry and system synthesis

in less than 1.9 s (wall-clock time); in this case the Schur complement system formation is several times more costly than the Schur complement system solution. The pressure reconstruction for visualization, for any given single wavenumber kvisa0 , is completed in less than 1 s. Hence for this example we again satisfy PDE App Req 2 and Req 3. The proximity of the actual timings to the (self-imposed) limits of the PDE App Requirements is not a coincidence: we choose sufficient GCE cores to accommodate the anticipated wavenumber sweeps. Toroidal Bend We next consider a circular duct with a toroidal bend. The model parameter is given by the 5-tuple μ ≡ (Lpre /a0 , Lpost /a0 , abend/a0 , θbend , ka0 ) in parameter domain P ≡ [1.5, 15] × [1.5, 15] × [1.2, 3] × [30◦, 180◦ ] × [0, 1.8412], where a0 is the inlet and outlet radius, Lpre and Lpost are the respective lengths of the straight ducts before and after the bend, abend is the duct axis radius of curvature in the bend, and θbend is the included angle of the bend (in degrees). Note here (ka0 )max = 1.8412, the first (azimuthal) duct mode cut-off wavenumber. We impose nonzero uniform velocity at the inlet and zero velocity at the outlet. We depict the model parameters and system synthesis in Fig. 9. As always, the black lines indicate global ports. The particular SYSTEM(μ) in Fig. 9 consists of a single velocity inlet component (inlet depicted in green), three duct components (including the zero velocity outlet component), and three bend components. We note that, although for exposition of the SCRBE port reduction procedure we assumed that all ports are paired, in actual practice we permit unpaired ports in particular for imposition of zero normal velocity; the latter is included in the port training procedure as a single-component subsystem. We emphasize that the models of our previous sections in fact correspond to axisymmetric domains, but are treated as fully three-dimensional problems in our PDE App: several components in the library are not axisymmetric, and hence in the training—since we do not prejudge the composition of future systems— all components, even geometrically axisymmetric components, must be treated as fully three-dimensional. The toroidal bend is an example of a necessarily three-dimensional model: the bend component induces three-dimensional behavior quite independent of any symmetries present in the other (e.g., straight circular duct) components in the system. Necessarily three-dimensional models of course

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more clearly illustrate the advantage of model order reduction relative to classical discretization approaches. The self-consistent verification proceeds following the template described in the context of the flanged horn. We thus turn to verification and validation with results reported in the literature, and in particular Félix et al. (henceforth FDN) [10]. The experimental configuration of FDN corresponds in our language to (velocity inlet and outlet boundary conditions and) parameter value Lpre = 20/7(= 2.857), Lpost = 12/7(= 1.714), abend/a0 = 9/7(= 1.286), and θ = 180◦. The translation ka0 → f (in Hz) is based on c0 = 343 m/s and (from FDN) radius a0 = 0.0175 m; the cut-off frequency for the first (azimuthal) duct mode, corresponding to ka0 = 1.8412, is given by fc = 5.76 kHz. In fact, we present results in terms of f/fc , hence independent of speed of sound and radius. FDN develops results based on a multimodal expansion (henceforth “Theory”), a Finite-Difference (FD) approach, and Experiment. FDN also discusses the simple equivalent-length approximation. We present in Fig. 10a, derived from Fig. 10a of FDN, our PDE App results for M = 10, FDN Theory, and FDN Experiment for the modulus of the inlet impedance as a function of normalized frequency f/fc for 0 ≤ f ≤ fc ; we also present the inlet impedance predicted by the equivalent-length theory. (Note we interpret the FDN impedance as specific impedance furthermore normalized by Z0 , as defined in Eq. (4) (II).) We note very good agreement between the PDE App and FDN Theory and FDN Experiment: the location of the resonances is certainly accurate to our 5% criterion; the amplitude of the inlet impedance is accurate to our 5% criterion except in the immediate vicinity of resonances. (For this problem, without any resistive impedances or radiative boundary conditions, only quasi-physical numerical dissipation for the PDE App, and approximate viscothermal losses for the FDN multimodal theory, regularize the Helmholtz problem.) In this three-dimensional geometry with frequencies close to the cut-off we will of course require port truncation M > 6 since otherwise our port space shall be absent modes with azimuthal dependence and in particular the first propagating mode. We present in Fig. 10b the PDE App results for M = 6, 10, and 20 for the inlet impedance as a function of normalized frequency f/fc for 0 ≤ f ≤ fc . As expected, for M = 6 the error is substantial; on the other hand, for M = 10 we include sufficient modes to very faithfully replicate the FDN results. We also observe that for M = 20 there is no further change in the PDE App predictions—a crude form of verification. We caution that “no change with increasing M” can be a misleading error indicator in particular since different models can excite different modes which may appear earlier or later in the sequence of port modes. We now turn to computational performance for port truncation M = 10. The sweep of Fig. 10a—nsweep = 200—is effected in ≈ 3 s (wall-clock time). In this case the formation of the reduced Schur complement linear system is roughly twice as costly as the solution of the reduced Schur complement linear system. The pressure reconstruction, for any given single wavenumber kvis a0 , is completed in ≈ 0.8 s. Here we clearly honor Req 2 and Req 3. We observe that the timings for this (necessarily) three-dimensional model are similar to the timings for our effectively axisymmetric problems (in fact, slightly better, as there are fewer components): in

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Fig. 10 Modulus of inlet impedance of a circular duct with toroidal bend as a function of normalized frequency for the experimental configuration of FDN: (a) Comparison of PDE App (M = 10) and FDN Theory, FDN Experiment, and Equivalent Straight Duct. (b) PDE App convergence with increasing M

general, model reduction will depend on the parametric variation of the pressure field but not on spatial dimensionality per se. Finally, in this necessarily three-dimensional geometry we can perform a meaningful timing comparison between the PDE App and the truth FE approximation. The following software libraries are incorporated in the truth FE implementation: the libMesh FE framework [17], the PETSc linear algebra package [4], and the parallel sparse direct solver MUMPS [2]. We find for our bend, the FDN configuration, that NSYS (μ) = 201,970; the latter in turn yields a truth FE computation time (not including the stitching together of the component meshes) on 8 GCE cores of a little over 13 s—for a single wavenumber. The PDE App thus provides online savings of

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well over two orders of magnitude (respectively, over an order of magnitude) for the QoI wavenumber sweep (respectively, for the pressure visualization) even for this modest system; we would expect even more substantial savings for larger systems with many instantiated components. We note we choose our FE spaces somewhat conservatively to ensure sufficient accuracy (yet not so conservatively as to incur prohibitive offline costs); as a result, our comparison here is slightly biased in favor of SCRBE. We have chosen for this work rather classical parametrized models for which there is a wealth of results—theoretical, computational, and experimental. However, the library includes many necessarily three-dimensional components, from which we can develop complicated three-dimensional parametrized models with dozens of model parameters; examples of the latter include woodwind and brass instruments, and the vocal tract. Acknowledgements We thank Professor Masayuki Yano of University of Toronto for his contributions to the formulation and verification of the PDE Apps, Dr Sylvain Vallaghé of Akselos for his generous assistance in the FE verification of SCRBE resonances, Professor Kathrin Smetana of the University of Twente for valuable discussions related to the transfer eigenproblem, Professor Peter Dahl of University of Washington and Professor Jer-Ming Chen of Singapore University of Technology and Design (SUTD) for insightful reviews of earlier acoustic PDE Apps, Thomas Leurent of Akselos for his strong support of PDE Apps for education, and Thuc Nguyen of Akselos for his contributions to the web platform. This work was supported by the Swiss Confederations Innovation Promotion Agency (CTI) under Grant 17802.1 PFIW-IW (JB), ONR Contracts N00014-11-1-0713 and N00014-17-1-2077, OSD/AFOSR Grant FA955009-0613, SUTD International Design Center, and an MIT Ford Professorship (ATP).

References 1. B. Almroth, P. Stern, F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA J. 16(5), 525–528 (1978) 2. P.R. Amestoy, A. Guermouche, J.Y. L’Excellent, S. Pralet, Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006) 3. I. Babuška, R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Model. Simul. 9, 373–406 (2011) 4. S. Balay, W.D. Gropp, L. Curfman McInnes, B.F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, ed. by E. Arge, A.M. Bruaset, H.P. Langtangen (Birkhäuser Press, Boston, 1997), pp. 163–202 5. M. Barrault, Y. Maday, N. Nguyen, A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. CR Acad. Sci. Paris Ser. I 339, 667–672 (2004) 6. P. Binev, A. Cohen, R. Dahmen, G. Petrova, P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43(3), 1457–1472 (2011) 7. D.T. Blackstock, Fundamentals of Physical Acoustics, 1st edn. (Wiley, Hoboken, 2000) 8. R.J. Craig, M. Bampton, Coupling of substructures for dynamic analyses. AIAA J. 3(4), 678– 685 (1968) 9. J.L. Eftang, A.T. Patera, Port reduction in component-based static condensation for parametrized problems: approximation and a posteriori error estimation. Int. J. Numer. Methods Eng. 96(5), 269–302 (2013)

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10. S. Félix, J.-P. Dalmont, C.J. Nederveen, Effects of bending portions of the air column on the acoustical resonances of a wind instrument. J. Acoust. Soc. Am. 131(5), 4164–4172 (2012) 11. U. Hetmaniuk, R. Lehoucq, A special finite element method based on component mode synthesis. Math. Model. Numer. Anal. 44(3), 401–421 (2010) 12. W.C. Hurty, On the dynamics of structural systems using component modes. AIAA Paper No. 64–487 (1964) 13. D.B.P. Huynh, D.J. Knezevic, J.W. Peterson, A.T. Patera, High-fidelity real-time simulation on deployed platforms. Comput. Fluids 43(1), 74–81 (2011) 14. P. Huynh, D.J. Knezevic, A.T. Patera, A static condensation reduced basis element method: approximation and a posteriori error estimation. Math. Model. Numer. Anal. 47(1), 213–251 (2013) 15. P. Huynh, D.J. Knezevic, A.T. Patera, A static condensation reduced basis element method: complex problems. Comput. Methods Appl. Mech. Eng. 259, 197–216 (2013) 16. F. Ihlenburg, I. Babuška, Finite element solution of the Helmholtz equation with high wave number. Part I: The h-version of the FEM. Comput. Math. Appl. 30(9), 9–37 (1995) 17. B.S. Kirk, J.W. Peterson, R.M. Stogner, G.F. Carey, libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 23(3–4), 237–254 (2006) 18. Y. Maday, E.M. Rønquist, The reduced basis element method: application to a thermal fin problem. SIAM J. Sci. Comput. 26(1), 240–258 (2004) 19. K.J. McMahon, A comparison of the transfer matrix method and the finite element method for the claculation of the transmission loss in a single expansion chamber muffler. Master’s thesis, RPI Hartford, December 2014 20. J.L. Munjal, Acoustics of Ducts and Mufflers, 2nd edn. (Wiley, Hoboken, 2014) 21. A.K. Noor, J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 455–462 (1980) 22. A Pinkus, N-Widths in Approximation Theory (Springer Science and Business Media, New York, 1985) 23. J.T. Post, E.L. Hixson, A modeling and measurement study of acoustic horns. Ph.D. Thesis, University of Texas at Austin, May 1994 24. G. Rozza, D.B.P. Huynh, A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008) 25. A. Selamet, Z.L. Ji, Acoustic attenuation performance of circular expansion chambers with extended inlet/outlet. J. Sound Vib. 223(2), 197–212 (1999) 26. A. Selamet, P.M. Radavich, The effect of length on the acoustic attenuation performance of concentric expansion chambers: an analytical, computational and experimental investigation. J. Sound Vib. 201(4), 407–426 (1997) 27. K. Smetana, A.T. Patera, Optimal local approximation spaces for component-based static condensation procedures. SIAM J. Sci. Comput. 38(5), A3318–A3356 (2016) 28. K. Veroy, C. Prud’homme, D.V. Rovas, A.T. Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. AIAA Paper No. 2003–3847 (2003), pp. 1–18 29. E.L. Wilson, The static condensation algorithm. Int. J. Numer. Methods Eng. 8(1), 198–203 (1974)

Sub-voxel Perfusion Modeling in Terms of Coupled 3d-1d Problem Karl Erik Holter, Miroslav Kuchta, and Kent-André Mardal

Abstract We study perfusion by a multiscale model coupling diffusion in the tissue and diffusion along the one-dimensional segments representing the vasculature. We propose a block-diagonal preconditioner for the model equations and demonstrate its robustness by numerical experiments. We compare our model to a macroscale model by Tofts [Modelling in DCE MRI, 2012].

1 Introduction The micro-circulation is altered in diseases such as cancer and Alzheimer’s disease, as demonstrated with modern perfusion MRI. In cancer, the so-called enhanced permeability and retention (EPR) effect describes the fact that the smaller vessels in a tumor are leaky, highly permeable vessels that enable the tumor cells to grow quicker than normal cells. In Alzheimer’s disease (AD), the opposite is alleged to happen. According to [8], hypoperfusion is a precursor to AD, and the cause of the pathological cell-level changes occuring in AD. This could also explain why various kinds of heart disease

K. E. Holter University of Oslo, Department of Informatics, Oslo, Norway Simula Research Laboratory, Fornebu, Norway e-mail: [email protected] M. Kuchta University of Oslo, Department of Mathematics, Oslo, Norway e-mail: [email protected] K.-A. Mardal () University of Oslo, Department of Mathematics, Oslo, Norway Simula Research Laboratory, Fornebu, Norway e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_2

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are risk factors for AD, as changes in the blood pressure would affect the perfusion of the brain. The vasculature (e.g. idealized as a system of pipes) and the surrounding tissue are clearly three-dimensional. However, the fact that in many applications the radii of the vessels are negligible compared to their lengths, permits reducing their governing equations to models prescribed on (one dimensional) curves where the physical radius R enters as a parameter. In this paper, we consider a coupled 3d-1d system ∂u = DΩ ΔΩ u + δΓ βλ ∂t ∂ uˆ = DΓ ΔΓ uˆ − βλ ∂t

in Ω, in Γ,

ˆ λ = β(ΠR u − u)

(1)

in Γ.

Here, δΓ is the Dirac measure on Γ ; u, uˆ are the concentrations in the tissue domain Ω and the one-dimensional vasculature representation Γ ; and DΩ , DΓ are conductivities on the respective domains. The last equation is then a generalized Starling’s law; relating uˆ and the value ΠR u of u averaged over the idealized cylindrical vessel surface centered around Γ .1 The equation thus represents a coupling between the domains. Variants of the system (1) have been used to study coupling between tissue and vasculature flow in numerous applications. In [7], a steady state limit of the system is used to numerically investigate oxygen supply of the skeletal muscles. Finite differences were used for the discretization. Using the method of Green’s functions, [17, 18] and [3] studied oxygen transport in the brain and tumors. Transport of oxygen inside the brain was also investigated by Linninger et al. [12] using the finite volume method (FVM) and [6] using the finite element method (FEM) for the 3d diffusion problem and FVM elsewhere. In these studies the 1d problem was transient. More recently, coupled models discretized entirely by FEM were applied to study cancer therapies, see e.g. [15] and references therein. The mathematical foundations of these works are rooted in the seminal contributions of [4, 5] where well-posedness of the following problem is analyzed −ΔΩ u + (uˆ − ΠR u)δΓ = f δΓ

in Ω

−ΔΓ uˆ − (uˆ − ΠR u) = g

in Γ.

(2)

x ∈ Γ and CR (x) be a circular crossection of the vessel surface with a plane  y ∈ R3 , (y − x) · dΓ ds (x) = 0 defined by the tangent vector of Γ at x. The surface avarage ΠR u of u is then defined by  (ΠR u)(x) = |CR (x)|−1 u(y) dy.

1 Let

CR (x)

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37

The presence of the measure term in (2) requires the use of non-standard spaces in the analysis. In [5], the variational formulation of the problem is proven to be well-posed using weighted Sobolev spaces. In particular, the solution u is sought in Hα1 (Ω), α > 0 while the test functions v for the first equation in (2) are taken from 1 (Ω). With this choice, the right-hand side f δ , v as well as the trace operator H−α Γ v → v|Γ are well defined, while the reduced regularity of u is sufficient to for the average ΠR u to make sense. As shown in [4], use of FEM for the formulation in weighted spaces yields optimal rates if the computational mesh is gradually refined towards Γ (graded meshes). Another approach to the analysis of (2) has recently been suggested in the numerical study [2]. Building on the analysis of [9] for the elliptic problem with a 0 dimensional Dirac right-hand side, the wellposedness of the problem was shown with trial spaces W 1,p (Ω), p = 3 − d2 and test spaces W 1,q (Ω), p−1 + q −1 = 1, and quasi-optimal error estimates for FEM shown in the norms which excluded a fixed neighborhood of Γ of radius R. In studying AD or EPR, the physical parameters may vary across several orders of magnitude while small or large time steps can be desirable depending on the time scales of interest. The solution algorithm for the employed model equations is thus required to be robust with respect to these parameters as well independent of the discretization. For (the transient version of) (2) the construction of such algorithms is complicated by the non-standard spaces on the domain Ω. A potential remedy for the problem can be introduction of a Lagrange multiplier which enforces the coupling between the domains with the goal of confining the non-standard spaces to the smaller domain Γ . This idea has been used by [11] to analyze robust preconditioners for 2d-1d coupled problems based on operators in fractional Sobolev spaces, in particular, (−ΔΓ )− , while numerical experiments reported in [10] suggest that for suitable exponents (−ΔΓ )s defines a preconditioner for the Schur complement of a 3d-1d coupled system with a trace constraint.2 We note that in both cases off-the-shelf methods were used as preconditioners for the operators on Ω. The system (1) includes an additional variable λ for the coupling constraint, cf. (2). We therefore aim to apply the techiques of [10, 11] to construct a meshindependent preconditioner for the problem, while the ideas of operator preconditioning [14] are used to ensure robustness with respect to the physical parameters and the time-stepping. The rest of the paper is organized as follows. Section 2 identifies the structure of the preconditioner. In Sect. 3 we discuss discretization of the proposed operator and report numerical experiments which demonstrate the robust properties. In Sect. 4 the system (1) is used to model tissue perfusion using a realistic geometry of the rat cortex. Conclusions are finally summarized in Sect. 5. 1 2

2 Note that in (2) and (1) the constraint/coupling is defined in terms of a surface averaging operator ΠR .

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2 Preconditioner for the Coupled Problem In the following we let Ω be a bounded domain in Rd , d = 2 or 3 and Γ be a subdomain of Ω of dimension 1. By L2 (D) we denote the space of square-integrable functions over D and H 1 (D) is the space of functions with first order derivatives in L2 (D). Discretizing (1) in time by backward-Euler discretization the problem to be solved at each temporal level is of the form Au = f with ⎤ 0 kβΠR∗ I − kDΩ ΔΩ ⎥ ⎢ A=⎣ 0 I − kDΓ ΔΓ kβI ⎦ kβI −k. kβΠR ⎡

(3)

and k being the time step size. Note that in order to obtain a symmetric problem the operator A uses the adjoint ΠR∗ of the averaging operator ΠR instead of the trace, cf. (1). The choice results in modeling error of order O(R). To motivate the structure of the preconditioner let us consider a 3 × 3 matrix ⎡

⎤ 1 + α1 0 β1 ⎢ ⎥ A = ⎣ 0 1 + α2 β2 ⎦ β1 β2 −γ where α1 , α2 , β1 , β2 , and γ are assumed to be positive. It can then be shown that with ⎤ ⎡ 0 0 (1 + α1 )−1 ⎥ ⎢ B=⎣ 0 0 (1 + α2 )−1 ⎦ 2 2 −1 2 2 −1 0 0 (γ + β1 + β2 ) + (γ + β1 /α1 + β2 /α2 ) the condition number of BA is bounded independent of the parameters. With this in mind we propose that A can be preconditioned by a block-diagonal operator ⎡

⎤ (I − kDΩ ΔΩ )−1 0 0 ⎢ ⎥ B=⎣ 0 (I − kDΓ ΔΓ )−1 0 ⎦ , 0 0 S

(4)

where S = S1−1 + S2−1

(5)

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39

and S1 = γ I + (kβ)2 ΠR ΠR∗ + (kβ)2I, S2 = γ I + (kβ)2 ΠR (−kDΩ ΔΩ )−1 ΠR∗ + (kβ)2(−kDΓ ΔΓ )−1 .

(6)

The operator B could be rigorously derived within operator preconditioning [14] as a Riesz map preconditioner for A viewed as an isomorphism from V (Ω) × Vˆ (Γ ) × Q(Γ ) to its dual space. In [11] the framework was applied to a system of two elliptic problems coupled by a 2d-1d constraint. For (1) extension of the analysis to parabolic problems would be required. In [13] robust preconditioners for time-dependent Stokes problem were analyzed as operators between sums of (parameter) weighted Sobolev spaces. Similarly, the structure of (5) suggests that Q = Q1 + Q2 with Q1 , Q2 being suitable interpolation spaces. However, here we shall not justify Q1 and Q2 (and the preconditioner B) theoretically. Instead, Q1 , Q2 are characterized and robustness of B is demonstrated by numerical experiments.

3 Discrete Preconditioner Considering (4), both (I − kDΩ ΔΩ )−1 and (I − kDΓ ΔΓ )−1 can be realized with off-the-shelf multilevel algorithms and the crucial question is thus how to construct S efficiently. Note that assembling S1 and in particular S2 might be too costly or even prohibitive, cf. (−ΔΩ )−1 in S2 . However, following (6) the preconditioner can be realized if operators spectrally equivalent to ΠR (−ΔΩ )−1 ΠR∗ and ΠR ΠR∗ are known and if the inverse (action) of the resulting approximations to S1 and S2 is inexpensive to compute.

3.1 Auxiliary Operators If Ω ⊂ R2 and ΠR is understood as the trace operator, the mapping properties of trace as a bounded surjective operator H 1 (Ω) → H (Γ ) can be used to show that ΠR (−ΔΩ )−1 ΠR∗ is spectrally equivalent with (−ΔΓ )− . At the same time ΠR ΠR∗ = ΠR (−Δ)0Ω ΠR∗ requires characterizing the space of traces of functions in L2 (Ω). We shall demonstrate by a numerical experiment that a spectrally equivalent operator is here provided by an operator h−1 Ih . Let Ωh , Γh be triangulations of Ω and Γ such that Γh consists of a subset of edges of the elements Ωh , cf. Fig. 1. Further let Vh , Qh be finite element spaces of continuous linear Lagrange elements on Ωh and Γh respectively. Finally we 1 2

1 2

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K. E. Holter et al.

(1, 1, 1)

(0, 0, 0)  d Fig. 1 Geometries used in preconditioning numerical experiments. The domain is Ω = 0, 1 while in order to prevent symmetries Γ (pictured in red) always features a branching point. Triangulation of Γ is made up of edges of the cells that triangulate Ω

consider the eigenvalue problem: Find λ ∈ R, u ∈ Vh , p ∈ Qh such that 





uv + Ω

pΠR v = λ Γ





(7) −1

qΠR u = λ Γ

∀v ∈ Vh ,

uv Ω

h

∀q ∈ Qh .

pq

Γ

Table 1 shows the spectral condition number κ = max |λ|/ min |λ| of the linear systems (7). For all the considered resolutions h the value of κ is bounded. As the mapping properties of ΠR ΠR∗ and ΠR (−ΔΩ )−1 ΠR∗ in case Ω ⊂ R3 are not trivially obtained from the continuous analysis, we again resort to finding the suitable approximations by numerical experiments. Similar to the two dimensional case, the first operator with ΠR having the constant radius R = 0.02 is found to be spectrally equivalent with h−1 Ih . Following [10], an approximation to ΠR (−ΔΩ )−1 ΠR∗ is searched for as a suitable power s < 0 of (−ΔΓ + IΓ ). More precisely, we look for the exponent yielding the most h-stable condition number of the eigenvalue problem: Find λ ∈ R, u ∈ Vh , p ∈ Qh such that 





∇u · ∇v + Ω

pΠR v = λ Γ

∇u∇v + uv 



qΠR u = λ Γ

∀v ∈ Vh ,

Ω

(8) p(−ΔΓ + IΓ ) q s

∀q ∈ Qh .

Γ

In (8) the powers are computed using the spectral decomposition of the operator −ΔΓ + IΓ .

Perfusion Modeling by Coupled 3d-1d System

41

Table 1 Spectral condition numbers of the eigenvalue problems related to approximations of ΠR ΠR∗ (Eq. (7)) and ΠR (−ΔΩ )ΠR∗ (Eq. (8)) Ω ⊂ R2 Ω ⊂ R3 1/ h 1/ h 32 64 128 256 512 1024 4 8 16 32 64 128 Equation (7) 4.36 4.33 4.35 4.36 4.36 4.35 4.26 5.22 5.58 4.44 4.85 4.85 Equation (8) 8.80 8.84 8.84 8.86 8.88 8.87 11.04 8.25 7.44 9.02 10.30 11.25 In the two-dimensional case (8) uses s = − 12 in agreement with the mapping properties of the continuous trace operator. Results for s = −0.55 are reported in the three dimensional case. On the finest triangulation dim Vh ∼ 106 and dim Qh ∼ 103 when d = 2 and dim Qh ∼ 102 for d =3

We shall not present here the results for the entire optimization problem and only report on the optimum which is found to be s = −0.55. For this value Table 1 shows the history of the condition numbers of system (8) using again R = 0.02. There is a slight growth by a constant increment on the finer meshes, however, the final condition number is comparable with that obtained on the coarsest mesh. We note that we have not investigated the behaviour of the exponent on different curves or with variable radius. This subject is left for future works.

3.2 Discrete Preconditioner for the Coupled Problem Applying the proposed preconditioner (4) of the coupled 3d-1d problem (3) requires evaluating the inverse of operators S1 and S2 in (6). The former is readily computed since, due to the suggested equivalence of ΠR ΠR∗ and h−1 Ih , the matrix representation of S1 is essentialy a rescaled mass matrix. For S2 we show that if (−ΔΓ + IΓ )s is computed from the spectral decomposition then the inverse S2−1 can be computed in a closed form. Let A, M be the n × n matrix representations of Galerkin approximations of −ΔΓ + IΓ and I in the space Qh ⊂ H 1 (Γ ). Following [11] the matrix representation of (−ΔΓ + IΓ )sh is Hs = MU Λs (MU ) where the matrices U , Λ solve the generalized eigenvalue problem AU = MU Λ such that U  MU = I . Using Hs it is easily established that the matrix representation of S2 is γ H0 + (kβ)2(kDΩ )−1 H− 1 + (kβ)2(kDΓ )−1 H−1 . 2

As Hs−1 = U Λ−s U  the inverse of the S2 matrix is given by −1  1 U γ Λ0 + (kβ)2(kDΩ )−1 Λ− 2 + (kβ)2(kDΓ )−1 Λ−1 U .

(9)

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Using the spectral decomposition, the cost of setting up the preconditioner S is determined by the cost of solving the generalized eigenvalue problem for U and Λ. This practically limits the construction to systems where dim Qh ∼ 103 . However, for the problems considered further, this limitation does not present an issue. In particular, the preconditioner can be setup on the discretization of vasculature of the cortex tissue used in Sect. 4 which contains approximately twenty thousand vertices. With a discrete approximation of S we can finally address the question of B being a robust preconditioner for (3). Motivated by (1) we do not vary all the paramaters, and instead, set γ = 1 in A. Morever, the conductivity on Ω is taken as unity and only variable DΓ > 1 is considered mimicking the expected faster propagation along the one-dimensional domain. Finally, the time step k and the coupling constant β shall take values between 10−4 and 10−8. For a fixed choice of parameters DΓ , β, k, the preconditioned problem BAx = Bf is considered on geometries from Fig. 1 and discretized with continuous linear Lagrange elements. The resulting linear system is then solved with the MinRes method where the iterations are stopped once the preconditioned residual norm is less than 10−10 in magnitude. The observed iteration counts are reported in Table 2. For both 2d-1d and 3d1d coupled problems the iterations can be seen to be bounded in the discretization parameter. Moreover the preconditioner performs almost uniformly in the considered ranges of DΓ and β while there is a clear boundedness in k as well. We note that these conclusions are not significantly altered if the ranges for k and β are extended to 1. Having demonstrated the numerical stability of our model, we next test it on the same problem as [19], namely a bloodborne tracer perfusing and later being cleared from tissue. While [19] considers this problem on a macroscopic scale, we model it on the micro-scale, where individual blood vessels can be resolved as part of our 1d domain.

4 Perfusion Experiment In [16], a 0.7 × 0.7 × 0.7 mm piece of mouse brain microvasculature was imaged using two-photon microscopy. To obtain a realistic geometry for our model, we used this data to generate a 3d mesh of the extravascular space in which vessel segments corresponded to 1d mesh edges (Fig. 2). The radius of the blood vessels is used as the radius R in the definition of the averaging operator ΠR , and ranged between 1 and 15 micron. To model a small region of tissue being perfused by a bloodborne tracer, we use initial conditions of u, uˆ = 0, and a boundary condition of uˆ = 1 mol L on the part of the boundary corresponding to inlet vessels. To model clearance, the inlet boundary condition was swapped from uˆ = 1 to uˆ = 0 after a third of the simulation time had passed.

Perfusion Modeling by Coupled 3d-1d System

43

Table 2 Number of iterations of MinRes method on (3) using (4) as preconditioner with S approximated using (9) DΓ

β

10−8

100

10−6

10−4

10−8

102

10−6

10−4

10−8

104

10−6

10−4

10−8

106

10−6

10−4

k 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4 10−8 10−6 10−4

2d-1d 1/ h 32 64 11 10 15 13 16 11 11 10 15 13 15 11 11 10 15 13 16 11 11 10 15 13 15 11 11 10 15 13 15 11 11 10 15 13 15 11 11 10 15 13 15 11 11 10 15 13 15 11 11 10 15 13 15 11 11 10 15 13 14 10 11 10 15 13 14 10 11 10 15 13 14 10

128 10 12 12 10 12 12 10 12 12 10 12 12 10 12 12 10 12 12 10 12 12 10 12 12 10 12 12 10 12 10 10 12 10 10 12 10

256 8 10 13 8 10 13 8 10 13 9 10 13 9 10 13 9 10 13 8 10 13 8 10 13 8 10 13 8 10 9 8 10 9 8 10 9

512 7 8 13 7 8 13 7 8 13 7 8 13 7 8 13 7 8 13 7 8 12 7 8 12 7 8 12 7 8 8 7 8 8 7 8 8

1024 7 8 14 7 8 14 7 8 14 7 8 13 7 8 13 7 8 13 7 8 10 7 8 10 7 8 10 7 8 5 7 8 5 7 8 5

3d-1d 1/ h 4 8 8 13 10 14 12 16 8 13 10 14 12 16 8 13 10 14 12 16 8 13 10 14 11 16 8 13 10 14 11 16 8 13 10 14 12 16 8 13 10 14 11 16 8 13 10 14 11 16 8 13 10 14 11 16 8 13 10 14 11 15 8 12 10 14 11 15 8 12 10 14 11 16

16 12 16 18 12 16 18 12 16 18 12 16 18 12 16 18 12 16 18 12 16 18 12 16 18 12 16 18 12 16 18 12 16 18 12 16 18

32 11 16 16 11 16 16 11 16 16 11 16 13 11 16 13 11 16 13 11 16 13 11 16 13 11 16 13 11 16 13 11 16 13 11 16 13

64 11 15 13 11 15 13 11 15 13 11 15 13 11 15 13 11 15 13 11 15 13 11 15 13 11 15 13 11 15 11 11 15 10 11 15 10

(Left) 2d-1d coupled problem and (right) 3d-1d coupled problem from Fig. 1 are considered

128 10 13 14 10 13 14 10 13 14 10 13 14 10 13 14 10 13 14 10 13 14 10 13 14 10 13 14 10 13 11 10 13 11 10 13 11

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Fig. 2 Example results shown on a (left) clip of the 3d domain and (right) on a slice. Notice the ‘halos’ of increased concentration immediately around the vessels

As parameters, we use DΓ = 6.926 × 107 μm2 /s, DΩ = 1.87 × 102 μm2 /s and β = 50 μm/s. We used k = 1s after verifying that reducing it did not significantly affect our results. In this experiment, it was unneccessary to use the preconditioner described in Sect. 3 since the problem size was small enough to allow for use of a direct solver. Tofts [19] assumes a relation Ktrans ∂Ct = (Cv − Ct ) ∂t ν

(10)

between the pixel tissue concentration Ct and the pixel vessel concentration Cv for some constant Ktrans . Here, ν is the vascular volume fraction, which in our geometry is about 0.76%. Our geometry is of a size comparable to a single pixel in [19], so Ct and Cv correspond to the normalized averages  u Ct = Ω Ω1

 πR 2 uˆ and Cv = Γ . 2 Γ πR

We computed Ktrans by solving for u, uˆ using our model, and then computing Ct , Cv as given above, and defining Ktrans such that Eq. (10) holds at each time point. This makes Ktrans a function of time, with units seconds−1 (Fig. 3).

Perfusion Modeling by Coupled 3d-1d System

3

45

·10−3

2.9

·10−3

Ktrans [min−1 ]

Ktrans [min−1 ]

2.8 2.6 2.4 2.2 2

2.8 2.7 2.6 2.5

1.8 0

1

2 3 t [seconds−1 ]

4

5

0

20

40 60 80 t [seconds−1 ]

100

Fig. 3 Behavior of Ktrans on short and long time scales

4.1 Discussion of Perfusion Experiment Our value for Ktrans is not entirely constant. There is a small variation in time, as perfusion seems to be somewhat faster when the extravascular space is completely empty of the tracer. As it starts getting saturated, perfusion slows down somewhat. This translates into Ktrans decreasing by about 20% over a period of about 5 min, from 0.0028 min−1 to 0.0023 min−1 . In [20], Ktrans was estimated in healthy and cancerous human brain tissue from MRI scans. In healthy tissue, they estimate Ktrans to be between 0.003 min−1 and 0.005 min−1 , that is, slightly higher than our results. There are several possible explanations for this difference. One might be that in our model, vascular transport is modeled as exceptionally fast diffusion for convenience, whereas in reality it occurs by convection. However, in both cases 1d transport is very fast compared to the 3d transport and the 1d-3d exchange. Further, Ktrans is defined in terms of the 1d-3d exchange alone, so non-extreme variations in the 1d transport seem unlikely to be relevant. Another possibility might be that the data of [20] are taken from human brain tissue, while our vasculature is taken from a mouse brain tissue, likely from a different region of the brain. A third reason might be our diffusion constants not exactly matching the tracer used by Zhang et al. [20]. In further work, it would be interesting to incorporate convective transport into the model and see if better agreement with the experimental data is observed. A suitable starting point here is [1], who derive a convection-diffusion type system (equations (3a), (3b)) by assuming that the blood flow qˆ in a segment is laminar and follows Poiseuille’s law qˆ = R 4 C∇ p. ˆ

46

K. E. Holter et al.

DΓ = 6.926 · 108 DΓ = 6.926 ·

0.8

0.6

mol/L

mol/L

0.8

1

μm2 s μm2 s 2 109 μm s

DΓ = 6.926 · 107

0.4

0.2 0

80

100

0.8

0.4

0

40 60 t [seconds−1]

DΩ = 1.87 ·

1

β = 5 · 100 s− 2 1

DΩ = 1.87 · 102

0.6

0.2 20

1

μm2 s μm2 s 2 103 μm s

DΩ = 1.87 · 101

mol/L

1

β = 5 · 101 s− 2 − 12

β = 5 · 10 s 2

0.6 0.4 0.2

20

40 60 t [seconds−1]

80

100

0

20

40 60 t [seconds−1]

80

100

Fig. 4 Plots of variation in Ct when different parameters are varied

4.2 Parameter Sensitivity Analysis We carried out a rudimentary parameter sensitivity analysis by varying the three parameters DΓ , DΩ , β by a factor 10 and seeing how that affected the tissue concentration Ct . Specifically, we started from a baseline of 2 2 μm2 , β = 5 · 101 s− , and for each DΓ = 6.926 · 108 μm s , DΩ = 1.87 · 10 s parameter, increased or decreased it by a factor 10. The results are shown in Fig. 4. They indicate that Ct depends more strongly on β and DΓ than on DΩ for the set of parameters considered here. 1 2

5 Conclusions A coupled 3d-1d system with an additional unknown enforcing the coupling between the domains was used as a model of tissue perfusion. For the system we proposed a robust preconditioner and demonstrated its properties through numerical experiments. Further, we have shown that the model can be applied to a physiological problem with reasonable results.

References 1. L. Cattaneo, P. Zunino, A computational model of drug delivery through microcirculation to compare different tumor treatments. Int. J. Numer. Methods Biomed. Eng. 30(11), 1347–1371 (2014) 2. L. Cattaneo, P. Zunino, Numerical investigation of convergence rates for the FEM approximation of 3D-1D coupled problems, in Numerical Mathematics and Advanced ApplicationsENUMATH 2013 (Springer, Berlin, 2015), pp. 727–734 3. S.J. Chapman, R.J. Shipley, R. Jawad, Multiscale modeling of fluid transport in tumors. Bull. Math. Biol. 70(8), 2334 (2008) 4. C. D’Angelo, Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: applications to one-and three-dimensional coupled problems. SIAM J. Numer. Anal. 50(1), 194–215 (2012)

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5. C. D’Angelo, A. Quarteroni, On the coupling of 1D and 3D diffusion-reaction equations: application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18(8), 1481–1504 (2008) 6. Q. Fang, S. Sakadži´c, L. Ruvinskaya, A. Devor, A.M. Dale, D.A. Boas, Oxygen advection and diffusion in a three dimensional vascular anatomical network. Opt. Express 16(22), 17530 (2008) 7. D. Goldman, A.S. Popel, A computational study of the effect of capillary network anastomoses and tortuosity on oxygen transport. J. Theor. Biol. 206(2), 181–194 (2000) 8. C. Jack, Cerebrovascular and cardiovascular pathology in Alzheimer’s disease. Int. Rev. Neurobiol. 84, 35–48 (2009) 9. T. Koppl, B. Wohlmuth, Optimal a priori error estimates for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 52(4), 1753–1769 (2014) 10. M. Kuchta, K.-A. Mardal, M. Mortensen, On preconditioning saddle point systems with trace constraints coupling 3D and 1D domains–applications to matching and nonmatching FEM discretizations. arXiv:1612.03574 (2016, preprint) 11. M. Kuchta, M. Nordaas, J.C. Verschaeve, M. Mortensen, K.-A. Mardal, Preconditioners for saddle point systems with trace constraints coupling 2D and 1D domains. SIAM J. Sci. Comput. 38(6), B962–B987 (2016) 12. A. Linninger, I. Gould, T. Marinnan, C.-Y. Hsu, M. Chojecki, A. Alaraj, Cerebral microcirculation and oxygen tension in the human secondary cortex. Ann. Biomed. Eng. 41(11), 2264–2284 (2013) 13. K.-A. Mardal, R. Winther, Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 98(2), 305–327 (2004) 14. K.-A. Mardal, R. Winther, Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18(1), 1–40 (2011) 15. M. Nabil, P. Decuzzi, P. Zunino, Modelling mass and heat transfer in nano-based cancer hyperthermia. R. Soc. Open Sci. 2(10), 150447 (2015) 16. S. Sakadži´c, E.T. Mandeville, L. Gagnon, J.J. Musacchia, M.A. Yaseen, M.A. Yucel, J. Lefebvre, F. Lesage, A.M. Dale, K. Eikermann-Haerter, et al., Large arteriolar component of oxygen delivery implies a safe margin of oxygen supply to cerebral tissue. Nat. Commun. 5, 5734 (2014) 17. T. Secomb, R. Hsu, N. Beamer, B. Coull, Theoretical simulation of oxygen transport to brain by networks of microvessels: effects of oxygen supply and demand on tissue hypoxia. Microcirculation 7(4), 237–247 (2000) 18. T.W. Secomb, R. Hsu, E.Y. Park, M.W. Dewhirst, Green’s function methods for analysis of oxygen delivery to tissue by microvascular networks. Ann. Biomed. Eng. 32(11), 1519–1529 (2004) 19. P.S. Tofts, T1-weighted DCE imaging concepts: modelling, acquisition and analysis. Signal 500(450), 400 (2010) 20. N. Zhang, L. Zhang, B. Qiu, L. Meng, X. Wang, B.L. Hou, Correlation of volume transfer coefficient Ktrans with histopathologic grades of gliomas. J. Magn. Reson. Imaging 36(2), 355– 363 (2012)

Iterative Linearisation Schemes for Doubly Degenerate Parabolic Equations Jakub Wiktor Both, Kundan Kumar, Jan Martin Nordbotten, Iuliu Sorin Pop, and Florin Adrian Radu

Abstract Mathematical models for flow and reactive transport in porous media often involve non-linear, degenerate parabolic equations. Their solutions have low regularity, and therefore lower order schemes are used for the numerical approximation. Here the backward Euler method is combined with a mixed finite element method, which results in a stable and locally mass-conservative scheme. At each time step one has to solve a non-linear algebraic system, for which one needs adequate iterative solvers. Finding robust ones is particularly challenging here, since the problems considered are double degenerate (i.e. two type of degeneracies are allowed: parabolic-elliptic and parabolic-hyperbolic). Commonly used schemes, like Newton and Picard, are defined either for nondegenerate problems, or after regularising the problem in the case of degenerate ones. Convergence is guaranteed only if the initial guess is sufficiently close to the solution, which translates into severe restrictions on the time step. Here we discuss an iterative linearisation scheme which builds on the L-scheme, and does not employ any regularisation. We prove its rigorous convergence, which is obtained for Hölder type non-linearities. Finally, we present numerical results confirming the theoretical ones, and compare the behaviour of the proposed scheme with schemes based on a regularisation step.

J. W. Both · I. S. Pop () University of Bergen, Department of Mathematics, Bergen, Norway Hasselt University, Faculty of Sciences, Diepenbeek, Belgium e-mail: [email protected]; [email protected] K. Kumar · F. A. Radu University of Bergen, Department of Mathematics, Bergen, Norway e-mail: [email protected]; [email protected] J. M. Nordbotten University of Bergen, Department of Mathematics, Bergen, Norway Princeton University, Department of Civil and Environmental Engineering, Princeton, NJ, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_3

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1 Introduction We consider the following non-linear, degenerate parabolic equation   ∂t b(u(t, x)) − ∇ · ∇u(t, x) = f (t, x),

t ∈ (0, T ], x ∈ Ω,

(1)

with given functions b : R → R and f : (0, T ] × Ω → R. Ω is a bounded domain in Rd , d ∈ {1, 2, 3}, having a Lipschitz continuous boundary ∂Ω and T is the final time. Initial and boundary conditions (for simplicity the latter are assumed to be of homogeneous Dirichlet type) complete the problem. Equation (1) is the transformed Richards equation after applying the Kirchhoff transformation in the absence of gravity (see e.g. [21]) or a diffusion equation with equilibrium sorption modelled by a Freundlich isotherm. Solving (1) is of interest for many applications of societal relevance, like environmental pollution, CO2 storage or geothermal energy extraction. A particular feature of (1) is that the problem may become degenerate, namely change its type from parabolic into elliptic or hyperbolic. One consequence of this is that the solutions typically lack regularity. Here we assume that b(·) is monotone increasing and Hölder continuous, which means that two types of degeneracy are allowed in (1). The first is when the derivative of b(·) vanishes (fast diffusion) and the second when it blows up (slow diffusion). In particular, here the vanishing of b (·) may occur on intervals. Since solutions to degenerate parabolic equations have low regularity (see [1]), low order discretisation methods are well suited for the numerical approximation of the solution. Here we combine the backward Euler (BE) method for the time discretisation with the mixed finite element method (MFEM). For the rigorous convergence analysis of the method we refer to [21] and the references therein. The resulting is a scheme that is both stable and locally mass-conservative. In this paper we discuss iterative solvers for the non-linear algebraic systems arising at each time step after the complete discretisation of (1). Observe that although referring specifically to the MFEM approach, the non-linear solvers presented here can be also applied to other spatial discretisations, like finite volumes, conforming or discontinuous Galerkin finite elements. The literature on non-linear solvers for (1) is very extensive, but covers in particular non-degenerate problems, or the case when b(·) is Lipschitz continuous. We refer to [3, 18] for Newton’s scheme, and to [6] for the modified Picard scheme. A combination of both is discussed in [12, 17]. Also, the Jäger-Kaˇcur scheme was introduced in [11]. We refer to [20] for the analysis of the Newton, modified Picard and the Jäger-Kaˇcur schemes for BE/MFEM discretisations. Recently, in [4] the capillary pressure and the saturation are expressed both in terms of a new variable, by respecting the original saturation-capillary pressure dependency. If the new variable is properly chosen, the Richards equation receives a character that is more suited for Newton’s scheme, in the sense that all non-linearities are Lipschitz continuous. We refer to [10] for a review detailing on such aspects.

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The scheme analysed here builds on the L-scheme, a robust fixed point scheme, which does not involve the computations of any derivatives or a regularisation step. The convergence, proved rigorously in [19, 25, 27], holds in the H 1 norm and regardless of the initial guess, but is linear. To improve this convergence, a combination between the L− and Newton schemes was discussed recently in [13]. By performing first a number of L-scheme iterations, one obtains an approximation that is close enough to the solution. After a switch to the Newton iterations, the convergence becomes quadratic. Compared to the literature cited above, here we adopt a more challenging setting: b(·) is only Hölder continuous and not necessarily strictly increasing. Whenever b (·) is unbounded, neither Newton nor Picard schemes can be applied directly. The common way to overcome this is to regularise b(·) (see [14]), e.g. to approximate it by a Lipschitz continuous function bε (·). Nevertheless, a regularisation will also imply a perturbation of the solution, which affects the accuracy of the method. Here, we propose an L-scheme for the degenerate Eq. (1), which is adapted to the Hölder continuous non-linearity. The linear convergence of the scheme is proved rigorously, and its performance is compared with the ones of the standard L- and Newton schemes, applied for the regularised problems. The paper is organised as follows. In the next section the fully discrete variational approximation of (1) is given and the assumptions are stated. Section 3 discusses different iterative schemes. First the modified L-scheme together with the convergence proof are given. Then the approach based on regularisation is discussed, with particular emphasis on the Newton scheme. Finally, in Sect. 5 a comprehensive comparison between the L-schemes and the Newton scheme are presented. The paper is concluded with final remarks.

2 The Fully Discrete Approximation Throughout this paper we will use common notations in the functional analyp sis. we mean the p-integrable functions with the norm f p :=   By L (Ω) 1/p , whereas H (div; Ω) := {f ∈ (L2 (Ω))d |∇ · f ∈ L2 (Ω)}. Further, f (x) dx) Ω we denote by ·, · the inner product on L2 (Ω) and by σ (Ω) the volume of Ω. Similarly, by H 1 (Ω) we mean the L2 (Ω) functions having the first order weak derivatives in L2 . To define the discretisation we let Th be a regular decomposition of the domain Ω (h is the mesh size) and 0 = t0 < t1 < . . . < tN = T , N ∈ N, is a partition of the time interval [0, T ] with constant time step size τ = tk+1 − tk , k ≥ 0. The lowest-order Raviart-Thomas elements (see e.g. [5]) are used for the discretisation in space. The spaces Wh × Vh ⊂ L2 (Ω) × H (div; Ω) are defined as Wh := {p ∈ L2 (Ω)| p|T (x) = pT ∈ R for all T ∈ Th }, Vh := {q ∈ H (div; Ω)|q|T (x) = aT + bT x, aT ∈ Rd , bT ∈ R for all T ∈ Th }. The lemma below (see [9]) will be used in the proof of Theorem 4.

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Lemma 1 There exists a constant CΩ > 0 not depending on the mesh size h, such that given an arbitrary wh ∈ Wh there exists vh ∈ Vh , satisfying ∇ · vh = wh and vh  ≤ CΩ wh . As mentioned, (1) is completed with homogeneous Dirichlet boundary conditions, and with the initial condition u(0, x) := u0 (x), with u0 ∈ L2 (Ω). Furthermore, the source term is f ∈ L2 (Ω). We make the following assumptions on b(·). (A1) The function b : R → R, with b(0) = 0, is non-decreasing and Hölder continuous: there exist Lb > 0 and α ∈ (0, 1] such that |b(x) − b(y)| ≤ Lb |x − y|α

for all

x, y ∈ R.

(2)

Remark 2 The case α = 1 corresponds to a Lipschitz continuous b(·), a case which is relatively well-understood [3, 11–13, 18–20, 25, 27]. The case α ∈ (0, 1) is encountered for the Richards equation under physically relevant parametrisations (the van Genuchten curves [15], see Remark 1.1 in [21]). Also, if Freundlich rates are used for modelling reactive transport, one has b(u) = u + φ(u), with φ increasing but non-Lipschitz. Then there exists an m ∈ R such that b ≥ m > 0, which simplifies the analysis of the iterative schemes. Remark 3 Non-linear convection q(·) can be added, however, if being Lipschitz continuous. The numerical schemes can be then easily modified to include such changes: one can deal with such non-linearities by using either the outcome at the last iteration, or by including this term in the Newton iteration, depending on the method used. For the ease of presentation, such cases are not considered here. In view of the lacking regularity, the solutions to (1) are weak. We refer to [1, 16] for existence and uniqueness results. Also, the equivalence between the conforming and mixed formulation, for both time continuous and time discrete problems, is being discussed in [21] (see also [22] for the case of a two-phase flow model). Such results provide the existence and uniqueness of a solution for the mixed formulation, and can be used for obtaining the rigorous convergence of the discretisation. Finally, for each time step, the backward Euler-MFEM discretisation of (1) reduces to a nonlinear, fully discrete variational problem (n ≥ 1). Problem Phn (The Non-Linear Fully Discrete Problem). Let un−1 ∈ Wh be given. h Find unh ∈ Wh and qnh ∈ Vh such that for any wh ∈ Wh and vh ∈ Vh there holds n b(unh ) − b(un−1 h ), wh  + τ ∇ · qh , wh  = τ f, wh ,

qnh , vh  − unh , ∇

· vh  = 0.

(3) (4)

Clearly, for n = 1, u0h can be taken as the L2 -projection of the initial condition u0 onto Wh (see also [21]).

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Here we assume that a solution to Problem Phn exists and is unique. For α = 1, i.e. when b is Lipschitz continuous, Theorem 4 below guarantees that the iterative scheme (5)–(6) is H 1 -contractive. This immediately provides the existence of a solution. For α ∈ (0, 1), the existence can be proved by using Brouwer’s fixed point theorem (see e.g. Lemma 1.4, p. 140 in [26]). We refer to [2, 7, 8, 24] for similar results in the context of two-phase porous media flow models. Finally, since b is monotone, uniqueness can be proved by comparison. The main challenge in solving the non-linear Problem Phn is to construct a linearisation scheme that is converging also for the case when b(·) is only Hölder continuous, implying that b (·) may become unbounded. The scheme is discussed in the section below. Typically, iterative approaches like the Newton, (modified) Picard, or the L-schemes are applied to the regularised problem, with a Lipschitz continuous approximation bε replacing b (see [3, 6, 13, 18, 19, 23, 25]). This will be detailed in Sect. 4.

3 A Robust Iterative Scheme Below we define a robust iterative scheme for (3)–(4), which does not involve regularisation, or computing any derivatives. We let the time step n ≥ 1 be fixed and assume un−1 ∈ Wh be given. Also, let L = 1δ , where δ > 0 is a small parameter h that will be chosen later to guarantee that the error decreases below a prescribed threshold. With i ∈ N, i > 0 being the iteration index, the iteration step is introduced through n,i Problem Phn,i (The L-Scheme). Let un,i−1 ∈ Wh be given. Find (un,i h h , qh ) ∈ Wh × Vh s.t. for all wh ∈ Wh and vh ∈ Vh one has n,i−1 n−1 )+b(un,i−1 ), wh  + τ ∇ · qn,i L(un,i h − uh h h , wh  = b(uh ) + τf, wh , (5) n,i qn,i h , vh  − uh , ∇ · vh  = 0.

(6)

As will be seen below, the convergence is obtained without imposing restrictions on n−1 the initial guess un,0 h ∈ Wh , but a natural choice is uh . As for Problem Phn , the uniqueness of a solution for Problem Phn,i follows by standard techniques. Specifically, assuming that Problem Phn,i has two solution pairs n,i (un,i h,k , qh,k ) ∈ Wh × Vh (k = 1, 2) and with (duh , dqh ) denoting their difference it holds Lduh , wh  + τ ∇ · dqh , wh  = 0, qh , vh  − duh , ∇ · vh  = 0,

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for all wh ∈ Wh and vh ∈ Vh . Taking in the above wh = duh , respectively vh = τ duh , and adding the resulting equations gives Lduh 2 + τ qh 2 = 0,

(7)

which immediately implies uniqueness. Moreover, since Problem Phn,i is linear and finite dimensional, uniqueness also implies the existence of a solution. To show the convergence of the scheme we define the errors n eun,i = un,i h − uh ,

n eqn,i = qn,i h − qh ,

and

where (unh , qnh ) is the solution pair of Problem Phn . We use in the next the elementary (in)equalities, holding for any c, d ≥ 0 and p, q > 1 s.t. p1 + q1 = 1 c(c − d) =

 1 2 c − d 2 + (c − d)2 , 2

and

cd ≤

cp dq + . p q

(8)

1 With fixed δ > 0 and n ∈ N, n ≥ 1, let L = and assume un−1 ∈ Wh known. h δ The main result supporting the convergence is Theorem 4 Assuming (A1) and α ∈ (0, 1), let i ∈ N, i ≥ 1 and un,i−1 ∈ Wh h n,i n and P n,i be given. If (unh , qnh ) and (un,i , q ) are the solutions of Problems P h h h h respectively, there holds 2

eun,i 2 + τ δR(δ, τ )eqn,i 2 ≤ R(δ, τ )eun,i−1 2 + 2C(α)R(δ, τ )δ 1−α .  Here R(δ, τ ) =

τδ 1+ 2 CΩ

 (1−α)  Lb (2α)α 1−α (1 + 2 2

(9)

−1 , CΩ being the constant in Lemma 1, and C(α) =

α)− 1−α σ (Ω). 1+α

Proof Subtracting (3) and (4) from (5), respectively (6), one gets for all wh ∈ Wh and vh ∈ Vh L(eun,i − eun,i−1 ) + b(un,i−1 ) − b(unh ), wh  + τ ∇ · eqn,i , wh  = 0, h

(10)

eqn,i , vh  − eun,i , ∇ · vh  = 0.

(11)

By taking wh = eun,i ∈ Wh , respectively vh = τ eqn,i ∈ Vh , adding the resulting equations and after some algebraic calculations one gets  L  n,i 2 eu  + eun,i − eun,i−1 2 + b(un,i−1 ) − b(unh ), eun,i−1  + τ eqn,i 2 h 2 L = eun,i−1 2 − b(un,i−1 ) − b(unh ), eun,i − eun,i−1 . h 2

(12)

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55

By (A1), it holds −1

1+α

α b(un,i−1 ) − b(unh ), eun,i−1  ≥ Lb α b(un,i−1 ) − b(unh ) 1+α . h h

(13)

α

Using now the inequality in (8) with p = 1

1+α α ,

q = 1 + α, c =

|b(un,i−1 )−b(unh )| h 1

and

α

2α 1+α Lb1+α ( 1+α )

α

2α 1+α n,i ) |eu − eun,i−1 | one gets d = Lb1+α ( 1+α

|b(un,i−1 ) − b(unh ), eun,i − eun,i−1 | h ≤

1

1+α

1 α

α b(un,i−1 ) − b(unh ) 1+α + h α

2Lb

(2α)α Lb en,i − eun,i−1 1+α 1+α . (α + 1)(α+1) u

(14)

From (12), (13) and (14) one obtains L 2

  ) − b(unh ), eun,i−1  + τ eqn,i 2 eun,i 2 + eun,i − eun,i−1 2 + 12 b(un,i−1 h

≤

L n,i−1 2  2 eu

+

(2α)α Lb eun,i − eun,i−1 1+α 1+α . (α + 1)(α+1) 2 2 1+α , q = 1−α , 1+α 1−α b ( 1+α ) 2 σ (Ω) 2 (α+1)(α+1) L

Using again Young’s inequality, but with p = L eun,i−1 1+α 1+α ( 1+α )

1+α 2

σ (Ω)

α−1 2

and d =

(2α)α L

c = eun,i − gives

(2α)α Lb eun,i − eun,i−1 1+α 1+α (α + 1)(α+1) α−1 1+α L ≤ σ (Ω) 1+α eun,i − eun,i−1 21+α + C(α)L α−1 2 1+α L ≤ eun,i − eun,i−1 2 + C(α)L α−1 , 2 where C(α) is defined in the formulation of the theorem. Above we used the 1−α inequality f 1+α ≤ σ (Ω) 2(1+α) f 2 , valid for any f ∈ L2 (Ω) and α ∈ (0, 1] since Ω is bounded. Now, from the last two estimates it follows that 1+α L n,i 2 1 L e  + b(un,i−1 ) − b(unh ), eun,i−1  + τ eqn,i 2 ≤ eun,i−1 2 + C(α)L α−1 . h 2 u 2 2

From (11) and using Lemma 1, a Poincare type inequality eun,i  ≤ CΩ eqn,i  can be obtained. Using this in the above, since L = 1/δ, one obtains (9). Remark 5 Observe that since R(δ, τ ) < 1 whereas δ has a positive power in the last term on the right of (9), this theorem gives the convergence of the scheme.

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More precisely, for any chosen tolerance T OL, one can chose δ such that the 2 R(δ,τ ) 1 term 2C(α)δ 1−α 1−R(δ,τ ) < 2 T OL. Since this is the sum of the last terms on the right in (9), this can be seen as the total error being accumulated while iterating in one time step. On the other hand, the first term in the right is showing how the error is contracted in one iteration. Thus, choosing i ∗ ∈ N large enough s.t. ∗ R(δ, τ )i eun,0 2 ≤ 12 T OL and applying (9) successively for i = i ∗ , i ∗ − 1, . . . , 1 one obtains that eun,i 2 < T OL. Nevertheless, the convergence rate is worsened with the decrease of δ, as R(δ, τ ) approaches 1 in this case. From theoretical point of view, this results in an increased number of iterations for obtaining the desired accuracy. This is a rather pessimistic interpretation, as the numerical examples studied in Sect. 5 indicate that the actual number of iterations is frequently better than what the theorem guarantees. Remark 6 If b is Lipschitz continuous, the problem reduces to the one studied in [13, 21]. In fact, for α = 1 the last step in the proof above is superfluous, and the estimate (9) holds with C(α) = 0. In this case, the iteration is a contraction, so the convergence is unconditional for any L ≥ Lb , the Lipschitz constant of b. Remark 7 Observe that the convergence can be achieved without requiring that the R(δ,τ ) time step size τ is sufficiently small. In fact, when calculating the ratio 1−R(δ,τ ) one sees that τ appears in the denominator, so the larger it is, the better the convergence 2 R(δ,τ ) of the iterative scheme. Further, the term 2C(α)δ 1−α 1−R(δ,τ ) is practically small without taking a too small δ. For example, if α = 0.5, the power of δ in this term 1+α 1−α = 10−6 . Also, becomes 1+α 1−α = 3. Taking δ = 0.01 (hence L = 100) gives δ the number C(α) is small too. In the situation above, if Lb = 0.5, and σ (Ω) = 1, C(α) ≈ 0.0046.

4 Iterative Schemes Based on Regularisation As follows from the above, the iterations introduced through Problem Phn,i converge also for the case of a Hölder continuous b and do not involve computing any derivatives. However, the iterations only converge linearly. A natural question appears: what is the performance of the new scheme in comparison with the Newton or the L-scheme, but applied for the regularised problems. To study this aspect we first present below these schemes and discuss their convergence. For simplicity we consider the function b : R → R, b(u) = (max{u, 0})α . Observe that b is not Lipschitz for arguments approaching 0 from above. For regularising it we let ε > 0 and consider the function bε : R → R, bε (u) = αεα−1 u + (1 − α) α , if u ∈ (0, ε), whereas bε (u) = b(u) everywhere else. Clearly, bε (·) is nondecreasing, and both bε (·), bε (·) are Lipschitz continuous with the Lipschitz

Iterative Schemes for Degenerate Equations

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constants Lbε = αεα−1 , respectively Lbε = α(1 − α)εα−2 . Moreover, it holds α 0 ≤ b(x) − bε (x) ≤ (1 − α)α 1−α εα . As before, for given ε > 0 and un−1 h,ε ∈ Wh (observe the dependency of the solution on ε), and with i ∈ N, i > 0 being the iteration index, the Newton iterations for Problem Phn are defined through Problem NEW T ONhn,i . n,i ∈ Wh be given. Find (un,i Let un,i−1 h,ε h,ε , qh,ε ) ∈ Wh × Vh s. t. for all wh ∈ Wh and v h ∈ Vh n,i n,i−1  n,i−1 bε (un,i−1 h,ε ) + bε (uh,ε )(uh,ε − uh,ε ), wh  n−1 +τ ∇ · qn,i h,ε , wh  = bε (uh,ε ) + τf, wh , (15) n,i qn,i h,ε , vh  − uh,ε , ∇ · vh  = 0.

(16)

Remark 8 (Regularised L-Scheme) A L-scheme for the regularised problem is obtained by replacing bε (un,i−1 h,ε ) with L ≥ 0 in (15). The resulting scheme is convergent for L ≥ Lbε /2, as proved in [13, 19, 20]. Moreover, the convergence holds in H 1 and for any initial guess, under very mild restrictions on the time step, but it is only linear. It is worth emphasising on the difference between the L-scheme in Sect. 3, designed for Hölder continuous non-linearities, and the L-scheme for the regularised problems. In the former case the errors at each iteration step consist of two components, one that is contracted, and another that accumulates. The choice of the L parameter is driven by these two: first, the accumulated errors should remain below a threshold 12 T OL, and second the contracted ones reduce to the same threshold. For the latter the problem is regularised so that the non-linearities become Lipschitz continuous, and then the L parameter is taken as Lbε . Remark 9 (Convergence of the Regularised Newton Scheme) Two issues concerning the convergence appear in this case. First, the solution uε of the regularised problem should not be too far from u, the solution to the original problem. This means that ε should be sufficiently small. On the other hand, the advantage of the Newton scheme is its quadratic convergence. Guaranteeing it requires typically a small τ because the scheme is only locally convergent, so the initial guess of the iteration should not be too far from the solution and the choice at hand is the solution at the previous time step. However, τ and ε are correlated. So satisfying both requirements might be quite challenging, if not impossible in certain computations. If one assumes additionally that b ≥ m > 0, which rules out the fast diffusion case, the sufficient condition for convergence is to choose τ = O(εa hd/2 ), with a depending on the Hölder exponent (see [20]). In the case b ≥ 0, one can further perturb b so that bε is bounded away from 0, e.g. by taking bεnew (u) = εu + bε (u) with bε (u) given before. Then the convergence is guaranteed for similar constraints, possibly with a different exponent a.

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To summarise, the convergence of Newton’s scheme depends on the choice of the discretisation and regularisation parameters. Fixing two parameters, e.g. h and ε, only a sufficiently small τ will guarantee the convergence. Alternatively, for fixed τ and ε, the mesh size can not be too small, and if the Newton scheme diverges, refining the mesh will not help. In other words, to achieve a certain accuracy, e.g. by letting ε  0, the convergence condition for the Newton scheme might become very restrictive.

5 Numerical Examples In this section we provide numerical examples to illustrate the performance of the scheme. We use the example mentioned in Sect. 4, b(u) = max{u, 0}α , for α = 0.5. The domain is the square Ω = (0, 1) × (0, 1), and the time interval is t ∈ (0.0, 0.5]. To evaluate the convergence we choose the source term, the boundary conditions and the initial condition such that the exact solution is 1 u(t, x, y) = − + 16 x(1 − x) y(1 − y)(t + 0.5). 2

(17)

For the discretisation we consider a 32 × 32 mesh with different time step sizes τ ∈ {0.05, 0.025, 0.0125}, resulting in 10, 20, respectively 40 time steps. To differentiate between the errors brought by the discretisation itself and those related to the iterative solver, we first compute a very accurate approximation of the nonlinear, fully discrete systems. Specifically, with Δui and Δqi denoting the difference between two iterates, the reference solution is the iteration satisfying Δui L2 (Ω) + Δqi L2 (Ω) < 10−8 , and

Δui L2 (Ω) ui L2 (Ω)

+

Δqi L2 (Ω) qi L2 (Ω)

< 10−8 .

This solution, called below uh , was computed with the L-type scheme in Sect. 3 to avoid additional regularisation errors. Having obtained uh we proceed by testing the three schemes discussed here, the L-scheme in the framework discussed in Sect. 3 (called H L), and the two (Newton and L) in Sect. 4, involving a regularisation step. In agreement with the result stated in Theorem 4 we choose an admissible tolerance T OL to be used as stopping criterion for the different iteration schemes. Specifically, u!h is accepted as numerical solution if it satisfies u!h − uh L2 (Ω) < T OL where uh is the (accurate) solution from above. We consider different tolerances, namely T OL ∈ {10−3 , 10−4 , 10−5 }. For the regularisation based schemes, the problem is first regularised by taking ε ∈ {10−3 , 10−4 , 10−5 }. For the L-scheme we take L = εα−1 , the Lipschitz constant of bε . For the H L-scheme we take L = 1δ where δ is such that the condition in Remark 5 on the accumulated error is met.

Iterative Schemes for Degenerate Equations Table 1 Results for the Newton scheme

TOL 1e−3 1e−3 1e−3 1e−4 1e−4 1e−4 1e−5 1e−5 1e−5

59

ε 1e−3 1e−4 1e−5 1e−3 1e−4 1e−5 1e−3 1e−4 1e−5

N-iterations per time step τ = 0.05 τ = 0.025 τ = 0.0125 1.7 1.2 1.2 1.6 1.3 nc 1.6 1.3 nc 2.2 2.1 nc 2.3 2.4 nc 2.3 2.3 nc nc nc nc 3.1 3.0 nc 3.1 3.2 nc

The scheme does not converge (nc) for the smallest time step and if ε is not in agreement with T OL

Table 1 presents the total number of Newton iterations and the corresponding, average number of iterations per time step for given different tolerances T OL, regularisation parameters ε and time step sizes τ . Observe that the parameters T OL and ε should be correlated to avoid that the regularisation error becomes dominating. In other words, a smaller T OL requires a smaller ε for obtaining the convergence. In the same spirit, a smaller τ requires smaller T OL and ε. For τ = 0.0125, it becomes almost impossible to obtain solutions within the required accuracy by using the Newton scheme, as ε has to be very small and then the condition number of the Jacobian becomes very high. This is evidenced by the appearance of cases where the Newton scheme did not converge, which are mentioned as nc. In summary, the Newton scheme fails to converge if either the regularisation parameter ε is too large for the chosen tolerance T OL, or if ε is too low, which makes the problem very badly conditioned. Clearly, if convergent, the Newton scheme requires the least number of iterations among all schemes. Similar experiments have been performed for the standard L-scheme, applied after regularising the problem. Recalling bε is Lipschitz, we set L = Lbε = αεα−1 . The actual values are given in Table 2. Table 3 presents the convergence results. As for the Newton scheme, one needs to correlate the parameters T OL, ε, and τ . To ensure convergence, if T OL is small ε should be small enough, otherwise the regularisation error will dominate and the convergence criterion will not be met. This is the reason why the L scheme, though unconditionally convergent in theory, is marked as not convergent for the case ε = 10−3 , if T OL = 10−4 or 10−5 . Also, observe that L = Lbε blows up with ε  0, while the convergence rate approaches 1 if L is large, or τ is small (see [19]). Therefore if ε and τ are small, combined with the finite precision arithmetic may lead to the divergence of the L-scheme. This also explains why the number of L-scheme iterations increases drastically with the decrease of the regularisation parameter. Compared to the Newton scheme, the number of L-iterations is much larger. On the other hand, the L-scheme is more robust than the Newton scheme, allowing to compute the solution for small time steps τ or for small regularisation parameters ε.

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Table 2 L values for the standard L-scheme, obtained for different values of ε ε L

1e−3 16

Table 3 Results for the standard L-scheme

1e−4 50

TOL 1e−3 1e−3 1e−3 1e−4 1e−4 1e−4 1e−5 1e−5 1e−5

ε 1e−3 1e−4 1e−5 1e−3 1e−4 1e−5 1e−3 1e−4 1e−5

1e−5 159

L-iterations per time step τ = 0.05 τ = 0.025 τ = 0.0125 30.5 38.9 48.4 96.9 124.6 155.2 305.8 394.6 492.8 47.9 nc nc 150.5 202.9 273 475.1 643.7 870.7 nc nc nc 204.5 281.5 nc 645.9 889.1 1247.9

The scheme does not converge (nc) if ε is not in agreement with T OL Table 4 The L parameters for the H L-scheme, computed for different values of T OL and τ

TOL 1e−3 1e−4 1e−5

L-parameters for the HL-scheme τ = 0.05 τ = 0.025 τ = 0.0125 19 23 29 40 50 62 84 106 134

The total iteration error is guaranteed below T OL (see Remark 5)

Finally we draw our attention to the H L-scheme, where the parameter L is chosen as mentioned in Remark 5, depending on T OL. Since the domain is the unit square one has CΩ = σ (Ω) = 1 and thus R(δ, τ ) = (1 + τ δ)−1 . For α = 0.5, 1 to reduce the accumulated errors below 12 T OL one needs to take δ < 32 (τ T OL) 3 , while L = 1δ . The corresponding values are given in Table 4. Observe that the values of L in this case are similar to the ones for the standard L scheme, except for the smallest tolerance. Also, the L values increase for smaller T OL and smaller time steps τ , which was not the case for the standard L scheme. The convergence results are given in Table 5. Since the L parameters have similar values for both L-type schemes, the number of iterations in both schemes is comparable whenever the standard L-scheme converges. However, for the H Lscheme, L can be chosen automatically, based on the required tolerance T OL and on the time step size τ . This leads to faster convergence rates, based on the theoretically results. Nevertheless, decreasing the tolerance T OL implies an

Iterative Schemes for Degenerate Equations Table 5 Results for the standard H L-scheme

61

TOL 1e−3 1e−4 1e−5

HL-iterations per time step τ = 0.05 τ = 0.025 τ = 0.0125 37.0 57.2 89.5 120.4 202.5 338.2 343.3 596.2 1057.4

The scheme converges for all values of T OL and all time steps τ

increasing L, which deteriorates the convergence rate. However, the H L-scheme converged for all combinations of parameters. When comparing the three schemes, it becomes clear that the Newton scheme requires the least number of iterations whenever it converges. On the other hand, the Newton scheme was the one which did not converge in the most of the cases considered here, so it is least robust. Also, the convergence criterion is not always met for the standard L-scheme due to regularisation. Both schemes require a regularisation step. Instead, no regularisation is needed for the H L-scheme. Clearly, it requires more iterations than the Newton scheme, but generally less than the standard L-scheme. Most important, it displayed a robust behaviour, as it converged in all experiments. In fact, this convergence can be achieved for any tolerance T OL and time step τ . It is worth mentioning that, next to the number of iterations, the total execution time is influenced by two factors: the time for solving the linear systems at each iteration, and the time for assembling the discretisation matrices. Among all three schemes, the Newton scheme is closest to generate ill conditioned matrices, if not singular. Therefore the linear solvers are more expensive than in the case of the L-type schemes. Moreover, the linear system needs to be reassembled completely every iteration, as the Jacobian depends on the current iteration, and involves many function evaluations. The L-type schemes behave better in this respect. For the example presented above, the emerging linear systems involve the discrete Laplacian and the discretisation of the identity operator multiplied by L. This not only generates better conditioned matrices, but these matrices remain unchanged for every iteration. In this case, a solver based on the LU -decomposition is an effective approach, as this decomposition needs to be performed only once.

6 Conclusion We discuss iterative schemes for solving the fully discrete non-linear systems obtained by a backward Euler—lowest order Raviart-Thomas mixed finite element discretisation of a class of doubly degenerate parabolic problems. Appearing as models of practical relevance, the non-linear function involved in the model must be increasing and Hölder continuous, but may remain constant over intervals. In consequence, two kinds of degeneracy are allowed, slow and fast diffusion. This

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leads to fully discrete systems that have singular Jacobians, which brings difficulties in finding robust iterative solvers. We present here an approach inspired by the L-scheme, which is suited for the case of Hölder continuous non-linearities. To apply the Newton scheme or the standard L-scheme in such a case, one needs to regularise first the problem, i.e. to approximate the non-linearity by a Lipschitz continuous one. This step is associated with additional errors. If highly accurate approximations of the exact, fully discrete solutions are needed, the regularisation step may be the cause of the fact that the convergence is very slow, if not impossible. The scheme discussed here makes no use of any regularisation. Instead, the parameter L is chosen not as the Lipschitz constant of the non-linearity, but in such a way that the error has a guaranteed decay below any chosen tolerance. We provide a rigorous proof for this decay, which also gives a practical way to choose the parameter L. We present numerical experiments where we compare the behaviour of the three schemes: Newton, standard L, and the L-variant proposed here. As resulting from these experiments, the Newton scheme requires the least number of iterations, but is also the least robust of all as there were the most cases where it did not converge. The standard L-scheme is more robust, at the expense of a high number of iterations. Also, convergence could not be achieved in all cases, in particular if the regularisation parameter is not in agreement with the required tolerance. The new scheme is improving these aspects: it shows convergence for any required tolerance, and any choice of the time step size. Nevertheless, an optimisation of the choice of L and possibly in combination with an optimal linear solver can make the proposed scheme an effective alternative to the traditional ones. Acknowledgements The research is partially supported by the Norwegian Research Council (NFR) through the NFR-DAAD grant 255715, the VISTA project AdaSim 6367 and the project Toppforsk 250223, Lab2Field 811716, by Statoil through the Akademia Grant and by the Research Foundation-Flanders (FWO) through the Odysseus programme (project G0G1316N).

References 1. H.W. Alt, S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311– 341 (1983) 2. T. Arbogast, The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow. J. Nonlinear Anal. Theory Methods Appl. 19, 1009– 1031 (1992) 3. N. Bergamashi, M. Putti, Mixed finite elements and Newton-type linearizations for the solution of Richards’ equation. Int. J. Numer. Methods Eng. 45, 1025–1046 (1999) 4. K. Brenner, C. Cances, Improving Newton’s method performance by parametrization: the case of the Richards equation. SIAM J. Numer. Anal. 55, 1760–1785 (2017) 5. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, New York, 1991) 6. M. Celia, E. Bouloutas, R. Zarba, A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res. 26, 1483–1496 (1990) 7. Z. Chen, Degenerate two-phase incompressible flow. Existence, uniqueness and regularity of a weak solution. J. Differ. Equ. 171, 203–232 (2001)

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8. L. Cherfils, C. Choquet, M.M. Diedhiou, Numerical validation of an upscaled sharp-diffuse interface model for stratified miscible flows. Math. Comput. Simul. 137, 246–265 (2017) 9. J. Douglas Jr., J. Roberts, Global estimates for mixed methods for second order elliptic problems. Math. Comput. 45, 39–52 (1985) 10. M.W. Farthing, F.L. Ogden, Numerical solution of Richards equation: a review of advances and challenges. Soil Sci. Soc. Am. J. (2017). https://doi.org/10.2136/sssaj2017.02.0058 11. W. Jäger, J. Kaˇcur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. Math. Model. Numer. Anal. 29, 605–627 (1995) 12. F. Lehmann, P. Ackerer, Comparison of iterative methods for improved solutions of the fluid flow equation in partially saturated porous media. Transp. Porous Med. 31, 275–292 (1998) 13. F. List, F.A. Radu, A study on iterative methods for Richards’ equation. Comput. Geosci. 20, 341–353 (2016) 14. R.H. Nochetto, C. Verdi, Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. 25, 784–814 (1988) 15. J.M. Nordbotten, M.A. Celia, Geological Storage of CO2. Modeling Approaches for LargeScale Simulation (Wiley, Hokoben, 2012) 16. F. Otto, L1 -contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differ. Equ. 131, 20–38 (1996) 17. C. Paniconi, M. Putti, A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems. Water Resour. Res. 30, 3357–3374 (1994) 18. E.J. Park, Mixed finite elements for non-linear second-order elliptic problems. SIAM J. Numer. Anal. 32, 865–885 (1995) 19. I.S. Pop, F.A. Radu, P. Knabner, Mixed finite elements for the Richards’ equations: linearization procedure. J. Comput. Appl. Math. 168, 365–373 (2004) 20. F.A. Radu, I.S. Pop, P. Knabner, On the convergence of the Newton method for the mixed finite element discretization of a class of degenerate parabolic equation, in Numerical Mathematics and Advanced Applications ed. by A. Bermudez de Castro, D. Gomez, P. Quintela, P. Salgado (Springer, Berlin, 2006), pp. 1192–1200 21. F.A. Radu, I.S. Pop, P. Knabner, Error estimates for a mixed finite element discretization of some degenerate parabolic equations. Numer. Math. 109, 285–311 (2008) 22. F.A. Radu, K. Kumar, J.M. Nordbotten, I.S. Pop, A convergent mass conservative numerical scheme based on mixed finite elements for two-phase flow in porous media. arHiv: 1512.08387 (2015) 23. F.A. Radu, J.M. Nordbotten, I.S. Pop, K. Kumar, A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015) 24. F.A. Radu, K. Kumar, J.M. Nordbotten, I.S. Pop, A robust, mass conservative scheme for twophase flow in porous media including Hölder continuous nonlinearities. IMA J. Numer. Anal. 38, 884–920 (2018) 25. M. Slodicka, A robust and efficient linearization scheme for doubly non-linear and degenerate parabolic problems arising in flow in porous media. SIAM J. Sci. Comput. 23, 1593–1614 (2002) 26. R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (AMS Chelsea Publishing, Providence, 2001) 27. W.A. Yong, I.S. Pop, A numerical approach to porous medium equations. Preprint 95–50 (SFB 359), IWR, University of Heidelberg, 1996

Mathematics and Medicine: How Mathematics, Modelling and Simulations Can Lead to Better Diagnosis and Treatments Erik A. Hanson, Erlend Hodneland, Rolf J. Lorentzen, Geir Nævdal, Jan M. Nordbotten, Ove Sævareid, and Antonella Zanna

Abstract Starting with the discovery of X-rays by Röntgen in 1895, the progress in medical imaging has been extraordinary and immensely beneficial to diagnosis and therapy. Parallel to the increase of imaging accuracy, there is the quest of moving from qualitative to quantitative analysis and patient-tailored therapy. Mathematics, modelling and simulations are increasing their importance as tools in this quest. In this paper we give an overview of relations between mathematical modelling and imaging and focus particularly on the estimation of perfusion in the brain. In the forward model, the brain is treated as a porous medium and a two compartment model (arterial/venous) is used. Motivated by the similarity with techniques in reservoir modelling, we propose an ensemble Kalman filter to perform the parameter estimation and apply the method to a simple example as an illustrative example.

E. A. Hanson · A. Zanna () Department of Mathematics, University of Bergen, Bergen, Norway e-mail: [email protected]; [email protected] E. Hodneland Christian Michelsen Research, Bergen, Norway e-mail: [email protected] R. J. Lorentzen · G. Nævdal · O. Sævareid International Research Institute of Stavanger, Stavanger, Norway e-mail: [email protected]; [email protected]; [email protected] J. M. Nordbotten Department of Mathematics, University of Bergen, Bergen, Norway Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_4

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1 Introduction The twentieth century produced such a plethora of discoveries and advances that in some ways the face of medicine changed out of all recognition. Life expectancy at birth is a primary indicator of the effect of health care on mortality. In 1901 in Europe, for instance, the life expectancy at birth was 48 years for males and 51.6 years for females. After steady increases, by the 1980s the life expectancy had reached 71.4 years for males and 77.2 years for females and continues increasing at the same rate (source: Encyclopædia Britannica). The rapid progress of medicine and health care in this era has been reinforced by the enormous improvement in communication between the scientists, but also the systematic use of statistics to develop more precise diagnostic tests and more effective therapies, and the spectacular advances in imaging techniques. This is development expected to continue at an increased pace (Fig. 1). These new advances within imaging have contributed to shifting focus from a mere qualitative image analysis (for instance whether there is or not a tumor in an organ) to a quantitative analysis, like measuring volumes and shapes, blood flow, perfusion etc. The quantification of functional features is built on a

1952: Bloch and Purcell develop MR

discovers X-Rays

clinical use of MRI

1920s: development of XRays

1898: Marie and Pierre Curie discovers polonium and radium

1950s: First works with ultrasound

1977: 1st human MR image 1980: 1.5 T

1987: Real time MRI of heart 1993: Functional MRI of brain

3T, 5T, 7T human 9T, 21.1T animal Functional MRI Diffusion tensor MRI PET-MR

1970s: Real time ultrasound machines

1999: PET/CT developed, by Townsend and Nutt

Fig. 1 A timeline of non-invasive imaging techniques, starting from Röntgen’s discovery of Xrays

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combination of advanced imaging and mathematical modelling. While the imaging technology, partly driven by advances in nuclear physics and computer science, is experiencing a rapid development, the modelling part of the quantification models is not progressing at a similar pace. Apparent modelling challenges addressed already in the early 1990s [9] have yet to be fully understood [20]. As an example of the future possibilities in the interaction between mathematical modelling and medical image analysis, in this paper we will focus on the problem of estimating perfusion from dynamic imaging. By perfusion we refer to the transfer of blood from the arterial to the venal system, while by dynamic imaging we refer to functional methods that aim to the analysis of concentration-time profiles of an indicator or tracer1 that induces signal changes in an organ of interest. By rapid dynamic image acquisitions, these signal changes are then converted into concentration-time curves. These curves are the starting point for parameter estimation techniques. We apply estimation techniques developed in relation to a completely different application: complex geophysical modelling. The paper is organized as follows: in Sect. 2 we briefly introduce compartment models and tracer kinetic which are currently used in diagnostics and discuss some of the limitations. These methods focus on the analysis of tracer concentrations over time and do not take into account the underlying anatomical structure or the forces driving the flow. In Sect. 3 we introduce simple geometrical assumptions allowing for a dynamical description of perfusion (forward model) using pressure gradients as the main forces for blood circulation. This is a novel PDE-framework for perfusion in organs described as porous media flow. Thereafter, the model parameters need to be tuned to match the dynamic imaging data (observations), independently of the choice of the model. The parameter estimation for the PDE system in Sect. 3 has clear similarities with identification of spatial distributed parameters in flow models for reservoir modelling and will be discussed in Sect. 4. A small illustrative example is provided in Sect. 5, before we conclude with an outlook in Sect. 6.

2 A Brief Introduction to Compartment Models and Tracer Kinetic One of the most widespread methods for analysing concentration-time curves is based on the technique of compartment modelling. Compartment models were introduced by Forrester [8] in economics/industrial decision making to describe a dynamical system in which a measurable quantity (indicator) flows between system components called compartments. A compartment is a well-mixed space in which the indicator has a uniform concentration.

1 Like a magnetic contrast agent in Magnetic Resonance (MR), a radioactive tracer in Computerized Tomography (CT) or Positron Emission Tomography (PET), microbubbles in Ultrasound (US).

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The use of compartments in modelling is tailored for stationary regimes and is based on some fundamental simplifying assumptions removing both a temporal and a spatial complexity from the problem: instantaneous mixing and uniform concentration in the compartment. Compartment models are widely used in medicine, with application especially to pharmacokinetic and the study of tumours. Dating back to George N. Stewart in late nineteenth century, investigations on the circulatory system have been carried out by monitoring the distribution of an indicator substance injected into the blood stream. Early efforts include measurements of blood volumes in heart and lungs, and attempts to characterise the volume of a vascular bed by utilising the concept of mean transit time (MTT). Using either a constant or a perfectly localized (δdistribution) concentration profile at the inlet, Meier and Zierler [15] formalized these efforts under what is known as central volume theorem, namely: Volume = Flow × MTT. In Dynamic Contrast-Enhanced MRI (DCE-MRI), the compartment framework was consolidated in the early 1990s by Larsson et al. [13], Tofts et al. [22] and Brix et al. [5]. Compartment models used in tracer kinetic consist typically of up to two or three compartments, but higher number of compartments have been used. In a generic setting, consider a system described by N compartments. Let Vj be the volume of the j th compartment and Cj (t) the concentration of the indicator (tracer) at time t. The change of tracer mass must be balanced by the amount of indicator that flows in and out of the compartment, Vj

d Cj (t) = Fi,j Ci (t) − dt i∈Inlets

˜ j (t), Fj,o Cj (t) + G

o∈Outlets

where the last term accounts for sources/sinks in the compartments. The system reduces to a linear system of differential equations, C˙ = AC + G(t), where A = V −1 F , V being the diagonal matrix of compartment volumes and F the matrix ˜ This constant coefficients linear system has a closed of fluxes, and G = V −1 G. t form solution C(t) = eAt C0 + 0 e(t −s)AG(s) ds. Assuming C0 = 0 (no indicator at initial conditions,  t a typical experimental setup), and introducing the convolution operator f ∗ g = 0 f (t − s)g(s) ds, we see that the concentration curve is C(t) = et A ∗ G(t) = R(t) ∗ G(t), where R(t) is a residue function. In medical applications like perfusion or filtration [4, 10], G(t) typically is strongly related to the arterial input function (AIF). The AIF tracer concentration is measured from some larger arteriole or artery close to the tissue under consideration and is often treated as a single inlet. The A-parameter and the corresponding residue function et A from the exact solution of the system above tells us about volumes and flow rates, but has poor tissue-specific properties otherwise. Therefore it is often preferred to use indicator concentration curves of the type 

∞

C(t) = R(t) ∗ G(t), 0

R(t) dt = 1,

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where the residue function is either empirically modelled to reflect the tissue properties (model-based) [21] and in this case will depend on tissue-specific parameters p, or has to be estimated altogether (model-free) usually by deconvolution using regularized forms of Fast Fourier Transform (FFT) or Singular Value Decomposition (SVD) [4]. Once the compartment model has been set up and the parameters p identified (either volumes and flow rates or organ specific parameters), the parameters are computed by matching the model to observations (dynamic images) either for a region of interest (ROI) or individually to image voxels by solving argminp (D(C(x, t, p), C o (x, t)) + S(C, C o , p)), where D is a distance function, C o is the reference (observation) and S(C, C o , p) is a regularization term. With the emergence and refinement of modern imaging technology, the focus has gradually shifted from average properties of large scale tissue structures, or complete organs, to localised tissue properties at an increasingly higher spatial and temporal resolution. This raises some methodological challenges: 1. Voxels or regions-of-interest (ROIs) are assumed to represent isolated systems which receive indicator through a single inlet, the AIF, with a known concentration. This in not necessarily the case in a high resolution imaging setting, as a region can have multiple inlets carrying different concentrations, and none of them can be measured directly [9]. 2. These isolated systems fail in exploiting the additional spatial structure and connectivity with the neighbouring regions. The above shortcomings have been shown to lead to well-known systematic errors, hence major limitations of the classical compartment model tracer-kinetic theory and difficulties in reproducibility of results [7, 11, 20]. In the quest for better models, these challenges are addressed in a theoretical work by Sourbron [20], where a systematic approach is formulated in terms of global spatial-temporal conservation relations for the flow of contrast agent between voxels. The procedure readily incorporates multi-compartment models, where the total flowing volume of each voxel is partitioned into a set of distinct but interacting compartments reflecting local tissue structure. Each compartment type potentially connects globally throughout the voxel-lattice thus constituting a set of interacting flow networks. Connection coefficients governing the flow between voxels and exchange between compartments can then in principle be identified from the evolving contrast agent distribution. We expect that forthcoming studies in this direction further will address issues like computational feasibility, possibilities, and limitations with respect to local parameter estimation in this framework. In a recent work by Nævdal et al. [16] Sourbron’s two-compartment model for “blood flow and perfusion” was augmented by introducing constitutive relations for the flow system in terms of a dual porosity, dual permeability formulation known from porous media modelling of fractured reservoirs for petroleum and groundwater

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applications, see e.g., [3, 19]. Assigning one pore system to represent the arterial network and the other acting as venous network, local transfer between the two systems can be attributed to blood flow actually feeding the local tissue. In [14] a similar dual model formulation was applied to study oxygen transport in tissue.

3 A Spatial Two-Compartment Model for Brain Perfusion Consider a patch of the capillary system containing a large number of capillaries. The width of a single capillary is in the range of a few microns, and in vivo detection of single capillaries is therefore much below any resolution found in current imaging devices. Instead, we want to model the average flow response of all capillaries within a voxel using models for flow in porous media. This approach has previously been explored by several authors for the task of modelling blood flow in live tissue. Flow modelling takes into account two basic principles, conservation of mass and conservation of momentum. We describe the governing equations below.

3.1 Conservation of Fluid Mass Mass balance of fluid flow is ensured by the continuity equation, expressed in global form as    d ˜ Qdx (1) φρdx + ρ(u · n)dA = dt Ωi ∂Ωi Ωi for a geometric control volume Ωi with boundaries ∂Ωi . Here, n is the outer unit normal vector of ∂Ωi , u : Ω × T → R3 is the flux per unit area [m3 s−1 m−2 ], ρ : Ω × T → R is the fluid density [kg m−3], and Q˜ : Ω × T → R is a fluid source term [kg s−1 m−3 ]. The volume fraction available for flow is given by 0 < φ < 1 (known as porosity in the geo-sciences). Equation (1) must be valid for every geometric control volume Ωi , hence, by the divergence theorem, one obtains the local form ∂ ˜ (φρ) + ∇ · (ρu) = Q. ∂t

(2)

For incompressible fluid and constant fluid density, this equation is equivalent to ∇ ·u=Q ˜ has units [m3 s−1 m−3 ]. where Q = Q/ρ

(3)

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3.2 Balance of Forces Associated with the arterial and venous pore systems, we define a label β ∈ {a, v}. The two pore systems are spatially identical, and their relative volume within a voxel is defined by the respective porosities. Within the capillary brain tissue, valid for each pore system, we model a low velocity flow according to Darcy’s law, providing the relation k u = − ∇p μ

(4)

between the flux u and the pressure p when neglecting the gravitational acceleration, k = k(x) [m2 ] is a permeability tensor, and p = p(x) [Pa] is the pressure. It is assumed that k is symmetric and positive definite with only nonzero diagonal elements kii = k, i = {1, 2, 3}. Now, assume that the flow from the arterial to the venous compartment, the perfusion P [m3 s−1 m−3 ], is proportional to the pressure difference between the arterial and venous compartment, P := α(pa − pv )

(5)

for a proportionality constant α = α(x) [m s kg−1]. The parameter α will reflect microstructural properties of the capillary tissue affecting its ability to mediate perfusion, mainly viscous resistance. Applying (5) yields the following system of partial differential equations in the pressure fields pa , pv within a capillary patch ΩC with boundary ∂ΩC ,  −∇ ·  −∇ ·

ka ∇pa μ

 = −P

x ∈ ΩC

=P

x ∈ ΩC

uβ · n = 0

x ∈ ∂ΩC \∂ΩA

kv ∇pv μ



pβ = p0

β

β β

x ∈ ∂ΩA

(6)

where n is the normal to ∂ΩC . An outer pressure is assigned as Dirichlet boundary conditions at the partial outer boundary ∂ΩA . The two equations are coupled via the perfusion term P , which is a negative sink term for the arterial pore system and a positive source term in the venous pore system. Neuman boundary conditions of no flow across boundaries are defined for ∂ΩC , separating the capillary patch from the surrounding tissue. The viscosity of the fluid μ is assumed to be constant everywhere. Our flow model for brain perfusion is entirely defined by (6). However, in order to develop a framework for parameter estimation valid for dynamic contrastenhanced acquisitions, we must dilute the tracer in the computed flow as a dynamic

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sequence. This dynamic sequence can then be used for estimation of physiological parameters.

3.3 Tracer Mass Balance and Indicator Dilution Tracer concentration C(t) = N/V [mol/m3 ] is the number of tracer molecules N within a region of interest (ROI) of volume V . The tracer distribution volume is typically different from the ROI volume, leading to the relation Cβ = φβ cβ

(7)

connecting fluid concentration cβ (x, t) [mol/m3] to the control volume concentration Cβ (x, t). The following criteria are assumed to hold: The injected tracer is homogeneously distributed in the incoming arterial input function (AIF), all physiological and structural parameters are stationary within the time of acquisition, and tracer transport by diffusion is not considered. For any of the two pore systems, the influx of tracer into a control volume Ωi , e.g. a voxel, is determined by the product of the fluid tracer concentration c(x, t) and the stationary vector fluid flux u(x) [m3 /s/m2 ]  c(u · n)dA

−

(8)

∂Ωi

where n is surface normal of Ωi pointing to the outward direction. The rate of change of tracer within the control volume equals d dt

 (9)

C(x, t)dx. Ωi

Combining (8) with (9) due to conservation of mass yields  φ Ωi

∂c dx = − ∂t

 c(u · n)dA.

(10)

∂Ωi

In addition, we must account for the source terms. Denote the fluid concentration of pore system β as cβ,k . Tracer mass balance for each of the pore systems yields  φa Ωi

 Ωi

∂ca dx = − ∂t



 ca (ua · n)dA −

∂Ωi

ca P dx Ωi

a ca = cAIF x ∈ ∂ΩA   ∂cv dx = − φv cv (uv · n)dA + ca P dx. ∂t ∂Ωi Ωi

(11)

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The total tracer mass within the control volume is linearly additive according to C(x, t) = Ca (x, t) + Cv (x, t).

(12)

The model for indicator dilution is governed by (7), (11), and (12).

4 Parameter Estimation In the previous section we formulated our forward model in terms of a set of partial differential equations with properties varying spatially. Several of the coefficients in these partial differential equations have biological interpretations, but their values are unknown. Therefore we are facing a large scale parameter estimation problem. Among the terms that are unknown in the continuous model for perfusion is the proportionality constant α in (5) and the permeabilities ka and kv in (6). The model that we have formulated for the forward problem has several similarities to modeling flow in an oil reservoir. The physical properties of the oil reservoirs as permeability and porosity are unknown and need to be estimated from available measurements. Since the forward models will be solved numerically, the models can be populated with spatially varying permeability values and estimated from available measurements. Within reservoir engineering, the ensemble Kalman filter (EnKF), and variants thereof, has been found to be a suitable technique to estimate such parameter fields, cf. [1, 17]. The EnKF is developed as a nonlinear extension of the Kalman filter [12]. The Kalman filter was originally developed to estimate the states s of a linear dynamical system which can be described as sn = F sn−1 + n ,

(13)

where the observable output y is given as yn = Gsn + ηn .

(14)

In the above equations, n ∼ N(0, CM ) is a model noise term, which has a zero-mean multinormal distribution and covariance matrix CM . Similarly, ηn is a measurement noise term satisfying ηn ∼ N(0, CD ). The Kalman filter has several interpretations. One interpretation is to view it as the solution of a recursive Bayesian estimation problem, where the initial prior distribution of the state vector s is a multinormal distribution with mean sˆ0 and covariance matrix C0 . At time step n, a new set of measurements (observations) yon becomes available. Since the system is linear, the posterior distribution will be multinormal, and its mean and covariance can be calculated recursively. Assuming that we have accounted for the measurements yo1 , . . . , yon−1 , the state s will be multinormal distributed with mean sˆn−1 and covariance matrix Cˆ n−1 , which we

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denote by s ∼ N(ˆsn−1 , Cˆ n−1 ). Since the forward model (13) is linear, we get that the state s is distributed as N(sn , Cn ) where sn = F sˆn−1 Cn = F Cˆ n−1 F T + Cˆ

(15)

before taking the measurement yon into account. Once the measurement yon is known, the posterior mean, sˆn , and covariance matrix, Cˆ n , can be obtained using the Kalman gain matrix Kn = Cn GT (GCn GT + CD )−1 , so that we get sˆn = sn + Kn (yon − Gsn ) Cˆ n = (I − Kn G)Cn −1 = (Cn−1 + GT CD G)−1 .

(16)

Here, I is an identity matrix of appropriate dimension. For our application, the forward model is nonlinear and certain adaptions are required. We write the forward model as sn = F (sn−1 ) + n

(17)

where the model noise n ∼ N(0, CM ). Our observations are assume to depend linearly on the states and are given as yn = Gsn + ηn where the measurement noise is given as ηn ∼ N(0, CD ). The ensemble Kalman filter can now be constructed by using an ensemble of N samples of the distribution  of the state vector s. At time n we can store these N samples in a matrix Sn = sn,1 . . . sn,N where sample i is denoted as sn,i . Initially the ensemble members (samples) are drawn from a prior distribution, which typically is a multinormal distribution. We use the ensemble to represent an approximation of the posterior distribution after assimilating the measurements yo1 , . . . , yon−1 . The change in the distribution by the forward model (17) is accounted for by running the forward model N times, i.e. by calculating sn,i = F (sn−1,i ) + n,i

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with n,i ∼ N(0, CM ). To get the approximation of the posterior distribution for the measurement yon , we use the relation yn,i = Gsn,i + ηn,i ,

i = 1, . . . , N,

(18)

to obtain yon,i . The ensemble is then updated using an approximation of the first and second order moment of the distribution of the state s. The first order moment, the mean, is s¯n = N1 N i=1 sn,i . The second order moment is given as the 1 N T (approximative) covariance matrix C¯n = N−1 i=1 (sn,i − s¯ n )(sn,i − s¯ n ) . Having computed the covariance matrix C¯ n , we can then compute the Kalman gain matrix which is given as Kn = C¯ n GT (GC¯ n GT + CD )−1 . The Kalman gain is then used to update each ensemble member individually by sˆn,i = sn,i + Kn (yon − yon,i ).

(19)

In practice, some care needs to be taken to avoid forming the full approximative covariance matrix C¯ n as this not feasible for large scale systems. Different approaches to handle this can be found in the literature. Some examples are given in [1, 6]. In the application we will present, our primary interest is to estimate parameters of the model. This can be handled by appending the unknown parameters p in the state vector and introducing an extended state vector   s s = . p e

(20)

The forward model can now be written as  sen = F (sen−1 ) =

 F (sn ) . pn

(21)

Here we have removed the model noise term as this will not be used in the example we are presenting. The ensemble based estimation of the parameters can then be performed on (20) using Eqs. (18), (19) and (21).

5 Numerical Example To illustrate the workflow described in the previous chapter we include an example, in which the goal is to estimate the proportionality constant α given in (5), a crucial parameter for quantifying perfusion. We assume that α is varying spatially on a small domain and that we have perfect knowledge about all other parameters of the model. In real images and medical applications, the size of the domain

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will be significantly larger, and increased number of voxels adds challenges to the estimation methodology. However, this is not usually a problem and large scale examples (in terms of number of grid-blocks/voxels) are already routinely used in different applications within the geosciences (e.g., oceanography, reservoir engineering).

5.1 Forward Model For a forward flow model we chose a field of view FOV=[3, 3, 3] mm, divided into [7, 7, 1] cells, hence providing a spatial resolution of [0.43, 0.43, 3.0] mm. The a arterial input ∂ΩA is assigned in the upper left corner of the domain with Dirichlet v ia assigned in the lower boundary conditions pa = 4 kPa. The venous outlet ∂ΩA right corner of the domain with Dirichlet boundary conditions pv = 1.4 kPa. The perfusion scaling parameter is set to α = 10−4 m s kg−1, and further scaled with the normalized Euclidean distance to the closest arterial inlet or venous outlet to create a continuously varying field for α (see Fig. 2 (upper left)). Constant permeability values ka = 1 × 10−13 m2 and kv = 2 × 10−13 m2 are assigned to the arterial and venal pore system, respectively. A porosity of φa = φv = 0.05 ia assigned to both pore systems. Fluid viscosity is set to μ = 3 × 10−3 Pa · s.

Fig. 2 Upper left: True value of α. Upper right: Estimated value of α (after 59 s). Lower left: Standard deviation in the estimation of α as a function of time (seconds). Lower right: Standard deviation of α at 59 s

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Equation (6) is discretized using two-point flux approximation [2]. The resulting linear system is solved using a direct solver, providing a steady-state pressure field pa and pv for each of the pore systems. The tracer evolution in (11) is integrated across each cell, and then approximated as a forward Euler time discretization, using first order upwinding for the tracer concentrations [18]. Hence, tracer concentration is taken upstream for incoming flow, and cell-centered for outgoing flow. A gamma a. function is chosen as arterial input function (AIF) at the inlet ∂ΩA

5.2 Solution of Inverse Problem We start by generating one hundred different spatially varying α-fields as follows. All the fields are generated from a multinormal distribution with constant mean 5 · 10−5 (which is close to the mean of the “true” α-field), but with different covariance matrices. All the covariance matrices C used for generating the ensemble members have size 49 × 49, corresponding to a domain of 7 × 7 × 1 voxels. The diagonal elements are set to σ 2 = (2 · 10−5 )2 . Off-diagonal entries Ci,j describe the covariance between voxels i and j and are assigned value σ 2 (1 − exp((d/a)2 )) where d is the distance (in blocks) between i and j . Since we do not assume much knowledge about a reasonable value for a, each spatially varying α-field is generated with its own a drawn from the uniform distribution on the interval [0, 7]. The mean of the initial ensemble varies slightly over the voxels, taking values in the range [4.8 · 10−5 , 5.4 · 10−5 ]. The true α that was use for generating the data set is shown in Fig. 2 (upper left). We simulated our forward model as described in Sect. 3 for 60 s and used the concentration values for each voxel from 3 s of simulation and further on with a time interval with 2 s between each sample to generate measurements. The measurements were generated by adding 10% noise to the simulated concentration values. Using a slight modification of the ensemble Kalman filter as described in Sect. 4, we estimated the α-field, obtaining the estimate shown in Fig. 2 (upper right) as the estimate. The modification that we did was to run our model from the initial time to the new set of measurements each time new data was assimilated. This means that we run the forward model with the initial ensemble of α values to 3 s, assimilate the observations and modified the parameters (the α-fields), run the forward simulation from initial time to time 5 s, assimilated new observations, updated the ensemble of α-fields, and so on, until all observations were assimilated. We can also quantify the uncertainty in the estimated values of α. We show how the standard deviations for each voxel develop as function of time in Fig. 2 (lower left). The spatial distribution of the final standard deviations for the estimated α is shown in Fig. 2 (lower right). In Fig. 3 we show two of the 49 concentration profiles that are used to estimate α. We can see that there is a significant spread in the simulations from the initial ensemble members (blue curves). Simulating with the final ensemble members (red) does not give much internal variability, which might be related to an underestimation of the uncertainty. A thick green line give the simulation with the correct α-field. It

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2.5

10 -4

1.2 1

2

0.8

1.5

0.6

1 0.4

0.5 0.2

0

0

20

40

seconds

60

0

0

20

40

60

seconds

Fig. 3 Measured concentrations (black crosses), true concentration profile (thick green curve), simulation with 100 initial ensemble members (blue curves) and 100 estimated ensemble members (red curves) for grid block (1,7) (left) and (7,7) (right)

can be seen that this simulation agrees with the simulations from the final ensemble, even though the α-fields does not coincide.

6 Outlook The goal of this paper was to present the feasibility of the application of parameterestimations techniques from reservoir modelling to a seemingly completely different problem: perfusion in organs, in this specific setting, the brain. The very preliminary results presented here are promising and at present we are extending the model to the human brain, introducing vascular trees and modelling the capillary blood distribution. Human organs are extremely complicated, and several layers of complexity can be considered, including respiratory effects on the blood flow, interstitial pressures, changing vessel diameters, transport across vessel walls, just to name a few. To counteract this behaviour we will lean on a Bayesian approach, fitting well with the ensemble Kalman filter method suggested here for solving the parameter estimation problem. Today’s therapies are decided on the base of diagnosis and statistics on patient groups. The increased availability of imaging techniques, physiological parameters, molecular markers, genetic data and other bio-markers, together with the increased computational power and the groundbreaking advances in the field of machine learning, are paving the way towards individually targeted therapies. Therapy will be decided not only based on the diagnosis, but also on the knowledge on how the individual patient reacts to specific treatments and drugs, in this way increasing effectiveness of treatment and reducing social health costs. Mathematics is going to play an increasingly important role in this process. Mathematical equations describe the processes and numerical simulations predict the outcomes. On one side, mathematics will become an indispensable tool to do

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research in life sciences; on the other side, complex biological systems that are poorly described in terms of today’s mathematical knowledge will stimulate the development of new mathematical theories. Acknowledgements This work is supported by the Norwegian Research Council project 262203 “Flow-based interpretation of Dynamical Contrast Enhanced Imaging data”.

References 1. S.I. Aanonsen, G. Nævdal, D.S. Oliver, A.C. Reynolds, B. Vallès, The ensemble Kalman filter in reservoir engineering – a review. SPE J. 14(3), 393–412 (2009) 2. J.E. Aarnes, T. Gimse, K.-A. Lie, An introduction to the numerics of flow in porous media using Matlab, in Geometric Modelling, Numerical Simulation, and Optimization (Springer, Heidelberg, 2007), pp. 265–306 3. G.I. Barenblatt, I.P. Zheltov, I.N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Mech. 24, 1286–1303 (1960) 4. A. Bjørnerud, K.E. Emblem, A fully automated method for quantitative cerebral hemodynamic analysis using DSC-MRI. J. Cereb. Blood Flow Metab. 30(5), 1066–1078 (2010) 5. G. Brix, W. Semmler, R. Port, L.R. Schad, G.L.G, W.J. Lorenz, Pharmacokinetic parameters in CNS Gd-DTPA enhanced MR imaging. J. Comput. Assist. Tomogr. 15, 621–628 (1991) 6. G. Evensen, The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 343–367 (2003) 7. A. Fieselmann, M. Kowarschick, A. Ganguly, J. Horneggerand, R. Fahrig, Deconvolutionbased CT and MR brain perfusion measurement: theoretical model revisited and practical implementation details. Int. J. Biomed. Imaging Article ID 467563, 20 p. (2011) 8. J.W. Forrester, Industrial dynamics: a major breakthrough for decision makers. Harv. Bus. Rev. 36(4), 37–66 (1958) 9. R.M. Henkelman, Does IVIM measure classical perfusion? Magn. Reson. Med. 16(3), 470– 475 (1990) 10. E. Hodneland, Å. Kjørestad, E. Andersen, J. Monssen, A. Lundervold, J. Rørvik, A. Zanna, In vivo estimation of glomerular filtration in the kidney using DCE-MRI, in Image and Signal Processing and Analysis (IEEE, Piscataway, NJ, 2011), pp. 755–761. ISSN 1845–5921 11. K. Jafari-Khouzani, K.E. Emblem, J. Kalpathy-Cramer, A. Bjørnerud, M.G. Vangel, E.R. Gerstner, K.M. Schmainda, K. Paynabar, O. Wu, P.Y. Wen, T. Batchelor, B. Rosen, S.M. Stufflebeam, Repeatability of cerebral perfusion using dynamic susceptibility contrast MRI in glioblastoma patients. Transl. Oncol. 8(3), 137–146 (2015) 12. R.E. Kalman, A new approach to linear filtering and prediction problems. Trans. AMSE J. Basic Eng. (Ser. D) 82, 34–45 (1960) 13. H. Larsson, M. Stubgaard, J.L. Frederiksen, M. Jensen, O. Henriksen, O.B. Paulson, Quantitation of blood-brain barrier defect by magnetic resonance imaging and gadolinium-DTPA in patients with multiple sclerosis and brain tumors. Magn. Reson. Med. 16, 117–131 (1990) 14. A. Matzavinos, C.-Y. Kao, J.E.F. Green, A. Sutradhar, M. Millerand, A. Friedman, Modeling oxygen transport in surgical tissue transfer. Proc. Natl. Acad. Sci. USA 29, 12091–12096 (2009) 15. P. Meier, K.L. Zierler, On the theory of the indicator-dilution method for measurement of blood flow and volume. J. Appl. Physiol. 6(12), 731–744 (1954) 16. G. Nævdal, O. Sævareid, R.J. Lorentzen, Data assimilation using MRI data, in Proceedings, VII European Congress on Computational Methods in Applied Sciences and Engineering (2016) 17. D.S. Oliver, Y. Chen, Recent progress on reservoir history matching: a review. Comput. Geosci. 15, 185–221 (2010)

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18. S. Patankar, Numerical Heat Transfer and Fluid Flow, 1st edn. (Hemisphere Publishing Corporation, Washington, 1980) 19. M. Presho, S. Wo, V. Ginting, Calibrated dual porosity, dual permeability modeling of fractured reservoirs. J. Pet. Sci. Eng. 77, 326–337 (2011) 20. S. Sourbron, A tracer-kinetic field theory for medical imaging, IEEE Trans. Med. Imaging 33(4), 935–946 (2014) 21. S.P. Sourbron, D.L. Buckley, Trace kinetic modelling in MRI: estimating perfusion and capillary permeability. Phys. Med. Biol. 57(2), R1–R33 (2012) 22. P. Tofts, A.G. Kermode, Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. 1. Fundamental concepts. Magn. Reson. Med. 17, 357–367 (1991)

Part II

Kernel Methods for Large Scale Problems: Algorithms and Applications

Convergence of Multilevel Stationary Gaussian Convolution Simon Hubbert and Jeremy Levesley

Abstract In this paper we give a short note showing convergence rates for periodic approximation of smooth functions by multilevel Gaussian convolution. We will use the Gaussian scaling in the convolution at the finest level as a proxy for degrees of freedom d in the model. We will show that, for functions in the native space of ln(d)

the Gaussian, convergence is of the order d − ln(2) . This paper provides a baseline for what should be expected in discrete convolution, which will be the subject of a follow up paper.

1 Introduction Approximation by convolution involves selecting a suitable integrable function K(x) (the convolution kernel) satisfying R K(x)dx = 1. A parameterized family of convolution kernels is generated from K by setting Kh (x) = h−1 K(x/ h) where h > 0. Then, for a given target function f, its convolution approximation f ∗ Kh converges to f , as h → 0. The rate of convergence depends upon the smoothness of f and the polynomial reproduction properties of the underlying convolution kernel. In this paper we consider the approximation of 1-periodic continuous functions by convolution with the Gaussian kernel. In this case it is only possible to reproduce the constant function and so, as we will see, convergence is limited to O(h2 ), regardless of additional smoothness requirements. However, if we employ a multilevel iterative refinement scheme we see that we get very rapid convergence. If the width of the Gaussian is halved at each iteration, then at the nth level of refinement we have

S. Hubbert Birkbeck, University of London, London, UK e-mail: [email protected] J. Levesley () Department of Mathematics, University of Leicester, Leicester, UK e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_5

83

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essentially a d = 2n scaling in the Gaussian. For a discrete scheme we would apply a quadrature at d = 2n equally spaced points, giving d degrees of freedom. If the target function is taken from a certain periodic Sobolev space, whose order prescribes the smoothness of the functions, then we see improved but saturated convergence rates. If we consider functions in the native space of the Gaussian, a ln(d)

space of infinitely differentiable functions, then the order of convergence is d − ln(2) . This work is motivated by the desire to prove convergence of the multilevel sparse grid quasi-interpolation introduced by Levesley and Usta [5]. Multilevel sparse grid algorithms using smooth functions were introduced by the author and collaborators [4]. Our focus is on the Gaussian kernel 1 ψ(x) = √ exp(−x 2 /2) 2π and we define ψh (x) = h−1 ψ(x/ h). We note the Fourier transform of ψ is  ψ(x) =

∞ −∞

ψ(y) exp(−2πixy)dy = exp(−2π 2 x 2 ).

We wish to approximate a continuous 1-periodic function f =

∞

where ek (x) = exp(2πikx), (k ∈ Z),

fk ek ,

k∈Z

by  f ∗ ψh (x) =

=

∞ −∞

f (t)ψh (x − t)dt =

∞ 

j +1

f (t)ψh (x − t)dt

j =−∞ j

∞ 

1

 f (t)ψh (x − t − j )dt =

j =−∞ 0

1

f (t)φh (x − t)dt,

0

where φh (x) =

∞

ψh (x − j ).

j =−∞

We note that φh is 1-periodic and a straightforward computation shows that φh (x) =

∞

k=−∞

ψ(hk)ek (x),

(1)

Convergence of Multilevel Stationary Gaussian Convolution

85

In view of this and the well-known convolution formula for periodic functions (see e.g. [3]) we have that f ∗ ψh =

∞

fk ψ(hk)ek ,

k=−∞

so that the error in convolution approximation is Eh f = f − f ∗ ψh =

∞

fk (1 − ψ(hk))ek .

(2)

k=−∞

This representation immediately shows that the convolution reproduces the constant, but not any other trigonometric polynomial. In Sect. 2 we will examine the convergence of the convolution approximation to target functions taken from certain periodic Sobolev spaces. We will show that no non-constant periodic function can have an approximation order smaller than O(h2 ). In view of this we explore a multilevel iterative refinement, halving h in the convolution approximation at each level. In Sect. 3 we examine the error for this scheme and show that we can improve upon O(h2 ) for functions with additional but finite smoothness. Specifically, we see that rapid improvements in the accuracy are exhibited in the early iterations but once the number of iterations passes a certain level, related to the smoothness of the function, the algorithm settles to converge at a polynomial rate. In Sect. 4 we introduce the native space for Gaussian approximation. The native space is a subspace of infinitely smooth functions and, for such functions, we show that the algorithm exhibits rapid improvements in accuracy at every iteration.

2 Convergence of the Convolution Approximation The functions we wish to approximate are continuous 1-periodic and taken from a periodic Sobolev space ⎧ ⎫ ⎛ ⎞1/2 ⎪ ⎪ ∞ ⎨ ⎬

Nβ = f = fk ek : f β = ⎝|f0 |2 + k 2β |fk |2 ⎠ < ∞ . ⎪ ⎪ ⎩ ⎭ k=−∞ k∈Z The Sobolev embedding theorem [2] ensures that if β > 12 then all functions in Nβ will be continuous. The following result gives error bounds for Gaussian convolution approximation of such functions.

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Proposition 1 Let f ∈ Nβ , where β > 12 . Then

Eh f ∞

⎧ ⎪ ⎪ C1 h2 ⎪ ⎪ (   ⎨ 2 ≤ h (C2 ln h1 + C3 ) ⎪ ⎪ ⎪ ⎪ ⎩C4 hβ− 12

for β > 52 ; for β = 52 ; 1 2

for

< β < 52 ,

where Ci i = 1, 2, 3, 4, are positive constants independent of h. Proof Since ψ(0) = 1 and ψ(−k) = ψ(k), (k ∈ Z+ ), we have, from (2), that Eh f  ≤

∞

(1 − ψ(hk))(|fk | + |f−k |).

(3)

k=1

Suppose that β =

5 2

+ α, where α > 0. Using the elementary bound 1 − exp(−x) ≤ x

for x > 0,

(4)

we have that 1 − ψ(hk) ≤ 2π 2 h2 k 2 . This yields ⎛ Eh f ∞ ≤ 2π 2 h2 ⎝

∞

k=1

k 2 |fk | +

∞

⎞ k 2 |f−k |⎠

k=1

⎛ ⎞ ∞ ∞

5 5 1 1 ⎠ = 2π 2 h2 ⎝ k 2 +α |fk | + k 2 +α |f−k | . α+ 21 α+ 12 k k k=1 k=1 Applying the Cauchy Schwarz inequality we have Eh f ∞ ≤ ⎛ 2π 2 h2 ⎝

∞

k=1

⎡ ⎤ ⎛ ⎞1 ⎛ ⎞1 2 2 ∞ ∞ 5 5

  2 +α ⎢  2  2 +α ⎥ 1 ⎠ ⎢⎝ k k2 |fk |2 ⎠ + ⎝ |f−k |2 ⎠ ⎥ ⎣ ⎦ 2α+1 k ⎞1 2

k=1

k=1

≤ 2π 2 h2 Cf  5 +α . 2

Now assume that β = 52 − α where 0 ≤ α < 2. In the following development we will partition the right hand side of (3) as follows; Eh f  ≤

mh ∞

(1 − ψ(hk))(|fk | + |f−k |) + (1 − ψ(hk))(|fk | + |f−k |), k=1

k=mh +1

Convergence of Multilevel Stationary Gaussian Convolution

87

where mh denotes the integer satisfying mh ≤ h1 ≤ mh + 1. For k ≤ mh we will, as before, use 1 − ψ(hk) ≤ 2π 2 h2 k 2 , whereas for k ≥ mh + 1 we use 1 − ψ(hk) ≤ 1. This leads to Eh f  ≤ 2π 2 h2

mh

k=1

)

∞

k 2 |fk | +

|fk | + 2π 2 h2

k=mh +1

*+

∞

k 2 |f−k | +

k=1

)

,

mh

*+

E+

|f−k | .

k=mh +1

,

E−

Focusing on E+ we can write: E+ = 2π 2 h2

mh

k 2 −α |fk |k α− 2 + 5

1

∞

k 2 −α |fk |

k=mh +1

k=1

1

5

k

5 2 −α

.

Applying the Cauchy-Schwarz inequality to the first sum, in the case where 0 < α < 2, we have

2π 2 h2

mh

k 2 −α |fk |k α− 2 5

1

⎛ ⎞1 ⎛ ⎞1 2 2 mh   5 mh

−α 2 2⎝ 2 2 2⎠ ⎝ 2α−1 ⎠ k ≤ 2π h |fk | k

k=1

k=1

≤ 2π h

2 2

k=1

Cf  5 −α mαh 2

≤ 2π Ch 2

2−α

f  5 −α . 2

For α = 0 we have 2π 2 h2

mh

5

1

k 2 |fk |k − 2

k=1

⎞1 ⎛ ⎞1 ⎛ 2 2 mh   5 mh

1⎠ 2 k 2 |fk |2 ⎠ ⎝ ≤ 2π 2 h2 ⎝ k k=1

k=1

   1 2 √ 2 2 1 ≤ 2π h f  5 (2 ln(mh )) ≤ 2π h ln f  5 . 2 2 h 1 2

2 2

Applying the Cauchy-Schwarz inequality to the second sum we have ∞

k=mh +1

⎛

k 2 −α |fk | 5

1 k 2 −α 5

⎞1 ⎛ 2 ∞ ∞   5 −α

2 ≤⎝ |fk |2 ⎠ ⎝ k2 k=mh +1

k=mh +1

 ≤ f  5 −α C 2

1 (mh + 1)4−2α

⎞1 1 k 5−2α

2

⎠

1 2

≤ Cf  5 −α h2−α . 2

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S. Hubbert and J. Levesley

2−α for 0 < α < 2 These two bounds √ allow us to deduce that E+ ≤ Ch 2 and E+ ≤ h (A ln(1/ h) + B). The equivalent bound for E− , the contribution involving Fourier coefficients of negative index, follows in the same fashion. Hence we conclude that

5 1 12 and 0 < h < after the j th iterative refinement satisfies:

1 2π .

Then the multilevel error

Convergence of Multilevel Stationary Gaussian Convolution

Mj f ∞

89

⎧  2j ⎪ ⎪ ⎪ 2πh ⎪C1 ⎪ j ⎪ ⎪ 22 ⎛ / ⎞ ⎪ ⎪ ⎪ 0  j  2j ⎨ ⎜ 0 ⎟ 2πh 22 ≤ + C3 ⎠ ⎝C2 1ln 2πh j ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪   2β−1 ⎪ 1 ⎪ 4 ⎪ ⎩C4 (2πh)β− 2 21j

if j


2β−1 4 ,

where Ci i = 1, 2, 3, 4, are positive constants independent of h. Proof The proof follows the same pattern as that of Proposition 1 and so rather than repeat the details we briefly outline the steps. We begin by noting that, since ψ(0) = 1 and ψ(−k) = ψ(k), (k ∈ Z+ ), we have Mj f ∞ ≤

∞

bj (k)(|fk | + |f−k |).

(8)

k=1

To reach the bound for the case where j < we note that (4) allows us to deduce that

2β−1 4

(or equivalently β > 2j + 1)

j j . . 2π 2 (2k)2 h2 23 π 2 k 2 h2 = = bj (k) ≤ 22 22 =1

=1



2πh j

2j k 2j .

(9)

22

Using this we replicate the steps from Proposition 1 (associated to the β > 52 case) to reach the required bound. For the remaining cases captured by j ≥ 2β−1 4 (or equivalently β ≤ 2j + 12 ) we follow Proposition 1 again and define mj to be the integer satisfying j

22 ≤ mj + 1. mj ≤ 2πh Using this, the error expression is split into a finite sum (including the first mj terms) and the remaining infinite sum. For bounding purposes we employ (9) for the finite sum and bj (k) ≤ 1 for the infinite sum. Once again by mimicking the steps from Proposition 1 (associated to the β ≤ 52 case) one can establish the stated bounds. We comment here that for functions of finite smoothness the speed at which the multilevel iterative refinement converges is restricted by the smoothness of the function. The error is reduced significantly in the early iterations, when j < 2β−1 4 but beyond this point the error decays asymptotically at a polynomial rate.

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S. Hubbert and J. Levesley

4 Native Space for Gaussian Approximation It is more natural to approximate functions from the so called native space for Gaussians: ⎧ ⎫ ⎛ ⎞1/2 ⎪ ⎪ ∞ ⎨ ⎬

Nψ = f : f ψ = ⎝ ψ(k)−1 |fk |2 ⎠ < ∞ . ⎪ ⎪ ⎩ ⎭ k=−∞ If we approximate such very smooth functions we have the following result. Theorem 3 Let f ∈ Nψ and 0 < h < 1. Then  Mj f ∞ ≤ C

4h2 j e

j 

1 2j

j f ψ .

Proof As in Proposition 2 we can use (9) to develop the multilevel error as follows Mj f ∞ ≤

∞

bj (k)(|fk | + |f−k |)

k=1

 ≤

4π 2 h2 2j

j

j

∞

  k 2j |fk | + |f−k |

k=1



 |fk | |f−k | += ψ(k)k ψ(k) ψ(k) k=1 ⎞ ⎛/ / / j 0 ∞  0∞ 0∞ 2 2 2 2

0 0 0 4π h |fk | |fk | ⎟ ⎜ 1 ≤ +1 ψ(k)k 4j ⎝1 ⎠ 2j ψ(k) ψ(k) k=1 k=1 k=1 

4π 2 h2 2j

∞ 2

2j

(10)

/ ∞ j  1 j 0  0 1 ψ(k)k 4j f ψ . ≤ 2 4π 2 h2 2j k=1

Concerning the infinite sum appearing in the bound above, we observe that for each j√ = 1, 2, . . . the non-negative function x → ψ(x)x 4j is increasing for 0 ≤ √ x ≤ πj and decreasing for x ≥ πj . Let mj denote the integer satisfying mj ≤ √ j π

≤ mj + 1. Then we can write

Convergence of Multilevel Stationary Gaussian Convolution ∞

ψ(k)k 4j =

k=1

≤

mj

1

ψ(k)k 4j

k=mj +1

k=1

 mj

∞

ψ(k)k 4j +

91

ψ(x)x 4j dx + ψ(mj )(mj )4j + ψ(mj + 1)(mj + 1)4j +

 ∞ mj +1

ψ(x)x 4j dx

 √   √ 4j     ∞ ∞ 2 2 j 2j j j + + e−2π x x 4j dx. ≤ 2ψ ψ(x)x 4j dx = 2 2 π π π e 0 0

Examining the integral we have 

∞

e 0

−2π 2 x 2 4j



∞

x dx =

e

−s



0

s 2π

2j

ds √ 2 2s



1

= √ 2(2π)(2π 2 )2j

∞

e−s s

2j − 12

0

  Γ 2j + 12 ds = √ . 2(2π)(2π 2 )2j

Using Stirling’s formula for the Gamma function [1], Formula 6.1.39, we have   2j  √ 1 2j Γ 2j + . ≤ C 2π 2 e Substituting this into the bound above we see that ∞

k=1

 ψ(k)k

4j

≤C

j eπ 2

2j .

Taking the square root and substituting into (10) we conclude that j  1 j  j 2j  Mj f ∞ ≤ 2 4π 2 h2 f ψ 2j eπ 2  j   4h2 j 1 j ≤C f ψ . e 2j We remark that, setting d = 2j , we get a convergence rate of O where α = 1/ ln(2) which is faster than any polynomial.



4h2 α ln d ed

α ln d ,

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References 1. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1964) 2. R.A. Adams, Sobolev Spaces (Academic, New York, 1975) 3. R. Bracewell, The Fourier Transform and Its Applications (2nd edn.) (McGrawHill, New York, 1986) 4. E.H. Georgoulis, J. Levesley F. Subhan, Multilevel sparse kernel-based interpolation. SIAM J. Sci. Comput. 35, 815–831 (2013) 5. F. Usta, J. Levesley, Multilevel quasi-interpolation on a sparse grid with the Gaussian. Numer. Algorithms 77, 793–808 (2017)

Anisotropic Weights for RBF-PU Interpolation with Subdomains of Variable Shapes R. Cavoretto, A. De Rossi, G. E. Fasshauer, M. J. McCourt, and E. Perracchione

Abstract The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has already been proved to be an effective tool for solving interpolation or collocation problems when large data sets are considered. It decomposes the original domain into several subdomains or patches so that only linear systems of relatively small size need to be solved. In research on such partition of unity methods, such subdomains usually consist of spherical patches of a fixed radius. However, for particular data sets, such as track data, ellipsoidal patches seem to be more suitable. Therefore, in this paper, we propose a scheme based on a priori error estimates for selecting the sizes of such variable ellipsoidal subdomains. We jointly solve for both these domain decomposition parameters and the anisotropic RBF shape parameters on each subdomain to achieve superior accuracy in comparison to the standard partition of unity method.

R. Cavoretto · A. De Rossi Department of Mathematics “Giuseppe Peano”, University of Torino, Torino, Italy e-mail: [email protected]; [email protected] G. E. Fasshauer Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO, USA e-mail: [email protected] M. J. McCourt SigOpt, Inc., San Francisco, CA, USA e-mail: [email protected] E. Perracchione () Department of Mathematics “Tullio Levi-Civita”, University of Padova, Padova, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_6

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1 Introduction Radial basis function (RBF)-based methods [4] find their natural applications in various fields, such as image reconstruction, resolution of partial differential equations and population dynamics. Two common computational issues arising when we deal with real situations involve creating approximations from a large number of scattered points and the one of producing accurate approximations despite ill-conditioned linear systems. In this article, we attack the first issue with an efficient computation by means of the Partition of Unity (PU) method [8]. It enables us to decompose the original interpolation problem (involving a matrix of the same size as the amount of data) into many small ones defined on subdomains/patches of the original domain. However, the design of these subdomains (and consequently the number of points lying on each patch) affects the accuracy of the approximation. Generally, when the PU method is used for scattered data interpolation, the PU subdomains are assumed to be balls of a fixed radius [2]. In [3], a local approach was proposed via the PU method that selects optimal local approximants: both the shape parameter and the patch radius were selected such that error estimates were minimized. This strategy was more effective at dealing with points that were inconsistently distributed throughout the domain. This previous work allowing varying patch radii was limited to only spherical patches. Such a scheme is not completely suitable for particular data distributions, such as track data [1]. To address this situation we propose to use ellipsoidal subdomains, allowing one free domain parameter per dimension. This adaptation requires careful selection of the PU weights to guarantee consistency, which we detail in our proposal. To match the anisotropic structure of these subdomains, we use anisotropic Wendland’s functions [7] to form our local approximants. The values of the shape parameters and of the semi-axes of patches are selected by minimizing theoretical error estimates. In particular, as in [3], we focus our attention on the Leave One Out Cross Validation (LOOCV) scheme [6]. By jointly optimizing for both the domain and RBF parameters, we are able to improve on the computational cost from [3]. We also factor in the impact of ill-conditioning during this optimal parameter search to balance accuracy and stability. The outline of the paper is as follows. In Sect. 2, we briefly review the main theoretical aspects of the RBF-PU method. Section 3 is devoted to the presentation of the proposed scheme which makes use of ellipsoidal patches. Numerical experiments are presented in Sect. 4. Section 5 deals with conclusions and work in progress.

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2 The RBF-Based Partition of Unity Method The approximation problem considered in this paper is formulated as follows. Consider a set XN = {x i , i = 1, . . . , N} ⊆ Ω of distinct data points (or data sites or nodes), arbitrarily distributed on a domain Ω ⊆ RM , with an associated set FN = {fi = f (x i ), i = 1, . . . , N} of data values (or measurements or function values), which are obtained by sampling some (unknown) function f : Ω −→ R at the nodes x i . The scattered data interpolation problem consists of finding a function R : Ω −→ R such that R (x i ) = fi , i = 1, . . . , N. To this end, we take R ∈ HΦ (XN ) = span{Φ(·, x i ), x i ∈ XN }, where Φ : Ω ×Ω −→ R is a strictly positive definite and symmetric kernel. More specifically, we take RBFs (radial kernels), and thus, we suppose that there exist a function φ : [0, ∞) → R and a shape parameter ε > 0 such that for all x, y ∈ Ω we have Φ(x, y) = φε (||x − y||2 ) := φ(r). In Table 1, we list the strictly positive definite RBFs that will be used later. Note that the RBFs depend on a shape parameter ε > 0 that significantly affects the accuracy of the approximation. We will later refer to the functions reported in Table 1 as isotropic kernels, meaning that ε is a scalar. By using RBFs, the interpolant assumes the form R(x) =

N

αk φ(||x − x k ||2 ),

x ∈ Ω.

(1)

k=1

The coefficients α = (α1 , . . . , αN )T in (1) are determined by solving the linear system Aα = f , where the entries of the matrix A ∈ RN×N are given by (A)ik = φ(||x i − x k ||2 ), i, k = 1, . . . , N, and f = (f1 , . . . , fN )T . The uniqueness of the solution is ensured by the fact that the kernel Φ is strictly positive definite and symmetric. One drawback of this method is the computational cost associated with the solution of potentially large linear systems. The PU method, presented below, enables us to overcome such issue. At first, we cover the domain Ω with d overlapping subdomains Ωj . To be more precise, we require a regular covering, i.e., {Ωj }dj=1 must fulfill the following properties: 1. for each x ∈ Ω, the number of subdomains Ωj , with x ∈ Ωj , is bounded by a global constant C1 , 2. each subdomain Ωj satisfies an interior cone condition, 3. the local fill distances hXNj are uniformly bounded by the global fill distance hXN , where XNj = XN ∩ Ωj . Table 1 Examples of strictly positive definite isotropic radial kernels

RBF Inverse MultiQuadric C ∞ (IMQ) Mat´ern C 2 (M2) Wendland C 2 (W2)

φ(r) (1 + ε2 r 2 )−1/2 e−εr (εr + 1) max (1 − εr, 0)4 (4εr + 1)

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Once we select weight functions Wj , j = 1, . . . , d, the PU interpolant can be defined as I (x) =

d

Rj (x) Wj (x) ,

j =1

with Rj (x) =

Nj

j

j

αk φ(||x − x k ||2 ),

k=1

where Rj is defined on the subdomain Ωj , Nj indicates the number of points on j Ωj and x k ∈ XNj , with k = 1, . . . , Nj . Therefore, the problem leads to solving j

j

d linear systems of the form Aj α j = f j , where α j = (α1 , . . . , αNj )T , f j = (f1 , . . . , fNj )T and the entries of Aj ∈ RNj ×Nj are given by (Aj )ik = φ(||x i − j

j

j

j

x k ||2 ), i, k = 1, . . . , Nj . Since the coefficients of the local interpolants are determined by imposing the local interpolation conditions, the functions Wj , j = 1, . . . , d, must form a partition of unity. Moreover, we also require that such partition of unity is k-stable (see, e.g. [7, Def. 15.16, p. 276]), which in particular implies that supp(Wj ) ⊆ Ωj . For instance, such conditions are satisfied for the well-known Shepard’s weights; refer e.g. to [8] for further details.

3 Optimal Local Interpolants for the RBF-Based PU Method We now focus on the selection of the PU patches, and we remove the standard assumption that they consist of balls of a fixed radius. Therefore we consider ellipj j soidal patches, i.e., each Ωj is defined through its semi-axes δ j = (δ1 , . . . , δM ). Moreover, in what follows, we use anisotropic kernels. We remark that any isotropic radial kernel can be turned into an anisotropic one by using a weighted 2-norm instead of an unweighted one. Thus, to fix the ideas, on a subdomain Ωj it is enough to replace the scalar value of the shape parameter εj with a symmetric positive definite matrix Ej . More precisely, we consider the special case for which j j Ej = diag(ε1 , . . . , εM ). This allows us to choose a different scaling along the dimensions of the problem. j j Our parametrization strategy consists of selecting both εj = (ε1 , . . . , εM ) and δ j such that the error estimates on Ωj are minimized. This study is motivated by the fact that the PU approximation error is governed by the local ones [7, Th. 15.19, p. 277].

3.1 Local Error Estimates For a general overview about error estimates refer e.g. to [4]. Here we focus on error predictions that are popular in statistics, and precisely on cross validation schemes.

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We describe the cross validation algorithm that is applied on a given Ωj to get a local a priori error estimate [4]. At first, we split the set XNj into two disjoint subsets: a training set TN t and a validation set VNjv such that Njt + Njv = Nj . The set TN t j j is used to construct a surrogate or partial approximation that is validated via the set VNjv . To simplify the following discussion, on Ωj we introduce the following block decomposition of the local interpolation matrix  Aj =

Atjt Atjv vv Avt j Aj

 ,

where, for example, the block Atjv is generated using training points to evaluate and validation data as centers of the kernels. Similarly, we partition α j = (α tj , α vj )T , f j = (f tj , f vj )T . With this notation, the prediction at the points in the validation v t t −1 t vt t t −1 t set using the training set is Avt j (Aj ) f j . In other words, |f j − Aj (Aj ) f j | provides information about the accuracy of the fit on the j -th subdomain. Then, we (q) take q partitions of VNjv , VNjv = {VN(1)v , . . . , VN v }, such that j

(k)

j

q

(i)

(k)

(k)

j

j

(k)

(k)

VN v ∩ VN v = ∅, for i = k, with ∪k=1 VN v = XNj , and TN t = XNj \VN v . j

j

j

Thus, as error estimate we consider the residual left over by the interpolants evaluated at the validation sets. The LOOCV scheme is a particular case of the j (k) general setting presented above for which q = Nj and each VN v = x k . Moreover j

−1 for the LOOCV scheme, Avv j is the diagonal element of Aj and thus, being a scalar, the computation simplifies. Indeed, as error estimate for the j -th subdomain we have (see also [6]) −1 ej = ||(α1 /(A−1 j )11 , . . . , αNj /(Aj )Nj Nj )||p , j

j

where in what follows we fix the index of the discrete norm to p = 2. In our PU context, using both anisotropic kernels and ellipsoidal patches, we have that ej = ej (εj , δj ). In fact, the shape parameter affects the accuracy of the RBF approximant and, for the PU method, the accuracy also depends on which points are involved in the computation of the local interpolants.

3.2 Description of the PU-LOOCV Method To minimize the LOOCV error estimates we use a multivariate optimization tool. This allows us to reduce the computational cost of the procedure presented in [3]. To be more precise, we consider the Nelder-Mead simplex algorithm [5]. Without

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going into details, we remark that on Ωj , given an initial guess (ε0j , δ 0j ), it provides an approximation of the optimal values by computing subsequent simplices and only needs function evaluations of the objective function [5]. In particular, for the implementation we use the MATLAB software and the fminsearch.m routine. Of course, we also need to impose the following constraints on the parameters we optimize: j

εk > 0,

and δ + ≥ δk ≥ δ ∗ , j

k = 1, . . . , M,

j = 1, . . . , d,

where δ ∗ ∈ R is chosen so that patches form a covering of the domain and δ + ∈ R is selected so that for each x ∈ Ω the number of subdomains Ωj , with x ∈ Ωj , is bounded. In this way, after optimizing the parameters (εj , δ j ), j = 1, . . . , d, we have a PU covering made of ellipsoids that is also regular. In fact, each of the patches satisfies an interior cone condition. This assumption is trivially verified for balls. However, it is also true for ellipsoids (see, e.g. [7, Pr. 11.26, p. 195]). Finally, in order to build a consistent PU setting, we also need to carefully choose the compactly supported functions for the PU Shepard’s weights, which are here constructed with the W2 function. Since we have ellipsoids, we need to select the anisotropic form of the W2 function. Thus, εj , which identifies the support of the compactly supported RBF, is taken so that supp(Wj ) = Ωj , j = 1, . . . , d.

4 Numerical Experiments Our experiments focus on bivariate interpolation. In Sect. 4.1, we show the numerical results obtained by considering known functions and artificial track data [1]. Then, in Sect. 4.2 we also take into account real data by analyzing an application to Earth’s topography.

4.1 Experiments with Artificial Data To illustrate the accuracy of the proposed method, we evaluate the interpolant on a grid of s = 402 points x˜ i , i = 1, . . . , s, on Ω = [0, 1]2 and we calculate Root Mean Square Error (RMSE) and Maximum Absolute Error (MAE). The patch centers are constructed as a grid of t 2 points on Ω, where t is the number of tracks. Of course this design will affect the accuracy of the approximation. Nevertheless, the scheme proposed here, allowing to choose variable subdomains, is consequently less sensitive to this choice. We show numerical results obtained by considering five sets of track data on Ω = [0, 1]2 sampled from the 2D Franke’s function, see e.g. [4]. In particular, the results

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Table 2 RMSEs, MAEs and CPU times obtained by using the PU-LOOCV and classical PU methods N 1000 (20 × 50) 2000 (25 × 80) 4000 (40 × 100) 8000 (50 × 160) 16, 000 (80 × 200)

Method PU-LOOCV PU PU-LOOCV PU PU-LOOCV PU PU-LOOCV PU PU-LOOCV PU

RMSE 8.02E–5 3.27E–3 2.58E–5 1.08E–3 4.67E–6 1.28E–3 1.25E–6 3.48E–4 4.55E–7 3.49E–4

MAE 1.56E–3 2.23E–2 3.33E–4 1.64E–2 8.70E–5 1.89E–2 2.77E–5 8.08E–3 6.14E–6 7.39E–3

CPU 43.3 0.5 64.7 1.2 136.0 7.2 189.0 17.4 475.0 109.0

of using LOOCV to optimize the semi-axes of the patches and the shape parameters of the local basis functions are reported in Table 2. They are compared with the classical PU method obtained by taking a grid of t 2 points on Ω as PU centers and a fixed patch radius δ = δ ∗ . Furthermore, we consider the isotropic IMQ kernel with shape parameter equal to 1. We select such shape parameter arbitrarily. Indeed, one of the main advantages of the proposed method is the one of automatically choosing safe values for the shape parameters. Of course, different values might lead to more accurate approximations, but it is not possible to provide a priori optimal or safe shape parameters. Moreover, we also report the CPU times. Tests have been carried out with the MATLAB software on an Intel(R) Core(TM) i7-6500U CPU 2.59 GHz processor. In Fig. 1 (left), we show an example of 2000 track data [1] and the ellipsoidal patches obtained via PU-LOOCV. From the numerical experiments we can note that the classical PU method, which makes use of circular patches, is not able to accurately fit the data, especially when a large number of points is involved. This might be due to a non-optimal selection of the shape parameter and/or of the patch size for the classical PU. In this sense, the PU-LOOCV reveals its robustness, selecting optimal values for those parameters and providing accurate results also when N grows. Finally, we point out that the proposed scheme, besides extending the idea of the method presented in [3] to subdomains having different shapes and to anisotropic kernels, thanks to the use of an optimization routine for the minimization problem, it also speeds up the procedure. For instance, using the same scheme outlined in [3] would take about 300 s for 1000 data. Moreover, note that Table 2 shows that the PU-LOOCV with 1000 points is more than twice as accurate and more than twice as fast than the classical PU method with 16,000 points.

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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

Fig. 1 Left: an illustrative example with 2000 track data that shows how patches are selected by means of the PU-LOOCV method. Right: the Korea’s map and the extracted tracks

4.2 Experiments with Real Data To test the method with real data, we consider points extracted from maps. We take, as example, a map of Korea (plotted in the right frame of Fig. 1) and we extract 40 tracks containing 100 points. The function values, being real samples of the elevation above sea level, are truly oscillating and thus the interpolation problem is particularly challenging. We also extract a grid of 402 points to evaluate the error. By using the M2 kernel, the RMSE for the PU-LOOCV is equal to 2.58E–2. The classical PU with a fixed radius δ = δ ∗ completely fails. However, for the classical PU, taking anisotropic kernels with variable parameters and δ = 3δ ∗ as fixed size of the circular patches gives RMSE=4.21E–2. In other words, we reach about the same accuracy, but this is not completely satisfying. Indeed, if we take circular subdomains of radius 3δ ∗ , the average of points on each patch is about 52, while the PU-LOOCV only requires on average 16 data per patch. Therefore, especially with real data, the approximation by means of an optimized PU method becomes essential.

5 Conclusions In this paper we presented a scheme for the optimal selection of local approximants in the PU method. The scheme, based on ellipsoidal patches, is particularly suitable for track data. Work in progress consists in comparing the LOOCV scheme with other a priori error estimates and in selecting suitable locations for the patch centers.

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Acknowledgements This research has been accomplished within RITA (Rete ITaliana di Approssimazione) and partially supported by GNCS-INδAM. The first and second authors were partially supported by the 2016–2017 project Metodi numerici e computazionali per le scienze applicate of the Department of Mathematics of the University of Torino. The third author was partially supported by grant NSF-DMS #1522687. The last author is supported by the research project No. BIRD167404.

References 1. G. Allasia, R. Besenghi, R. Cavoretto, A. De Rossi, Scattered and track data interpolation using an efficient strip searching procedure. Appl. Math. Comput. 217, 5949–5966 (2011) 2. R. Cavoretto, A. De Rossi, E. Perracchione, Efficient computation of partition of unity interpolants through a block-based searching technique. Comput. Math. Appl. 71, 2568–2584 (2016) 3. R. Cavoretto, A. De Rossi, E. Perracchione, Optimal selection of local approximants in RBF-PU interpolation. J. Sci. Comput. 74, 1–22 (2018) 4. G.E. Fasshauer, M.J. McCourt, Kernel-Based Approximation Methods Using Matlab (World Scientific, Singapore, 2015) 5. J.C. Lagarias, J.A. Reeds, M.H. Wright, P.E. Wright, Convergence properties of the NelderMead simplex method in low dimensions. SIAM J. Optimiz. 9, 112–147 (1998) 6. S. Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math. 11, 193–210 (1999) 7. H. Wendland, Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 17 (Cambridge University Press, Cambridge, 2005) 8. H. Wendland, Fast evaluation of radial basis functions: Methods based on partition of unity, in Approximation Theory X: Wavelets, Splines, and Applications, ed. by C.K. Chui et al. (Vanderbilt University Press, Nashville, 2002), pp. 473–483

Radial Basis Function Approximation Method for Pricing of Basket Options Under Jump Diffusion Model Ali Safdari-Vaighani

Abstract Option contracts under actual market conditions which are more complex than a simple Black-Scholes model are important hedging strategies in the modern financial market. Basket options are attractive products which required the reliable pricing method to take all the beneficial characteristics of a basket option such as correlation effect of underlying assets. The focus of this paper is to present the radial basis function partition of unity method (RBF–PUM) for evaluation of basket options in which underlying assets price follow the Merton jump diffusion model. Numerical experiments are performed for the resulting partial integro-differential equation (PIDE). The resulting valuation method allow for an adaptive space discretization in region near the exercise price to reduce computational cost. The domain truncation effects on the computational error is investigated for the proposed numerical approach. Our numerical examples with two and three underlying assets show that the proposed scheme is accurate, capability of local adaptivity, and efficient in comparison of alternative methods for accurate option prices.

1 Introduction Pricing financial contracts on several underlying assets are received more and more interest as a demand for complex derivatives. The Black-Scholes PDE [1] and its extensions are the basic and the most well-known modeling for valuation of options with one underlying asset as well as basket options. Basket options provide a cheaper alternative to buying individual options on each asset to hedge against risk and also the transaction costs are greatly lowered when one only buys a single option rather than multiple options. Option contracts with several underlying assets

A. Safdari-Vaighani () Department of Mathematics, Faculty of Mathematical and Computer Sciences, Allameh Tabataba’i University, Tehran, Iran e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_7

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interpreted by extension of the Black-Scholes PDE which is potential problem for numerical studies. The direct radial basis function approximation method applied for twodimensional European option [9] and American option by Fasshauer et al. [3] where dynamics of asset prices follow pure diffusion models. Local adaptive RBF collocation method for American basket option under Black-Scholes PDE is introduced as an efficient numerical method [10, 11]. The finite difference method is also introduced for numerical solution of two assets option under Merton and Kou jump diffusion models [2]. RBF-PU method for pricing options under the jump diffusion model with local volatility for single underlying asset is studied on [8]. However, option pricing in the jump diffusion model with several underlying assets is potential problem in computational finance. The aim of this paper is to develop the radial basis function partition of unity method (RBF–PUM) for option pricing in the jump diffusion model, mainly on the Merton model with several underlying. The proposed method allows for local adaptivity, the sparse differentiation matrices, high order of the convergence rate with easy implementation in high number of the dimensions.

2 Basket Option Pricing Under Jump Diffusion Processes To reproduce a more realistic behavior of the underlying assets, we assume that the asset price Si , i = 1, . . . , d follows the risk-neutral processes dSi = μdt + σi dWi + (eJi − 1)Si dq, Si

(1)

where μ denotes a constant expected rate of return, and σi is volatility of the underlying ith-asset, respectively. Here, Wi is the standard Brownian motion where ρij is correlation between Wi , Wj . In Eq. (1), dq is Poisson process with the mean arrival rate λ > 0 and Ji is jump size of the ith-asset. Using the Ito’s formula for finite activity of jump processes, the contingent claim V (S, t) that depends on S = (S1 , . . . , Sd ) ∈ Ω˜ = Rd+ can be derived by taking the expectation under the risk neutral process. The resulting PIDE is given by

1

∂ 2V ∂V ∂V =− ρij σi σj Si Sj − rSi + rV ∂t 2 ∂Si ∂Sj ∂Si d

d

d

i=1 j =1

⎛

 −λ

Ω˜

⎝V (SeJ , t) − V (S, t) −

i=1

d

i=1

⎞ ∂V ⎠ g(J )dJ Si (eJi − 1) ∂Si

(2)

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where the jump magnitudes J = (J1 , . . . , Jd ) have some known probability density g(J ). In Merton model [7], the density function for the jump magnitudes follow the normal distribution with mean vector μ˜ and covariance matrix Σ as   1 ˜ t Σ −1 (J − μ) ˜ . g(J ) = (2π)−d/2 (det Σ)−1/2 exp − (J − μ) 2

(3)

Let Si = exi , τ = T −t and by the variable transformation, V (ex , T −τ ) = U (x, τ ) Eq. (2) rewritten as

σ2 1

∂ 2U ∂U ∂U = ρij σi σj + (r − i − κi λ) − (r + λ)U ∂τ 2 ∂xi ∂xj 2 ∂xi i=1 j =1 i=1 ) *+ , d

d

d

LU

 +λ )Ω where κi =



Ω (e

U (x + J, τ )g(J )dJ (x, τ ) ∈ Ω × (0, T ], *+ ,

Ji

(4)

IU

− 1)g(J )dJ .

3 Payoff and Boundary Conditions Using the facts of price scaling, we may consider the payoff and boundary conditions I (x, τ ) > 0 in log price scaling as a linear function of exi I (x, τ ) = c0 (τ ) +

d

cj (τ )exj

(5)

j =1

Substituting it into the governing PDE equation without jump term leads to Iτ = −rI and therefore ci (τ ) = e−rτ ci (0), i = 1, . . . , d and c0 (τ ) is a constant. If we take the ci (0) = −αi and c0 = E, for typical European basket put can be written as I (x, τ ) = max(E −

d

αj exj −rτ , 0).

(6)

j =1

The payoff and boundary conditions for pricing equation (4) may be given as follow U (x, 0) = I (x, 0) = max(E −

d

j =1

αj exj , 0),

x∈Ω

(7)

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lim U (x, t) = lim I (x, t) = 0,

xi →∞

xi →∞

U (x, τ ) = I (x, τ ) = max(E −

x ∈ Ω, i = 1, . . . , d. d

αj exj −rτ , 0),

(8) x ∈ Ωi

(9)

j =1,i=j

where Ωi = {x ∈ Ω|xi → −∞, xj > −∞, j = i}, i = 1, . . . , d is corresponded to the near field boundary condition and related to the reduced (d − 1)-dimensional basket option.

4 Radial Basis Function Collocation Schemes For scalar function values fj at scattered distinct node locations xj ∈ R, j = 1, . . . , N, the standard RBF interpolant takes the form s(x) =

N

λj φ(x − xj ),

(10)

j =1

where φ is a real-valued function such as the inverse multiquadric (IMQ) φ(r) = √ 1 coefficients λj ∈ R for j = 1, . . . , N, are determined by interpolation 2 2 ε r +1

conditions s(xi ) = fi , i = 1, . . . , N. In matrix form, the coefficient vector λ¯ = [λ1 , . . . , λN ]T can be obtained by solving linear system Aλ¯ = f,

(11)

where Aij = φ(xi − xj ), i, j = 1, . . . , N, and f = [f1 , . . . , fN ]T . If we define ¯ φ(x) = (φ(x−x1 ), . . . , φ(x−xN )), when λ¯ is known, the RBF interpolant (10) can be rewritten as −1 ¯ λ¯ = φ(x)A ¯ s(x) = φ(x) f.

(12)

We notice that matrix A is invertible for distinct node points where RBF function is positive definite such as IMQ. For the approximation proposes, we need to apply the linear operator L on Eq. (12) to evaluate the sL = [Ls(x1 ), . . . , Ls(xN )]T at the set of node points X = {xi }N i=1 which is leading to sL = ΦL A−1 f.

(13)

where ΦL = [Lφ(xi − xj )]i,j =1,...,N . The differentiation matrix DL under operator L is given by DL = ΦL A−1 .

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The arising differentiation matrix is based on the standard direct RBF interpolation method which is dense matrix. The differentiation matrices can be achieved based on the local properties of the RBF interpolation such as partition of unity RBF method in which the arising differentiation matrices are sparse [6, 10].

4.1 Radial Basis Function Partition of Unity Method Let Ω ⊂ Rd be an open set, {Ωi }M i=1 be an open cover of Ω satisfying a pointwise overlap condition. In the RBF–PUM, the global interpolation function s(x) is constructed as follow s(x) =

M

wj (x)sj (x),

(14)

j =1

where sj is an RBF interpolation on patch Ωj and wj : Ωj → R are compactly supported, nonnegative weight functions subordinate to the cover. The partition of unity weight functions wj satisfy the partition of unity property M

wj (x) = 1,

(15)

j =1

which is constructed using Shepard’s method as follows ϕj (x) wj (x) = M , k=1 ϕk (x)

j = 1, . . . , M,

(16)

where ϕj (x) are compactly supported functions such as Wendland functions with support on Ωj . In the RBF–PUM, a time-dependent problem is approximated by s(x, t) =

M

wj (x)sj (x, t),

(17)

j =1

where sj (x, t) is an RBF approximant of the type (12) on Ωj . For a more detailed discussion on construction of the differentiation matrices by radial basis function partition of unity method see [10].

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5 RBF Approximation Method for Basket Option Model Since numerical computations can only be performed on finite domains, the first step is to reduce the PIDE to a bounded domain. Assume that the computational bounded domain to the PDE part be the truncated domain ΩX ⊂ Ω and assume the I truncated integral domain is ΩJ . Let {xi }N i=1 be set of distinct interior node points and {xi }N i=NI +1 is boundary points of ΩX . Collocation the PIDE (4) at the node points leads to the linear system of ODEs UI (τ ) = DL UI (τ ) + λJ DI UI (τ ),

(18)

where UI (τ ) = [U1 (τ ), . . . , UNI (τ )]T . The DL UI (τ ) and DI UI (τ ) corresponds to the PDE part and integral part, respectively. From Eq. (13), the differentiation matrix is

σ2 1

DL UI (τ ) = ρij σi σj Dij,I UI (τ ) + (r − i − κi λ)Di,I UI (τ ) 2 2 d

d

d

i=1 j =1

i=1

− (r + λ)UI (τ ) + F (τ )

(19)

where D.,I contains the columns of the differentiation matrices corresponding to interior points, and

σ2 1

ρij σi σj Dij,b Fb (τ ) + (r − i )Di,b Fb (τ ) 2 2 d

F (τ ) =

d

i=1 j =1

d

(20)

i=1

forces the boundary conditions. In Eq. (20), the matrix D.,b contains the known boundary columns and Fb (τ ) = [U (xNI +1 , τ ), . . . , U (xN , τ )]T .

5.1 Approximation of the Integral Term For node points into the localized integral domain, xi ∈ ΩJ the integral term in Eq. (4) can be approximated by quadrature rules such as  IU (xi , τ ) =

g(J − xi )U (J, τ ) dJ ≈ ΩX

NI

wk g(xk − xi )U (xk , τ ),

x i ∈ ΩJ

k=1

(21) where wj is weight of the quadrature rules. In matrix form, the approximated integral term can be written as λJ DI UI (τ ) = λIJ W UI (τ ),

x i ∈ ΩX

(22)

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where the matrix W includes the weights and distribution function values, [Wi,k ] = wk g(xk − xi ),

i, k = 1, . . . , NI ,

(23)

and the diagonal matrix IJ is for localization of the integral domain to ΩJ [IJ i,j ] =

1 0

if i = j and xi ∈ ΩJ otherwise.

Substitution of Eq. (19) together Eq. (22) in Eq. (18) leads to the ODEs system. We solve the arising system of ODEs by unconditionally stable second order backward differentiation formula (BDF-2) [4, p. 401] which is described also in [9].

6 Numerical Experiments In whole of numerical experiments, IMQ is considered as a radial basis function, and compact support Wendland function for weight function in RBF–PUM case. The experiments are carried out on a machine with an Intel Core i7 processor, 2.7 GHz and 16 GB RAM. The approximation of the solution is performed by uniform and non-uniform discretization of the computational domains ΩX = [−3, 3]d and ΩJ = [−2, 2]d where payoff parameters are r = 0.05, E = 1, T = 1. In practice, the region near the exercise price is the most interesting one where one wishes to obtain option prices. Along the xi -directions we want to have a distribution of node points which is more dense in a neighborhood of the exercise price in log scaling. We would like to apply the non-uniform discretization that has recently be employed, e.g., in [5, 10]. In order to cluster nodes around the exercise price E in log price scaling, we define the node coordinates in each direction xi through xi = ln E + l sinh(ξj ),

1 ≤ j ≤ n,

(24)

where ξj ∈ [ξ1 , ξn ] are equidistant values and l is a parameter that determines the amount of node refinement in the xi direction near the exercise price. By the requirement that the nodes should fall in the interval [xmin, xmax ] we can compute the range of ξ to ξ1 = sinh−1 ((xmin − ln E)/ l) ξn = sinh−1 ((xmax − ln E)/ l). The centers of the patches are defined with a similar pattern as for the node distribution. In our numerical experiments we have used l = 0.4 for both nodes and partition centers.

A. Safdari-Vaighani

1

1

0.8

0.8 Option Value

Payoff

110

0.6 0.4 0.2 0 0

0.6 0.4 0.2 0 0

5

5 10 S1

15

0

10

5

15

10 S1

S

2

15

0

10

5

15

S

2

Fig. 1 The payoff function of put basket option of two asset (left), and RBF–PUM approximation solution with correlated assets and λ = 0.1 (right)

For the numerical illustrations, we consider both independent correlations of the assets, i.e., ρij = 1, i = j and correlated assets with ρij = 0.5, i = j . Furthermore, we use the same strategies to choose the parameter values from [2, 10] given by (σ1 , σ2 ) = (0.3, 0.3), (μ˜ 1 , μ˜ 2 ) = (0.1, 0.1), (σ˜ 1 , σ˜ 2 ) = (0.4, 0.4) and ρ˜ij = 0, i = j . Figure 1 shows the payoff function and approximated solution of put basket option by tailored node distribution. The shape parameter is not optimized in numerical results while it is scaled with respect to the node density in the patches. Let εj be the shape parameter of j th patch which is defined as εj = ε

h hj

(25)

where h is the global node distance correspond to the computational domain and hj is local node distance in j th patch. The numerical solutions for the two asset case have been computed by ε = 1.25. In Fig. 2, the behavior of error as function of n for RBF–PUM with uniform node and tailored nodes is compared with finite difference method (FDM) that uses equidistance node points. This means that same (fixed domain) PIDE is solved by all methods and the resulting ODEs system is solved by BDF-2 scheme. In the left Fig. 2 the assets are fully correlated and jump arrival rate is λ = 0.1 while in the right part of this figure λ = 0.2 and assets are correlated with ρij = 0.5, i = j . The reference solution is computed using the FDM with n = 141. The max error is estimated from evaluation at 25 equidistance point in the region [E − 0.1, E + 0.1]2 which consist the part of domain near the exercise price. This figure indicates that non-uniform RBF–PUM has significantly better accuracy in comparison of uniform RBF–PUM and FDM for the same number of node points. This capability helps to have a better performance for higher dimensions without using huge number of the discretized points. The visualization of the option price with three underlying assets is demonstrated in Fig. 3, where the axes are correspond to the assets price and the color bar denotes

Radial Basis Function Approximation Method for Pricing of Basket Options −2

−2

10

||Error||∞

∞

||Error||

10

tailored uniform FDM

−3

10

−4

tailored uniform FDM

−3

10

−4

10

20

111

10 25

30

35 n

40

45

50

20

25

30

35

n

40

45

50

55

Fig. 2 The error against the number of points per dimension n for ε = 1.25h/ hj , ρ12 = 1 and λ = 0.1 (left), and ρ12 = 0.5 and λ = 0.2 (right). In both cases, errors are computed against a reference solution computed with FDM for n = 140 and the error is evaluated at (S1 , S2 ) = (0.9 : 0.05 : 1.1, 0.9 : 0.05 : 1.1) 0.8 0.7 3

0.6

0.4 0.3

2

S

3

0.5

1

0.2 0.1

3

2 S2

1

2

1

3

S

1

Fig. 3 The RBF–PUM approximated solution of basket option of three assets where ρij = 0.5 i = j and λ = 0.1. The bars represent the option price. Right figure shows the node points

the option value. The parameters are the same as two asset case which is extended for three asset. For having the better vision, the larger values of assets price are skipped. For having the better sense of the error behavior, the trend of the error is plotted for five different case of the asset prices where the reference solution is computed for finnier grid points of adaptive RBF–PUM. The meaningful trend of the error for numerical solution is clear in Fig. 4.

112 −1

10

error

Fig. 4 Error as function of n where the reference solution is produced by n = 30. S = (0.9, 0.9, 1) solid line (open circle), S = (0.9, 1, 1) dash line (open square), S = (1, 1, 1) solid line (open triangle left), S = (1.1, 1, 1) dash line (open diamond), and S = (1.1, 1.1, 1) solid line (open triangle right), respectively

A. Safdari-Vaighani

−2

10

−3

10

14

16

18

20 n

22

24

26

Acknowledgements The author would like to thank Elisabeth Larsson, Uppsala University for a valuable discussion on the issues regarding RBF–PUM.

References 1. F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973) 2. S.S. Clift, P.A. Forsyth, Numerical solution of two asset jump diffusion models for option valuation. Appl. Numer. Math. 58(6), 743–782 (2008) 3. G. Fasshauer, A.Q.M. Khaliq, D.A. Voss, Using mesh free approximation for multi asset American options in Mesh Free Methods, ed. by C.S. Chen. J. Chin. Inst. Eng. 27, 563–571 (2004) 4. E. Hairer, S. Nørsett, G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edn. (Springer, Berlin, 2000) 5. K.J. In’t Hout, S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation. Int. J. Numer. Anal. Model. 7(2), 303–320 (2010) 6. E. Larsson, V. Shcherbakov, A. Heryudono, A least squares radial basis function partition of unity method for solving PDEs. SIAM J. Sci. Comput. 39(6), 2538–2563 (2017) 7. R.C. Merton, Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3(1–2), 125–144 (1976) 8. R. Mollapourasl, A. Fereshtian, H. Li, X. Lu, RBF-PU method for pricing options under the jump diffusion model with local volatility. J. Comput. Appl. Math. (2018). https://doi.org/10. 1016/j.cam.2018.01.002 9. U. Pettersson, E. Larsson, G. Marcusson, J. Persson, Improved radial basis function methods for multi-dimensional option pricing. J. Comput. Appl. Math. 222(1), 82–93 (2008) 10. A. Safdari-Vaighani, A. Heryudono, E. Larsson, A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications. J. Sci. Comput. 64(2), 341–367 (2015) 11. V. Shcherbakov, Radial basis function partition of unity operator splitting method for pricing multi-asset American options. BIT Numer. Math. 56(4), 1401–1423 (2016)

Greedy Algorithms for Matrix-Valued Kernels Dominik Wittwar and Bernard Haasdonk

Abstract We are interested in approximating vector-valued functions on a compact set Ω ⊂ Rd . We consider reproducing kernel Hilbert spaces of Rm -valued functions which each admit a unique matrix-valued reproducing kernel k. These spaces seem promising, when modelling correlations between the target function components. The approximation of a function is a linear combination of matrix-valued kernel evaluations multiplied with coefficient vectors. To guarantee a fast evaluation of the approximant the expansion size, i.e. the number of centers n is desired to be small. We thus present three different greedy algorithms by which a suitable set of centers is chosen in an incremental fashion: First, the P -Greedy which requires no function evaluations, second and third, the f -Greedy and f/P -Greedy which require function evaluations but produce centers tailored to the target function. The efficiency of the approaches is investigated on some data from an artificial model.

1 Matrix-Valued Kernels We will give a short overview on the theory of matrix-valued kernels and how they can be applied in the context of approximation/surrogate modelling. For further information and a more thorough introduction, we refer to literature, e.g. [1, 3]. For a compact set Ω ⊂ Rd a bivariate function k : Ω × Ω → Rm×m is called a matrix-valued kernel if k(x, y) = k(y, x)T . It is further denoted as (strictly) positive definite if for any finite set X = {x1 , . . . , xn } ⊂ Ω of pairwise distinct points the associated block Gramian matrix k(X, X) := (k(xi , xj ))i,j ∈ Rmn×mn is positive

D. Wittwar () · B. Haasdonk University of Stuttgart, Institute of Applied Analysis and Numerical Simulation, Stuttgart, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_8

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(semi-)definite. In this case, there exists a unique Hilbert space, the so called native space Hk of Rm -valued functions over the domain Ω such that the kernel k satisfies k(·, x)α ∈ Hk ,

∀ x ∈ Ω, ∀ α ∈ Rm

f, k(·, x)αHk = f (x) α, T

∀ x ∈ Ω, ∀ f ∈ Hk , ∀ α ∈ R

(1) m

(2)

where (2) is called the reproducing property. It follows that the directional kernel evaluations k(·, x)α are the Riesz representers of the directional point evaluation functionals δxα : Hk → R, δxα (f ) := f (x)T α. With the Cauchy-Schwarz inequality these δxα are bounded. Vice versa, if for a Hilbert space H of functions f : Ω → Rd all directional point evaluation functionals are bounded, there exists a unique positive definite kernel which satisfies (1)–(2). Hence, such a Hilbert space is referred to as reproducing kernel Hilbert space, or RKHS for short, and k is called its reproducing kernel. Given a function f : Ω → Rm and a set of n pairwise distinct points X := f {x1 , . . . , xn }, the kernel interpolant sX of f on the centers X can be defined via f

sX (x) :=

n

k(x, xi )αi ,

(3)

i=1

where the coefficient vectors αi ∈ Rm solve the linear system p

k(xj , xi )αi = f (xj ),

for j = 1, . . . , n.

(4)

i=1

We assume in the following, that k is a (not necessarily strictly) positive definite kernel and f ∈ Hk . In this case, (4) may have non-unique coefficient solutions, f which however all represent the unique interpolant sX . In the case of strictly positive definite kernels, even the coefficient vectors in (4) are unique for arbitrary finite sets of pairwise distinct points X ⊂ Ω, and interpolation is also well-posed if f ∈ / Hk . f Moreover, as a consequence of the reproducing property (2), the interpolant sX can be identified as the best approximation of f in the subspace N (X) ⊂ Hk given by N (X) := span{k(·, x)α| x ∈ X, α ∈ Rm }.

(5)

Since N (X) is a finite dimensional closed subspace, it is also an RKHS, since the directional point evaluation functionals δxα restricted to N (X) are still bounded. Hence, N (X) admits its own unique reproducing kernel kN (X). It can be shown, c.f. [6], that this reproducing kernel is given by kN (X)(x, y) = k(x, X)k(X, X)+ k(X, y),

(6)

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where k(X, X)+ denotes the Moore-Penrose pseudoinverse of the Gramian matrix k(X, X). Furthermore, the orthogonal projection operator ΠN (X) : Hk → N (X) is well defined and, therefore, we are able to define the Power-Function PX : Hk∗ → R via PX (λ) =

|λ(f ) − λ(ΠN (X) (f ))| , f Hk f ∈Hk \{0} sup

for λ ∈ Hk∗ .

(7)

Using the Cauchy-Schwarz inequality and the fact that ΠN (X) is self-adjoint, it can be shown, see [6], that for the directional point evaluation functional δxα the PowerFunction is given by PX (δxα )2 = α T (k(x, x) − kN (X)(x, x))α.

(8)

For notational convenience, we denote as PX (x) := k(x, x) − kN (X)(x, x)

(9)

the Power-Function matrix, which in general is positive semidefinite. Combining (7)–(9) we get the following directional error bound f

|(sX (x) − f (x))T α|2 ≤ α T PX (x)αf 2Hk .

(10)

2 Greedy Algorithm For a given kernel and target function, the quality of the interpolant is dependent on the choice of centers. For the selection of these centers we employ the kernel greedy algorithm, whose pseudo code is given in Algorithm 1, and which works as follows: We assume to have a given finite sampling ΩN ⊂ Ω of the input space, an initial set of centers X ⊂ Ω, this may be empty, a tolerance ε > 0 and an error indicator function E. Now, we iteratively select a point maximizing E, add it to the set of centers and compute the next approximant by interpolation on the small set of chosen centers. This is repeated until the tolerance ε is reached. Algorithm 1 General Kernel Greedy Algorithm Require: finite sampling of the input domain ΩN ⊂ Ω, kernel k : Ω × Ω → Rm×m , target function f : Ω → Rm , initial set of centers X, error indicator function E, tolerance ε > 0. 1: while max E(k, f, X, x) ≥ ε do x∈ΩN

2:

x ∗ = arg max E(k, f, X, x) x∈ΩN

3: X = X ∪ {x ∗ } 4: end while 5: return X

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In the following we consider three instantiations by different choices of E, resulting in the P -Greedy, f -Greedy and f/P -Greedy algorithms.

2.1 P -Greedy The P -Greedy uses an error indicator that depends on the Power-function but is independent of the target function. Hence, it results in a set of centers which are suitable for variety of different target functions. Furthermore, no (expensive) function evaluations are necessary in the selection process which can thus be performed in a rapid manner. Using the directional error bound (10) and taking the supremum over all directions α ∈ Rm of length one, we end up with the bound f

sX (x) − f (x)22 ≤ PX (x)2 f 2Hk ,

(11)

where  · 2 denotes the Euclidean norm for vectors and the spectral norm for matrices, respectively. The error indicator function E1 is then given as E1 (k, f, X, x) := E1 (k, X, x) := PX (x)2 .

(12)

We note, that by (6) we have kN (X)(x, x) = k(x, x) for all x ∈ X and thus E1 (k, X, x) = 0 for all x ∈ X. Moreover, as a direct consequence of (7) we have E1 (k, X, x) ≤ E1 (k, Y, x) for all x ∈ Ω and Y ⊂ X. In particular the algorithm terminates after a finite number of steps and no point is chosen a second time. In the scalar-valued case, see [2], this algorithm has recently been shown to result in quasi-optimal approximations for kernels of Sobolev spaces [4] and asymptotically uniformly distributed point sets.

2.2 f -Greedy For the f -Greedy the error indicator function E2 is given by f

E2 (k, f, X, x) := sX (x) − f (x)22 .

(13)

One can see, that the indicator relies on the evaluation of the target function and should therefore select a set of centers that is tailored to the target function. This is expected to lead to a smaller number of centers when compared to the P -Greedy. However, it involves all target values f (x), x ∈ ΩN which may not be cheaply available, and the resulting set of centers is individually suited to this particular target function. In contrast to the indicator E1 the indicator E2 is in general not decreasing, i.e. the inequality E2 (k, f, X, x) ≤ E2 (k, f, Y, x) for x ∈ Ω and Y ⊂ X does not necessarily hold. Nonetheless, we still have E2 (k, f, X, x) = 0 for all x ∈ X and no point is selected twice.

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2.3 f/P -Greedy By the reproducing property (2) we obtain f

f

f (x) − sX (x)2 ≤ k(x, x)2 f − sX Hk ,

(14)

thus, the error in the Euclidean norm can be bounded by a kernel dependent constant and the error in the Hilbert space norm. Hence, it seems reasonable to choose an indicator function E3 in such a way, that the error in the Hilbert space norm is f minimized. Since sX is the best approximation of f in N (X) we have f

f

f − sX 2Hk = f 2Hk − sX 2Hk

(15)

and, therefore, minimization of the left hand side is equivalent to maximizing f the Hilbert space norm of the interpolant sX . For this purpose the error indicator function E3 is chosen as E3 (k, f, X, x) := (sX (x) − f (x))T PX (x)+ (sX (x) − f (x)). f

f

(16)

For scalar strictly positive definite kernels this is equal to f

E3 (k, f, X, x) :=

|sX (x) − f (x)| PX (x)

thus a fraction of an “f” and “P” dependent term motivating the notion “f/P”-Greedy. The following lemma, which extends the results in [5] to matrix-valued kernels, shows that the right hand side in (16) is equal to the gain in the square of the Hilbert f space norm of the interpolant sX 2Hk , when the set of centers X is enriched by x: Lemma 1 (Local Optimality of the f/P -Greedy Selection Rule) Let k : Ω × Ω → Rm×m be a positive definite matrix-valued kernel, f ∈ Hk and f X = {x1 , . . . , xn } ⊂ Ω a finite set of pairwise distinct points. Let sX ∈ N (X) denote the unique interpolant of f on the centers X. Then it holds for all x ∈ Ω: sX∪{x} 2Hk = sX 2Hk + (sX (x) − f (x))T PX (x)+ (sX (x) − f (x)). f

f

f

f

(17)

Proof We restrict ourselves to the strictly p.d. case. For the non-strictly p.d. case technical consideration of the null spaces of the kernel matrices is required without major change of the main arguments. For suitable coefficients the interpolants can be expressed as f

sX∪{x} = k(·, X)α + k(·, x)αn+1 ,

α ∈ Rmn , αn+1 ∈ Rm

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and f

sX = k(·, X)β,

β ∈ Rmn . f

f

Furthermore, let A := k(X, X) and B := k(X, x). Since both sX∪{x} and sX interpolate f on X we have Aα + Bαn+1 = Aβ

⇐⇒

β = α + A−1 Bαn+1 .

(18)

f

For the norm of sX∪{x} it holds f

T sX∪{x} 2Hk = α T Aα + 2α T Bαn+1 + αn+1 k(x, x)αn+1

and using (18) we get f

sX 2Hk = β T Aβ T = α T Aα + 2α T Bαn+1 + αn+1 B T A−1 Bαn+1   f T k(x, x) − B T A−1 B αn+1 = sX∪{x} 2Hk − αn+1 f

T = sX∪{x} 2Hk − αn+1 PX (x)αn+1 .

(19)

For the difference between the target function and the interpolant on X we again have via (18) f

f

f

f (x) − sX (x) = sX∪{x} (x) − sX (x) = B T α + k(x, x)αn+1 − B T β   = k(x, x) − B T A−1 B αn+1 = PX (x)αn+1 .

(20)

Combining (19) and (20) concludes the proof as f

f

T sX 2Hk = sX∪{x} 2Hk − αn+1 PX (x)αn+1

= sX 2Hk + (sX (x) − f (x))T PX (x)−1 (sX (x) − f (x)). f

f

f

 Similar to the f -Greedy, the f/P -Greedy is more expensive than the P -Greedy and in general the indicator is not monotically decreasing. However, due to the interpolation property we have E3 (k, f, X, x) = 0 for all x ∈ X and thus the algorithm again terminates with a finite number of centers.

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3 Numerical Example In this section we want to investigate the effect of the different error indicator function on the quality of the approximation and the placement of the centers. For this purpose we consider the unit disc segment Ω = {x = (r cos(ϕ), r sin(ϕ))T ∈ ˜ with Ω˜ = [0, 1] × [ 1 π, 5 π], the target function f = (fi )8 : R2 | (r, ϕ)T ∈ Ω} i=1 3 3 Ω → R8 given by fi (x) :=

10

e−#(i+1)/2$x−xj  , 2

i = 1, . . . , 8,

j =1

with x1 = (0, 0)T and xj = 0.1(cos( j6 π), sin( j6 π))T , j = 2, . . . , 10 and the kernel k : Ω × Ω → R8×8 given by a diagonal Gaussian with decaying widths ⎧ ⎨ e−#(i+1)/2$x−y2 , i = j ki,j (x, y) := ⎩ 0, i = j By straightforward computation one can see that f (x) = k(x, Y )1 where Y = {x1 , . . . , x10 } and 1 ∈ R80 is the vector containing only ones. In particular we have f 2Hk = 1T k(Y, Y )1 ≈ 768.295. For the Greedy algorithm we choose ΩN by transforming 50 × 50 uniformly distributed points in Ω˜ to rectangular coordinates, which results in 2451 sample points and use the tolerance ε = 10−7 . The sets of centers which are generated are denoted by Xi , i = 1, 2, 3 where the index corresponds to the index of the respective error indicator function Ei , i = 1, 2, 3 from Sect. 2. In Fig. 1 the decay of the error indicator (maximum Ei over the training set ΩN ) and maximum training error for an increasing number of centers

Fig. 1 Error indicator decay (left) and maximum training error decay in the Euclidean norm (right) for increasing number of centers

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Fig. 2 Distributions of the centers X1 , X2 and X3

are depicted. As we can see, it takes 114 iterations for the P -Greedy algorithm to terminate, where only 35 (f -Greedy) and 29 (f/P -Greedy) are required for the other algorithms. This is caused by the slow decay in the Power-function which in itself is caused by the narrow Gaussians which model the last target function components. As we mentioned before in Sect. 2.1, we can see in Fig. 2 that the set X1 is somewhat uniformly distributed while X3 is clearly not space filling. In Fig. 3 the decay for the maximum test error in the Euclidean norm err2i on the ˜ test set ΩM generated by transforming 100 × 100 uniformly distributed points in Ω, k and Hilbert space norm erri , i = 1, 2, 3 are shown. While the error in the Hilbert space norm is monotically decaying for any choice of Ei . This is not the case for the Euclidean norm. However, in both cases the f/P -Greedy generates the best sets

Fig. 3 Test error decay in the Euclidean norm (left) and in the Hilbert space norm (right) for increasing number of centers

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with regards to the number of centers that are used in the interpolant expansion. For example, the P -Greedy algorithm takes about 70 iteration to reach a Euclidean error of order 10−4 , while the f/P -Greedy requires 29 to reach the same result. Overall, all three variants generate very sparse kernel-based models. Future work will aim at surrogate modelling for engineering applications with those techniques. Acknowledgements We thank Gabriele Santin for fruitful discussions.

References 1. M. Alvarez, L. Rosasco, N.D. Lawrence, Kernels for vector-valued functions: a review. Found. Trends Mach. Learn. 4(3), 195–266 (2012) 2. S. De Marchi, R. Schaback, H. Wendland, Near-optimal data-independent point locations for radial basis function interpolation. Adv. Comput. Math. 23(3), 317–330 (2005) 3. C.A. Micchelli, M. Pontil, Kernels for multi-task learning, in Advances in Neural Information Processing Systems (MIT, Cambridge, 2004) 4. G. Santin, B. Haasdonk, Convergence rate of the data-independent P-greedy algorithm in kernelbased approximation. Dolomites Res. Notes Approx. 10, 68–78 (2017) 5. R. Schaback, J. Werner, Linearly constrained reconstruction of functions by kernes with applications to machine learning. Adv. Comput. Math. 25, 237–258 (2006) 6. D. Wittwar, G. Santin, B. Haasdonk, Interpolation with uncoupled separable matrix-valued kernels. arXiv 1807.09111

GPU Optimization of Large-Scale Eigenvalue Solver Pavel Kus, ˚ Hermann Lederer, and Andreas Marek

Abstract We present a GPU implementation of a large-scale eigenvalue solver as a part of the ELPA library. We describe the methodology of utilizing the GPU accelerators within an already well optimized MPI-based code. We present numerical results using two different HPC systems equipped with modern GPU accelerators and show the performance benefits of the GPU version.

1 Introduction Solving large eigenvalue systems is, apart from being a classical problem of linear algebra with a broad range of applications, a substantial part of many important problems in materials science, computational chemistry, and namely electronic structure theory, where a key task is the solution of Schrödinger-like eigenproblems [1]. Since the solution of the eigenproblem scales as O(n3 ), where n is the size of the matrix, it can easily dominate the whole compute-time for largescale calculations. The ELPA library [1–3] is a well established eigensolver library used by many computational chemistry and materials science codes. It provides an efficient implementation in distributed memory with good scaling properties for many thousands of CPU cores as well as optimizations targeting various particular architectures. It also contains specific algorithmic advantages for problems where only a certain part of eigenvalues and eigenvectors are sought. Since the role of accelerated HPC systems with a high peak performance is of growing importance, their efficient usage should be ensured. For this reason, substantial efforts are taken to adapt HPC applications to GPU computing. In this contribution we describe our ongoing effort of improving the performance of the

P. K˚us () · H. Lederer · A. Marek Max Planck Computing and Data Facility, Garching bei München, Germany e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_9

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ELPA library on supercomputers with GPU-equipped nodes. We show multiple ways how GPU support can be employed within an already highly optimized parallel MPI-based code and we discuss their advantages and drawbacks. We comment on both algorithmic and technical aspects of the problem and show performance comparisons.

2 Eigenvalue Solver We look for the solution of (possibly generalized) eigenvalue problem AV = BV Λ. The steps of finding solution to this problem are well known and conceptually simple, see, e.g, [4]. First, if we are to compute a generalized eigenvalue problem, we start by computing the Cholesky decomposition B = LLT and by transforming the problem to a standard one, A˜ = V˜ Λ with A˜ = L−1 A(L−1 )T ,

V˜ = LT V .

The next step is the reduction of the matrix to a tridiagonal form ˜ T, T = QAQ where Q = Qn · · · Q2 Q1 and QT = QT1 QT2 · · · QTn are the successive Householder matrices reducing one column of A˜ at a time. The Householder matrices Qi = I − βi vi viT are never constructed explicitly, but are always represented only by the Householder vector vi . In each step, a new Householder vector is computed and stored in place of an eliminated column of A, reducing the memory requirements. The diagonal and sub-diagonal of the resulting matrix are stored separately. Applying transformations on A from both sides complicates blocking and usage of efficient BLAS level 3 kernels. This restriction is alleviated in the two-stage algorithm, which will be briefly described later. The next step is the solution of the tridiagonal eigenvalue problem T Vˆ = Vˆ Λ and the final step is the back transformation of the k required eigenvectors V˜ = QT Vˆ .

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If the original problem was of the generalized form, we have to transform the eigenvectors once more using V˜ = LT V . The two stage algorithm, featured in ELPA 2, differs from the previous description in performing the conversion to the tridiagonal matrix in two steps. In the first step, the matrix is reduced to a banded matrix. This allows usage of highly optimized BLAS level 3 functions. In the second step, the matrix is further reduced to the tridiagonal form. The two stage tridiagonalization is almost always faster, however, the price to pay is the need to also transform each eigenvector (found by the tridiagonal solver) twice, in order to find the eigenvector of the original system. This makes ELPA 2 an obvious choice when only a small part of the eigenvectors are sought. When, on the other hand, most or all of the eigenvectors are needed, the best choice might depend on other parameters (such as the matrix size, particular hardware, etc) as well. In the rest of this contribution, however, we use only results of the ELPA 1 algorithm.

3 The ELPA Library Although the basic algorithm is conceptually quite simple, a highly optimized, distributed and scalable implementation using MPI is very challenging. The ELPA library [3] uses a block-cyclic distribution of the matrix, which is well documented and allows codes using the Scalapack library [5] to easily switch to ELPA in order to gain a performance benefit. ELPA employs very efficient communication patterns in processor rows and columns, benefiting from the block-cyclic distribution of the matrix. Local operations are done by calls to BLAS (or cuBLAS) for parts of local matrices and vectors. Apart from the MPI communication routines the code also contains an OpenMP and a GPU implementation. The complexity is further increased by having both real and complex compute paths as well as single and double precision variants in one code. Moreover, different optimizations are needed for different architectures and thus the code has to be sometimes split on the algorithmic level as well.

4 GPU Computing Using general purpose graphical processing units (GP-GPU) to accelerate codes in the HPC field is a well established topic. The reason is obvious—the peakperformance of such devices is huge (for example, Nvidia Tesla P100 “Pascal” with a peak-performance of 5.7 TFlop/s is much more powerful than a typical Intel Xeon processor based node with 2 CPUs). It is thus very tempting to use GPUs in HPC calculations. There exist many ongoing projects within the field of linear algebra which try to bring the GPU performance to the users, such as [6] or [7], just to name a few.

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There are, however, obstacles caused by a specific computing model of those devices and it is very challenging to reach anything near the theoretical peakperformance of the GPUs. As they were conceived as graphical accelerators, even though they are much more general-purpose nowadays, the performance is still optimal for doing the same operation on large amount of data and it is usually impossible to run the whole HPC calculation on GPUs. The usual approach is to use GPUs as accelerators, when the application runs on CPUs and offloads computationally very intensive (but usually small in terms of code length) parts (called kernels) to the GPU. There are thus several aspects that have to be taken care of: first, data has to be moved from the CPU memory to the GPU memory (and back). These memories are physically separated and if the memory transfers are not done with care, they can easily degrade the performance of the application. Second, a programming model has to be selected. There are several options. CUDA C offers the best performance, but the code has to be hand-tuned to a specific architecture and is not portable. Furthermore, it works only for Nvidia GPUs. More generic approaches, such as OpenACC have advantages in higher portability and code maintainability (the code can be ported step by step by adding OpenACC directives). The performance, however, is quite often far from optimal. The last option, which is the easiest from the point of view of code development and maintenance is to use GPU kernels from a library, which has been already optimized by the hardware vendor. We use the latter approach. Our CPU optimization of ELPA works with a distributed matrix, it handles data communications among individual nodes by explicit calls to the MPI library and uses highly optimized BLAS kernels for operations on pieces of the locally stored matrix and vectors. Our GPU implementation is a natural extension of this approach, since we explicitly initiate the data transfers between the main memory and the device memory and then perform local calculations on pieces of the locally stored part of the matrix using calls to the highly optimized cuBLAS library[7]. By this approach, we do not carry the burden of optimizing our code for future GPU architectures, assuming, that such optimization will be done by the hardware vendor.

5 GPU Implementation of ELPA 1 The main focus of the ELPA library is on efficient large-scale calculations, which should remain true for the GPU version as well. We thus still use the MPI-based implementation with a block-cyclic distribution of the matrices. Locally, each MPI task communicates with the GPU to transfer data to and from its memory and to launch the compute kernels on the device. All the memory transfers are done explicitly to avoid unnecessary data movements. We mostly rely on the cuBLAS library, only in ELPA 2 we utilize one kernel, which has been hand-written in CUDA C.

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Nowadays, in a typical compute node, the number of CPU cores (several dozens) is much bigger then the number of GPU devices within the node. Thus in a typical setup, multiple MPI tasks have to use the same GPU. In such a setting, to be able to use the GPU efficiently, the Nvidia Multi-process Service [8] has to be used. This daemon has to be started before the run of the application. Its role is to dispatch the requests from individual MPI tasks to the GPU device and to use its streams efficiently. Without the MPS the performance of the application deteriorates significantly. In general, we tried to keep a single code path for both the CPU and the GPU version of ELPA, with the difference, that instead of calling BLAS operations on parts of the matrices, we call corresponding cuBLAS operations. Obviously, care has to be taken to synchronize the data in GPU memory with the main memory, while keeping the data transfers low. Sometimes, however, more substantial changes in the algorithm have to be done in order to obtain the best performance. The CPU version of ELPA has been highly optimized by keeping the cache reuse in mind. For this reason, many of the algorithms use explicit blocking and try to reuse pieces of the matrices, which are in cache, for multiple operations. This is often not favorable for a GPU, since it cannot benefit from caching, but, rather, it benefits from large amounts of data being processes in one run. Some of the blocking strategies thus had to be changed and the algorithm had to be altered to better suite the GPU.

6 Numerical Results The success of the ELPA library is caused by a combination of its good node-level performance and its efficient MPI-based communication patterns among multiple nodes within the HPC system. The development of the GPU version, which is described in this contribution, is in principle a node-level optimization, since we do not alter the MPI communication patterns, but only speed-up the local computations by offloading compute intensive kernels to the GPU accelerator. For this reason, we only present performance comparisons using one node, although it can also run on a large HPC system. We have tested the GPU implementation on two different systems representing two different architectures: the first one is a standard Intel-based system with two Intel Xeon Ivy Bridge processors with 20 cores in total, equipped with two Nvidia Tesla K40m GPUs. The second machine has two IBM Power8 processors and four powerful Nvidia Pascal P100 GPUs. Referred to as the “Minsky” system, it also features the new NVlink interconnect, which should speed-up data transfers between the main and GPU memory. On the Intel system we used the Intel 16 compilers, MKL 2017 (containing the BLAS functions) and CUDA 8 (containing the cuBLAS implementation). On the Minsky machine we used the GNU compilers version 5.3, the ESSL library version 5.5 (containing the BLAS functions) and CUDA 9. We always run our GPU code with the Nvidia Multi process service enabled.

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Fig. 1 Total solution times in seconds for CPU only and CPU + GPU versions of ELPA 1. Results from two different architectures are shown for comparison. The problem with the largest matrix is not always computed due to memory limitations

Figure 1 shows the comparison of the total run-time for problems with different matrix sizes. Plotted are the results for both mentioned machines and for both the CPU and the GPU versions of ELPA 1. Comparing the two CPU versions we can see, that the Power8 machine is generally faster, mostly due to higher frequency (4 GHz compared to 2.8 GHz of the Intel machine). For both machines when the matrix is small, the CPU only implementation is significantly faster, since there is not enough workload to saturate the GPUs. From certain threshold on (matrix size of around 5000), however, this behavior changes and the GPU version becomes much faster. In Table 1 we can see timings for a matrix of dimension 32768. We can also compare the achieved speed-ups (bold font) by going from the CPU version to the GPU version on Intel + K40m system (3.6×) and on the Minsky system (5.9× or 9.2×, using 2 or 4 GPUs, respectively). As it has been described previously, the ELPA 1 algorithm consists of three individual steps, whose performance dependencies are different. In Fig. 2 we show Table 1 Comparison of solution times for matrix of size 32768 × 32768

Intel + K40m Power8 + P100

2 CPUs only Time (s) 1423 798

2 CPUs + 2 GPUs Time (s) Speedup 396 3.6× 136 5.9×

Speed up is with respect to CPU only version

2 CPUs + 4 GPUs Time (s) Speedup – – 86.9 9.2×

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Fig. 2 Detailed comparison of individual parts of the ELPA 1 algorithm on the Minsky node (see in text) using CPU only and CPU and all 4 GPUs

the run times of the individual steps. We selected only two setups for this detailed study, the best performing CPU and GPU variants, i.e., the Minsky CPU and Minsky CPU + 4 GPUs versions. In all cases, the (one-stage) ELPA 1 compute-time is dominated by the tridiagonalization step. This step contains large BLAS level 2 operations (matrix-vector multiplications), which can not be very efficiently implemented in neither BLAS (CPU) nor cuBLAS (GPU) libraries. Still, since most of the work done in both the tridiagonal solver and the back substitution is hidden in BLAS level 3 operations, which are particularly efficient on GPU, we can see, that the speedup of those functions is higher than the speed-up of the tridiagonalization step and thus the GPU version is even more limited by the BLAS level 2 dominated tridiagonalization. The solution times for large matrices are listed in detail in Table 2. The first two cases are intended for a comparison between the two tested architectures, since the same matrix size is used, and only 2 GPUs are utilized for the Minsky system. The last case shows the largest possible (due to memory limitations) computed matrix with the full Minsky node (using all 4 GPUs). It is worth noticing, that for the back substitution step we get almost 30× speed-up. This is caused by the efficient implementation using BLAS level 3 operations only. Indeed, even the total speed-up of almost 12× is very good.

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Table 2 Performance of individual stages of the ELPA 1 algorithm Machine Mat. size

Total Tridiag. Solve Back s.

2xIntel+2xK40m 32768 CPU CPU+GPU t (s) t (s) s-up 1424 396 3.6× 1108 333 3.3× 117 24.1 4.9× 198 36.9 5.4×

2xPower8+2xP100 32768 CPU CPU+GPU t (s) t (s) s-up 798 136 5.9× 555 113 4.9× 88.9 12.9 6.9× 154 10.3 15×

2xPower8+4xP100 65536 CPU CPU+GPU t (s) t (s) s-up 6139 514 11.9× 4263 422 10.1× 678 49.2 13.8×3 1198 42.1 28.5×

Speed up is with respect to CPU only version

7 Conclusions We have presented a GPU implementation of the ELPA library. We have described the methods used to utilize the potential of GPU accelerators to speed-up large-scale eigenvalue and eigenvector computation, while preserving the ELPA scalability by keeping the block-cyclic distribution of matrices and efficient MPI-based communication patterns. With this GPU implementation, codes using ELPA can significantly reduce their time to solution. We show, that the achieved speed-up depends on the matrix size and the hardware, but can reach up to 12× on the Minsky system for a large matrix. From the presented results the performance relationships for different hardware can be derived. We also show that the current bottleneck of the ELPA 1 algorithm lies in the tridiagonalization step, because of the necessary matrix-vector multiplications, which can not be implemented very efficiently on neither CPU nor GPU. However, since the other two steps, namely the solution of the tridiagonal system and the back substitution can be computed very efficiently on the GPU, this bottleneck is (relatively) even more severe in the GPU version. Acknowledgements Part of this work is co-funded by BMBF grant 01IH15001 of the German Government.

References 1. T. Auckenthaler, V. Blum, H.-J. Bungartz, T. Huckle, R. Johanni, L. Krmer, B. Lang, H. Lederer, P.R. Willems, Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations. Parallel Comput. 37, 783–794 (2011) 2. A. Marek, V. Blum, R. Johanni, V. Havu, B. Lang, T. Auckenthaler, A. Heinecke, H.-J. Bungartz, H. Lederer, The ELPA library - scalable parallel eigenvalue solutions for electronic structure theory and computational science. J. Phys. Condens. Matter 26, 213201 (2014) 3. ELPA Library, http://elpa.mpcdf.mpg.de 4. G.H. Golub, C.F.V. Loan, Matrix Computations (John Hopkins University Press, Baltimore, 2013)

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ScaLAPACK - Scalable Linear Algebra PACKage, http://netlib.org/scalapack Matrix Algebra on GPU and Multicore Architectures, http://icl.utk.edu/magma CuBLAS Library, https://developer.nvidia.com/cublas Multi-Process Service, https://docs.nvidia.com/deploy/pdf/CUDA_Multi_Process_Service_ Overview.pdf

Part III

Advanced Discretization Methods for Computational Wave Propagation

On the Efficiency of the Peaceman–Rachford ADI-dG Method for Wave-Type Problems Marlis Hochbruck and Jonas Köhler

Abstract The Peaceman–Rachford alternating direction implicit (ADI) method is considered for the time-integration of a class of wave-type equations for linear, isotropic materials on a tensorial domain, e.g., a cuboid in 3D or a rectangle in 2D. This method is known to be unconditionally stable and of conventional order two. So far, it has been applied to specific problems and is mostly combined with finite differences in space, where it can be implemented at the cost of an explicit method. In this paper, we consider the ADI method for a discontinuous Galerkin (dG) space discretization. We characterize a large class of first-order differential equations for which we show that on tensorial meshes, the method can be implemented with optimal (linear) complexity.

1 Introduction In this paper, we investigate the efficiency of the Peaceman–Rachford scheme applied to a directional splitting for a central fluxes dG space discretization of the split operators. We characterize a class of wave-type problems for which we show that one timestep of the fully discrete scheme can be performed in linear complexity w.r.t. the total number of spatial degrees of freedom. We start by providing definitions and results used to describe the aforementioned class of problems, which is then introduced in Sect. 2. In Sect. 3, we review the methods used for discretization and Sect. 4 is devoted to the efficiency of this discretization. Section 5 then provides some numerical tests to confirm the theoretical results.

M. Hochbruck · J. Köhler () Karlsruhe Institute of Technology, Karlsruhe, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_10

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1.1 Notation Throughout the paper, we denote the ith canonical unit vector by ei and the ith component of a vector v by vi . By (·, ·)S , we denote the standard L2 inner product over a set S and by δij the Kronecker delta. Further, if S is a countable set, we denote the number of its elements by |S|.

1.2 Operators with Decoupled Partial Derivatives In order to characterize problems enabling a splitting for which the Peaceman– Rachford method can be performed in linear complexity we start with some definitions. Definition 1 Let M1 , . . . , Md ∈ Rm×m be symmetric matrices and denote by Ii = {j ∈ {1, . . . , m} | Mi ej = 0} the set of indices of non-zero columns (or rows) in Mi , i = 1, . . . , d. Then we call M1 , . . . , Md ∈ Rm×m decoupled block-diagonal if Ii ∩ Ij = ∅ for all i = j . Hence, d symmetric and decoupled block-diagonal matrices have pairwise disjoint non-zero rows and columns. The name decoupled block-diagonal is motivated by the following property. Theorem 2 Let M1 , . . . , Md ∈ Rm×m be symmetric and decoupled blockdiagonal. Then there is a permutation matrix P ∈ Rm×m s.t. for all i = 1, . . . , d, the matrix P T Mi P is block diagonal with at most one non-zero diagonal block which vanishes in all other matrices P T Mj P , j = i. Proof The assertion follows from the symmetry of the matrices Mi if we reorder the rows and columns by the indices in I1 , then I2 , . . . , Id , and last the indices of those columns which vanish in all matrices. & % Using this notion, we characterize first order differential operators, whose partial derivatives completely decouple. d Definition 3 Let M = i=1 Mi ∂i be a first order differential operator with symmetric matrices Mi ∈ Rm×m , i = 1, . . . , d. We say that M has decoupled partial derivatives if M1 , . . . , Md are decoupled block-diagonal.

2 Framework Let Ω ⊂ Rd be a bounded paraxial tensorial domainwith boundary ∂Ω let n and d d be the outer unit normal on ∂Ω. Further, let L = L ∂ , A = A i i i=1 i=1 i ∂i , d m×m . We consider B = B ∂ , with symmetric matrices L , A , B ∈ R i i i i=1 i i

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homogeneous first order wave-type equations of the form ∂t u(t) = Lu(t) = (A + B)u(t),

u(0) = u0 ,

t ∈ [0, T ],

(1)

where A and B have decoupled partial derivatives. This class of problems includes, e.g., advection and wave equations in 2D and Maxwell’s equations in 3D. These examples are given as follows. 2D Advection Equation Here, we have m = 1, Li = αi for i = 1, 2 with the advection velocity vector α. We consider homogeneous inflow boundary conditions, i.e., u(t) = 0 on the inflow boundary ∂Ω + = { x ∈ ∂Ω | α · n > 0 }. The split operators are given by A1 = α1 , A2 = 0 and B1 = 0, B2 = α2 and the boundary conditions are given by n1 u(t) = 0 on ∂Ω + for A and n2 u(t) = 0 on ∂Ω + for B, respectively 2D Wave Equation Here, we have m = 3 and ⎛ ⎞ p ⎜ ⎟ u = ⎝ q1 ⎠ , q2

L1 = e1 e2T + e2 e1T ,

L2 = e1 e3T + e3 e1T .

We consider homogeneous Dirichlet boundary conditions, i.e., p(t) = 0 on ∂Ω. The split operators are given by A1 = L1 , A2 = 0 and B1 = 0, B2 = L2 with boundary conditions given by n1 p(t) = 0 on ∂Ω for A and n2 p(t) = 0 on ∂Ω for B, respectively. 3D Maxwell’s Equations Here, we have m = 6 and   E u= , H



0 L˜ T Li = ˜ i Li 0

 ,

where L˜ 1 = e2 e3T − e3 e2T , L˜ 2 = e3 e1T − e1 e3T and L˜ 3 = e1 e2T − e2 e1T . We consider  perfectly conducting boundary conditions, i.e., 3i=1 L˜ i ni E = 0 on ∂Ω. The split operators are given by (cf. [5, 7]) 

0 A˜ T Ai = ˜ i Ai 0



 ,

0 B˜ T Bi = ˜ i Bi 0



with A˜ 1 = −B˜ 1T = e2 e3T A˜ 2 = −B˜ 2T = e3 e1T A˜ 3 = −B˜ 3T = e1 e2T and we subject   A to 3i=1 A˜ i ni E = 0 and B to 3i=1 B˜ i ni E = 0 on ∂Ω. Remark 4 For ease of presentation, we omit material parameters in this paper. However, in the case of isotropic materials, all statements apply with only minor changes: the operator D −1 L with D = diag(δ1 , . . . , δm ), δ1 , . . . , δm ∈ L∞ (Ω), takes over the role of L (and analogously for A and B) and the average in the

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dG-discretization (see below) is replaced by a weighted average, taking possible jumps in the material parameters into account. Further, these parameters have to be incorporated into the mass matrix. Because of the diagonal structure of D, no further coupling is introduced, and the efficiency analysis can be performed completely analogously.

3 Discretization In this section, we review the Peaceman–Rachford scheme for the temporal discretization [6] and the central flux discontinuous Galerkin (dG) scheme [2, 3] used for the spatial discretization.

3.1 Temporal Discretization The Peaceman–Rachford scheme [6] applied to (1) reads (I − τ2 A)un+1/2 = (I + τ2 B)un , (I − τ2 B)un+1

= (I + τ2 A)un+1/2 .

This scheme is of (conventional) order two and unconditionally stable if A and B are dissipative operators (see, e.g., [4]). It requires the solution of two linear systems whose coefficient matrices are given by the spatially discrete counterparts of I − τ2 A or I − τ2 B. However, if the operators A and B have decoupled partial derivatives (cf. Definition 3), we will show that this can be achieved in optimal (linear) complexity w.r.t. the total number of spatial degrees of freedom. Remark 5 If the spatial dimension d exceeds two, the advection and wave equation do not admit a splitting into two operators with decoupled partial derivatives. To preserve the linear complexity for solving the occurring linear systems, a splitting with d split operators would have to be employed. However, a straightforward generalization of the Peaceman–Rachford method seems to lack either stability or accuracy in general. Alternatively, one can recover linear complexity for higher spatial dimensions d by employing a Lie-type scheme of the form un+1 = (I − τ Ad )−1 · · · (I − τ A2 )−1 (I − τ A1 )−1 un , for L = A1 + . . . + Ad . This scheme is unconditionally stable and of (conventional) order one.

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3.2 Spatial Discretization We use a central flux dG method to discretize the split differential operators in space [2, 3]. For this, we equip Ω with a mesh T = {K} consisting of paraxial tensorstructured elements. We gather the faces of T in the set F = {F }, which is further decomposed into the set of interior faces F int and the set of boundary faces F bnd . Due to the tensorial structure of the mesh, normal vectors to the faces in F are ±ej for some j ∈ {1, . . . , d}. For F ∈ F we denote the unit normal vector to F in positive coordinate direction by nF . Hence, we have Fα =

3 ˙ d i=1

F α,i ,

F α,i = { F ∈ F α | nF = ei },

α ∈ { int, ext },

(2)

where F α,i are the sets of faces with normals pointing in the ith direction. For each interior face F ∈ F int , we additionally denote the two elements containing F as K1F and K2F , where the numbering is done s.t. nF is the outer normal to K1F . To approximate functions in space, we use the broken polynomial space Vh = { v ∈ L2 (Ω) | v|K ∈ Pk for all K ∈ T },

(3)

where Pk denotes the set of polynomials of degree at most k in each variable. We could also allow the polynomial degree k to depend on K, but for the sake of presentation we do not pursue this further in this paper. For the efficiency analysis, we consider the basis 3 K Vh = { φ1K , . . . , φN } k K∈T

of Vh , where supp(φiK ) ⊂ K for i = 1, . . . , Nk , e.g., a standard discontinuous Lagrange basis. Since functions in the space Vh may be discontinuous across the faces of the mesh, we define the average and the jump of a (possibly vector-valued) function v over an interior face F ∈ F int as {{v}}F =

(v|K F )|F + (v|K F )|F 1

2

2

,

vF = (v|K F )|F − (v|K F )|F . 1

2

Let uh , ϕh ∈ Vh . We define the central flux dG-discretization ∂i,h of ∂i as (∂i,h uh , ϕh )Ω =

(∂i uh , ϕh )K −

K∈T

=

K∈T

(nFi uh F , {{ϕh }}F )F

F ∈F int

(∂i uh , ϕh )K −

F ∈F int,i

(uh F , {{ϕh }}F )F ,

(4)

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where the second equality follows by the definition of F int,i in (2). With this, we define the dG-discretization of the split operators for uh , ϕh ∈ Vhm as (Ah uh , ϕh )Ω =

d

(Ai ∂i,h uh , ϕh )Ω − bA (uh , ϕh ), i=1

(Bh uh , ϕh )Ω

d

= (Bi ∂i,h uh , ϕh )Ω − bB (uh , ϕh ),

(5)

i=1

where ∂i,h is meant to act componentwise and bA , bB model the boundary conditions of the corresponding operators, respectively. The concrete boundary terms for the examples in Sect. 2 are as follows. 2D Advection Equation (Homogeneous Inflow Boundary Conditions) For uh , ϕh ∈ Vh , we have bA (uh , ϕh ) =

(α1 uh , ϕh )F ,

bB (uh , ϕh ) =

bnd,1 F ∈F+

(α2 uh , ϕh )F ,

bnd,2 F ∈F+

where F ∈ F+bnd,i = { F ∈ F bnd,i | F ∩ ∂Ω + = ∅ }. 2D Wave Equation (Homogeneous Dirichlet Boundary Conditions) For uh = (ph , q1,h , q2,h )T , ϕh = (φh , ψ1,h , ψ2,h )T ∈ Vh3 , we have bA (uh , ϕh ) =

(ph , ψ1,h )F ,

bB (uh , ϕh ) =

F ∈F bnd,1

(ph , ψ2,h )F .

F ∈F bnd,2

3D Maxwell’s Equations (Perfectly Conducting Boundary Conditions) For uh = (EhT , HhT )T , ϕh = (ΦhT , ΨhT )T ∈ Vh6 , we have bA (uh , ϕh ) =

3



(A˜ i Eh , Ψh )F ,

i=1 F ∈F bnd,i

bB (uh , ϕh ) =

3



(B˜ i Eh , Ψh )F .

i=1 F ∈F bnd,i

4 Efficiency In this section, we investigate the efficiency of the Peaceman–Rachford dG scheme, which is mainly determined by the cost to solve linear systems involving the discrete counterparts of I − τ2 A and I − τ2 B, respectively. We show that, using a

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suitable ordering of the degrees of freedom, the corresponding matrices have blocktridiagonal structure, where the block-sizes only depend on the polynomial degree k and the number of indices in the corresponding set Ii , but is independent of the total number of degrees of freedom. Hence, the corresponding systems can be solved in linear time. The mass matrix resulting from the discretization of I is block-diagonal if the degrees of freedom are ordered elementwise, which is well-known for dG-methods. Hence, it suffices to investigate the non-zero patterns of the matrices corresponding to Ah and Bh , respectively. As these are defined in terms of the discrete partial derivatives ∂i,h , i = 1, . . . , d, we begin by investigating them.

4.1 Structure of ∂i,h To investigate the non-zero pattern of the discrete partial derivatives, we insert the basis functions in Vh into the bilinear form (4). For K1 = K2 , we have

(∂i φjK1 , φK2 )K = 0,

j,  = 1, . . . , Nk ,

K∈T

since supp(φiK ) ⊂ K. Hence, if we order the basis functions elementwise, these terms only contribute to the blockdiagonal with block-width Nk . For the sum over the interfaces, we obtain contributions outside of the blockdiagonal. However, for F ⊂ ∂K, we have {{φjK }}F = 0,

φjK F = 0.

/ F int,i , i.e., K1 and K2 not sharing a common Hence, for K1 and K2 with K1 ∩ K2 ∈ face with normal in the ith direction, it holds

(φjK1 F , {{φK2 }}F )F = 0, j,  = 1, . . . , Nk . F ∈F int,i

Thus, these terms only contribute to off-blockdiagonal entries if the corresponding basis functions are non-zero on elements sharing such a face. If we, in addition to ordering the degrees of freedom elementwise, order the elements of the mesh along these normal vectors, the only additional entries appear in the first sub- and super-blockdiagonals. Altogether, with this ordering of the degrees of freedom, the discretized partial derivative ∂i,h is represented by a block-tridiagonal matrix.

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4.2 Structure of the Discrete Split Operators To investigate the non-zero pattern of the discrete split operators, we insert the basis functions in Vhm into the bilinear forms (5). We only consider Ah , since for Bh one can proceed completely analogously. According to the index sets from Definition 1 corresponding to A, we decompose 4d m , where the basis Vhm into Vhm = ˙ i=1 Vh,i m = Vh,i

3 ˙ j ∈ Ii

{ φ ej | φ ∈ Vh }.

m and ψ ∈ V m , there exist  ∈ I ,  ∈ I and φ , φ ∈ V s.t. For ψ1 ∈ Vh,i 2 1 i 2 j 1 2 h h,j ψ1 = φ1 e1 , ψ2 = φ2 e2 . This implies d d

(Ar ∂r,h ψ1 , ψ2 )Ω = eT2 Ar e1 (∂r,h φ1 , φ2 )Ω = δij eT2 Ai e1 (∂i,h φ1 , φ2 )Ω , r=1

r=1

where the last equality follows as a consequence of Theorem 2, since A1 , . . . , Ad are symmetric and decoupled block diagonal. Hence, basis functions belonging to m , i = 1, . . . , d completely decouple. By ordering the basis functions different Vh,i according to these sets, we thus obtain (up to) d diagonal blocks. The structure of these blocks is determined by the structure of (∂i,h φ1 , φ2 ), which was analyzed in m Sect. 4.1. Therefore, by ordering the elements, and thus the basis functions in Vh,i belonging to them, according to the applied partial derivative, we obtain a blocktridiagonal structure for each i = 1, . . . , d and thus globally, since these blocks decouple. However, in contrast to Sect. 4.1 the block-size is |Ii |Nk , i = 1, . . . , d, since per element we have Nk basis functions for each index in Ii which are coupled through Ai . For the boundary conditions used in the examples above, no further coupling is introduced by the boundary terms bA and bB , respectively. This can be seen with a similar argument as for the interfaces. Hence, these terms do not change the blocktridiagonal structure. Remark 6 It is possible to generalize the method to domains consisting of a union of paraxial tensorial domains without losing its linear complexity. This can be seen with the same arguments as above if a tensorial mesh is used to discretize each subdomain. The tensorial structure of the mesh (or the submeshes), however, is indispensable for the efficiency of the method. Without this structure, the normal vectors of the element faces have multiple non-zero entries. This results in coupling between neighboring elements w.r.t. more than one face, which destroys the linear complexity of the overall scheme. Remark 7 One can further speed up the method by using a tensorial basis for the space Vh . This is due to the fact that the inner products in (4) then reduce to a

On the Efficiency of the Peaceman–Rachford ADI-dG Method 103

time [s]

Fig. 1 Runtimes of the ADI and the Verlet method (including assembling of the matrices). Computations are carried out on 14 uniform grids ranging from 8 to 34 elements per unit length with polynomial degree k = 1 on the grid elements. Time stepsize is τ = 0.01 and 200 steps are performed

143

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ADI-dG Verlet-dG unstable O(N )

101 105

N = |Vhm |

106

product of one-dimensional integrals. This leads to a Kronecker-product structure of the resulting matrices, which can be exploited to solve the occurring linear systems more efficiently. Note, however, that in the case of general isotropic materials, this is only possible if the material parameters have product structure as well. The reason for this is that inner products weighted by these parameters have to be used to compute the mass matrix. The reduction of these inner products to one-dimensional integrals is only possible, if the weights have product structure.

5 Numerical Results We implemented the method in deal.ii [1] for Maxwell’s equations to verify the theoretical results. Upon request, the code to conduct these experiments will be provided. The computational domain is Ω = [0, 2] × [0, 1]2 with material parameters chosen to be constant. To solve the linear systems in each timestep, a block-LU solver is used. For comparison, runtimes of the explicit Verlet or leap-frog method with the same configurations are shown. The runtimes illustrated in Fig. 1 clearly show that the method takes only about twice as long as the explicit Verlet method, which is unstable on the three finest meshes. A rigorous error analysis showing temporal order two independent of the spatial mesh will be presented in a separate paper. Acknowledgements We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.

References 1. W. Bangerth, R. Hartmann, G. Kanschat, deal.II – a General Purpose Object Oriented Finite Element Library, ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007)

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2. D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Springer Mathematics and Applications (Springer, Berlin, 2012) 3. J. S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods. Springer Texts in Applied Mathematics (Springer, Berlin, 2008) 4. M. Hochbruck, T. Jahnke, R. Schnaubelt, Convergence of an ADI splitting for Maxwell’s equations. Numer. Math. 129, 535–561 (2015) 5. T. Namiki, A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microwave Theory Tech. 47, 2003–2007 (1999) 6. D.W. Peaceman, H.H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955) 7. F. Zhen, Z. Chen, J. Zhang, Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method. IEEE Trans. Microwave Theory Tech. 48, 1550– 1558 (2000)

Trefftz-Discontinuous Galerkin Approach for Solving Elastodynamic Problem Hélène Barucq, Henri Calandra, Julien Diaz, and Elvira Shishenina

Abstract Methods based on Discontinuous Finite Element approximation (DG FEM) are basically well-adapted to specifics of wave propagation problems in complex media, due to their numerical accuracy and flexibility. However, they still lack of computational efficiency, by reason of the high number of degrees of freedom required for simulations. The Trefftz-DG solution methodology investigated in this work is based on a formulation which is set only at the boundaries of the mesh. It is a consequence of the choice of test functions that are local solutions of the problem. It owns the important feature of involving a space-time approximation which requires using elements defined in the space-time domain. Herein, we address the Trefftz-DG solution of the Elastodynamic System. We establish its well-posedness which is based on mesh-dependent norms. It is worth noting that we employ basis functions which are space-time polynomial. Some numerical experiments illustrate the proper functioning of the method.

1 Introduction Among the different possible approaches to solve partial differential equations there exists a distinct family of methods based on the use of trial functions in the form of exact solution of the governing equations (but not the boundary conditions). The

H. Barucq · J. Diaz Magique-3D, Inria-UPPA E2S-CNRS, Pau, France H. Calandra Total S.A., Houston, TX, USA e-mail: [email protected] E. Shishenina () Magique-3D, Inria-UPPA E2S-CNRS, Pau, France Total S.A., Houston, TX, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_11

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idea was first proposed by Trefftz in 1926 [1], and since then it has been largely developed and generalized. The main step for its implementation as an efficient computational tool was achieved in 1978 when Jirousek and his collaborators proposed the Hybrid-Trefftz (HT) finite element model [2]. The results of their work allowed solving different boundary value problems, thus giving roots to multiple applications in different fields such as potential problems, plane elasticity, plate bending (thin, thick, post-buckling), heat conduction as well as advective-diffusive transport (see [3] and the references therein). Trefftz type methods have been widely used with time-harmonic formulations by Farhat, Tezaur, Harari, Hetmaniuk (2003–2006) (see [4, 5]), Gabard (2007) (see [6]), Badics (2014) (see [7]), Hiptmair, Moiola, Perugia (2011–2013) (see [8–10]) and others, while studies are still limited for reproducing temporal phenomena. Only few papers are interested in Maxwell equations in time [11–14]. They are mostly devoted to a theoretical analysis of the method, showing the convergence and stability. To the best of our knowledge, numerical tests involving plane wave approximation are restricted to 1D + time dimensional case. Space-time Trefftz approximation by Lagrange multipliers for the second order formulation of the transient wave equation was explored in [15, 16]. A Trefftz-DG method for the first-order transient acoustic wave equations in arbitrary space dimensions has been introduced in a recent paper of Moiola and Perugia [17]. It is an extension of the one-dimensional scheme of Kretzschmar et al. [14]. The authors provide a complete a priori error analysis in both mesh-dependent and mesh-independent norms. In this work we develop a theory for Trefftz-DG method applied to the Elastodynamic System (ES) of wave propagation. We confirm well-posedness of the variational problem based on the estimates in mesh-dependent norms. We consider a space-time Trefftz polynomial basis and provide some numerical results for 2D ES. We give a short conclusion and discuss the perspectives in the end of this paper.

2 Trefftz-DG Formulation for Elastodynamics The Elastodynamic System is based on three fundamental laws of continuum mechanics: movement equations, constitutive equations (Hooke’s law), and geometric equations (infinitesimal strain tensor definition) [18]. We consider a global space-time domain Q = Ω × I , where Ω ⊂ R d is a bounded Lipschitz physical space domain and I = (0, T ) is a time interval. The Lamé coefficients λ ≡ λ(x), μ ≡ μ(x) and solid density ρ ≡ ρ(x) are the solid parameters, assumed to be piecewise constant and positive.

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We consider the first order ES in terms of velocity v ≡ v(x, t) and stress σ ≡ σ (x, t) fields: ⎧ ⎪ in Q, ⎪ ⎪ ∂t σ − C ε(v) = 0 ⎪ ⎪ ⎪ ⎨ ρ∂t v − divσ = 0 in Q, ⎪ v(·, 0) = v0 , σ (·, 0) = σ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ = gD

in Ω, in ∂Ω × I.

Here C is the elastic coefficient tensor, ε(v) = (∇v + ∇v T )/2 is the infinitesimal strain tensor. The boundary conditions gD ≡ gD (x, t), the velocity v0 and the stress σ 0 are the initial data. By symmetry and positivity of the tensor C, the application ε −→ C ε is an isomorphism in the symmetrical tensor space [18]. Thus, we may consider the corresponding inverse application A, verifying the same properties of symmetry and positivity: ⎧ ⎪ A∂t σ − ε(v) = 0 in Q, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ∂t v − divσ = 0 in Q, (1) ⎪ v(·, 0) = v0 , σ (·, 0) = σ 0 in Ω, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ = gD in ∂Ω × I.

2.1 Space-Time Trefftz-DG Formulation We choose a Lipschitz sub-domain K ⊂ Q such that λ, μ and ρ are constant in K. We define nK ≡ (nxK , ntK ) as the outward pointing unit normal vector on ∂K and v, 2 (v, σ ) ∈ H 1 (K)d × H 1 (K)d , where d is the dimension of the physical space Ω. Multiplying both equations of (1) by the test functions (ω, ξ ) ∈ H 1 (K)d × 2 1 H (K)d respectively, and integrating by part in space and time we obtain:  5 6 σ (A∂t ξ − ε(ω)) + v · (ρ∂t ω − divξ ) dv − K (2)  5 6 t x t x + A σ : ξ nK − v · nK ξ + ρvnK · ω − σ nK · ω ds = 0. ∂K

Without losing generality with respect to the classical space DG methods, we introduce a non-overlapping mesh Th on Q, whose elements are right prisms, with vertical sides parallel to the time axis. All solid parameters inside the elements K are supposed to be constant, so that all discontinuities lie on the inter-element boundaries.

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The mesh skeleton Fh = ∪K∈Th ∂K can be decomposed into the families of element faces: FhI

Internal I -faces

(x-fixed)

FhΩ

Internal Ω-faces

(t-fixed)

FhD

External Dirichlet boundary faces

(∂Ω × [0, T ])

Fh0

External initial time faces

(Ω × {0})

FhT

External final time faces

(Ω × {T })

The space-time DG formulation for (1) consists in searching (vhp , σ hp ) ∈ 2 V(Th ) ⊂ H 1 (Th )d × H 1 (Th )d such that, for all K ∈ Th and for all (ω, ξ ) ⊂ V(Th ) the following identity holds true: −

 5 

K

+ ∂K

6 σ hp : (A∂t ξ − ε(ω)) + vhp · (ρ∂t ω − divξ ) dv

5

6 A σˆ hp : ξ ntK − vˆhp · nxK ξ + ρ vˆhp ntK · ω − σˆ hp nxK · ω ds = 0.

(3)

The numerical fluxes vˆhp , σˆ hp are defined in the standard DG notations [19] on the mesh skeleton Fh as follows:  FhI

 FhΩ

 FhD FhT Fh0

vˆhp σˆ hp vˆhp σˆ hp

 ≡ 

vˆhp · nxK σˆ hp nxK   vˆhp σˆ hp   vˆhp σˆ hp

≡  ≡ ≡ ≡

  {v { hp}} − δ[[σ hp ]]x {σ { hp}} − γ [[vhp ]]x   vhp − σ hp −   vhp · nxK − δ(σ hp − gD )nxK gD nxK   vhp σ hp   v0 σ0

Here δ ∈ L∞ (FhI ∪ FhD ) and γ ∈ L∞ (FhI ) are positive penalty parameters. This choice is recommended in order to improve the numerical stability of the scheme. We define the Trefftz approximation space, such that the chosen test functions (ω, ξ ) satisfy the initial elastodynamic system in the homogeneous sense (without source term and boundary conditions):   T(Th ) ≡ (ω, ξ ) ∈ V(Th ) s. t. ρ∂t ω − divξ = A∂t ξ − ε(ω) = 0 in all K ∈ Th .

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Thanks to this choice of discrete space we remove a volume integration term in (3). Summing over all elements K ∈ Th , we obtain a space-time Trefftz-DG formulation for (1): Seek (vhp , σ hp ) ∈ V(Th ) such that, for all (ω, ξ ) ∈ T(Th ), it holds true:  −  + 

FhI

FhΩ

5 6 {σ hp}[[ω]]x + {vhp}[[ξ ]]x − γ [[vhp ]]x [[ω]]x − δ[[σ hp ]]x [[ξ ]]x ds 5

 6 A σ hp − : [[ξ ]]t + ρvhp − [[ω]]t ds − 

FhD

6 vhp · nxK ξ − δσ hp : ξ ds

5 6 6 1 A σ hp : ξ + ρvhp · ω ds − A σ hp : ξ + ρvhp · ω ds = 2 Fh0 FhT   5 5 6 6 1 A σ hp : ξ + ρvhp · ω ds + gD nxK · ω + δgD : ξ ) ds, 2 Fh0 FhD

+

5

5

or, by introducing bilinear AT DG (· ; ·) and linear T DG (·) operators: Seek (vhp , σ hp ) ∈ V(Th ) such that, for all (ω, ξ ) ∈ T(Th ), it holds true: AT DG ((vhp , σ hp ); (ω, ξ hp )) = T DG (ω, ξ hp ).

(4)

It is worth mentioning that in addition to its setting at the boundaries of the elements only, the Trefftz-DG formulation does not involve differential operators. It is thus straightforward to implement.

2.2 Well-Posedness of Trefftz-DG Formulation In this section we show the coercivity and continuity estimates proving wellposedness of the obtained Trefftz-DG method for ES in mesh-dependent norms. We refer to the Appendix B in [20] for more details. The analysis is carried out inside the framework developed in [14] for the time-dependent Maxwell problem. We introduce two mesh-dependent norms in T(Th ): 7 1 72 72 72 1 17 17 7 7 7 1 7 7 7 7(A) /2 [[ξ]]t 7 2 Ω + 7ρ /2 [[ω]]t 7 2 Ω + 7γ /2 [[ω]]x 7 2 I L (Fh ) L (Fh ) L (Fh ) 2 2 7 1 7 7 7 7 7 7 72 1 17 17 1 7 72 72 7 1 72 7 + 7δ /2 [[ξ ]]x 7 2 I + 7(A) /2 ξ 7 2 T + 7ρ /2 ω7 2 T + 7δ /2 ξ 7 2 D , L (Fh ) L (Fh ) L (Fh ) L (Fh ) 2 2

|||(ω, ξ )|||2T DG ≡

1 |||(ω, ξ )|||2T DG∗ ≡ |||(ω, ξ )|||2T DG + ρ /2 ω− 2 2

L (FhT )FhΩ

1 + δ − /2{ω} { }2 2

L (FhI )

1 + γ − /2{ξ { } } 22

L (FhI )

1 + δ − /2 ξ 2 2

1 + (A) /2 ξ − 2 2

L (FhT )FhD

L (FhT )FhΩ

.

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Thus, for the bilinear AT DG (·, ·) and linear T DG (·) forms we obtain the following coercivity AT DG ((ω, ξ ); (ω, ξ )) = |||(ω, ξ )|||2T DG , ∀(ω, ξ ) ∈ T(Th ), and continuity properties with respect to the chosen norms. |AT DG ((v, σ ); (ω, ξ ))| ≤ 2 |||(v, σ )|||T DG∗ |||(ω, ξ )|||T DG , 61 / 2 √ 5 1 1 |T DG (ω, ξ )| ≤ 2 ρ /2 v0 2L2 (F 0 ) + A /2 σ 0 2L2 (F 0 ) (gD ≡ 0). h

h

The above estimates confirm the well-posedness of the Trefftz-DG problem for ES, moreover: |||(v − vhp , σ − σ hp )|||T DG ≤ 3

inf

(ω,ξ )∈V(Th )

|||(v − ω, σ − ξ )|||T DG∗ .

3 Numerical Implementation In this Section we discuss the choice of the discrete approximation space, and provide some numerical tests for 2D + time elastic model.

3.1 Polynomial Basis The flexibility in the choice of basis functions is one of the advantages of Trefftztype methods. The main condition is to satisfy the governing equations inside each element. The natural choice in the case of harmonic problems can be the plane wave trigonometric basis. However, when applied to the space-time formulations, it demands a high number of trigonometric functions of different frequencies, in order to provide a better approximation order. Thus, it increases the number of degrees of freedom, and as a result—the global numerical cost of the algorithm. We have computed a space-time polynomial basis, using generating exponential functions—local solutions of the initial systems of equations (see [20, 21] for more details). Numerically speaking, this basis generates a lower computational bound than the standard trigonometric ones. It contains the couples of polynomial functions for velocity—stress, of degrees less or equal to n (n = 0, 1, 2, 3), satisfying the initial ES, to provide an approximation of order n.

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a

151

b

Fig. 1 Convergence of velocities vP and vS in function of cell size h = Δx. (a) Velocity vP . (b) Velocity vS

3.2 Numerical Results In order to explore the method and its algorithm in general, and to perform some R basic numerical tests for its validation, we have developed a prototype MATLAB' code, which is, technically speaking, quite limited. We consider the elastic medium, represented by a unit square with the Dirichlet conditions at the boundaries. The final time of propagation is T = 1. The medium parameters are ρS = 1, λ = 1, μ = 2. All model parameters are dimensionless quantities. We introduce a source term in the center of the medium. The source signal is represented by the Gaussian function, so that it takes approximately five elements per wavelength. Zero initial conditions are imposed for the tests. Figure 1 shows some results of convergence of the (a) P -velocity and (b) Svelocity. The convergence curves have been computed for different approximation orders (n = 0, 1, 2, 3), and they represent the numerical error as a function of cell size in logarithmic scale. Even though the initial model is limited (large mesh of 30 × 30 × 30 elements, Dirichlet boundaries, which causes many reflections), the numerics reproduce the expected propagation characteristics quite well, and the convergence in both cases is of order higher than the corresponding approximation order.

4 Conclusion We have applied the theory for Trefftz-DG method to the Elastodynamic System, and we have studied the well-posedness of the problem. The new polynomial basis has been computed for numerical implementation of the method. The computed

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numerical solutions have been validated by the analytical ones. The convergence results are of higher order, compared with the classical DG methods. Even though the obtained results are very promising, from the optimisation point of view, it seems necessary to study the alternative to a global matrix inversion, which brings the main computational cost. We have also in perspective to pass from simple rectangular meshes to more complicate forms (in space domain)—which is one of main advantages of Trefftz-DG methods. It gives the possibility of developing a hybrid method, based on numerical coupling of Trefftz-DG method in Elastics, with less expensive Finite Volume Method (FVM) in Acoustics, in order to create a software able to solve more realistic problems. Acknowledgements This work is supported by the Inria—Total S.A. strategic action “Depth Imaging Partnership” (http://dip.inria.fr).

References 1. E. Trefftz, Ein Gegenstuck zum Ritzschen Verfahren, in Proceedings of the 2nd International Congress of Applied Mechanics, Zurich (1926), pp. 131–137 2. J. Jirousek, Basis for development of large finite elements locally satisfying all field equations. Comput. Methods Appl. Mech. Eng. 14(1), 65–92 (1978) 3. O.C. Zienkiewic, Trefftz type approximation and the generalized finite element method history and development. Comput. Assist. Mech. Eng. Sci. 4, 305–316 (1997) 4. C. Farhat, I. Harari, U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng. 192(11), 1389–1419 (2003) 5. R. Tezaur, C. Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int. J. Numer. Methods Eng. 66(5), 796–815 (2006) 6. G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225(2), 1961–1984 (2007) 7. Z. Badics, Trefftz-discontinuous Galerkin and finite element multi-solver technique for modeling time-harmonic EM problems with high-conductivity regions. IEEE Trans. Magn. 50(2), 401–404 (2014) 8. R. Hiptmair, A. Moiola, I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49(1), 264–284 (2011) 9. A. Moiola, R. Hiptmair, I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 62(5), 809–837 (2011) 10. R. Hiptmair, A. Moiola, I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. 82(281), 247–268 (2013) 11. S. Petersen, C. Farhat, R. Tezaur, A space-time discontinuous Galerkin method for the solution of the wave equation in the time domain. Int. J. Numer. Methods Eng. 78(3), 275–295 (2009) 12. F. Kretzschmar, S.M. Schnepp, I. Tsukerman, T. Weiland, Discontinuous Galerkin methods with Trefftz approximations. J. Comput. Appl. Math. 270, 211–222 (2014) 13. H. Egger, F. Kretzschmar, S.M. Schnepp, T. Weiland, A space-time discontinuous Galerkin Trefftz method for time dependent Maxwell’s equations. SIAM J. Sci. Comput. 37(5), B689– B711 (2015) 14. F. Kretzschmar, A. Moiola, I. Perugia, S. Schnepp, A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems. IMA J. Numer. Anal. 36(4), 1599–1635 (2015)

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15. D. Wang, R. Tezaur, C. Farhat, A hybrid discontinuous in space and time Galerkin method for wave propagation problems. Int. J. Numer. Methods Eng. 99(4), 263–289 (2014) 16. L. Banjai, E. Georgoulis, O. Lijoka, A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation. SIAM J. Numer. Anal. 55(1), 63–86 (2017) 17. A. Moiola, I. Perugia, A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math. 138(2), 389–435 (2018) 18. P. Le Tallec, Modélisation et calcul des milieux continus (Editions Ecole Polytechnique, Palaiseau, 2009) 19. J.S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer Science & Business Media, Berlin, 2007) 20. H. Barucq, H. Calandra, J. Diaz, E. Shishenina, Space-time Trefftz - discontinuous Galerkin approximation for elasto-acoustics. Inria Bordeaux Sud-Ouest, LMAP CNRS, Total EP, RR9104 (2017) 21. A. Maciag, J. Wauer, Solution of the two-dimensional wave equation by using wave polynomials. J. Eng. Math. 51(4), 339–350 (2005)

Part IV

Unfitted Finite Element Methods: Analysis and Applications

FETI-DP Preconditioners for the Virtual Element Method on General 2D Meshes Daniele Prada, Silvia Bertoluzza, Micol Pennacchio, and Marco Livesu

Abstract We analyze the performance of a state-of-the-art domain decomposition approach, the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) method (Toselli and Widlund, Domain decomposition methods—algorithms and theory. Springer series in computational mathematics, vol 34, 2005), for the efficient solution of very large linear systems arising from elliptic problems discretized by the Virtual Element Method (VEM) (Beirão da Veiga et al., Math Models Methods Appl Sci 24:1541–1573, 2014). We provide numerical experiments on a model linear elliptic problem with highly heterogeneous diffusion coefficients on arbitrary Voronoi meshes, which we modify by adding nodes and edges deriving from the intersection with an unrelated coarse decomposition. The experiments confirm also in this case that the FETI-DP method is numerically scalable with respect to both the problem size and number of subdomains, and its performance is robust with respect to jumps in the diffusion coefficients and shape of the mesh elements.

1 Introduction Polytopic meshes allow the treatment of complex geometries, a crucial task for many applications in computational engineering and scientific computing. We consider here the problem of preconditioning the Virtual Element Method (VEM), which can be viewed as an extension of the Finite Element Method to handle such a kind of meshes. In view of a possible parallel implementation of the method, we consider a state-of-the-art domain decomposition approach, the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) method. It has been proved

D. Prada () · S. Bertoluzza · M. Pennacchio Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, Pavia, Italy e-mail: [email protected]; [email protected]; [email protected] M. Livesu Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, Genova, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_12

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that the FETI-DP method is still scalable when dealing with VEM, under the assumptions that the subdomains, obtained by agglomerating clusters of polygonal elements, are shape regular [1]. Such an assumption can be quite restrictive. In practice, it reduces to asking that the fine tessellation is built as a refinement of the previously existing coarse subdomain decomposition. This, of course, does not generally hold, so, in order to apply FETI-DP in a more general case, we propose to build the coarse decomposition independently from the tessellation, and modify the latter by inserting nodes and edges deriving from “cutting” it with the macroedges of the subdomains. Of course, the resulting modified tessellation will possibly contain nasty elements with very small edges. Numerical tests do however show that FETI-DP is quite robust in this respect, providing satisfactory results also in this framework. This paper is organized as follows. A basic description of VEM is given in Sect. 2. The FETI-DP method is introduced in Sect. 3, whereas the algorithm for partitioning Ω into subdomains and modifying the mesh is given in Sect. 4. Numerical experiments that validate the theory are presented in Sect. 5.

2 The Virtual Element Method (VEM) In this paper we focus on the numerical solution of the following model elliptic boundary value problem of second order discretized with VEM − ∇ · (ρ∇u) = f in Ω,

u = 0 on ∂Ω,

(1)

with f ∈ L2 (Ω), where Ω ⊂ R2 is a polygonal domain. We assume that the coefficient ρ is a scalar such that for almost all x ∈ Ω, α ≤ ρ(x) ≤ M for two constants M ≥ α > 0. The variational formulation of Eq. (1) reads as follows: find u ∈ V := H01 (Ω) such that a(u, v) = (f, v) ∀v ∈ V ,

(2)

with 

 ρ(x)∇u(x) · ∇v(x) dx,

a(u, v) = Ω

(f, v) =

f (x)v(x) dx. Ω

We consider a family {Th }h of tessellations of Ω into a finite number of simple polygons K, and let Eh be the set of edges e of Th . For each tessellation Th , we assume there exist constants γ0 , γ1 , α0 , α1 > 0 such that: – each element K ∈ Th is star-shaped with respect to a ball of radius ≥ γ0 hK , where hK is the diameter of K; – for each element K ∈ Th the distance between any two vertices of K is ≥ γ1 hK ;

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– Th is quasi-uniform, that is, for any two elements K and K  in Th we have α0 ≤ hK / hK  ≤ α1 . For each polygon K ∈ Th we define a local finite element space V K,k as V K,k = {v ∈ H 1 (K) : v|∂K ∈ C 0 (∂K), v|e ∈ Pk (e) ∀e ∈ ∂K, Δv ∈ Pk−2 (K)}, with P−1 = {0}. Then, the global virtual element space Vh is defined as Vh = {v ∈ V : w|K ∈ V K,k ∀K ∈ Th }. We will consider the following degrees of freedom, uniquely identifying a function vh ∈ Vh : – the values of vh at the vertices of the tessellation; – for each edge e, the values of vh at the k − 1 internal points of the k + 1 points Gauss-Lobatto quadrature rule on e; – for each element K, the moments up to order k − 2 of vh in K. Due to the definition of the discrete space Vh , the bilinear form a in Eq. (2) is not directly computable on discrete functions in terms of the degrees of freedom. The VEM stems from replacing a with a suitable approximate bilinear form ah . Thus, the discrete form of problem (1) reads as follows: find uh ∈ Vh such that ah (uh , vh ) = fh (vh )

∀ vh ∈ Vh .

(3)

Further details on how to compute the bilinear form ah , as well as the study of the convergence, stability and robustness properties of the method can be found in [2–4]. For further details on the implementation of the method we refer to [5].

3 The FETI-DP Domain Decomposition Method for the VEM Since the degrees of freedom corresponding to the edges of the polygons in the tessellation are nodal values, the FETI-DP method is defined as in the finite element case. More precisely let Ω be split as Ω = ∪ Ω  , with Ω  = ∪K∈T  K, where h

Th are disjoint subsets of Th . In view of the quasi uniformity assumptions on the tessellation Th and assuming that also the decomposition into subdomains is quasi uniform, we can introduce global mesh size parameters H and h such that for all  and for all K we have hK ( h and diam(Ω  ) ( H . We let Γ = ∪ ∂Ω  \ ∂Ω denote the skeleton (or interface) of the decomposition. 8h ⊃ Vh denote the space obtained by dropping the continuity constraint Let V at all nodes interior to the macro edges of the decomposition (which we will call dual nodes), while retaining continuity at cross points. Problem (3) is equivalent to

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8h satisfying finding u˜ h ∈ V ⎧ ⎨

 1 ah (v˜h , v˜h ) − Ω f v˜h , 2 ⎩ such that u˜ is continuous across the interface. h J (u˜ h ) = minv˜h ∈V8h J (v˜h ), with J (v˜h ) =

(4)

8h , we denote by v˜ ∈ RM the corresponding vector coefficient, where For each v˜h ∈ V 8h . Let B be a matrix whose entries assume value in the set M is the dimension of V {−1, 0, 1}. The continuity constraints across the interface can then be expressed as B u˜ = 0. By introducing a set of Lagrange multipliers λ ∈ range(B) to enforce the continuity constraints, we obtain a finite dimensional saddle point formulation of (4) 8u˜ + B T λ = f˜, A

B u˜ = 0,

8 and f˜ are the finite dimensional representation of ah (v˜h , w˜ h ) and where A  8h , respectively. Since A 8 is symmetric and positive definite [6], f v ˜ , ˜h ∈ V h ∀ v˜ h , w Ω we can eliminate u, ˜ and obtain a linear system for the Lagrange multiplier. This linear system is solved with a conjugate gradient method with a preconditioner that takes the same form as in the finite element case [6]. In [1] the authors proved that, as in the finite element case, the condition number of the preconditioned matrix increases at most as (1 + log(k 2 H / h))2 , under the assumption that the Ω  are shape regular.

4 Subdomain Partitioning by Conformal Meshing In general, subdomains Ω  obtained as the union of polygonal elements of a tessellation are not shape regular, unless this is constructed in two stages: first, a decomposition into shape regular subdomains is defined; then, each subdomain is refined to obtain the final tessellation. An alternative is to define the subdomains independently of the tessellation and to modify the latter by “cutting” it with the edges of the subdomains. By construction, the subdomains will then be the union of elements of the new tessellation. We provide here details of our meshing algorithm, which we implemented in C++ using CinoLib [7]. Given a tessellation Th of Ω and a set of L polygonal subdomains that jointly cover Ω without overlaps, we output a domain tessellation with matching interfaces along subdomain boundaries. In the general case the edges of Th will intersect subdomain boundaries at many points. We start by splitting all the edges that are crossed by subdomain boundaries, thereby producing an edge soup. For any pair of edges in the soup, either the two edges are completely disjoint or they meet at a common endpoint (Fig. 1, middle). When computing intersections, exact arithmetics should be used to handle pathological cases due to roundoff errors introduced by finite machine arithmetics [8].

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Fig. 1 Schematic representation of our meshing algorithm

In order to generate the output, we process the edge soup and produce a list of polygons, each one represented as an ordered list of vertices. To this end, for each oriented edge in the soup, we trace polygons by iteratively visiting the leftmost edge until hitting the starting edge at its opposite endpoint, thus closing the polygon (Fig. 1, middle right). If all the edges are visited twice (both orientations count) the result is a list of polygons, with vertices ordered counterclockwise. The only polygon with clockwise winding order will be the one covering the whole domain and containing all and only the boundary edges of Th , which is discarded during the post-processing phase. The leftmost edge is found as follows: given an oriented edge e (from node vi to node vj ), the set of edges incident to it at vj are first classified as being either on the left or on the right of e. Let e (vj → vk ) be one of these edges and A the 2 × 2 matrix having (vj − vi ) and (vk − vj ) as rows. e is at the left of e if det(A) > 0, at the right of e if det(A) < 0 and collinear with e if det(A) = 0. Computing det(A) is known to be a critical issue in finite floating point arithmetics. For this reason, Shewchuck predicates [9] are used to estimate its sign robustly. If there exist some edges at the left of e, then the polygon is locally convex and the leftmost turn is the one that passes through the edge e that minimizes the dot product between e /e  and e/e. Conversely, if all the edges incident to vj are at the right of e, the polygon is locally concave and the leftmost turn is the one that passes through the edge e that maximizes the dot product between e /e  and e/e.

5 Numerical Results Here we consider the performance of the preconditioned FETI-DP method. In each experiment, the domain Ω = [0, 1]2 has been discretized using Voronoi cells of arbitrary shape. Every tessellation Th of Ω is partitioned into L squared subdomains using the approach described in Sect. 4. This approach can introduce new polygons with very small edges, as shown in Fig. 2. In order to test the robustness of FETIDP, we consider two different types of data: i) ρ = 1, f = sin(2πx) sin(2πy); ii) for each subdomain, ρ = 10α , α random integer in [−5, 5], f uniform random in [−1, 1].

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Fig. 2 (Left) General Voronoi mesh partitioned into L = 16 subdomains. (Right) Zoomed view of the boundary between two subdomains, where a tiny triangle is generated Table 1 Results obtained with the first type of data and polynomial order k = 1 on two Voronoi meshes L D.o.f. 1/ h hmin voro1, 40, 000 initial polygons 64 86202 55.93 1.04 × 10−5 144 90561 55.93 1.04 × 10−5 256 94954 55.93 1.04 × 10−5 voro2, 160, 000 initial polygons 64 330104 113.38 1.01 × 10−5 144 338451 113.38 1.04 × 10−5 256 346829 113.38 1.00 × 10−5

γ0

γ1

λmin

λmax

It.

5.73 × 10−3 3.97 × 10−3 2.94 × 10−3

1.32 × 10−3 1.32 × 10−3 1.13 × 10−3

1.05 1.05 1.06

5.78 5.37 4.90

14 14 13

4.11 × 10−4 4.11 × 10−4 4.11 × 10−4

2.47 × 10−3 2.10 × 10−3 1.87 × 10−3

1.06 1.05 1.05

7.32 6.56 6.25

16 15 15

Table 1 shows that, with the first type of data, by fixing the initial mesh Th and increasing the number of subdomains, thereby fixing h while decreasing H , the condition number κ decreases as expected. The smallest edge in the mesh hmin , as well as the parameters γ0 and γ1 introduced in Sect. 2 are also listed to stress that the presence of arbitrarily shaped Voronoi polygons does not hinder the convergence of the method. In Fig. 3, κ 1/2 is plotted as a function of the degrees of freedom of the whole problem, for varying number of subdomains L. The size of the problem is increased by taking five different meshes with 10, 000, 20, 000, 40, 000, 80, 000, and 160, 000 polygons, respectively, as initial tessellations of Ω. Solid and dashed lines correspond to the first and second types of data, respectively. With the first type, for fixed L, κ 1/2 clearly grows as log(degrees of freedom), in agreement with the theoretical bound. This behavior does not seem to be affected by jumps in the diffusion coefficient, especially for high L. Finally, we present some computations performed with high order elements (see Table 2 and Fig. 4). In Fig. 4 (Left), it is possible to verify the polylogarithmic dependence of κ on the polynomial order k. Indeed, for H / h fixed, the expected bound (1 + log(k 2 H / h)2 ∼ (1 + log(k 2 ))2 ∼ (1+2 log(k))2 is confirmed. In Fig. 4 (Right), we keep the polynomial order fixed to k = 3 and k = 5 and increase the dimension of the problem by using the five meshes

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2.6

2.4

2.2

2

L = 64 L = 144 L = 256

1.8

104.5

105 Dof

105.5

Fig. 3 Plots of κ 1/2 as a function of the global degrees of freedom for fixed polynomial order k = 1 but increasing number of subdomains L. Solid and dashed lines correspond to the first and second types of data, respectively

Table 2 κ 1/2 and number of iterations of the preconditioned FETI-DP system with the first type of data, for a fixed starting mesh (voro2, see Table 1) but increasing number of subdomains L and polynomial order k L\k 64 144 256

2 3.13 2.98 2.97

19 18 18

3 3.51 3.37 3.37

4 3.75 3.60 3.57

21 21 21

23 22 22

5 3.91 3.78 3.74

23 23 23

6 4.05 3.93 3.88

24 24 24

4 4

κ1/2

3.8 3.5

3.6 3.4

3

3.2

k=3 k=5

L = 64 L = 256

3 2

3

4 k

5

6

105.5

106 Dof

106.5

Fig. 4 High order elements. Solid and dashed lines correspond to the first and second types of problem data, respectively. (Left) Plot of κ 1/2 as a function of the polynomial order k, initial mesh voro2 (see Table 1). We have H / h ≈ 20.04 for L = 64 and H / h ≈ 10.02 for L = 256. (Right) Plot of κ 1/2 as a function of the global degrees of freedom, H / h ≈ 10.02, L = 256

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mentioned above. The expected bound (1 + log(k 2H / h))2 ∼ (1 + log(H / h))2 (for fixed k) is confirmed for both types of problem data. All these experiments demonstrate that the performance of the preconditioned FETI-DP method is robust with respect to jumps in the diffusion coefficients and shape of the mesh elements. Acknowledgements This paper has been realized in the framework of the ERC Project CHANGE, which has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 694515). The authors would also like to thank the members of the Shapes and Semantics Modeling Group at IMATI-CNR for fruitful discussions on conformal meshing.

References 1. S. Bertoluzza, M. Pennacchio, D. Prada, BDDC and FETI-DP for the virtual element method. Calcolo 54(4), 1565–1593 (2017) 2. L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013) 3. L. Beirão da Veiga, A. Chernov, L. Mascotto, A. Russo, Basic principles of hp virtual elements on quasiuniform meshes. Math. Models Methods Appl. Sci. 26(8), 1567–1598 (2016) 4. L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, Virtual element method for general secondorder elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016) 5. L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24, 1541–1573 (2014) 6. A. Toselli, O. Widlund, Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2005) 7. M. Livesu, CinoLib: a generic programming header only C++ library for processing polygonal and polyhedral meshes (2017). https://github.com/maxicino/cinolib 8. M. Attene, Direct repair of self-intersecting meshes. Graph Models 76(6), 658–668 (2014) 9. J.R. Shewchuk, Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete Comput. Geom. 18(3), 305–363 (1997)

Modeling Flow and Transport in Fractured Media by a Hybrid Finite Volume: Finite Element Method Alexey Chernyshenko and Maxim Olshanskii

Abstract We develop a hybrid finite volume—finite element method for solving a coupled system of advection-diffusion equations in a bulk domain and on an embedded surface. The method is applied for modeling of flow and transport in fractured porous medium. Fractures in a porous medium are considered as sharp interfaces between the surrounding bulk subdomains. We take into account interaction between fracture and bulk domain. The method is based on a monotone nonlinear finite volume scheme for equations posed in the bulk and a trace finite element method for equations posed on the surface. The surface of fracture is not fitted by the mesh and can cut through the background mesh in an arbitrary way. The background mesh is an octree grid with cubic cells and we get a polyhedral octree mesh with cut-cells after grid-surface intersection. The numerical properties of the hybrid approach are illustrated in a series of numerical experiments.

1 Introduction The development of methods for the numerical solving of systems of coupled bulksurface PDEs has become very popular recently. Different approaches can be used successfully depending on how the surface is recovered and equations are treated. Basic approach is to introduce finite element spaces in the volume and on the induced triangulation of the surface, if a tetrahedral tessellation of the volume is available that fits the surface. The resulting fitted bulk-surface finite element method was studied for various problems. Unfitted finite element methods allow the surface to cut through the background tetrahedral mesh. In the class of finite element methods also known as cutFEM or traceFEM, standard background finite element A. Chernyshenko () Institute of Numerical Mathematics, Moscow, Russia M. Olshanskii University of Houston, Houston, TX, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_13

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spaces are employed, while the integration is performed over cut domains and over the embedded surface [2]. The benefits of the unfitted approach are the efficiency in handling implicitly defined surfaces, complex geometries, and the flexibility in dealing with evolving domains. The hybrid method described in this paper belongs to the general class of unfitted methods. If the finite element method is used for a problem in a bulk domain, then it is natural to consider a finite element method for a problem posed on surface. However, depending on application, available software or personal experience, other discretizations such as finite volume or finite difference methods can be preferred for the PDE posed in the volume. In this paper we present a numerical method based on the sharp-interface representation, which uses a finite volume method to discretize the bulk PDEs. The method allows the surface to overlap with the background mesh in an arbitrary way, avoids regular triangulating the surface and any extension of the surface PDE to the bulk domain. It combines the monotone (i.e. satisfying the discrete maximum principle) finite volume method on general meshes [5, 6] with the trace finite element method on octree meshes [4]. In the octree TraceFEM one considers the bulk finite element space of piecewise trilinear continuous functions and uses the restrictions (traces) of these functions to the surface. These traces are further used in a variational formulation of the surface PDE. Effectively, this results in the integration of the standard polynomial functions over the reconstructed surface. For the surface problem only degrees of freedom from the cubic cells cut by the surface are active. Surface parametrization is not required, no PDE extension of the surface is needed. The resulting hybrid FV–FE method is robust with respect to the position of surfaces against the background mesh and is well suited for handling non-smooth surfaces and implicit surfaces. This technique can be also applied for tetrahedral or more general polyhedral tessellations of the bulk domain, but we use octree grid with cubic cells here. The Cartesian structure and built-in hierarchy of octree grids makes mesh adaptation, reconstruction and data access fast and easy. The drawback is that an octree grid provides only the first order (staircase) approximation of a general geometry. Allowing the surface to cut through the octree grid in an arbitrary way overcomes this issue, but challenges us with the problem of building efficient bulk-surface discretizations. The hybrid TraceFEM–FV method complements the advantages of using octree grids by delivering the higher order accuracy for both bulk and surface numerical solutions.

2 Mathematical Model Consider the bulk domain Ω ⊂ R3 and a piecewise smooth surface Γ ⊂ Ω. The surface Γ may have several connected components. If Γ has a boundary, for simplicity we assume that ∂Γ ⊂ ∂Ω, but the model can be extended to immersed surfaces. Thus, we have the subdivision Ω = ∪i=1,...,N Ω i into simply connected

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subdomains Ωi such that Ω i ∩ Ω j ⊂ Γ , i = j . Consider the flow in the bulk domain and along the fracture. The fracture is modeled by the surface Γ . We denote by pi , w i , pΓ , wΓ the pressures and the Darcy velocities in Ωi and on the surface. Note that wΓ has the physical meaning of the flow rate through the cross-section of the fracture. The coupled flow can be modeled by the following system of equations [1], in subdomains, divwi = qi

in Ωi ,

wi = −Ki ∇pi

in Ωi ,

pi = pΓ

on Γ,

(1)

and on the surface divΓ wΓ = qΓ + (w1 · n − w2 · n) wΓ = −dKΓ ∇Γ pΓ

on Γ,

(2)

on Γ,

where we employ the following notations: ∇Γ , divΓ denote the surface tangential gradient and divergence operators, n is a unit normal vector at Γ , d > 0 is the fracture width coefficient; qi , qΓ are given source terms and Ki , KΓ are permeabilities in the subdomains and in the fracture. Also we add Dirichlet’s and Neumann’s boundary conditions. Consider an agent that is soluble in the fluid and transported by the flow in the bulk and along the fractures. The solute volume concentration is denoted by u, ui = u|Ωi . The solute surface concentration along Γ is denoted by v. Change of the concentration happens due to convection by the velocity fields w i and wΓ , diffusive fluxes in Ωi , diffusive flux on Γ , as well as the fluid exchange and diffusion flux between the fractures and the porous matrix. These coupled processes can be modeled by the following system of equations [1], in subdomains, ⎧ ⎪ ⎨ φi ∂ui + div(wi ui − Di ∇ui ) = fi in Ωi , ∂t ⎪ ⎩ ui = v on ∂Ωi ∩ Γ,

(3)

and on the surface, φΓ

∂v + divΓ (wΓ v − dDΓ ∇Γ v) = FΓ (u) + fΓ ∂t

on Γ,

(4)

where FΓ (u) stands for the net flux of the solute per surface area due to fluid leakage and hydrodynamic dispersion; fi and fΓ are given source terms in the subdomains and in the fracture; Di denotes the diffusion tensor in the porous matrix; the surface diffusion tensor is DΓ . Both Di , i = 1, . . . , N, and DΓ are symmetric and positive

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definite; φi > 0 and φΓ > 0 are the constant porosity coefficients for the bulk and the fracture. The total surface flux FΓ (u) represents the contribution of the bulk to the solute transport in the fracture. The mass balance at Γ leads to the equation FΓ (u) = [−Dn · ∇u + (n · w)u]Γ ,

(5)

where [w(x)]Γ denotes the jump of w across Γ in the direction of n. If Γ is piecewise smooth, we need additional conditions on the edges, assuming the continuity of concentration, conservation of fluid mass and solute flux. Also we add Dirichlet’s boundary conditions for the concentration u and v on ∂ΩD and ∂Γ D and homogeneous Neumann’s boundary conditions on ∂ΩN and ∂Γ N , respectively. Initial conditions are given by the known concentration u0 and v0 at t = 0.

3 Hybrid Finite Volume: Finite Element Method Consider a Cartesian background grid with cubic cells. To produce a mesh with an octree hierarchical structure we allow local refinement of the grid by sequential division of any cubic cell into eight cubic subcells. This mesh gives the tessellation Th of the computational domain Ω, Ω = ∪T ∈Th T . The surface Γ ⊂ Ω cuts through the mesh in an arbitrary way. Instead of Γ we will consider Γh , a given polygonal approximation of Γ . We assume that the reconstructed surface Γh divides Ω into N subdomains Ωi,h , and ∂Γ h ⊂ ∂Ω. Also we do not imply any restrictions on how Γh intersects the background mesh. The reconstructed surface Γh is a C 0,1 surface that can be partitioned in planar triangular elements: Γh =

3

K,

K∈Fh

where Fh is the set of all triangular segments K. In practice, we construct Γh using Multi-material cubical marching squares algorithm [3]. The induced tessellation of Ωi,h can be considered as a subdivision of the volume into general polyhedra. Let Ti,h be the tessellation of Ωi,h into non-intersected polyhedra. For the advection and diffusion in the matrix we apply a non-linear FV method devised on general polyhedral meshes in [5], which is monotone and has compact stencil. The trace of the background mesh on Γh induces a ‘triangulation’ of the fracture, which is very irregular, and so we do not use it do build a discretization method. To handle advection and diffusion along the fracture, we first consider finite element space of piecewise trilinear functions for the volume octree mesh Th . We further, formally, consider the restrictions (traces) of these background functions on Γh and use them in a finite element integral form over Γh . Thus the irregular triangulation of Γh is used for numerical integration only, while the trial and test functions are tailored to the background regular mesh. It

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appears that the properties of this trace finite element method are driven by the properties of the background mesh, and they are independent on how Γh intersects Th . The TraceFEM was devised and first analysed in [7] and extended for the octree meshes in [4]. A natural way to couple two approaches is to use the restriction of the background FE solution on Γh as the boundary data for the FV method and to compute the FV two-side fluxes on Γh to provide the source terms for the surface discrete equation. Further we provide details of the coupling between discrete bulk and surface equations. For brevity we consider only the system (3). The PDEs in the bulk and on the surface are coupled through the boundary condition ui = v on ∂Ωi,h ∩ Γh (second equation in (3)) and the net flux FΓh (u) on Γh , which stands as the source term in the surface equation (4). On Γh the solution vh is defined as a trace of the background finite element piecewise trilinear function. The averaged value of vh is computed on each surface triangle K ∈ Fh using a standard quadrature formula. These values assigned to the barycenters of K from Fh serve as the Dirichlet boundary data for the FV method on Γh . The discrete diffusive and convective fluxes are assigned to barycenters of all faces on Ti,h , i = 1, . . . , N. Since each triangle K ∈ Fh is a face for two neighbouring cells Ti ∈ Ti,h and Tj ∈ Tj,h , i = j , the diffusive and convective fluxes are assigned to K from both sides of Γh . The discrete net flux FΓh (uh ) at the barycenter of K is computed as the jump of the fluxes over K. In the TraceFEM this value is assigned to all x ∈ K, and numerical integration is done over all surface elements K ∈ Fh to compute the right-hand side of the algebraic system. We iterate between the bulk FV and surface FE solvers on each time step to satisfy all discretized equations and boundary conditions. We assume an implicit time stepping method. This results in the following system on each time step: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

˜ + div(wu − D∇)u = fˆ in Ω \ Γ, Lu := φu u=v

on Γ,

n∂Ω · ∇u = 0 on ∂Ω N , u = uD on ∂Ω D , ⎪ ⎪ ⎪ ⎪ ⎪ LΓ v := φ˜ Γ v + divΓ (wΓ v − dDΓ ∇Γ v) = FΓ (u) + fˆΓ on Γ, ⎪ ⎪ ⎪ ⎪ ⎩ n · ∇v = 0 on ∂Γ , v = v on ∂Γ , ∂Γ

N

D

(6)

D

the right hand sides fˆ, fˆΓ account for the solution values at the previous time step. We solve the coupled system (6) by the fixed point method. For the sake of brevity we will not describe the details and continue with numerical experiments.

4 Numerical Results This section shows several numerical examples, which demonstrate the accuracy and capability of the hybrid method for flow and steady transport problems.

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Fig. 1 The figure illustrates calculated velocities for a symmetric (left) and nonsymmetric flow (right)

4.1 Flow in a Bulk with Planar Fracture This is a 3D extension of the first numerical test from [1]. Consider the domain Ω = [0, 1.5] × [0, 1] × [0, 1] divided into two equally subdomains by the planar surface x = 0.75. The permeability in Ω is equal to 1, the permeability in the fracture × the width of the fracture is equal 2. Flow in the fracture is driven by the pressure drop of 10 (and 5 for the nonsymmetric case) the upper boundary to the bottom boundary. On the left and right boundaries of Ω the pressure is given, and there are no flow conditions on the others boundaries. We consider two cases. A symmetric case where pressures on the left and on the right boundaries of the domain are equal to the pressure pΓ on the upper boundary. In the nonsymmetric case there is a pressure drop of 10 from the right boundary to the left boundary. Numerical results are illustrated in Fig. 1. Arrows represent the velocity field. The color scale shows the magnitude of velocity. We can observe that there is a flow interaction between the fracture and the bulk domain. For instance some fluid is coming out of the fracture and then is coming back into it. In the nonsymmetric case most of the flow is attracted into the fracture, but is still some flow towards the left boundary.

4.2 Steady Analytical Solution for a Triple Fracture Problem Next consider the coupled surface-bulk diffusion problem in the domain Ω = [0, 1]3 with an embedded piecewise planar Γ . We design Γ to model a branching fracture. In the basic model, Γ = Γ (0) consists of three planar pieces, Γ (0) = Γ12 ∪Γ13 ∪Γ23 , Γij = Ωi ∩Ωj i = j , such that Ω1 = {x ∈ Ω | x < 0.5 and y > x}, Ω2 = {x ∈ Ω | x > 0.5 and y > x − 1}, Ω3 = Ω \ (Ω 1 ∪ Ω 2 ). To model a generic situation when Γ cuts through the background mesh in an arbitrary way, we consider the tessellations of Ω = [0, 1]3 into three subdomains by a surface Γ (α). The surface Γ (α) is obtained from Γ (0) by applying the clockwise

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Fig. 2 The figure illustrates the bulk domain with uniform mesh and the surface mesh on the fracture, rotated by 40 degrees

rotation by the angle α around the axis x = z = 0.5. We show the results with α = 40o . The resulting tessellation of Ω and surfaces mesh are illustrated in Fig. 2. To define the solution solving the stationary equations (3), we introduce 9 ψ1 =

16(y − 12 )4 , y > 0, y≤

1 2 1 2

,

ψ2 = x − y,

ψ3 = x + y − 1.

We define the solution of the basic model problem (α = 0) ⎧ ⎪ ⎨ sin(2πz) · ψ2 (x) · φ3 (x) x ∈ Ω1 , u(x) = x ∈ Ω2 , sin(2πz) · ψ1 (x) ⎪ ⎩ sin(2πz)2x · ψ (x) x ∈ Ω3 , 1

v = u|Γ (0) .

The solution for the problem with rotated fracture is obtained by applying the same rotation. Other parameters are set to be w = wΓ = 0, φi = φΓ = 0, Di = I , 0.1 DΓ,i = 10I for i = 1..3, and d23 = 0.1, d13 = d12 = √ . An interesting feature 2 of this problem is that the surface Γ is only piecewise smooth. The bulk grid is not fitted to the internal edge E = Γ12 ∩Γ13 ∩Γ23 , and hence the tangential derivatives of v are discontinuous inside certain cubic cells from ThΓ . This is well-known to result in the 12 -reduction of convergence order. This suboptimal order for a sequence of uniform background meshes is demonstrated by the results in Table 1.

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Table 1 The error in the numerical solution for the problem with triple fracture Ω

Γ

#d.o.f. 991 7996 64046 512258 353 1932 8766 36676

L2 -norm 5.934e−3 1.700e−3 4.907e−4 1.503e−4 8.167e−3 2.146e−3 7.115e−4 2.538e−4

Rate 1.80 1.80 1.82 1.66 1.59 1.49

H 1 -norm 4.080e−1 1.621e−1 6.263e−2 2.541e−2 2.709e−1 1.275e−1 6.279e−2 3.121e−2

Rate 1.33 1.37 1.39 1.09 1.02 1.01

L∞ -norm 3.783e−2 1.276e−2 3.515e−3 1.237e−3 2.696e−2 5.566e−3 2.063e−3 7.251e−4

Rate 1.56 1.86 1.61 1.85 1.31 1.51

5 Conclusions The paper presented a hybrid FV–FE method for the coupled bulk-surface systems of PDEs. The distinct feature of the method is that the same background mesh is used to solve equations in the bulk and on the surfaces, there is no need to fit this mesh to the embedded surfaces. This makes the approach attractive to treat problems with complicated embedded structures like those occurring in the simulations of flow and transport in fractured porous media. Acknowledgements This work has been supported by the Russian Science Foundation Grant 1771-10173.

References 1. C. Alboin, J. Jaffré, J.E. Roberts, C. Serres, Modeling fractures as interfaces for flow and transport, in Fluid Flow and Transport in Porous Media. Mathematical and Numerical Treatment, vol. 295 (American Mathematical Society, Providence, 2002), p. 13 2. E. Burman, S. Claus, P. Hansbo, M.G. Larson, A. Massing, Cutfem: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104(7), 472–501 (2015) 3. A. Chernyshenko, Generation of octree meshes with cut cells in multiple material domains (in russian). Numer. Methods Progr. 14, 229–245 (2013) 4. A.Y. Chernyshenko, M.A. Olshanskii, An adaptive octree finite element method for PDEs posed on surfaces. Comput. Methods Appl. Mech. Eng. 291, 146–172 (2015) 5. A. Chernyshenko, Y. Vassilevski, A finite volume scheme with the discrete maximum principle for diffusion equations on polyhedral meshes, in Finite Volumes for Complex Applications VIIMethods and Theoretical Aspects. Springer Proceedings in Mathematics and Statistics, ed. by J. Fuhrmann, M. Ohlberger, C. Rohde, vol. 77 (Springer International Publishing, Cham, 2014), pp. 197–205 6. K. Lipnikov, D. Svyatskiy, Y. Vassilevski, Minimal stencil finite volume scheme with the discrete maximum principle. Russ. J. Numer. Anal. Math. Model. 27(4), 369–385 (2012) 7. M. Olshanskii, A. Reusken, J. Grande, A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47, 3339–3358 (2009)

A Cut Cell Hybrid High-Order Method for Elliptic Problems with Curved Boundaries Erik Burman and Alexandre Ern

Abstract We design a Hybrid High-Order method for elliptic problems on curved domains. The method uses a cut cell technique for the representation of the curved boundary and imposes Dirichlet boundary conditions using Nitsche’s method. The physical boundary can cut through the cells in a very general fashion and the method leads to optimal error estimates in the H 1 -norm.

1 Introduction The Hybrid High-Order (HHO) method has been recently introduced in [5, 6]. The idea is to approximate the solution using cell and face unknowns. The cell unknowns can be eliminated locally in each mesh cell, leading to a global problem coupling only the face unknowns. The HHO method is devised from a local reconstruction operator and a stabilization operator coupling the cell and face unknowns in each mesh cell. This leads to a discretization method that supports general meshes (with polyhedral cells and non-matching interfaces), is locally conservative, and delivers energy-norm error estimates of order (k + 1) (and L2 -norm error estimates of order (k + 2) under full elliptic regularity) using polynomials of order k ≥ 0 for the face unknowns and of order l ∈ {k − 1, k, k + 1} ∩ N for the cell unknowns. The use of polyhedral meshes can simplify the meshing problem in many situations. Nevertheless, in some cases it can still be convenient to avoid building meshes fitted to the domain boundary. This is typically the case when the boundary

E. Burman Department of Mathematics, University College London, London, UK e-mail: [email protected] A. Ern () Université Paris-Est, CERMICS (ENPC), Marne-la-Vallée Cedex, France INRIA Paris, Paris, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_14

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changes during the computation, as in free-boundary problems or optimisation problems, or when the boundary is curved and high-order accuracy is to be retained. In classical finite element methods, fictitious domain approaches [7], where the computational mesh does not respect the domain boundary, are often efficient, but at the expense of accuracy. In order to improve the accuracy, unfitted finite element methods were introduced [1, 8]. A well-known difficulty for unfitted finite element methods is that the conditioning of the linear system resulting from discretization has a strong dependence on the geometry of the intersection of the physical boundary with the mesh cells. Unfavorably cut cells, that is, cells having a small intersection with the physical domain, lead to ill-conditioning. In the case of H 1 -conforming methods, this problem is typically solved by adding a penalty term that weakly couples the polynomial approximation in adjacent cells [3]. When using discontinuous approximation, another approach was proposed in [9] in the context of elliptic interface problems, where any unfavorably cut cell is merged with a neighboring cell having a favorable cut. This local agglomeration procedure generally leads to meshes having polyhedral cells. In this work, we consider a (simple) elliptic PDE posed on an open, bounded, Lipschitz set Ω  in Rd with a non-homogeneous Dirichlet condition on Γ := ∂Ω  : −∇·(κ∇u) = f,

in Ω  ,

(1a)

u = gD

on Γ.

(1b)

1

Here f ∈ L2 (Ω  ), gD ∈ H 2 (Γ ) and κ is a positive, piecewise constant coefficient. We assume that Ω  has a curved boundary Γ , assumed to be of class C 2 and non self-intersecting, and we want to avoid meshing Ω  using cells with curved faces. One possible way to avoid this is to consider a fictitious domain approach. We embed Ω  into a larger polyhedron Ω that can be meshed easily (by definition, a polyhedron is a finite union of simplices and has therefore a piecewise planar boundary). This mesh of Ω can be composed of cells having simple shapes (simplices, prisms, hexahedra), but can also contain more general polyhedral shapes. Our goal is to devise a cut-cell HHO method to approximate (1) using an unfitted mesh of the fictitious domain Ω. For the imposition of the Dirichlet boundary condition, we use Nitsche’s method, and robustness with respect to the cuts is ensured by using a local mesh agglomeration procedure reminiscent of that proposed in [9]. Owing to the local cell agglomeration, the resulting mesh of Ω is generally of polyhedral type. The present cut-cell HHO method and its analysis are adapted from the method devised in [4] for elliptic interface problems. An alternative HHO method for elliptic problems on curved meshes has been proposed recently in [2] using nonlinear mappings for the face unknowns and increasing the order of the reference face polynomials. An outline of the paper is as follows. In the next section, we present the discrete framework, technical results, and the main assumptions. In Sect. 3, we introduce the cut-cell HHO method. Finally, we present an error estimate in Sect. 4.

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2 Discrete Setting Let (Th )h>0 be a shape-regular family of matching meshes covering Ω exactly, where the index h refers to the maximal diameter of the mesh cells. Mesh cells are closed sets by convention. For simplicity, we assume that the diffusion coefficient is piecewise constant on Th and we let κT denote its value on T ∈ Th . The meshes can have polyhedral cells, and the mesh cells are considered by convention to be open subsets of Rd . The shape-regularity criterion for polyhedral meshes is that they admit a matching simplicial sub-mesh that satisfies the usual shape-regularity criterion in the sense of Ciarlet and such that each sub-cell (resp., sub-face) belongs to only one mesh cell (resp., at most one mesh face) having uniformly the same diameter. The shape-regularity of the mesh sequence is quantified by a parameter ρ ∈ (0, 1). In what follows, B(y, a) denotes the open ball with center y and radius a. For a subset S ⊂ Rd , hS denotes the diameter of S.

2.1 Notation for Unfitted Meshes For coherence of our notation, let us set Ω \Γ := Ω  . Let us define the partition \Γ Th = Th ∪ ThΓ ∪ Thc , where the subsets \Γ

Th

:= {T ∈ Th | T ⊂ Ω \Γ },

ThΓ := {T ∈ Th | measd−1 (T ∩ Γ ) > 0},

(2a) (2b)

collect, respectively, the mesh cells inside the physical domain Ω \Γ and the mesh cells cut by the physical boundary Γ . The mesh cells in Thc do not play any role in what follows so that we can discard them without loss of generality. Similarly, the \Γ mesh faces are collected in the set Fh which is partitioned into Fh = Fh ∪ FhΓ , \Γ where Fh collects the mesh faces inside the physical domain Ω \Γ and FhΓ collects the mesh faces cut by the physical boundary. For any mesh cell T ∈ ThΓ , we define T \Γ := T ∩ Ω \Γ and T Γ := T ∩ Γ . The boundary of T \Γ is decomposed as ∂T \Γ = (∂T )\Γ ∪ T Γ , where (∂T )\Γ := ∂T ∩ Ω \Γ (see Fig. 2 below).

2.2 Admissible Meshes The cut-cell HHO method is to be deployed on meshes satisfying two assumptions. Assumption 1 means that the physical boundary is well resolved by the mesh; this assumption is quantified by a regularity parameter γ ∈ (0, 1). Assumption 2 means that all the mesh cells in ThΓ are cut favorably by the physical boundary; this property is quantified by a cut parameter δ ∈ (0, 1).

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Assumption 1 (Resolving Γ ) There is γ ∈ (0, 1) s.t. for all T ∈ ThΓ , there is a point xˆ T ∈ Rd so that, for all s ∈ T Γ , xˆ T − s2 ≤ γ −1 hT and d(xˆ T , Ts Γ ) ≥ γ hT where Ts Γ is the tangent plane to Γ at the point s. Assumption 2 (Cut Cells) There is δ ∈ (0, 1) such that, for all T ∈ ThΓ , there is x˜ T ∈ T \Γ so that B(x˜ T , δhT ) ⊂ T \Γ . Let T˜h be an initially given mesh from a shape-regular sequence of (polyhedral) meshes. It is shown in [4, Lemma 6.1] that the Assumption 1 is satisfied provided T˜h is fine enough. To ensure that the Assumption 2 is also satisfied, we identify those mesh cells in T˜hΓ (with obvious notation) for which the assumption fails and agglomerate them with neighboring elements satisfying the assumption. The agglomeration procedure is as follows, where for all T ∈ T˜h , we define the set of elements with non-empty intersection with T as Δ(T ) := {T  ∈ T˜h | T ∩T  = ∅}: 1. Partition T˜hΓ into T˜hKO ∪ T˜hOK so that T˜ ∈ T˜hOK iff there is x T˜ ∈ T˜ s.t. the ball B(x T˜ , δ0 hT˜ ) is in T˜ \Γ = T˜ ∩ Ω \Γ , whereas T˜ ∈ T˜hKO otherwise. 2. For all T˜ ∈ T˜hKO , we choose a neighboring mesh cell N(T˜ ) so that \Γ N(T˜ ) ∈ (T˜hOK ∪ T˜h ) ∩ Δ(T˜ ).

(3)

It is shown in [4] that this set is nonempty if the mesh T˜h is fine enough and if the initial cut parameter δ0 is small enough. \Γ 3. Let N(T˜hKO ) be the collection of all the cells in T˜hOK ∪T˜h that have been selected at least once in Step 2. For all T˜ $ ∈ N(T˜hKO ), we define the agglomerated cell T˜ $ ∪ {T˜ ∈ T˜hKO | N(T˜ ) = T˜ $ }. agglo 4. We collect the agglomerated cells in T˜h and define the new mesh as   \Γ agglo OK KO ˜ ˜ ˜ . Th := (Th ∪ Th ) \ N(Th ) ∪ T˜h

(4)

It is shown in [4] that the new mesh Th is still shape-regular (with possibly a smaller value of the parameter ρ), still satisfies the Assumption 1 (with possibly a smaller parameter γ ), and additionally satisfies the Assumption 2 (with possibly a parameter δ smaller than δ0 ).

2.3 Trace Inequalities The following two trace inequalities, which are crucial for the robustness of the error estimate (20) (see below), hinge on Assumptions 1 and 2. For a proof, we refer the reader to [4, Lemma 3.3 and 3.4].

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Lemma 3 (Multiplicative Trace Inequality) There are cmtr > 0 and θmtr ≥ 1, depending on ρ and γ , such that, for all T ∈ ThΓ , there is xˇ T ∈ T so that, for all v ∈ H 1 (T † ) with T † = B(xˇ T , θmtr hT ),  1  1 1 − vL2 (∂T \Γ ) ≤ cmtr hT 2 vL2 (T † ) + vL2 2 (T † ) ∇vL2 2 (T † ) .

(5)

Lemma 4 (Discrete Trace Inequality) Let l ∈ N, l ≥ 0. There is cdtr , depending on l, ρ, and δ, such that, for all T ∈ ThΓ , and all v ∈ Pld (T \Γ ), −1

vL2 (∂T \Γ ) ≤ cdtr hT 2 vL2 (T \Γ ) .

(6)

3 The Cut-Cell HHO Method In this section, we describe the cut-cell HHO method for the ficitious domain problem. Let k ≥ 0 be the polynomial degree for the face unknowns; the polynomial degree for the cell unknowns is (k + 1). For any mesh cell T ∈ Th , the set F∂T is \Γ the collection of the faces of T . Whenever T ∈ ThΓ , the set F∂T = {F \Γ = F ∩ Ω \Γ | F ∈ FT , measd−1 (F \Γ ) > 0} is the collection of the faces and cut faces of T that partition the boundary (∂T )\Γ .

3.1 Uncut Cells \Γ

Let T ∈ Th . The local unknowns form a pair composed of one polynomial of order (k + 1) in T and a piecewise polynomial of order k on ∂T . The local unknowns are generically denoted as k ˆ vˆT = (vT , v∂T ) ∈ Pk+1 d (T ) × Pd−1 (F∂T ) =: XT ,

(7)

with Pkd−1 (F∂T ) = F ∈F∂T Pkd−1 (F ). The placement of the discrete unknowns for the uncut cells is illustrated in Fig. 1.

Fig. 1 Uncut hexagonal cell. Left: k = 0; Right: k = 1. Each dot symbolizes one degree of freedom (not necessarily a pointwise evaluation)

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We define the following local bilinear form for all v, w ∈ H 1 (T ):  aT (v, w) =

κT ∇v·∇w.

(8)

T

There are two ingredients to devise the local HHO bilinear form. The first one is a reconstruction operator. Let vˆT = (vT , v∂T ) ∈ XˆT . Then, we reconstruct k+1 a polynomial rTk+1 (vˆT ) ∈ Pk+1 d (T ) by requiring that, for all z ∈ Pd (T ), the following holds true:  aT (rTk+1 (vˆT ), z)

= aT (vT , z) −

κT ∇z·nT (vT − v∂T ),

(9)

∂T

where nT is the unit outward-pointing normal to T . Note that rTk+1 (vˆT ) is uniquely defined  k+1 up to anadditive constant; one way to fix the constant is to prescribe T rT (vˆT ) = T vT . The second ingredient is the stabilization bilinear form defined so that, for all vˆT , wˆ T ∈ XˆT ,  k sT (vˆT , wˆ T ) = κT h−1 Π∂T (vT − v∂T )(wT − w∂T ), (10) T ∂T

k denotes the L2 -orthogonal projector onto the piecewise polynomial where Π∂T k space Pd−1 (F∂T ). Finally, the local HHO bilinear and linear forms to be used when assembling the global discrete problem are as follows: For all vˆT , wˆ T ∈ XˆT , \Γ

aˆ T (vˆT , wˆ T ) = aT (rTk+1 (vˆT ), rTk+1 (wˆ T )) + sT (vˆT , wˆ T ),  \Γ f wT . ˆT (wˆ T ) =

(11a) (11b)

T

3.2 Cut Cells: Fictitious Domain Problem Let T ∈ ThΓ . We use capital letters to denote a generic function V ∈ H 1 (T \Γ ). The unit outward-pointing normal to Γ is denoted n. We define the following Nitsche bilinear form for all V , W ∈ H s (T \Γ ), s > 32 : ;  : κT nT (V , W ) = κT ∇V ·∇W − (κT ∇V ·n)W + (κT ∇W ·n)V − η V W , hT T \Γ TΓ (12) 

2 where c where the user-specified parameter η is such that η ≥ 4cdtr dtr results from the discrete trace inequality (6) with polynomial degree l = k (this follows using standard arguments for the stability of Nitsche’s method, see [4, Lemma 5.1]). The

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Fig. 2 Cut hexagonal cell. The physical domain Ω \Γ (red/light grey) is located below the curved boundary Γ so that (∂T )\Γ is composed of two faces and two cut faces. Left: k = 0; Right: k = 1. Each dot symbolizes one degree of freedom (not necessarily a pointwise evaluation)

local HHO unknowns are \Γ \Γ VˆT = (VT , V∂T ) ∈ XˆT := Pk+1 ) × Pkd−1 (F∂T ), d (T

(13)

\Γ

where Pkd−1 (F∂T ) is the piecewise polynomial space of order k on (∂T )\Γ based \Γ

on the (sub-)faces in F∂T ; note that we do not introduce discrete unknowns on T Γ . The placement of the discrete HHO unknowns in the cut cells is illustrated in Fig. 2. As for the uncut cells, there are two key ingredients to devise the local HHO bilinear form: reconstruction and stabilization. Let VˆT ∈ XˆT . Then, we reconstruct \Γ ) by requiring that, for all Z ∈ Pk+1 (T \Γ ), a polynomial RTk+1 (VˆT ) ∈ Pk+1 d (T d the following holds true: nT (RTk+1 (VˆT ), Z) = nT (VT , Z) −

 (∂T )\Γ

κT ∇Z·nT (VT − V∂T ).

(14)

It follows from the stability of Nitsche’s method that RTk+1 (VˆT ) is uniquely defined by up to an additive constant; one way to fix the constant is to prescribe  (14)k+1 ˆ ˆ ˆ ˆ R T \Γ T (VT ) = T \Γ VT . Concerning stabilization, we set for all VT , WT ∈ XT , sT (VˆT , Wˆ T ) = κT h−1 T

 (∂T )\Γ

k Π(∂T (VT − V∂T )(WT − W∂T ), )\Γ

(15)

k where Π(∂T denotes the L2 -orthogonal projector onto the piecewise polynomial )\Γ \Γ

space Pkd−1(F∂T ). Finally, the local HHO bilinear and linear forms are as follows: For all VˆT , Wˆ T ∈ XˆT , aˆ TΓ (VˆT , Wˆ T ) = nT (RTk+1 (VˆT ), RTk+1 (Wˆ T )) + sT (VˆT , Wˆ T ),   Γ ˆ ˆ f WT + gD φT (WT ), T (WT ) = T \Γ

TΓ

with φT (WT ) = −κT ∇WT ·nΓ + ηκT h−1 T WT .

(16a) (16b)

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3.3 The Global Formulation \Γ

To formulate the global discrete problem, we introduce the sets Tˆh := Th \Γ Ω \Γ | T ∈ ThΓ } and Fˆ h := Fh ∪ {F ∩ Ω \Γ | F ∈ FhΓ }, and we set k Xˆh := T ∈Tˆ Pk+1 d (T ) × F ∈Fˆ Pd−1 (F ). h

h

∪ {T ∩

(17)

\Γ Let Vˆh ∈ Xˆh . For all T ∈ Th , we denote vˆT = (vT , v∂T ) ∈ XˆT the components of Vˆh attached to the uncut cell T . For all T ∈ ThΓ , we denote VˆT = (VT , V∂T ) ∈ XˆT the components of Vˆh attached to the cut cell T . The global discrete problem then reads as follows: Find Uˆ h ∈ Xˆh s.t.

∀Wˆ h ∈ Xˆh ,

aˆ h (Uˆ h , Wˆ h ) = ˆh (Wˆ h ),

(18)

with

aˆ h (Vˆh , Wˆ h ) =

\Γ T ∈ Th

ˆh (Wˆ h ) =

\Γ

aˆ T (vˆT , wˆ T ) +

aˆ TΓ (VˆT , Wˆ T ),

(19a)

T ∈ThΓ

\Γ ˆT (wˆ T ) +

\Γ T ∈ Th

ˆΓT (Wˆ T ),

(19b)

T ∈ThΓ

\Γ \Γ \Γ where aˆ T (·, ·) and ˆT (·) are defined by (11) for all T ∈ Th and aˆ TΓ (·, ·) and ˆΓT (·) are defined by (16) for all T ∈ ThΓ .

4 Main Result: Error Estimate We now state our main result on the error analysis. The proof follows by adapting the arguments of [4, Thm. 5.9]. Theorem 5 (Error Estimate) Assume that u ∈ H k+2 (Ω \Γ ), k ≥ 0, is the solution to (1). Let Uˆ h ∈ Xˆh solve (18). Then, the following bound holds true:

∇(u − uT )2T +

\Γ

∇(u − UT )2T \Γ

T ∈ThΓ

T ∈ Th

+

T ∈ThΓ

2 2(k+1) h−1 |u|2H k+2 (Ω \Γ ) . T gD − UT T Γ  h

(20)

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Acknowledgements The first author was partly supported by EPSRC Grant EP/P01576X/1. This work was initiated when the authors were visiting the Institut Henri Poincaré during the Fall 2016 Thematic Trimester “Numerical Methods for Partial Differential Equations”. The support of IHP is gratefully acknowledged.

References 1. J.W. Barrett, C.M. Elliott, Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces. IMA J. Numer. Anal. 7(3), 283–300 (1987) 2. L. Botti, D.A. Di Pietro, Assessment of hybrid high-order methods on curved meshes and comparison with discontinuous Galerkin methods (2017). HAL e-print hal-01581883 3. E. Burman, Ghost penalty. C. R. Math. Acad. Sci. Paris 348(21–22), 1217–1220 (2010) 4. E. Burman, A. Ern, An unfitted hybrid high-order method for elliptic interface problems (2017). ArXiv e-print 1710.10132 5. D.A. Di Pietro, A. Ern, A Hybrid High-Order locking-free method for linear elasticity on general meshes. Comput. Meth. Appl. Mech. Eng. 283(1), 1–21 (2015) 6. D.A. Di Pietro, A. Ern, S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Meth. Appl. Math. 14(4), 461–472 (2014) 7. V. Girault, R. Glowinski, Error analysis of a fictitious domain method applied to a Dirichlet problem. Jpn. J. Indust. Appl. Math. 12(3), 487–514 (1995) 8. A. Hansbo, P. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47–48), 5537–5552 (2002) 9. A. Johansson, M.G. Larson, A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numer. Math. 123(4), 607–628 (2013)

A Cut Finite Element Method with Boundary Value Correction for the Incompressible Stokes Equations Erik Burman, Peter Hansbo, and Mats G. Larson

Abstract We design a cut finite element method for the incompressible Stokes equations on domains with curved boundary. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche’s method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman et al. (Math Comput 87(310):633–657, 2018).

1 Introduction Let Ω be a domain in Rd with smooth boundary ∂Ω and exterior unit normal n. We will consider a cut finite element method (CutFEM) for Stokes problem on Ω with Dirichlet boundary conditions. See [6] and the references therein for an introduction to CutFEM. The Stokes problem takes the form: find u : Ω → Rd and p : Ω → R

E. Burman () Department of Mathematics, University College London, London, UK e-mail: [email protected] P. Hansbo Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden e-mail: [email protected] M. G. Larson Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_15

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satisfying the weak form of the system of equations −Δu + ∇p = f

in Ω,

(1)

∇ ·u=0

in Ω,

(2)

on ∂Ω,

(3)

u=g

 where f ∈ [H −1(Ω)]d and g ∈ [H 1/2(∂Ω)]d , ∂Ω g · n ds = 0, are given data. It follows from the Lax-Milgram Lemma that there exists a unique solution u ∈ [H 1(Ω)]d and from Brezzi’s Theorem that there exists a unique solution p ∈ L20 (Ω). We also have the elliptic regularity estimate uH s+2 (Ω) + pH s+1 (Ω)  f H s (Ω) + gH s+3/2 (∂Ω) ,

s ≥ −1.

(4)

See [2] for further details. Here and below we use the notation  to denote less or equal up to a multiplicative constant and we let (·, ·)X denote the L2 -scalar product over X with associated norm  · X . The objective of the present paper is to propose a cut finite element method for the problem (1)–(3). Unfitted finite element methods for incompressible elasticity was first discussed in [1], for the coupling over an internal (unfitted) interface. The fictitious domain problem for the Stokes equations was then considered in [4, 5, 11] and more recently the inf-sup stability for several different well-known elements on unfitted meshes was proved in [8]. Further work on the Stokes interface problem was presented in [9]. The upshot in the present contribution is that we, following [7], use a piecewise affine representation of the physical boundary and introduce a correction in the Nitsche formulation to correct for the low order geometry error. This allows us to use for instance a piecewise affine levelset for the geometry representation used in the integration over the cut elements, while retaining the accuracy of a (known) higher order representation of the boundary. This provides an alternative to representing curved boundaries using isoparametric mappings, see [10] for an application of this technique to the Stokes equations. We will focus herein on the derivation of the CutFEM boundary value correction method for the Stokes system, this is the topic of Sect. 2. We then state some fundamental results (without proof) in Sect. 3 and finally we report some numerical examples in Sect. 4.

2 The CutFEM for Stokes Equations: Derivation Here we will give the elements of the numerical modelling that leads to the cut boundary value correction method for Stokes equations.

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2.1 The Domain We let % be the signed distance function to ∂Ω, negative on the inside of the domain and positive on the outside, and we let Uδ (∂Ω) be the tubular neighborhood {x ∈ Rd : |%(x)| < δ} of ∂Ω. Then there is a constant δ0 > 0 such that the closest point mapping p(x) : Uδ0 (∂Ω) → ∂Ω is well defined and we have the identity p(x) = x − %(x)n(p(x)). Here we recall that n(p(x)) denotes the outward pointing normal of ∂Ω at the point p(x). We assume that δ0 is chosen small enough so that p(x) is well defined.

2.2 The Mesh, Discrete Domains, and Finite Element Spaces – Let Ω0 ⊂ Rd be a convex polygonal domain such that Uδ0 (Ω) ⊂ Ω0 , where Uδ (Ω) := Uδ (∂Ω) ∪ Ω. Let K0,h , h ∈ (0, h0 ], be a family of quasiuniform partitions, with mesh parameter h, of Ω0 into shape regular triangles or tetrahedra K. We refer to K0,h as the background mesh. – Given a subset ω of Ω0 , let Kh (ω) be the submesh defined by Kh (ω) = {K ∈ K0,h : K ∩ ω = ∅}, i.e., the submesh consisting of elements that intersect ω, and let Nh (ω) = ∪K∈Kh (ω) K, be the union of all elements in Kh (ω). Below the L2 -norm of discrete functions frequently should be interpreted as the broken norm. For example for norms over Nh we have

v2Nh (ω) := v2K . K∈Kh (ω)

– Let Ωh , h ∈ (0, h0 ], be a family of polygonal domains approximating Ω, possibly independent of the computational mesh. We assume neither Ωh ⊂ Ω nor Ω ⊂ Ωh , instead the accuracy with which Ωh approximates Ω will be important. – Let the active mesh Kh be defined by Kh := Kh (Ω ∪ Ωh ), i.e., the submesh consisting of elements that intersect Ωh ∪ Ω, and let Nh := Nh (Ω ∪ Ωh ), be the union of all elements in Kh .

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k – Let V0,h be the space of piecewise continuous polynomials of degree k defined on K0,h and let the finite element space Vhk be defined by k Vhk := {vh : vh := v˜h |Nh for v˜h ∈ V0,h }.

– To each Ωh we associate the outward pointing normal νh : ∂Ωh → Rd , |νh | = 1, and the distance from ∂Ωh to ∂Ω in the direction νh , %h : ∂Ωh → R, such that if ph (x, ς ) := x + ς νh (x) then ph (x, %h (x)) ∈ ∂Ω for all x ∈ ∂Ωh . We will also assume that ph (x, ς ) ∈ Uδ0 (Ω) for all x ∈ ∂Ωh and all ς between 0 and %h (x). For conciseness we will drop the second argument of ph below whenever it takes the value %h (x). We assume that the following assumptions are satisfied δh := %h L∞ (∂Ωh ) = o(hζ ),

h ∈ (0, h0 ],

(5)

and νh − n ◦ pL∞ (∂Ωh ) = o(hζ −1 ),

h ∈ (0, h0 ],

(6)

where ζ ∈ {1, 2}. We also assume that h0 is small enough for some additional geometric conditions to be satisfied, for details see [7, Section 2.3].

2.3 Numerical Modelling We now proceed to show how to obtain a boundary value correction formulation for the Stokes system. Derivation Let f = Ef , u = Eu, and p = Ep, be stable extensions of f , u and p from Ω to Uδ0 (Ω). For v ∈ Vh we have using Green’s formula (f, v)Ωh = (f + Δu − ∇p, v)Ωh − (Δu − ∇p, v)Ωh = (f + Δu − ∇p, v)Ωh \Ω + (∇u, ∇v)Ωh − (p, ∇ · v)Ωh − (νh · ∇u − νh p, v)∂Ωh , where we used the fact f + Δu − ∇p = Ef − ΔEu − ∇Ep, which is not in general equal to zero outside Ω. Now the boundary condition u = g on ∂Ω may be enforced weakly as follows (f, v)Ωh = (f + Δu − ∇p, v)Ωh \Ω + (∇u, ∇v)Ωh − (p, ∇ · v)Ωh − (νh · ∇u − νh p, v)∂Ωh − (u ◦ ph − g ◦ ph , νh · ∇v)∂Ωh + βh−1 (u ◦ ph − g ◦ ph , v)∂Ωh .

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Since we do not have access to u ◦ ph we use a Taylor approximation in the direction νh u ◦ ph (x) ≈ Tk (u)(x) :=

k j

Dνh u(x) j %h (x), j! j =0

j

where Dνh is the j th partial derivative in the direction νh . Thus it follows that the solution to (1)–(3) satisfies (f, v)Ωh = (f + Δu − ∇p, v)Ωh \Ω + (∇u, ∇v)Ωh − (p, ∇ · v)Ωh − (νh · ∇u − νh p, v)∂Ωh − (Tk (u) − g ◦ ph , νh · ∇v)∂Ωh + βh−1 (Tk (u) − g ◦ ph , v)∂Ωh − (u ◦ ph − Tk (u), νh · ∇v)∂Ωh + βh−1 (u ◦ ph − Tk (u), v)∂Ωh , for all v ∈ Vh . Rearranging the terms we arrive at (∇u, ∇v)Ωh − (p, ∇ · v)Ωh − (νh · ∇u − pνh , v)∂Ωh − (Tk (u), νh · ∇v)∂Ωh + βh−1 (Tk (u), v)∂Ωh + (f + Δu − ∇p, v)Ωh \Ω − (u ◦ ph − Tk (u), νh · ∇v)∂Ωh + βh−1 (u ◦ ph − Tk (u), v)∂Ωh = (f, v)Ωh − (g ◦ ph , νh · ∇v)∂Ωh + βh−1 (g ◦ ph , v)∂Ωh ,

(7)

for all v ∈ Vhk . The discrete method is obtained from this formulation by dropping the consistency terms of highest order, i.e. those on lines three and four of Eq. (7). Bilinear Forms We define the forms a0 (v, w) := (∇v, ∇w)Ωh

(8)

− (νh · ∇v, w)∂Ωh − (Tk (v), νh · ∇w)∂Ωh + βh−1 (Tk (v), w)∂Ωh , j±k (v, w) := γj

k



h2l±1 ([Dnl F v], [Dnl F w])F ,

(9)

F ∈Fh l=1

ah (v, w) := a0 (v, w) + j−k (v, w),

(10)

bκ (q, w) := (q, ∇ · v)Ωh − κ(q, v · νh )∂Ωh ,

(11)

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s(y, q) := j+m (y, q) + γp

h3 ([nF · ∇y], [nF · ∇q])F ,

(12)

F ∈Fh

lh (w) := (f, w)Ωh − (g ◦ ph , νh · ∇w)∂Ωh + βh−1 (g ◦ ph , w)∂Ωh ,

(13)

where γj , γp , and β, are positive constants. Here we used the notation: – Fh is the set of all internal faces to elements K ∈ Kh , i.e. faces that are not included in the boundary of the active mesh Kh , that intersect the set Ω \ Ωh ∪ ∂Ωh , and nF is a fixed unit normal to F ∈ Fh . – Dnl F is the partial derivative of order l in the direction of the normal nF to the face F ∈ Fh . – [v]|F = vF+ −vF− , with vF± = lims→0+ v(x ∓snF ), is the jump of a discontinuous function v across a face F ∈ Fh . – The stabilizing terms j±k (v, w) are introduced to extend the coercivity of a0 (·, ·), to all of Nh and similarly for the pressure. Thanks to this property one may prove that the condition number is uniformly bounded independently of how Ωh intersects the mesh following the ideas of [3, 11]. – Observe the presence of the penalty coefficient β in (8) and (13). In order to guarantee coercivity β has to be chosen large enough and due to the Taylor expansions we also have to require that h ∈ (0, h0 ] with h0 sufficiently small. The Method Find: (uh , ph ) ∈ Wh := [Vhk ]d × Vhm such that ah (uh , v) − b1 (ph , v) + b0 (q, uh ) + s(ph , q) = lh (v),

∀(v, q) ∈ Wh ,

(14)

where ah is defined in (10), b0 and b1 in (11), s in (12) and finally lh in (13). For the analysis below it will be convenient to use the compact formulation: find: (uh , ph ) ∈ Wh such that Ah [(uh , ph ), (v, q)] + s(ph , q) = lh (v),

∀(v, q) ∈ Wh ,

where Ah [(uh , ph ), (v, q)] := ah (uh , v) − b1 (ph , v) + b0 (q, uh ). Remark 1 Note that different forms b· (·, ·) are used in the moment and mass equations and that −b1 (ph , uh ) + b0 (ph , uh ) = (ph , uh · νh )∂Ωh . It follows that the velocity pressure coupling term is skew-symmetric only up to a boundary term that is essential for consistency. The reason this term is omitted in the mass equation is that it is not consistent and must either be improved using a special boundary value correction, or omitted. For simplicity we here chose the latter option.

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3 Theoretical Results In this section we will report on some fundamental theoretical results that hold for the formulation (14). Due to space limitations the proof will be given elsewhere. For stability we will use the following norm defined for functions restricted to 1 1 the discrete space Wh or (v, q) in [H k+ 2 + (Ω0 )]d × H m+ 2 + (Ω0 ),  > 0, 1

|||(v, q)||| := vH 1 (Ωh ) + h− 2 v∂Ωh 1

1

+ h∇qΩh + qΩh + j−k (v, v) 2 + s(q, q) 2 .

3.1 Inf-Sup Stability Key to discrete well-posedness and to the error analysis is the following inf-sup stability result that is robust with respect to how the discrete domain intersects the mesh. The main difficulty in the proof of this result is to handle the lack of skew symmetry between the terms b1 and b0 . It follows however that the perturbation can be absorbed by the L2 -norm of the pressure and the boundary penalty term when β is sufficiently large and h sufficiently small. Proposition 2 Let either m = k and γp > 0 or k = 2 and m = 1 and γp = 0 and assume that (5)–(6) hold with ζ = 1. Then there exists α > 0, h0 > 0 such that for all (v, q) ∈ Wh , when h < h0 , there holds α|||(v, q)||| ≤

sup (w,y)∈Wh

Ah [(v, q), (w, y)] + s(q, y) . |||(w, y)|||

3.2 A Priori Error Estimate In this section we will present an optimal error estimate in the natural energy norm of the problem on the physical domain. The proof of the estimate follows the ideas of [7], this time combining inf-sup stability on the discrete error, Galerkin orthogonality, continuity of Ah , estimation of geometry errors and finally approximability. The error on the physical domain is then controlled by adding and subtracting an interpolant of the exact solution and using the triangle inequality. If ih denotes the Lagrange interpolant on [Vhk ]d , u − uh H 1 (Ω) ≤ u − ih EuH 1 (Ω) + ih Eu − uh H 1 (Nh )  u − ih EuH 1 (Ω) + |||(ih Eu − uh , 0)|||.

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Theorem 3 Let (u, p) ∈ [H s (Ω)]d × H s−1(Ω), with s ≥ 2 be the solution to (1)– (3). Assume that the hypothesis of Proposition 2 are satisfied and that in addition (4) holds with s ≥ 1 and (5)–(6) hold with ζ = 2. Let (uh , ph ) ∈ [Vhk ]d × Vhm be the solution of the finite dimensional problem (14). Then there holds u − uh H 1 (Ω) + p − ph Ω  hσ (|u|H σ +1 (Ω) + |p|H σ (Ω) ), where σ = min{k, s − 1}.

4 Numerical Example In our numerical example we consider a two dimensional problem discretized by the lowest order (inf–sup stable) Taylor–Hood element: piecewise quadratic, continuous, approximation of the velocity and piecewise linear, continuous, approximation of the pressure, together with a piecewise linear approximation of the domain. We shall study the convergence with and without boundary modification. We consider a problem from [4] with exact solution (with f = 0), u1 = 20xy 3,

u2 = 5x 4 − 5y 4 ,

p = 60x 2y − 20y 3.

Our computational domain is a disc with center at the origin. The exact velocities are used as Dirichlet data on the boundary of the domain. Note that since the exact solution is given everywhere, setting Dirichlet data on the approximate boundary is not a problem in this (special) case; to simulate the knowledge of data on the boundary only, we take the boundary data from the edge of the exact domain and use as boundary conditions on the approximate boundary, using the closest point projection. We choose the method parameters γj = 10−3 , γp = 0 and β = 100. In Fig. 1 we show the convergence obtained with and without boundary modification. We note that without boundary modification we lose optimal convergence in velocities but retain optimal convergence for pressure, which is expected since the approximation of the boundary is piecewise linear leading to an O(h2 ) geometric consistency error. With boundary modification we recover optimal order convergence also for the velocity.

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Pressure, L2, no mod., rate 2 Pressure, L2, mod., rate 2 Velocity L2, no mod., rate 2

-4

Velocity L2, mod., rate

3.5

-6

log of error

-8

-10

-12

-14

-16 -10

-5

0

5

log of meshsize

Fig. 1 Convergence results

Acknowledgements This research was supported in part by EPSRC grant EP/P01576X/1, the Swedish Foundation for Strategic Research Grant No. AM13-0029, the Swedish Research Council Grants Nos. 2013-4708, 2017-03911, and the Swedish Research Programme Essence.

References 1. R. Becker, E. Burman, P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Eng. 198(41–44), 3352–3360 (2009) 2. D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44 (Springer, Heidelberg, 2013) 3. E. Burman, Ghost penalty. C. R. Math. Acad. Sci. Paris 348(21–22), 1217–1220 (2010) 4. E. Burman, P. Hansbo, Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem. ESAIM Math. Model. Numer. Anal. 48(3), 859–874 (2014) 5. E. Burman, S. Claus, A. Massing, A stabilized cut finite element method for the three field Stokes problem. SIAM J. Sci. Comput. 37(4), A1705–A1726 (2015) 6. E. Burman, S. Claus, P. Hansbo, M.G. Larson, A. Massing, CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104(7), 472–501 (2015) 7. E. Burman, P. Hansbo, M.G. Larson, A cut finite element method with boundary value correction. Math. Comput. 87(310), 633–657 (2018) 8. J. Guzmán, M. Olshanskii, Inf-sup stability of geometrically unfitted Stokes finite elements. Math. Comput. 87, 2091–2112 (2018). http://dx.doi.org/10.1090/mcom/3288 9. P. Hansbo, M.G. Larson, S. Zahedi, A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85, 90–114 (2014).

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10. P. Lederer, C.-M. Pfeiler, C. Wintersteiger, C. Lehrenfeld, Higher order unfitted FEM for Stokes interface problems. Proc. Appl. Math. Mech. 16, 7–10 (2016) 11. A. Massing, M.G. Larson, A. Logg, M.E. Rognes, A stabilized Nitsche fictitious domain method for the Stokes problem. J. Sci. Comput. 61(3), 604–628 (2014)

Numerical Integration on Hyperrectangles in Isoparametric Unfitted Finite Elements Fabian Heimann and Christoph Lehrenfeld

Abstract We consider the recently introduced idea of isoparametric unfitted finite element methods and extend it from simplicial meshes to quadrilateral and hexahedral meshes. The concept of the isoparametric unfitted finite element method is the construction of a mapping from a reference configuration to a higher order accurate configuration where the reference configuration is much more accessible for higher order quadrature. The mapping is based on a level set description of the geometry and the reference configuration is a lowest order level set approximation. On simplices this results in a piecewise planar and continuous approximation of the interface. With a simple geometry decomposition quadrature rules can easily be applied based on a tesselation. On hyperrectangles the reference configuration corresponds to the zero level of a multilinear level set function which is not piecewise planar. In this work we explain how to achieve higher order accurate quadrature with only positive quadrature weights also in this case.

1 Introduction In the endeavor to develop numerical schemes for PDEs which are very flexible with respect to changes in the geometry, methods that allow to embed complex and changing PDE domains into a static background mesh have received a lot of attention in recent years. Here, we mention the extended finite element method [7], unfitted discontinuous Galerkin method [1], TraceFEM [15], CutFEM [2] and the finite cell method [16]. These unfitted finite element methods most often allow for higher order finite element formulations. However, to achieve higher order accuracy also with respect to the geometrical accuracy in the numerical integration is challenging due to the flexible—and often implicit—geometry description.

F. Heimann · C. Lehrenfeld () Institut für Numerische und Angewandte Mathematik, University of Göttingen, Göttingen, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_16

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Different approaches based on moment fitting [14], matching curved subtriangulations [4, 5, 8] or boundary value corrections [3] have been considered in the literature with different advantages and disadvantages, see also the literature overview in [12]. In the following we consider a different approach which is based on a piecewise multilinear approximation of the geometry combined with a parametric mapping of the underlying mesh. For simplices this method has been proposed in [12] and applied and analysed for different PDE problems in [9, 10, 13]. In this contribution we extend the technique to hyperrectangles.

2 Level Set Descriptions and Parametric Mappings In this section we introduce basic notations for domains, explain the basis configuration in approximating the geometry and explain how high order geometry approximations are obtained from this.

2.1 A Piecewise Multilinear Approximation of the Geometry Let Ω be a polygonal background domain with a quasi-uniform mesh Th that consists of hyperrectangles only (quadrilaterals in 2D, hexahedra in 3D). On this mesh a level set function φ is described, e.g. by a higher order finite element approximation. We denote the geometries that are implicitly defined through φ as Ω− := {φ < 0}, Ω+ := {φ > 0} and Γ := {φ = 0}. As a basis for the geometry approximation we use a piecewise multilinear approximation φhmlin ∈ Vh1 := {v ∈ C 0 (Ω) | v|T ∈ Q1 (T ), ∀T ∈ Th } where Q1 (T ) is the space of (mapped) multilinear functions on the hyperrectangle T . φhmlin defines an approximation Γ mlin := {x ∈ Ω | φhmlin(x) = 0} of Γ . In contrast to the cases treated in [10–13] where the mesh consists only of simplices, Γ mlin is not mlin and Ω mlin are defined correspondingly. polygonal, cf. Fig. 1. Ω+ − The approximation of the domains with Γ mlin is only second order accurate, i.e. dist(Γ mlin , Γ ) ∼ h2 . However, the handling of integrals on domains described by Γ mlin is significantly easier than directly dealing with the geometry of Γ . To obtain higher order accuracy we combine this low order representation with a parametric mapping Θh that we introduce next.

Integration in Isoparametric Unfitted FEM

triangles (uncurved) triangles (curved) Θh (Γ lin ) Γ lin

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quads (uncurved) Γ mlin

quads (curved) Θh (Γ mlin )

Fig. 1 Different explicit approximations of a level set geometry. The region in green (blue) is the approximation of Ω− (Ω+ ). Note that Γ lin (zero level of a piecewise linear approximation of φ) is piecewise straight while Γ mlin is not

2.2 Improved Geometrical Accuracy with a Parametric Mapping As in isoparametric finite element methods for the approximation of curved boundaries, the idea presented and analyzed in [9, 12, 13] is to construct a mapping Θh : Ω → Ω which improves the only second order accurate approximation of Γ mlin towards Γ so that Θh (Γ mlin ) is a higher order approximation of Γ , cf. Fig. 1. The mapping is constructed such that (locally on every cut element) the following relation is solved for: φ(y) = φhmlin (x) for y := Θ(x) and x close to Γ mlin .

(1)

Note that (1) implies that for x ∈ Γ mlin we have φhmlin (x) = 0 = φ(Θ(x)) which means that Θ(x) is on Γ . We sacrifice the exactness of (1) by projecting Θ into a finite element space, resulting in Θh , a continuous, piecewise polynomial mapping, with dist(Θh (Γ mlin ), Γ )  hk+1 where k is the polynomial degree in the approximation. Away from cut elements the mapping makes an element-wise smooth transition to the identity. The construction of the mapping Θh consists of element-local operations and can be realized efficiently. For details we refer to [12, 13], where only the case of simplex meshes is treated. The method however translates to hyperrectangles.

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2.3 Reduction to Integration Problems on the Reference Element We make use of the improved geometry approximation Γh = Θh (Γ mlin ) in an unfitted finite element method. This means that we map finite element spaces and integrals according to the mapping. In this study we assume an integrand f which allows for a smooth extension into a neighborhood which we also denote as f . mlin ) and We only discuss the treatment of integrals and have with Ωh± = Θh (Ω± mlin Γh = Θh (Γ ) the approximations  f dx ≈



Ω±

 f dx =

(f ◦ Θh )J d x¯ =

Ωh±



 f ds ≈ Γ

 f ds = Γh

mlin Ω±

(f ◦ Θh )JΓ d s¯ =

Γ mlin



(f ◦ Θh )J d x, ¯

(2a)

(f ◦ Θh )JΓ d s¯,

(2b)

mlin T ∈Th T ∩Ω±



mlin T ∈Th T ∩Γ

where J and JΓ are the ratios of measures between the domains of integration, with J = det(DΘh ) and JΓ = J · DΘh−T · n where n is the normal to Γ mlin . As usual in finite elements, the integrals in (2) on (parts of) one element T are transferred to integrals on a reference element Tˆ . Let ΦT : Tˆ → T be the multilinear mapping from the reference hyperrectangle Tˆ (unit square or cube) to T ∈ Th and the composition of ΦT and Θh be Ψ : Tˆ → Θh (T ), Ψ = Θh ◦ ΦT . Then, with the mlin ) and Tˆ := Tˆ ∩ Φ −1 (Γ mlin ), we have reference cut domains Tˆ± := Tˆ ∩ ΦT−1 (Ω± Γ T 



mlin T ∩Ω±



T ∩Γ mlin

(f ◦ Θh )J d x¯ =

Tˆ±



(f ◦ Θh )JΓ d s¯ =

TˆΓ

(f ◦ Ψ )Jˆ d x, ˆ

Jˆ = det(DΨ ),

ˆ (f ◦ Ψ )JˆΓ d sˆ, JˆΓ = Jˆ · DΨ −T · n,

(3a) (3b)

where nˆ is the normal to TˆΓ . We notice that the functions Ψ and hence Jˆ and JˆΓ are explicitly and efficiently evaluable on every point in Tˆ . Further, the domains Tˆ± and TˆΓ are prescribed by a multilinear level set function φˆ hmlin . Summing up, we have reduced the integration problem to finding sets of quadrature points and positive weights Q± = {(xl , ωl )}, respectively QΓ = {(xl , ωl )} so that for a given order k we have with g := Jˆ(Γ ) · (f ◦ Ψ )  Tˆ±

g d xˆ =

 ω · g(x) + O(hk ) and

(x,ω)∈Q±

TˆΓ

g d xˆ =

ω · g(x) + O(hk ),

(4)

(x,ω)∈QΓ

where Tˆ± := {φˆ hmlin ≶ 0} and TˆΓ := {φˆ hmlin = 0}. In the following, we assume that the integrand g is (sufficiently) smooth in the domain of integration.

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Remark 1 Only in the resolved case, where the curvature of TˆΓ on the reference element is small, the parametric mapping Θh (and hence Ψ ) leads to higher order geometrical accuracy and shape-regular elements at the same time. In the underresolved case however, a limitation of the mesh transformation is applied which results in an only second order geometry approximation, cf. [12, Section 2.6]. In this case any robust quadrature methods which is second order accurate w.r.t. the geometry is applicable, e.g. the topology preserving marching cube proposed in [6]. Hence, in the sequel we will only treat the resolved case which justifies Assumptions A and B. The construction of quadrature rules is based on a simple hierarchy. We start with quadrature rules on the unit interval, cf. Sect. 3. Based on these, we construct rules on the unit square, cf. Sect. 4, which are again exploited in the construction of rules on the unit cube, cf. Sect. 5. We notice that the overall approach is similar in virtue to the one presented in [17], but applies a different strategy with respect to surface integrals.

3 Integration on the Unit Interval We consider Tˆ = (0, 1) and φˆ hmlin(x) = a(1 − x) + bx. W.l.o.g. we assume sgn a = −1 (the other cases follow similar lines). If there is a cut in Tˆ , i.e. sgn b = 1, the zero of φˆ hmlin is x ∗ = a/(a − b). Hence, TˆΓ = {x ∗ } and Tˆ− = (0, x ∗ ), Tˆ+ = (x ∗ , 1). If there is no cut, TˆΓ = ∅ and Tˆ− = (0, 1), Tˆ+ = ∅. Let G k (yl , yr ) = {(x, ˜ w)} ˜ denote the mapped Gauss rule with exactness degree k on the interval (yl , yr ). Applying these on the cut intervals we obtain (Q− , Q+ , QΓ ) = (G q (0, x ∗ ), G q (x ∗ , 1), {(x ∗ , 1)}) in the case of a cut and otherwise (Q+ , Q− , QΓ ) = (G q (0, 1), ∅, ∅). The property (4) then follows from the corresponding accuracy of the Gauss rule on the unit interval.

4 Integration on the Unit Square On the unit square Tˆ = (0, 1)2 we exploit the idea of iterated integrations, i.e. we rewrite an integral over Tˆ as an integral over integrals of the form treated in Sect. 3. Bulk Integrals We first discuss the case of bulk integrals and w.l.o.g. assume that we are interested in an integral of g over Tˆ− ⊂ (0, 1)2 . With I− (y) := {x ∈ (0, 1) |

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y

y

TˆΓ h(y)

Tˆ−

y0∗

(1−h(y))dy

y1∗

0

h(y) dy

Tˆ+

y

φˆmlin h= 1

y

φˆmlin h= 1

0

y2∗

φˆmlin h =−1

y

φˆmlin h= 1

x

∂y2

1 dx dy y

0

I± (y )

y 0

1 dx dy y

I± (y)

Fig. 2 Example of a cut configuration of a bilinear level set function (left), the corresponding outer integral to g = 1 (middle) and its 2nd partial derivative (right)

φˆ hmlin (x, y) < 0}, the integration on Tˆ− can be reformulated as 

 Tˆ−

1

g(x) ˆ d xˆ = 0





1

g(x, y) dx dy =

G(y) dy,

G(y) :=

0

I− (y)

g dx. I− (y)

Before we apply a quadrature rule for the y-integral we notice that G(y) can have discontinuities in the first derivative whenever the edges (0, y), (1, y) are cut by the interface, cf. Fig. 2 for an illustration for g = 1. Therefore, we first subdivide the ∗ , y ∗ ) ⊂ (0, 1) interval (0, 1) into N ∈ {1, 2, 3} disjoint sub-intervals Ii = (yi−1 i ∗ ∗ ∗ with y0 = 0, yN = 1 and yi the (sorted) cut positions on the vertical edges for i = 1, . . . , N − 1. Afterwards we apply iterated quadrature which yields  Tˆ−

g(x) ˆ d xˆ =

N 

i=1

yi∗ ∗ yi−1

G(y) dy ≈

N



i=1

(y,ωy )∈G k(Ii )

ωy · G(y).

To compute G(y) we apply the numerical integration from Sect. 3 as φˆ hmlin (·, y) is a linear function. Surface Integrals Next, we consider the case of surface integrals, i.e. we are interested in an integral of g over TˆΓ ⊂ (0, 1)2 . To apply an adapted version of the approach of iterated integrals we have to take into account the different measure of the integral. Let us assume that there is a graph representation {(h(y), y) | ∗ , y ∗ )} for Tˆ for exactly one l ∈ {1, . . . , N} (if Tˆ extends over y ∈ (yl−1 Γ Γ l two slices, we sum their contributions). Then h(y) is defined implicitly through φˆ hmlin (h(y), y) = 0. From the implicit function theorem we obtain h (y) = −∂y φˆ hmlin /∂x φˆ hmlin and

2

1 + h (y)2 = ∇ φˆhmlin /|∂x φˆ hmlin | =: qx .

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199

Fig. 3 Three bilinear level sets and the quadrature rules (Q± , QΓ and the maximum of qx /qy ) obtained from the strategy in Sect. 4

This gives rise to the following quadrature rule: 

 TˆΓ

g(x) ˆ d sˆ =

yl∗

G(y) dy ≈

∗ yl−1

ωy · G(y), G(y) = g(h(y), y) · qx (h(y), y).

(y,ωy )∈G k(Il )

Here h(y) does not need to be represented explicitly but is evaluated in the dimension reduction procedure. The applicability of this approach obviously depends on the factor qx := ∇ φˆ hmlin /|∂x φˆ hmlin | which is in general not bounded. However, if the geometry is well-resolved, cf. Remark 1, there is an upper bound on the curvature of the level sets φˆ hmlin which justifies the following assumption. Assumption A There is a direction ξ ∈ {x, y} and a constant c < 1, s.t. max pξ ≤ c with pξ := |∂ξ φˆ hmlin |/∇ φˆ hmlin. x∈ ˆ Tˆ

(5)

2 1 ˆ ≤ C := 1−c ˆ ∈ Tˆ . This implies for η = ξ that there holds qη (x) 2 < ∞ ∀ x We can interpret pξ as cos(θξ ) where θξ is the angle between the tangent of a level set and the unit axis ξ . Assumption A excludes level sets that are (arbitrary close to) parallel to both unit axes. Hence, we can either execute the previously discussed strategy directly and have the bound qx ≤ C (if ξ = y) or we swap the integration directions (x ↔ y) and can make use of qy ≤ C. In Fig. 3 examples of −1 quadrature points and maxx∈ ˆ Tˆ qξ , ξ ∈ {x, y} are shown. Note that qx (and qy ) has at most five extremal points in Tˆ s.t. a numerical determination of a suitable integration direction is simple.

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5 Integration on the Unit Cube For the bulk integrals on the unit cube we can apply the idea of iterated integrals in the same ways as in the 2D case with possibly more (at most 5) sub intervals due to the larger number of vertical edges. Surface Integrals Let z be the outer integration direction for the approach of iterated integrals. With similar arguments as in the 2D case one can show that there holds 

 TˆΓ

where qxy :=

1

g(x) ˆ d sˆ =

g(x, y, z) qxy ds dz, 0

IΓ (z)

 − 1 2 1 − pz2 , pz := |∂z φˆ hmlin|/φˆhmlin  and IΓ (z) := {(x, y) ∈

(0, 1)2 | φˆ hmlin (x, y, z) = 0}. To bound qxy we again make use of the assumption of sufficient resolution which justifies the following assumption. Assumption B There is a direction ξ ∈ {x, y, z} and a constant c < 1, s.t. max pξ = max |∂ξ φˆ hmlin|/∇ φˆhmlin  ≤ c < 1. x∈ ˆ Tˆ

x∈ ˆ Tˆ

(6)

This implies that for η, ζ ∈ {x, y, z}, η, ζ = ξ , η = ζ , there holds qηζ = qζ η ≤ 1 C := (1 − c2 )− 2 < ∞ ∀xˆ ∈ Tˆ . In the resolved case, we can choose at least one direction so that qηζ < C for a constant C with a reasonable bound. To check a sufficient (and close to necessary) condition if (6) is fulfilled for a chosen direction ξ and a chosen constant c requires only the evaluation of the vertex values of φˆ hmlin . Let us assume w.l.o.g. that (6) is fulfilled with ξ = z. Then, we make the following approximation  TˆΓ

g(x) ˆ d sˆ =

l

where G(y) approximates described in Sect. 4 (Note





yl∗

G(y) dy ≈

∗ yl−1

ωy · G(y),

(y,ωy )∈G k(Il )

IΓ (z) g(x, y, z)qxy ds with the numerical mlin (·, ·, z) is again a bilinear function). that φˆ m

integration

6 Numerical Example As an application of the numerical integration we consider the interface problem as in [13] with the same isoparametric unfitted Nitsche formulation considered in that paper. The only difference is that we consider a uniform 8 × 8 quadrilateral initial background mesh. In Fig. 4 the PDE and the convergence rates in the L2 norm under

Integration in Isoparametric Unfitted FEM 1.5

Interface problem: ⎧ −div(α± ∇u) = f± in Ω± , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [[α∇u · n]]Γ = 0 on Γ, [[u]]Γ = 0

⎪ ⎪ ⎪ ⎪

201

φ= x

0

on Γ,

Ω−

4

−1

L u − uh

Γ

u = uD on ∂Ω, with (α− ,α+ ) = (1, 2).

−1.5 −1.5

Ω+ 0

0 1 2 3 4

L2 (Ω)

eoc

1.177 × 10−3 2.709 × 10−4 2.003 × 10−5 1.359 × 10−6 6.982 × 10−8

— 1.84 3.76 3.88 4.26

1.5

Fig. 4 PDE problem to be discretized with the isoparametric unfitted Nitsche method with order k = 3 (left), geometrical configuration (middle) and convergence table (right) for the numerical example

successive uniform refinements are shown. For details on the parameters in the PDE and the discretization we refer to [13]. We observe the same optimal order that has been seen in [13] on a triangular mesh. Acknowledgements The authors gratefully acknowledge funding by the German Science Foundation (DFG) within the project “LE 3726/1-1” and suggestions on a former version of this paper by Hans-Georg Raumer and an anonymous reviewer.

References 1. P. Bastian, C. Engwer, An unfitted finite element method using discontinuous Galerkin. Int. J. Numer. Meth. Eng. 79, 1557–1576 (2009) 2. E. Burman, S. Claus, P. Hansbo, M.G. Larson, A. Massing, CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Meth. Eng. 104, 472–501 (2015) 3. E. Burman, P. Hansbo, M. Larson, A cut finite element method with boundary value correction. Math. Comput. 87, 633–657 (2018) 4. K.W. Cheng, T.-P. Fries, Higher-order XFEM for curved strong and weak discontinuities. Int. J. Numer. Meth. Eng. 82, 564–590 (2010) 5. K. Dréau, N. Chevaugeon, N. Moës, Studied X-FEM enrichment to handle material interfaces with higher order finite element. Comput. Meth. Appl. Mech. Eng. 199, 1922–1936 (2010) 6. C. Engwer, A. Nüßing, Geometric integration over irregular domains with topologic guarantees (2016). arXiv:1601.03597 7. T.-P. Fries, T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Meth. Eng. 84, 253–304 (2010) 8. T.-P. Fries, S. Omerovi, Higher-order accurate integration of implicit geometries. Int. J. Numer. Meth. Eng. 106, 323–371 (2016) 9. J. Grande, C. Lehrenfeld, A. Reusken, Analysis of a high order trace finite element method for PDEs on level set surfaces (2016). arXiv:1611.01100 10. P. Lederer, C.-M. Pfeiler, C. Wintersteiger, C. Lehrenfeld, Higher order unfitted FEM for Stokes interface problems. Proc. Appl. Math. Mech. 16, 7–10 (2016) 11. C. Lehrenfeld, A higher order isoparametric fictitious domain method for level set domains (2016). arXiv:1612.02561 12. C. Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings. Comput. Methods Appl. Mech. Eng. 300, 716–733 (2016) 13. C. Lehrenfeld, A. Reusken, Analysis of a high-order unfitted finite element method for elliptic interface problems. IMA J. Numer. Anal. (2017).

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14. B. Müller, F. Kummer, M. Oberlack, Highly accurate surface and volume integration on implicit domains by means of moment-fitting. Int. J. Numer. Meth. Eng. 96, 512–528 (2013) 15. M.A. Olshanskii, A. Reusken, J. Grande, A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47, 3339–3358 (2009) 16. J. Parvizian, A. Düster, E. Rank, Finite cell method. Comput. Mech. 41, 121–133 (2007) 17. R. Saye, High-order quadrature method for implicitly defined surfaces and volumes in hyperrectangles. SIAM J. Sci. Comput. 37, A993–A1019 (2015)

Part V

Advances in Numerical Linear Algebra Methods and Applications to Partial Differential Equations

On a Generalization of Neumann Series of Bessel Functions Using Hessenberg Matrices and Matrix Exponentials A. Koskela and E. Jarlebring

Abstract The Neumann expansion of Bessel functions (of integer order) of a function g : C → C corresponds to representing g as a linear combination of ∞ basis functions ϕ0 , ϕ1 , . . ., i.e., g(s) = w ϕ   (s), where ϕi (s) = Ji (s), =0 i = 0, . . ., are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

1 Introduction Let g : C → C be a function analytic in a neighborhood of the origin containing s ∈ C. We consider expansions of the form g(s) =

∞

(1)

w ϕ (s),

=0

where the coefficients w0 , w1 , . . . ∈ C and the basis functions ϕ0 , ϕ1 , . . . satisfy ⎡

⎡ ⎤ ⎤ ϕ0 (s) ϕ0 (s) ⎢ ⎥ ⎥ d ⎢ ⎢ϕ1 (s)⎥ = H∞ ⎢ϕ1 (s)⎥ , ⎣ ⎣ ⎦ ds .. .. ⎦ . .

⎡ ⎡ ⎤ ⎤ ϕ0 (0) 1 ⎢ ⎢ ⎥ ⎥ ⎢ϕ1 (0)⎥ = e1 := ⎢0⎥ . ⎣ . ⎦ ⎣.⎦ .. ..

(2)

A. Koskela · E. Jarlebring () KTH Royal Institute of Technology, Stockholm, Sweden e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_17

205

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The real-valued matrix H∞ is an infinite upper Hessenberg matrix with nonzero subdiagonal entries. Due to the Picard-Lindelöf theorem, the differential equation (2) defines a unique sequence of functions ϕ0 , ϕ1 , . . . under the condition that H∞ 2 ≤ C < ∞.

(3)

This implies that also Hn 2 ≤ C, where Hn ∈ Rn×n is the leading submatrix of H∞ . For certain choices of H∞ (2) defines well-known basis functions. We will show in Sect. 2 that this class of basis functions includes monomials, Bessel functions and modified Bessel functions (of the first kind) and the expansion corresponds to wellknown expansions (e.g. Taylor, Bessel function Neumann expansion). Although our work is of general character, it originated from the need of such expansions in iterative methods. More precisely, the expansion occurs in the infinite Arnoldi method both for linear ODEs [7, 8] and similarly for nonlinear eigenvalue problems [6], where the fact that H∞ is a Hessenberg matrix leads to the property that the m does not have to be determined a priori. We wish to point out that our proposed generalization of Bessel functions, is not the only generalization, cf., e.g., [4], but to our knowledge this work is the only approach using Hessenberg matrices and the infinite dimensional ODE (2). In this work we study this expansion, both from a computational and theoretical perspective. – We derive procedures to compute the coefficients w0 , w1 , . . . , wn−1 , based on the derivatives of g and Hn . – We derive a procedure to compute the basis functions based on a truncation of H∞ using the matrix exponential [9], and we provide a convergence analysis of the corresponding problem. – We provide an error analysis for the truncation of the series, i.e., for the error of n−1  the approximation g(s) ≈ w ϕ (s). =0

The results are illustrated by comparing different choices of the basis functions, i.e., different choices of the matrix H∞ , for a given a function g(s).

2 Well-Known Basis Functions We first illustrate that the class of expansions (1) includes several well-known expansions. The scaled monomials form the simplest example of such a sequence of basis functions. If we define ϕ (s) := s  /!,  = 0, . . ., then (2) is satisfied with

Generalization of Neumann Series Bessel Expansion

207

H∞ given by a transposed Jordan matrix ⎡

H∞

0 ⎢ ⎢1 0 =⎢ ⎢ 1 0 ⎣ .. .. . .

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

(4)

In this case, the expansion (1) corresponds to a Taylor expansion and the coefficients are given by w = g () (0),  = 0, . . .. It turns out that also Bessel functions and modified Bessel functions of the first kind satisfy (2). It can be seen as follows.  π The Bessel functions of the first kind are defined by (see e.g. [1]) J (s) := π1 0 cos(τ − s sin(τ )) dτ , for  ∈ N, and they satisfy J (s) =

1 (J−1 (s) − J+1 (s)). 2

J− (s) = (−1) J (s),  > 0 ⎧ ⎨1 if  = 0 J (0) = . ⎩0 otherwise

(5)

5 6T ∈ Cn , i.e., a vector of Bessel functions Let J¯n (s) = J0 (s) J1 (s) . . . Jn−1 (s) with non-negative index. Moreover, let ⎡

0 −1 ⎢1 ⎢ 2 0 − 12 ⎢ . . . ⎢ Hn = ⎢ . . . . . . ⎢ .. ⎢ . 0 − 12 ⎣ 1 0 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ∈ Rn×n . ⎥ ⎥ ⎦

(6)

From the relations (5), we easily verify that the Bessel functions of the first kind are solutions to the infinite-dimensional ODE of the form (2), with H∞ defined by (6). More precisely,  J¯∞ (s) = H∞ J¯∞ (s),

J¯∞ (0) = e1 .

With similar reasoning we can establish an ODE (2) also for the modified Bessel functions of the first kind, which are defined by I (s) := (−i) J (is).

(7)

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The definition (7) and the properties (5) lead to the infinite-dimensional ODE  I¯∞ (s) = H∞ I¯∞ (s),

I (0) = e1 ,

6T 5 where I¯n (s) = I0 (s) I1 (s) . . . In−1 (s) and ⎡

0 1 ⎢1 ⎢ 2 0 12 ⎢ . . . ⎢ Hn = ⎢ . . . . . . ⎢ .. ⎢ . 0 ⎣ 1 2

⎤

1 2

⎥ ⎥ ⎥ ⎥ ⎥ ∈ Rn×n . ⎥ ⎥ ⎦

(8)

0

Therefore the Bessel functions and the modified Bessel functions of the first kind satisfy (2). The Euclidean norm bound of the operator (C) can be derived for these choices as proven with recurrence relations of Chebyshev polynomials in [7, Lemma 2]. Lemma 1 (Basis Functions. Lemma 2 in [7]) The conditions for the basis functions in (2) are satisfied with C = 2 for, (a) scaled monomials, i.e., ϕi (s) = s!/i!, with H∞ defined by (4); (b) Bessel functions, i.e., ϕi (s) = Ji (s), with H∞ defined by (6); and (c) modified Bessel functions, i.e., ϕi (s) = Ii (s), with H∞ defined by (8).

3 Computation of Coefficients wi Assume that an expansion of the form (1) exists and let 6 5 Wn = w0 w1 . . . wn−1 . By considering the th derivative of g(s) and using the properties of basis functions (2) we have that  g () (0) = W∞ H∞ e1 = Wn Hn e1

for all  < n.

(9)

In the last equality we used the fact that H∞ is a Hessenberg matrix, and that all  e except the first +1 elements will be zero. The non-zero elements elements of H∞ 1 will also be equal to Hn e1 . We now define the upper-triangular Krylov matrix 6 5 Kn (Hn , e1 ) = e1 Hn e1 . . . Hnn−1 e1 ,

Generalization of Neumann Series Bessel Expansion

209

and the matrix Gn as 6 5 Gn = g(0) g  (0) . . . g (n−1) (0) . From these definitions and the property (9) it follows that Wn = Gn Kn (Hn , e1 )−1

for all n ≥ 1.

(10)

Notice that Kn (Hn , e1 ) is invertible if and only if the subdiagonal elements of Hn are non-zero. In a generic situation, the relation (10) can be directly used to compute the coefficients w ,  ∈ N, given the derivatives of g. The above approach reduces to standard procedures of evaluating coefficients in Bessel-Neumann series (for integer order) when ϕi = Ji or ϕi = Ii . The formalization is omitted due to space limitation, but can be found in the technical report [7, Lemma 3]. The relation with Chebyshev polynomials (in particular [1, pp. 775] and [10]) is pointed out in [7, Remark 4]. See also other works on the Bessel-Neumann series coefficients in [5], not necessarily of integer order.

4 Computation of Basis Functions In order to use the expansion in practice, we need the possibility to compute the basis functions. We propose and study a natural approach of approximating the basis functions with a vector of functions generated by the truncated Hessenberg matrix Hn , i.e., ⎤ ϕ0 (s) ⎢ . ⎥ ⎥ ϕ¯n (s) := ⎢ ⎣ .. ⎦ ≈ exp(sHn )e1 . ϕn−1 (s) ⎡

(11)

The term exp(sHn ) can be efficiently and accurately computed using well-known methods for the matrix exponential [9]. Let ε¯ n denote the error in the basis functions, i.e., the difference between the basis functions and the functions generated by the truncated Hessenberg matrix, ε¯ n (s) := ϕ¯ n (s) − exp(sHn )e1 , = [In , 0] exp(sH∞ )e1 − exp(sHn )e1 ,

(12)

where [In , 0] gives the first n rows of an (infinite) identity matrix. Since H∞ and Hn are Hessenberg matrices, for i = 0, . . . , n − 1 we have i [In , 0]H∞ e1 = Hni e1

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such that n first terms in the Taylor expansions of the matrix exponentials in (12) cancel. We can therefore explicitly bound the basis function error as ¯εn (s) = [In , 0]Rn (sH∞ )e1 − Rn (sHn )e1  ≤ Rn (sH∞ ) + Rn (s(Hn )) ≤ 2Rn (sH∞ ) ≤

2(sC)n sC e , n!

 z where Rn (z) = ∞ =n ! , i.e., the remainder of the truncated Taylor expansion. This shows that the basis function approximations converge superlinearly as n → ∞, for a fixed value of t. Sharper bounds can be derived for special cases. Sharper results for Bessel functions can be found in [7, Section 4.1.1–4.1.2].

5 Truncation of the Expansion We now provide sufficient conditions for the convergence of the expansion by giving bounds for the truncation error en := g(s) − Wn ϕ¯n (s).

5.1 Truncation Bound with Matrix Exponential Remainder Term The truncation error can now be expressed as infinite matrices as follows: en =

∞

p=n

 wp ϕp (s) = W∞

 0n 0 exp(sH∞ )e1 . 0 I

(13)

i e = 0, when j > i +1. The fact that H∞ is a Hessenberg matrix implies that ejT H∞ 1 Therefore     ∞

0n 0 (sH∞ )i 0n 0 en = e1 = W∞ W∞ Rn (sH∞ )e1 0 I 0 I i! i=0

Generalization of Neumann Series Bessel Expansion

211

A simple sufficient condition for the convergence of the series is obtained by assuming that the sequence |w0 |, |w1 |, . . . is bounded, since then |en | ≤W∞ 2 Rn (sH∞ )e1 2 ≤ W∞ 1 Rn (|s|H∞ 2 ) ≤W∞ 1 Rn (|s|C) → 0,

as n → ∞.

The above reasoning can be generalized by scaling with a bounding sequence and leads to the following theorem. The proof follows analogous to the derivation of (13). Theorem 2 Suppose the w0 , w1 , . . . is bounded in modulus by d0 , d1 , . . . ∈ R+ , |wi | ≤ di , i = 0, 1, . . . . Let D = diag(d0 , d1 , . . .). Then, |en | ≤DRn (sH∞ )e1 2 = d0 Rn (sDH∞ D −1 )e1 2 . As a consequence of the theorem, if we have DH∞ D −1 2 ≤ ∞, we may use the bound and obtain en ≤|d0 | max i

≤|d0 | max i

|wi | Rn (|s|DH∞ D −1 2 ) |di | |wi | |s|DH∞ D −1 2 |s|n DH∞ D −1 n2 e . |di | n!

5.2 Bounds Using Decay of Elements in the Matrix Exponential Equation (13) can be reformulated as |en | ≤

∞

j =n+1

|wj ||ejT exp(sH∞ )e1 |

(14)

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or with scaling as in Theorem 2, |en | ≤

∞

d0 |ejT exp(sDH∞ D −1 )e1 |.

(15)

j =n+1

Note that DH∞ D −1 is banded if H∞ is banded. The elements of matrix functions of banded matrices are bounded in a ridge shaped way. There are bounds on the absolute value of elements which show that elements far from the diagonal have to be small. These can be used here to analyze the bounds (14) and (15) as they consist of elements (j, 1) of the matrix exponential. As an example we mention a result by Iserles [3, Theorem 2.2] which is applicable for a tridiagonal H∞ and that of Benzi and Razouk [2, Theorem 3.5] which is also used in [7, Lemma 9, Theorem 11].

6 Illustrating Example The following example illustrates how a generalized Bessel-Neumann expansion of the form (1) can lead to a better approximation of the function when the matrix H∞ is chosen appropriately. We consider the function     g(s) = eαs sin s/3 + cos(s) (16) with α = 1/2. We use the Hessenberg matrices corresponding to scaled monomials (4), Bessel functions (6), modified Bessel functions (8), and an artificially constructed Hessenberg matrix given by H + αI where H is given by (6). The infinite matrix H + αI is bounded in Euclidean norm since H + αI  ≤ H  + |α|. The coefficients and basis functions are computed as described in Sects. 3 and 4. In Figs. 1 and 2 we clearly see that the generalized Bessel-Neumann expansion converges faster for this (specific) example than the other alternatives. Fig. 1 Relative truncation error as a function of truncation parameter, when s = 1 for the function (16)

10 4 10 1 10 -2 10 -5 10 -8 10 -11 10 -14 10 -17 0

5

10

15

20

Generalization of Neumann Series Bessel Expansion Fig. 2 Relative truncation error as a function of truncation parameter, when s = 10 for the function (16)

213

10 4 10 1 10 -2 10 -5 10 -8 10 -11 10 -14 10 -17 10 -20

0

10

20

30

40

50

7 Concluding Remarks We have here proposed the use of generalized Bessel-Neumann expansions, and shown both computational and theoretical properties for this approach. The main purpose of this work is to show that it is possible use these expansions and that the convergence speed depends on the choice of the matrix H∞ which determines both the basis functions and the coefficients of the expansion. We have illustrated numerically that different choices of H∞ lead to different performance, and it is not at all obvious which basis functions are the best, and in which sense they are good for a particular function.

References 1. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Applied Mathematics Series, vol. 55 (National Bureau of Standards, Washington, 1964) 2. M. Benzi, N. Razouk, Decay bounds and O(n) algorithms for approximating functions of sparse matrices. Electron. Trans. Numer. Anal. 28, 16–39 (2007) 3. A. Iserles, How large is the exponential of a banded matrix? N. Z. J. Math. 29(2), 177192 (2000) 4. F.H. Jackson, A generalization of Neumann’s expansion of an arbitrary function in a series of Bessel’s functions. Proc. Lond. Math. Soc. s2-1(1), 361–366 (1904) 5. D. Jankov, T.K. Pogány, E. Süli, On the coefficients of Neumann series of Bessel functions. J. Math. Anal. Appl. 380(2), 628–631 (2011) 6. E. Jarlebring, K. Meerbergen, W. Michiels, A Krylov method for the delay eigenvalue problem. SIAM J. Sci. Comput. 32(6), 3278–3300 (2010) 7. A. Koskela, E. Jarlebring, The infinite Arnoldi exponential integrator for linear inhomogeneous ODEs. Technical Report. KTH Royal Institute of Technology (2015). http://arxiv.org/abs/1502. 01613

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8. A. Koskela, E. Jarlebring, M.E. Hochstenbach, Krylov approximation of linear ODEs with polynomial parameterization.. SIAM J. Matrix Anal. Appl. 37(2), 519–538 (2016) 9. C. Moler, C.V. Loan, Nineteen dubious ways to compute the exponential of a matrix, twentyfive years later. SIAM Rev. 45(1), 3–49 (2003) 10. G. Watson, A Treatise on the Theory of Bessel Functions, 2nd edn. (Cambridge University Press, Cambridge, 1995)

Influence of the SIPG Penalisation on the Numerical Properties of Linear Systems for Elastic Wave Propagation Uwe Köcher

Abstract Interior penalty discontinuous Galerkin discretisations (IPDG) and especially the symmetric variant (SIPG) for time-domain wave propagation problems are broadly accepted and widely used due to their advantageous properties. Linear systems with block structure arise by applying space-time discretisations and reducing the global system to time-slab problems. The design of efficient and robust iterative solvers for linear systems from interior penalty discretisations for hyperbolic wave equations is still a challenging task and relies on understanding the properties of the systems. In this work the numerical properties such as the condition number and the distribution of eigenvalues of different representations of the linear systems coming from space-time discretisations for elastic wave propagation are numerically studied. These properties for interior penalty discretisations depend on the penalisation and on the time interval length.

1 Introduction The accurate and efficient simulation of time-domain first- and second-order hyperbolic elastic and acoustic wave propagation phenomena is of importance in many engineering fields with e.g. electromagnetic, acoustic and seismic applications as well as for non-destructive structural health monitoring of light-weighted fibre reinforced materials; cf. e.g. [1, 2] and references therein. Recently there is again an increased interest in the simulation of multi-physics systems including coupled elastic wave propagation together with fluid flow in porous media; cf. [3, 4]. Such models appear for instance in battery engineering and for biomedical applications.

U. Köcher () Helmut-Schmidt-University, University of the Federal Armed Forces Hamburg, Department of Mechanical Engineering, Hamburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_18

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The ability of the efficient high-order approximation of the space-time wavefield is of fundamental importance for time-domain numerical simulations of wave phenomena. Especially discontinuous Galerkin methods (dG) as spatial discretisations exhibit as favourable over continuous finite element methods in terms of lesser numerical dispersion at the same discretisation order such as polynomial degree and degrees of freedom per wavelength; cf. [5]. Reed and Hill introduced in 1973 the first discontinuous Galerkin method for first-order hyperbolic steady-state neutron transport. Meanwhile there exist many discontinuous Galerkin methods for the spatial discretisation of elliptic, parabolic and hyperbolic problems; cf. e.g. [6]. Interior penalty discontinuous Galerkin discretisations (IPDG) and especially the symmetric interior penalty variant (SIPG) for time-domain wave propagation problems are broadly accepted and widely used due to their advantageous properties and their ability for parallel numerical simulations. The SIPG discretisation of the second-order wave equations is convergent of optimal order in the energy- and L2 -norm for any polynomial approximation degree in space; cf. [7]. All interior penalty discontinuous Galerkin families include additional terms combined of trace operators on the interior and exterior boundaries between mesh elements. Interior penalty methods include a stabilisation term which penalises jumps of the trial and test function traces on the boundaries to ensure coercivity of the bilinear form corresponding to the Laplacian operator. The needed weighting depends on spatial mesh parameters such as the cell diameter, the cell anisotropy and the local polynomial degree as well as on material parameters but not on the time discretisation parameters; cf. [6–8]. The estimation of the local minimal choice of the penalisation is of fundamental importance but it turns out to be difficult in physically relevant problems. An over-penalisation should be avoided since such results in linear systems with higher condition numbers than necessary. The overpenalisation may result in inefficient iterative system solves or breakdowns if it is not take into account in the solver design. The common drawback of discontinuous Galerkin discretisations compared to their continuous Galerkin counterparts is that they need more degrees of freedom to obtain the same analytic convergence order. In physically relevant problems the drawback of the higher number of degrees of freedom is not that present compared to analytic test problems. To the contrary, physically relevant problems gain advantages, such as avoiding linear-elasticity locking phenomena and the ability to capture incompatible but relevant boundary and initial conditions, from the lesser grid stiffness of discontinuous approaches in space and time; cf. e.g. [1]. Linear systems with block structure arise by applying space-time discretisations and reducing the global system to time-slab or time-interval problems; cf. [1]. The design of efficient and robust iterative solvers for linear systems from interior penalty discretisations for hyperbolic wave equations is still a challenging task and relies on understanding the properties of the systems. This work presents the numerical properties such as the condition number and the distribution of eigenvalues and their dependency on the penalisation and on the time interval length of different representations of the linear systems coming from

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space-time discretisations for elastic wave propagation. To the best knowledge of the author such results are not present in the literature. The following structure of the paper is the brief introduction of the elastic wave equation and its space-time discretisation in Sect. 2, the presentation of different representations of the fully discrete systems which are further studied in Sect. 3, the numerical experiments section providing the results in Sect. 4 and summarising and concluding remarks in Sect. 5.

2 Elastic Wave Equation and Space-Time Discretisation The elastic wave equation in its first-order in time, or displacement-velocity, representation reads as the following system. The primal variables are the displacement u and the velocity v. The volume is represented by the domain Ω ⊂ Rd , with dimension d = 2, 3, and the time domain is given by I = (0, T ) for some finite final time T . Find the pair {u, v} from ρs (x)∂t u(x, t) − ρs (x)v(x, t) = 0

in Ω × I ,

ρs (x)∂t v(x, t) − ∇ · σ (u(x, t)) = f (x, t)

in Ω × I ,

(1)

with boundary conditions u(x, t) = g(x, t) on ΓD × I , σ (u(x, t)) n = h(x, t) on ΓN × I , and initial conditions u(x, 0) = u0 (x) in Ω × {0}, v(x, 0) = v 0 (x) in Ω × {0}. The linearised stress tensor is given by σ (u) = C : (u) and is composed of the action of the (possibly anisotropic) linear elasticity tensor C on the linearised strain (u) = (∇u + ∇uT )/2. The solid mass densities inside the volume are denoted by ρs . Acting internal volume forces are denoted by f . A detailed derivation of the system can be found in [1]. Standard notation for function spaces, norms and inner products is used. Let H = L2 (Ω)d , V = H01 (ΓD ; Ω)d , V ∗ = H −1 (ΓD ; Ω)d , H = L2 (I ; H ), V = {v ∈ L2 (I ; V ) | ∂t v ∈ H} and W = L2 (I ; V ). The initial-boundary value problem with purely homogeneous Dirichlet boundary condition g = 0 admits an unique weak solution u ∈ V ∩ C(I/; V ), v ∈ H ∩ C(I/; H ) and ∂t v ∈ L2 (I ; V ∗ ) as given by the literature; cf. e.g. [9, Sect. 2]. A variational space-time discretisation is briefly presented in the sequel; for all details consider [1]. A piecewise polynomial continuous Galerkin approximation of polynomial degree r is used as semi-discretisation in time. Hereby a piecewise discontinuous space of polynomial degree r − 1 is used as test function space which allows the truncation of the global space-time system to time interval or time slab problems. Let 0 = t0 < · · · < tN = T the partition of the temporal domain I = (0, T ) into N subintervals In = (tn−1 , tn ), n = 1, . . . , N. The length of the subinterval In is defined by τn = tn − tn−1 and let τ = max1≤n≤N τn the global time discretisation parameter. Let Pr (In ; X) denote the space of polynomials of degree r or less on the interval In ⊂ I with values in some Banach space X. Introducing

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ξ ∈ C(I/; R), with ξ |In ∈ Pr (I/n ; R), and ζ ∈ L2 (I ; R), with ζ |In ∈ Pr (I/n ; R), as global piecewise polynomial trail and test basis functions. The semi-discrete system cG reads as: Find the pairs {ucG τ |In , v τ |In } ∈ Pr (In ; V ) × Pr (In ; V ), with coefficients ι ι U n , V n ∈ V for ι = 1, . . . , r, such that r  

ακ,ι (ρs U ιn , 8 ω ) − βκ,ι (ρs V ιn , 8 ω) = 0 ,

∀8 ω∈V,

ι=0

r  r 

βκ,ι (F ιn , ω) , ακ,ι (ρs V ιn , ω) + βκ,ι a(U ιn , ω) = ι=0

(2) ∀ω ∈ V ,

ι=0

for all κ = 1, . . . , r, and with  (t), ζ ακ,ι := (ξn,ι n,κ (t))In

 = 

βκ,ι := (ξn,ι (t), ζn,κ (t))In =

I

ξι (t) ζκ (t) dt

=

ξι (t) ζκ (t) τn dt = I

r

μ=0 r

wμ ξι (tμ ) ζκ (tμ ) , (3) τn wμ ξι (tμ ) ζκ (tμ ) ,

μ=0

using quadrature weights wμ and quadrature points tμ from QGL(r+1) such that the continuity conditions between neighbouring subintervals hold. For the discretisation in space, let the partition Th of the space domain Ω into some finite number of disjoint elements K. Denoting by hK the diameter of the element K and the global space discretisation parameter as h = maxK∈Th hK . The spatial mesh is allowed to be anisotropic but non-degenerated. Let the partition /h = F /B ∪ F I into disjoint parts F /B , F I for the set of boundary faces and the set F h h h h /B := F B(ΓD ) ∪ F B(ΓN ) , of interior faces. The set of boundary faces is divided as F h h h FhB := FhB(ΓD ) , into disjoint parts FhB(ΓD ) , FhB(ΓN ) coinciding with the Dirichlet B(Γ ) boundary ΓD and the Neumann boundary ΓN . We let Fh := Fh D ∪ FhI = FhB ∪ FhI . The jump trace operators, which enforce weak Dirichlet boundary conditions, are defined by v0 :=

⎧ ⎨v|

F+

− v|F − ,

F ∈ FhI ,

⎩v|

F+

,

F ∈ FhB ,

⎧ ⎨v| + − v| − , F ∈ F I , F F h v := ⎩v| + − v|Γ , F ∈ F B . F D h (4)

The average trace operator is defined by ⎧   ⎨ 1 t v| +  + t v| −  , F F F F {{t F (v)}} := 2  ⎩t F v| +  , F     using the traction vector t F v|F ± := σ v|F ± n± .

F ∈ FhI , F ∈ FhB ,

(5)

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219

To be concise as possible, we give directly the bilinear form ah corresponding to the weak Laplacian as ah (un,ι h , ωh ) =

 K∈Th

+

K

σ (un,ι h ) : (ωh ) −



F ∈Fh

F

un,ι h ·

 F

{{t F (un,ι h )}} · ωh 0

F ∈Fh   γF ωh 0 − S {{t F (ωh )}} ,

(6)

where S ∈ {1, −1, 0} denotes the consistency parameter. The choice S = 1 leads to SIPG, S = −1 to NIPG and S = 0 to IIPG. Note that the non-homogeneous Dirichlet boundary terms can be shifted efficiently to the right hand side and thus only a stiffness matrix for homogeneous Dirichlet boundaries has to be assembled. The value of the interior penalty parameter γF is determined by an inverse estimate to balance the terms involving numerical fluxes on the element boundaries F ∈ Fh in Eq. (6) and to ensure coercivity of the bilinear form ah . We let γF = γ0 γF,C γF,K , where γF,K denotes a parameter depending only the polynomial approximation degree p in space and the shape of the elements K ± , γF,C denotes a (scalar) parameter depending the material parameters in K ± and γ0 denotes an additional tuning parameter; cf. for details [1, 8].

3 Fully Discrete Systems In this work only the linear systems of continuous piecewise linear approximations in time, that are SIPG(p)–cG(1) and FEM(p)–cG(1), are studied. The results can be used for higher order time discretisations, since their efficiency depends strongly on solving additional problems corresponding to the following systems; cf. for details [1]. To derive fully discrete systems from Eq. (2), the basis functions in time are specified as Lagrange polynomials, which are defined via the quadrature points t0 = 0 and t1 = 1 of the two-point Gauß-Lobatto quadrature rule QGL(2) on the time reference interval I = [0, 1]. ξ0 (t) = 1 − t ,

ξ0 (t) = −1 ,

ξ1 (t) = t ,

ξ1 (t) = 1 ,

ζ1 (t) = 1 ,

and, with that, the coefficients ακ,ι and βκ,ι from Eq. (3) are evaluated as α1,0 = −1 , α1,1 = 1 ,

β1,0 = τ2n , β1,1 = τ2n .

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We recast the arising algebraic system by the following linear system with block structure: Find the coefficient vectors u1In , v 1In ∈ RNDoF from L x = b given by ⎡ − τn M ⎣ 2 M

⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎤ ⎡ τn v0 0 M v 1In M M ⎦⎣ ⎦ = ⎣ ⎦ ⎣ In⎦ , ⎦ + ⎣2 τn 0 τn 1 u1In M − τ2n A u0In 2 (bIn + bIn ) 2A

(7)

and denoting by M the mass matrix and by A the stiffness matrix. The assemblies b0In and b1In include the contributions from the forcing terms and inhomogeneous Dirichlet boundary from Eq. (6) using Eq. (4). The flexible GMRES method with an inexact Krylov-preconditioner can be used for example to solve the linear system (7) efficiently. With some algebraic steps, the block system can be condensed to K u1In = 8 b and a postprocessing as given by (M +

τn2 1 4 A) uIn

=

τn2 0 4 (bIn

+ b1In ) + (M −

τn2 0 4 A)uIn

+ τn Mv 0In ,

v 1In = τ2n (u1In − u0In ) − v 0In .

(8)

The linear system Eq. (8) is comparable to a classical discretisation by employing a Crank-Nicolson scheme for the time discretisation. Hence, optimised solvers for linear systems with the matrix K can be re-used.

4 Numerical Experiments In this section we study some numerical properties of the linear systems as of Eqs. (8) and (7). Therefore we approximate the analytic solution ⎡ uE ([x1 , x2 ]T , t) = ⎣

sin((t + x1 ) · 2π) sin((t + x2 ) · 2π)

⎤ ⎦

(9)

on Ω × I = (0, 1)2 × (0, 1) with ∂Ω = ΓD . The right hand side, initial and E boundary values √ are derived by plugging u into Eq. (1). We set the mesh size to −1 h = 10 /(2 2) and use p = 2 elements in space. The material is isotropic with Young’s E modulus E = 70, Poisson’s ratio ν = 0.34 and density ρs = 2.8. The numerical simulations are done with the DTM++/ewave frontend solver of the author for the deal.II library; cf. [1]. The solutions of the displacement and velocity are illustrated by Fig. 1. The convergence for the error eu = uE − ucG(1) τ,h and the dependency on the penalisation γ0 to obtain a certain accuracy is presented in Fig. 2. For γ0 = 106 the experimental order of convergence in time is 2.00 in the L2 (I ; L2 (Ω))-norm and the calculated errors of the SIPG and FEM discretisations are comparable. Figure 3 shows the

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Fig. 1 Visualisation of the solution Eq. (9) at t = 1 for the magnitude of displacement (left) and the velocity (right) with SIPG(2)–cG(1) and τn = 1.25 × 10−2 10

10

10

10

1

2

SIPG(2)–cG(1), SIPG(2)–cG(1), SIPG(2)–cG(1), SIPG(2)–cG(1), FEM(2)–cG(1)

3

4

10

2

10 τn

1

1e+3 1e+4 1e+5 1e+6 100

experimental condition number

Fig. 2 Interior penalty parameter γF influence on the experimental convergence behaviour in time for approximating uE from Eq. (9) for Sect. 4

107 105

SIPG(2)–cG(1), SIPG(2)–cG(1), FEM(2)–cG(1)

1e+6 1e+5

103 101

low accuracy 10

5

10

4

10

3

10

2

10

1

τn

Fig. 3 Influence of the interior penalty parameter γF and the global time discretisation parameter τn on the experimental condition number for Sect. 4

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τn

10

10

10

2

4

6

0

0.2

0.4

0.6

0.8

1

Fig. 4 Distribution of normalised eigenvalues of matrices K(τn ) for SIPG(2) with γ0 = 106 (solid blue) and for FEM(2) with τn = 10−6 (dotted red) for Sect. 4

effects of the penalisation and the choice of τn on the condition numbers. Roughly, the SIPG condition numbers scale with τn and γ0 down to a certain limit comparable to FEM condition numbers. Figure 4 illustrates the distribution of the normalised eigenvalues of the matrix K for SIPG(2)–cG(1) with γ0 = 106 for several values of τn and the normalised eigenvalues of the matrix K for FEM(2)–cG(1) for τn = 10−6 . The eigenvalues collect in small number of clusters for SIPG by choosing τn ≈ 1/γ0 . The effect of clustering could not be reproduced for the comparable FEM system or for the block system.

5 Conclusions The influence of the penalisation of a SIPG discretisation is analysed numerically for fully discrete linear systems of different representations and compared with their standard finite element counterparts. It is shown that the condition number of the condensed SIPG system matrix scales with the penalisation and the time subinterval length with a challenging numerical experiment and with the resulting effect of eigenvalue clustering for τn ≈ 1/γ0 . A (much) faster convergence behaviour of the conjugate gradient method of the condensed SIPG system for small τn was noticed but not analysed by the author for several higher-order three-dimensional problems with physically relevance before; cf. for details [1]. Now, a faster convergence behaviour of the conjugate gradient method can be explained due to the clustering effect of the eigenvalues. This work clearly help to design preconditioners for SIPG discretisations of the elastic wave equation, but keep it as future work. Acknowledgements The author was partially supported by E.ON Stipendienfonds (Germany) under the grant T0087/29890/17 while visiting University of Bergen.

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References 1. U. Köcher, Variational space-time methods for the elastic wave equation and the diffusion equation. Ph.D. thesis, Helmut-Schmidt-University Hamburg (2015), pp. 1–188. urn:nbn:de: gbv:705-opus-31129 2. U. Köcher, M. Bause, Variational space-time discretisations for the wave equation. J. Sci. Comput. 61(2), 424–453 (2014). https://doi.org/10.1007/s10915-014-9831-3 3. A. Mikelic, M.F. Wheeler, Theory of the dynamic Biot-Allard equations and their link to the quasi-static Biot system. J. Math. Phys. 53(123702), 1–16 (2012). https://doi.org/10.1063/1. 4764887 4. M.A. Biot, The influence of initial stress on elastic waves. J. Appl. Phys. 11(8), 522–530 (1940). https://doi.org/10.1063/1.1712807 5. J.D. De Basabe, M.K. Sen, M.F. Wheeler, The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion, Geophys. J. Int. 175(1), 83–93 (2008). https://doi. org/10.1111/j.1365-246X.2008.03915.x 6. D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002). https://doi.org/ 10.1137/S0036142901384162 7. M.J. Grote, A. Schneebeli, D. Schötzau, Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44(6), 2408–2431 (2006). https://doi.org/10.1137/ 05063194X 8. R.H.W. Hoppe, G. Kanschat, T. Warburton, Convergence analysis of an adaptive interior penalty discontinuous Galerkin method, SIAM J. Numer. Anal. 47(1), 534–550 (2008). https://doi.org/ 10.1137/070704599 9. W. Bangerth, M. Geiger, R. Rannacher, Adaptive Galerkin finite element methods for the wave equation. Comput. Methods Appl. Math. 10(1), 3–48 (2010). https://doi.org/10.2478/cmam2010-0001

Function-Based Algebraic Multigrid Method for the 3D Poisson Problem on Structured Meshes Ali Dorostkar

Abstract Multilevel methods, such as Geometric and Algebraic Multigrid, Algebraic Multilevel Iteration, Domain Decomposition-type methods have been shown to be the methods of choice for solving linear systems of equations, arising in many areas of Scientific Computing. The methods, in particular the multigrid methods, have been efficiently implemented in serial and parallel and are available via many scientific libraries. The multigrid methods are primarily used as preconditioners for various Krylov subspace iteration methods. They exhibit convergence that is independent or nearly independent on the number of degrees of freedom and can be tuned to be also robust with respect to other problem parameters. Since these methods utilize hierarchical structures, their parallel implementation might exhibit lesser scalability. In this work we utilize a different framework to construct multigrid methods, based on an analytical function representation of the matrix, that keeps the amount of computation high and local, and reduces the memory requirements. This approach is particularly suitable for modern computer architectures. An implementation of the latter for the three-dimensional discrete Laplace operator is derived and implemented. The same function representation technology is used to construct smoothers of sparse approximate inverse type.

1 Introduction In scientific computations, Poisson’s or Laplace equation is among most studied, analysed and tested partial differential equations. Different forms of the Laplace equation appear in various applications such as the stationary heat transfer, the Navier-Stokes equations, in elasto-static problems, potential flow analysis, and many more.

A. Dorostkar () Department of Information Technology, Uppsala University, Uppsala, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_19

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The vast amount of application fields of the Laplace and Poisson’s equations has driven the need to develop fast and reliable solvers for the discrete versions of the equations. The topic is very well studied, however, we focus on a reason to seek yet a different approach to solve systems with the discrete Laplace operator. The largest interest lies in the three-dimensional case where the degrees of freedom become too large to use either of the direct solution methods, Fast Fourier transform-based methods and such. Experience shows that various versions of multilevel and multigrid methods are among the most efficient solvers for the problem at hand. Although in theory multigrid methods possess numerically and computationally optimal properties, in practice they exhibit some drawbacks. We consider the algebraic multigrid (AMG) as a reference. For very large problems, the construction of AMG may become prohibitive due to very high memory requirements, sometimes too large to fit in the distributed memory of a modern HPC cluster (cf. [16]). This is not in favour of the modern computer architectures, where computation is cheap and memory references and communication are expensive. This performance bottleneck makes the idea to use the AMG framework, based on a particular analytic function representation of the matrix, particularly plausible for 3D problems. The idea itself is not new. It has been introduced as early as in 1991 in [8]. Consider a class of structured matrices known as Generalized Locally Toeplitz (GLT) matrices which can be characterized by an analytical function, referred to as the symbol of the matrix, cf. [10]. Based on the symbol, one can construct all the ingredients of a multigrid method. This approach opens the possibility to obtain a matrixfree MG, saving memory usage and increasing the computational intensity of the corresponding implementation. The GLT-based MG technique has undergone developments in many directions (See for instance[1, 3, 5, 7, 9]). So far the case for constructing GLT-MG for the 3D Laplacian has not been completed and implemented. This is the topic of the current paper. Compared to the earlier work of the author on GLTMG [6] for elasticity, this paper introduces the use of sparse approximate inverses as a substitute smoother in the GLTMG implementation. The resulting algorithm can be implemented completely matrixfree. The contents of this report is structured as follows. In Sect. 2 we construct the symbol representation of the anisotropic Laplacian in 3D, discretized by standard conforming piecewise trilinear basis functions. In Sect. 3 we describe the implementation of the related GLT-AMG method. Further, we show in Sect. 4 that one can use the GLT technology to compute sparse approximate inverses of the level matrices, to be used as smoothers, or as preconditioners in the smoothing steps. Section 5 contains an analysis of computer resources demands for the GLT-AMG method and the possible savings in memory, together with increase in number of computations to better match the characteristics of multicore computer architectures or GPUs. In Sect. 6 some numerical illustrations of the GLT-AMG performance are shown, as well as a comparison with that of AGMG (cf. [14]).

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2 Function Representation of the Anisotropic Laplace Operator We consider Poisson’s equation in 3D with anisotropic Laplace operator, − ε1

∂ 2u ∂ 2u ∂ 2u − ε2 2 − ε3 2 = f 2 ∂x1 ∂x2 ∂x3

in Ω

(1)

where Ω ⊂ R3 and εi , i = 1, 2, 3 are positive constants. For simplicity, we assume that the coefficients are scaled appropriately and are less or equal to one. Without loss of generality, we assume Ω to be the unit cube, discretized with an equidistant structured mesh with stepsize h, and that homogeneous Dirichlet boundary conditions are imposed on the whole boundary ∂Ω of Ω. To discretize (1), we choose standard conforming trilinear FEM. The variational formulation is standard and is therefore omitted. The resulting algebraic linear system to solve is Au = f, where A is sparse, symmetric and positive definite. For our purposes, it is important to analyse the structure of A and how it is viewed as a Toeplitz (or nearly Toeplitz) matrix. To this end, we introduce major concepts from the GLT theory.

2.1 GLT and the Symbols of a Matrix We briefly introduce the notion of Toeplitz matrices and their symbols, for a more complete discussion see [10]. Denote by f (θ1 , · · · , θd ) a d-variate complex-valued integrable function, defined over the domain Qd = [−π, π]d , d ≥ 1. Denote by fk the Fourier coefficients of f , 1 fk = (2π)d

 Qd

f (θ )e−i (k,θ) dθ, k = (k1 , · · · , kd ) ∈ Zd , i 2 = −1,

(2)

 where (k, θ ) = dj=1 kj θj , n = (n1 , · · · , nd ), and N(n) = n1 · · · nd . Following the notation in [17], with each f we can associate a sequence of Toeplitz matrices {Tn }, Tn = {fk− }nk,=eT ∈ CN(n)×N(n) , e = [1, 1, · · · , 1] ∈ Nd . The function f is referred to as the generating function, also known as the symbol, of Tn . Using a more compact notation we write Tn = Tn (f ). An evaluation of the symbol f (θ ) over an equispaced grid gives an approximation of the eigenvalues of Tn (f ). Moreover, the error in the approximation of the eigenvalues approaches zero as n → ∞. A d-dimensional partial differential problem with constant coefficients, discretized on quadrilateral finite elements of degree p is a d-level block-valued Toeplitz matrix with blocks of size s = pd . In case of problems where the

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coefficients are non-constant and continuous, the matrix sequence Tn (f ) is of Generalized Locally Toeplitz type with symbol f . Remark 1 An important feature of the GLT sequences is that a matrix obtained by a sequence of algebraic operations on GLT matrices is itself a GLT matrix. The symbol of the new GLT matrix is a function obtained by applying the same algebraic operations on the corresponding symbols of the original matrices.

2.2 The Symbol of the 3D Anisotropic Laplacian To construct the symbol for the discrete counterpart of (1), we view the matrix A as the summation of three matrices Ai of the same size. The Ai matrices correspond to the case where εi = 1, εj = 0 j = i. We remove the effect of the grid size by multiplying A with h/9, which gives a simpler symbol expression. The matrices Ai are of GLT type with symbols fi where f1 (θ1 , θ2 , θ3 ) = (4 + 2cos(θ1))(1 + 0.5cos(θ2))(2 − 2cos(θ3)), f2 (θ1 , θ2 , θ3 ) = (4 + 2cos(θ1))(2 − 2cos(θ2))(1 + 0.5cos(θ3)), f3 (θ1 , θ2 , θ3 ) = (2 − 2cos(θ1))(4 + 2cos(θ2))(1 + 0.5cos(θ3)). Hence, the matrix A is a GLT sequence Tn (f ) where f (θ1 , θ2 , θ3 ) =

 h ε1 f1 (θ1 , θ2 , θ3 ) + ε2 f2 (θ1 , θ2 , θ3 ) + ε3 f3 (θ1 , θ2 , θ3 ) . 9

(3)

Figure 1 illustrates the quality of the symbol f . We see that the equispaced sampling of f gives a good approximation of the eigenvalues of A up to a finite number of outliers. The outliers correspond to the effect of the boundary conditions on A (c.f. [10] for more details). Fig. 1 Sorted equispaced sampling of the symbol f vs the true eigenvalues of A

1.2 Sampling of the symbol Eigenvalues of A

1 0.8 0.6 0.4 0.2 0

0

1000

2000

3000

4000

5000

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3 GLT Based Algebraic Multigrid Method The standard algebraic multigrid (AMG) method works on a hierarchy of matrices that are constructed based on the original matrix where the low frequency errors in the iterative solvers are mapped to higher frequencies on the coarser meshes. On the coarse levels these errors can be smoothed rapidly with simple, cheap iterative methods [15]. There are a few components necessary for the construction of AMG, namely, prolongation and restriction operators which are used to move through the constructed hierarchy, the matrices on each coarse level, a solver for the coarse grid which can be a direct or an iterative method, and pre-/post-smoothers that are cheap solution methods in charge of damping the error modes on each level. The prolongation operator Pi is defined as the product between a Toeplitz matrix Tni (pi ) and the transpose of a k-level cutting matrix Hni . Here, pi is a k-variate first order trigonometric polynomial and the cutting matrix Hni is obtained as the Kronecker product of k 1-level cutting matrices Hn1i ⎡

Hn1i

010 ⎢ 0 1 0 ⎢ =⎢ .. .. .. ⎢ . . . ⎣ 0 1

⎤ ⎥ ⎥ ⎥, ⎥ ⎦ 0

Hn1i ∈ Rni+1 ×ni .

Assuming that fi has a zero of order 2q in θi0 , we choose pi to be pi (θ ) = c ·

k .

[1 + cos(θj − (θi0 )j )]q .

j =1 H A We define the level matrices as Ai = Pi−1 i−1 Pi−1 , i = 1, . . . , m, A0 = A. T The restriction operator Ri = Pi (see [4] and the references therein). The symbol  fi of the matrix Ai is fi (θ ) = (1/2k ) η∈C (θ) pi2 (η)fi (η), where C(θ ) is the set of corner points defined as C(θ ) = {η|ηj ∈ {θj , θj + π}}. Having defined coarse level matrices, prolongation, and restriction operators, we are ready with the construction of GLT-MG.

4 Choosing a Smoother for the GLT-Based AMG Next, by taking advantage of the GLT theory, we provide a computationally cheap smoother for AMG. The question how to choose it for anisotropic problems is discussed in the literature, for instance in [18].

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The matrix A is symmetric positive definite (spd) in the anisotropic Poisson problem. Hence for the level i we choose the conjugate gradient (CG) as a smoother preconditioned with a sparse approximate inverse (SPAI) of Ai . SPAI is a branch of preconditioning techniques, where we construct D ≈ A−1 . The preconditioned system is then DAx = Db. The true inverse of a sparse matrix is usually dense. Thus, to overcome this limitation, a “good” sparsity pattern is often prescribed, restricting the amount and placement of the non-zero elements of D. How to define a “good” sparsity structure for the approximate inverse of a general matrix remains an open problem and has been studied for many years (cf. [2, 11– 13]). However, a rigorous way to determine it may not be a feasible task for general matrices. For the considered class of problems we know that the entries in the true inverse of the discrete Laplacian exhibit fast decay away from the main diagonal and, therefore, a band structure of the approximate inverse is an appropriate choice. Our idea to construct SPAI comes from the property of GLT sequences stated in Remark 1. We form a GLT sequence based on the following symbol, g(θ ) =

1 , f (θ ) + γ

G = Tn (g).

(4)

Here γ is a non-negative scalar which guarantees that g(θ ) is bounded and integrable. We obtain the elements of the inverse by using (4) in (2). The sparsity pattern is set by prescribing a drop tolerance, a bandwidth, or both. An optional additional step to improve the quality of G is to compute a diagonal matrix DG such 7 72 that G+DG minimizes the Frobenius norm 7I − (G + DG )A7W . In the numerical tests, W is the identity matrix.

5 Analysis of the Required Computer Resources for GLT Based AMG The properties of a Toeplitz matrix make it suitable for parallel computations. From the definition of a Toeplitz matrix we know that the elements along descending diagonals are constant. Moreover, from (2) we are able to compute these elements directly from the symbol of the Toeplitz matrix. Thus, computing the first element of each diagonal is enough to construct the matrix or its approximate inverse. Storing a Toeplitz matrix is flexible and can be performed in two ways. – Compute the first element of each diagonal in each level of a d-level Toeplitz matrix and store < it in an array. The memory consumption of the matrix is then of order O( dk=1 bk ) where bk is the bandwidth of the kth level of the Toeplitz matrix. – Avoid storing any entries and compute every element as soon as it is needed thus creating a matrix-free algorithm.

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Note that the element computation is performed at least once for the first storage strategy and many times in the case of the second method. Thus, the complexity of the numerical integration plays a role in the performance of the GLT-MG. Particularly for non-smooth, multidimensional symbols, the integration itself could make the second approach unsuitable.

6 Numerical Experiments In this section we provide a few numerical experiments to present the performance of the GLT-MG. In the first case we show a GLT-MG method with one Gauss-Seidel iteration as its pre-/post-smoothing step. The prolongation and restriction operators are constructed based on the discussions in the previous section. The number of iterations and the final error in the solution is compared to AGMG. Table 1 provides the results. We see that GLT-MG is robust and optimal and performs fewer iterations than AGMG. Next, we replace the smoother with ten iterations of PCG preconditioned with the symbol-based SPAI. Table 2 shows the number of iterations compared to AGMG. We see that although the number of iterations are growing with size, still our method needs less iterations than AGMG. Table 1 GLTMG with GS as a smoother compared to AGMG

Size 125 729 4913 35937 274,625

ε = (1, 1, 1) GLT-MGGS Iter Error 2 1.26e−09 2 6.22e−09 2 1.06e−08 2 1.24e−08 2 1.30e−08

Table 2 GLTMG + SPAI as smoother compared to AGMG; ε = (1, 1, 1)

AGMG Iter Error 1 2.67e−16 7 8.70e−09 7 1.42e−08 11 8.09e−08 11 3.75e−07

ε = (1, 1, 10−4 ) GLT-MGGS Iter Error 3 2.62e−06 3 1.47e−06 3 7.11e−05 3 4.92e−05 3 3.96e−05

Size 125 729 4913

AGMG Iter Error 1 1.10e−15 11 4.33e−08 13 2.98e−07 13 1.35e−06 14 1.85e−06

GLTMG 2 3 5

AGMG 1 7 7

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7 Conclusions In this paper we construct an algebraic multigrid preconditioner based on the GLT theory and solve the three-dimensional anisotropic Poisson problem. Taking advantage of the Toeplitz structure of the so-constructed matrices, we provide an algebraic multigrid preconditioner with a low memory footprint and high computational intensity. we design a sparse approximate inverse preconditioner for the coarse level matrices and use it in the pre-/post-smoothers. The construction of SPAI is completely parallelizable and the memory consumption is controllable. We show that our proposed AMG method is numerically robust and competitive with the AGMG method.

References 1. A. Aricó, M. Donatelli, S. Serra-Capizzano, V-cycle optimal convergence for certain (multilevel) structured linear systems. SIAM J. Matrix Anal. Appl. 26(1), 186–214 (2004) 2. O. Axelsson, Iterative Solution Methods (Cambridge University Press, Cambridge, 1996) 3. M. Donatelli, S. Serra-Capizzano, Multigrid methods for (multilevel) structured matrices associated to a symbol and related applications. Bollettino dell Unione Matematica Italiana 9(6), 319–347 (2013) 4. M. Donatelli, S. Serra-Capizzano, D. Sesana, Multigrid methods for Toeplitz linear systems with different size reduction. BIT Numer. Math. 52(2), 305–327 (2012) 5. M. Donatelli, M. Molteni, V. Pennati, S. Serra-Capizzano. Multigrid methods for cubic spline solution of two point (and 2D) boundary value problems. Appl. Numer. Math. 104(Supplement C), 15–29 (2016). Sixth International Conference on Numerical Analysis – Recent Approaches to Numerical Analysis: Theory, Methods and Applications (NumAn 2014) 6. M. Donatelli, A. Dorostkar, M. Mazza, M. Neytcheva, S. Serra-Capizzano, Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem. Comput. Math. Appl. 74(5), 1015–1028 (2017). SI: SDS2016 – Methods for PDEs 7. M. Donatelli, C. Garoni, C. Manni, S. Serra-Capizzano, H. Speleers, Symbol-based multigrid methods for galerkin b-spline isogeometric analysis. SIAM J. Numer. Anal. 55(1), 31–62 (2017) 8. G. Fiorentino, S. Serra-Capizzano, Multigrid methods for Toeplitz matrices. Calcolo 28(3–4), 283–305 (1991) 9. G. Fiorentino, S. Serra-Capizzano, Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions. SIAM J. Sci. Comput. 17(5), 1068– 1081 (1996) 10. C. Garoni, S. Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications, vol. I (Springer, 2017) 11. T. Huckle, A. Kallischko, Frobenius norm minimization and probing for preconditioning. Int. J. Comput. Math. 84(8), 1225–1248 (2007) 12. S. Kharchenko, L. Y. Kolotilina, A. Nikishin, A.Y. Yeremin, A robust ainv-type method for constructing sparse approximate inverse preconditioners in factored form. Numer. Linear Algebra Appl. 8(3), 165–179 (2001) 13. L.Y. Kolotilina, A.Y. Yeremin, Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl. 14(1), 45–58 (1993) 14. A. Napov, Y. Notay, An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34(2), A1079–A1109 (2012)

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15. Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, Philadelphia, 2003) 16. E. Turan, P. Arbenz, Large scale micro finite element analysis of 3d bone poroelasticity. Parallel Comput. 40(7), 239–250 (2014). 7th Workshop on Parallel Matrix Algorithms and Applications 17. E.E. Tyrtyshnikov, A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl. 232, 1–43 (1996) 18. J. van Lent, S. Vandewalle, Multigrid waveform relaxation for anisotropic partial differential equations. Numer. Algorithms 31(1), 361–380 (2002)

Part VI

Numerical Methods in Biophysics

Mathematical Modelling of Phenotypic Selection Within Solid Tumours Mark A. J. Chaplain, Tommaso Lorenzi, Alexander Lorz, and Chandrasekhar Venkataraman

Abstract We present a space- and phenotype-structured model of selection dynamics between cancer cells within a solid tumour. In the framework of this model, we combine formal analyses with numerical simulations to investigate in silico the role played by the spatial distribution of oxygen and therapeutic agents in mediating phenotypic selection of cancer cells. Numerical simulations are performed on the 3D geometry of an in vivo human hepatic tumour, which was imaged using computerised tomography. Our modelling extends our previous work in the area through the inclusion of multiple therapeutic agents, one that is cytostatic, whilst the other is cytotoxic. In agreement with our previous work, the results show that spatial inhomogeneities in oxygen and therapeutic agent concentrations, which emerge spontaneously in solid tumours, can promote the creation of distinct local niches and lead to the selection of different phenotypic variants within the same tumour. A novel conclusion we infer from the simulations and analysis is that, for the same total dose, therapeutic protocols based on a combination of cytotoxic and cytostatic agents can be more effective than therapeutic protocols relying solely on cytotoxic agents in reducing the number of viable cancer cells.

M. A. J. Chaplain · T. Lorenzi School of Mathematics and Statistics, University of St Andrews, St Andrews, UK e-mail: [email protected]; [email protected] C. Venkataraman () Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton, UK e-mail: [email protected] A. Lorz CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, Paris, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_20

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1 Introduction In this work, we extend a space- and phenotype-structured model of selection dynamics in a solid tumour that we proposed in [4, 5]. Our model consists of a nonlinear integro-differential equation (IDE) for the spatiotemporal evolution of the phenotypic distribution of cancer cells coupled to a system of nonlinear partial differential equations (PDEs) for the dynamics of abiotic factors. A novel feature of this work, over [4], is that we consider multiple spatially distributed therapeutic agents. Through the coupling of formal analyses and numerical simulations, we show that spatial variations in abiotic factors promote the formation of local niches and lead to the selection of different phenotypic variants, in agreement with the viewpoint of cancer as an eco-evolutionary process [6]. The modelling carried out in the present work allows us to consider optimisation of treatment protocols. In particular, our results suggest that, for the same total dose, therapeutic protocols based on the delivery of cytotoxic drugs in combination with cytostatic agents can be more effective than therapeutic protocols relying solely on cytotoxic drugs in reducing the number of viable cancer cells.

2 Model Description We identify the tumour geometry with a spatial domain Ω ⊂ R3 . At any time instant t ≥ 0, we characterise the state of each cancer cell in the tumour by means of a pair (x, y) ∈ Ω × [0, 1]. The vector x ∈ Ω identifies the spatial position of the cell and the scalar variable y ∈ [0, 1] ⊂ R stands for the normalised expression level of a hypoxia-responsive gene [8]. Cells within the tumour proliferate and die due to competition for limited space. Moreover, both a cytotoxic drug, which acts by increasing the cell death rate, and a cytostatic drug, which acts by reducing the cell proliferation rate, can be administered. We assume increasing values of the phenotypic state y to be correlated with a progressive switch towards a hypoxic phenotype which, in turn, implies a progressive reduction in the proliferation rate [1]. Additionally, given that cytotoxic agents target mostly rapidly proliferating cells, we assume that higher values of the phenotypic state y correspond with higher levels of resistance to the cytotoxic drug [3]. Given the local population density n(t, x, y), we compute the cell density and the mean cell phenotypic state at time t and position x as follows  ρ(t, x) =

1

n(t, x, y) dy 0

1 and μ(t, x) = ρ(t, x)



1

y n(t, x, y) dy.

(1)

0

Finally, we introduce the functions s(t, x) ≥ 0, c1 (t, x) ≥ 0 and c2 (t, x) ≥ 0 to model the local concentration of oxygen, cytotoxic drug and cytostatic drug at position x and time t, respectively.

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2.1 Dynamics of Cancer Cells The dynamics of the local population density n(t, x, y) is governed by the following nonlinear IDE   ∂n (t, x, y) = R y, ρ(t, x), s(t, x), c1 (t, x), c2 (t, x) n(t, x, y). (2) ∂t   In Eq. (2), the functional R y, ρ, s, c1 , c2 represents the fitness of cells in phenotypic state y at position x and time t (i.e. the fitness landscape of the tumour), given the local environmental conditions determined by the cell density ρ(t, x) and the concentrations of abiotic factors s(t, x), c1 (t, x) and c2 (t, x). We define the fitness landscape of the tumour as   R y, ρ, s, c1 , c2 =

  + r(y, s) 1 − k2 (c2 ) ) *+ ,

f (y) ) *+ , proliferation in hypoxic conditions

proliferation in oxygenated environments

− k1 (y, c1 ) − ) *+ ,

d ρ(t, x). ) *+ ,

death due to cytotoxic drug

death due to competition for space

(3)

Building upon the considerations and the modelling strategies presented in [4], we introduce the following definitions: 5 6 f (y) = ζ 1 − (1 − y)2 , k1 (y, c1 ) = γc1

r(y, s) = γs

c1 (t, x) (1 − y)2 , αc1 + c1 (t, x)

 s(t, x)  1 − y2 , αs + s(t, x)

k2 (c2 ) = γc2

c2 (t, x) . αc2 + c2 (t, x)

(4)

(5)

The biological meaning of the different parameters are summarised in Table 1. The function 0 ≤ k2 (c2 ) ≤ 1 measures the percentage reduction in the proliferation rate of cancer cells caused by the cytostatic drug. Since the efficacy of many cytostatic drugs is directly linked with adequate oxygen tension [2], we make the prima facie assumption that the cytostatic drug reduces the proliferation rate in oxygenated environments, whereas it does not affect the proliferation rate under For this reason, the term f (y) is not multiplied by the factor hypoxic conditions.  1 − k2 (c2 ) . Moreover, we assume the function k2 to be increasing in the drug dose c2 . A detailed discussion of the biological assumptions that underlie the definitions of the other functions can be found in [4].

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2.2 Dynamics of Abiotic Factors The abiotic factors diffuse in space, decay over time and are consumed by cells. We note that the dynamics of abiotic factors is faster than cellular proliferation and death [9]. From a mathematical viewpoint, this means that we can assume oxygen and the drugs to be in quasi-stationary equilibrium. In this setting, the dynamics of the functions s(t, x), c1 (t, x) and c2 (t, x) are described by the following elliptic PDEs which are coupled to the IDE (2) 

1

βs Δs(t, x) = ηs

  r y, s n(t, x, y) dy + λs s(t, x),

(6)

  k1 y, c1 n(t, x, y) dy + λc1 c1 (t, x),

(7)

  k2 c2 n(t, x, y) dy + λc2 c2 (t, x).

(8)

0

 βc1 Δc1 (t, x) = ηc1

1 0

and  βc2 Δc2 (t, x) = ηc2

1 0

The biological meanings of the different parameters are summarised in Table 1 and further details can be found in [4]. Concentrating on the biological scenario whereby the tumour is avascular and the concentrations of abiotic factors in the medium surrounding the tumour are constant in time, we choose the following boundary conditions for Eqs. (6)–(8) s(·, x) = S(x),

c1 (·, x) = C1 (x)

and c2 (·, x) = C2 (x) ∀ x ∈ ∂Ω.

(9)

3 Formal Analysis of Phenotypic Selection We denote the local cell density and the dominant phenotypic state (i.e. the phenotypic state y that maximises the cellular fitness at position x) at equilibrium as ρ(x) and y(x), respectively. Moreover, we denote the steady-state distributions of abiotic factors as s(x), c1 (x) and c2 (x). By means of the formal arguments used in [4], one finds ⎡ ρ(x) =



ζ + Ac1 (x)

2

⎤

⎥ 1⎢ ⎢As,c (x) − Ac (x) + ⎥ 2 1 ⎣ d ζ + As,c2 (x) + Ac1 (x) ⎦

(10)

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and ζ + Ac1 (x) , ζ + As,c2 (x) + Ac1 (x)

y(x) =

(11)

where   As,c2 (x) = γs 1 − k2 (c2 )

s(x) c1 (x) and Ac1 (x) = γc1 . αs + s(x) αc1 + c1 (x)

Such formal results are consistent with the asymptotic results presented in [7]. The expressions given by Eqs. (10) and (11) demonstrate that the local cell density ρ and the dominant phenotypic state y at a certain position are determined by the concentrations of oxygen s, cytotoxic drug c1 and cytostatic agent c2 at the same position. This is illustrated by the heat-maps in Fig. 1, which relate to the parameter values given in Table 1 and c1 = K Ct ot ,

c2 = (1 − K) Ct ot ,

K ∈ [0, 1].

(12)

The heat-maps display the values of ρ¯ and y¯ as functions of the parameter K in Eq. (12) and the oxygen concentration s¯ . We observe that, for sufficiently high oxygen levels, the local cell density is minimised for values of K between 0.4 and 0.5, which corresponds to the situation whereby the cytotoxic drug and the cytostatic agent are used in combination.

Fig. 1 Heat-maps showing the local cell density ρ¯ and dominant phenotypic state y¯ at equilibrium as functions of the parameter K [cf. Eq. (12)] and the local concentration of oxygen s¯ . For sufficiently high oxygen levels, the local cell density is minimised for values of K between 0.4 and 0.5, which corresponds to the situation whereby the cytotoxic drug and the cytostatic agent are used in combination

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Table 1 Nondimensionalised parameter values Parameter αc1 αc2 αs βc1 , βc2 βs γc1 γc2 γs ζ d η c1 , η c2 ηs λc λs Ctot S0

Biological meaning Michaelis-Menten constant of cytotoxic drug Michaelis-Menten constant of cytostatic drug Michaelis-Menten constant of oxygen Diffusion coefficients of cytotoxic and cytostatic drug Diffusion coefficient of oxygen Maximum cell death rate due to cytotoxic drug Maximum percentage reduction in cell proliferation rate due to cytostatic drug Maximum cell proliferation rate in oxygenated environments Maximum cell proliferation rate under hypoxic conditions Rate of cell death due to competition for space Scaling factors for cell consumption of the drugs Scaling factor for cell consumption of oxygen Decay rate of cytotoxic drug Decay rate of oxygen Total drug dose delivered Oxygen concentration in the surrounding environment

Value 0.2 10−2 0.2344 5 20 1.8 × 10−4 0.8 1.5 × 10−5 1 × 10−6 2 × 10−5 400 3125.2 0.1 0.3 0.01 1

4 Numerical Solutions In this section we report on simulations of the model system (2), (6), (7) and (8). For the spatial domain Ω we consider a tetrahedral discretisation of the real geometry of a human hepatic tumour obtained from the 3D-IRCADb-01 database (http://www. ircad.fr/). We make use of the following boundary conditions for s, c1 and c2 : S(x) = S0

∀ x ∈ ∂Ω

and C1 (x) = K Ct ot ,

C2 (x) = (1−K) Ct ot

∀ x ∈ ∂Ω

with

K ∈ [0, 1].

(13)

The parameter Ct ot measures the total delivered drug dose. To carry out numerical simulations, we tune the value of the parameter K and we use the values given in Table 1 for the other parameters of the model. Further details on the tumour geometry and the model parametrisation can be found in [4]. For the elliptic PDEs, we use a P1 finite element method and treat the nonlinear terms explicitly using the values from the previous time-step. For the IDE, we use an IMEX Euler method in which the nonlinear terms are treated explicitly. The mesh that we employ has 9932 DOFs and the time-step we use correspond to 1 × 103 with a final time of 5 × 106 , by which time the numerical solutions are at steady state values in all the cases under consideration. The numerical method employed is detailed in [4].

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Fig. 2 Upper row: computed equilibrium distributions of oxygen (Sh ), cytotoxic drug (C1h ) and cytostatic drug (C2h ) for K = 0.6. Lower row: computed mean phenotypic state (μh ), predicted ¯ local cell density at equilibrium for K = 0.6. dominant trait (y), ¯ computed (ρh ) and predicted (ρ) The results of the formal analysis agree with the results of numerical simulations

Figure 2 summarises the numerical solutions at equilibrium for K = 0.6 [cf. Eq. (13)]. We show a slice through the tumour in order to visualise the interior. We also post-process the computed values of Sh , C1,h and C2,h to evaluate the corresponding dominant phenotypic state y¯ and the local cell density ρ¯ according to Eqs. (10)–(11), which we compare with the computed mean phenotypic state μh and local cell density ρh . There is an excellent agreement between the numerical solutions at equilibrium and the predictions of our formal analyses. As the qualitative features of the numerical solutions for the other values of K are similar to those of the numerical solutions obtained for K = 0.6, we do not display the equilibrium spatial distributions for other values of K. However, in order to illustrate the variation in cell density and phenotypic state at equilibrium as we change K, in Fig. 3 we report on the computed total cell number and mean phenotypic state at equilibrium in the tumour and on the tumour surface. The total number of cells and the surface cell number at equilibrium are minimised for K ≈ 0.8 and K ≈ 0.4, respectively. We note that this agrees with the results of the formal analysis illustrated by the heat-map in the left panel of Fig. 1 as the concentrations

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Fig. 3 Reading from left to right, computed total cell number, surface cell number, mean phenotypic state over the tumour and surface mean phenotypic state at equilibrium for different values of the parameter K. The total cell number over the tumour is minimised amongst the considered values by taking K ≈ 0.8, whilst the surface cell number is minimised for K ≈ 0.4

of therapeutic factors and oxygen correspond to those used to make the plots of Fig. 1 only at the boundary of the tumour. Our results suggest that therapeutic protocols based on the delivery of lower doses of cytotoxic drugs in combination with cytostatic agents may be more effective than therapeutic protocols relying solely on higher doses of cytotoxic drugs in reducing the number of viable cancer cells. Acknowledgements CV wishes to acknowledge partial support from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642866. AL was supported by King Abdullah University of Science and Technology (KAUST) baseline and start-up funds (BAS/1/1648-01-01 and BAS/1/1648-01-02). MAJC gratefully acknowledges support of EPSRC grant no. EP/N014642/1.

References 1. J.D. Gordan, J.A. Bertout, C.-J. Hu, J.A. Diehl, M.C. Simon, Hif-2α promotes hypoxic cell proliferation by enhancing c-Myc transcriptional activity. Cancer Cell 11(4), 335–347 (2007) 2. C. Legendre, S. Avril, C. Guillet, E. Garcion, Low oxygen tension reverses antineoplastic effect of iron chelator deferasirox in human glioblastoma cells. BMC Cancer 16(1), 51 (2016) 3. M.C. Lloyd, J.J. Cunningham, M.M. Bui, R.J. Gillies, J.S. Brown, R.A. Gatenby, Darwinian dynamics of intratumoral heterogeneity: not solely random mutations but also variable environmental selection forces. Cancer Res. 76(11), 3136–3144 (2016) 4. T. Lorenzi, C. Venkataraman, A. Lorz, M.A. Chaplain, The role of spatial variations of abiotic factors in mediating intratumour phenotypic heterogeneity. Preprint available from http://hdl. handle.net/10023/10685 (2017) 5. A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame, Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull. Math. Biol. 77(1), 1–22 (2015) 6. L.M. Merlo, J.W. Pepper, B.J. Reid, C.C. Maley, Cancer as an evolutionary and ecological process. Nat. Rev. Cancer 6(12), 924–935 (2006)

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7. S. Mirrahimi, B. Perthame, Asymptotic analysis of a selection model with space. J. de Mathématiques Pures et Appliquées 104(6), 1108–1118 (2015) 8. S. Strese, M. Fryknäs, R. Larsson, J. Gullbo, Effects of hypoxia on human cancer cell line chemosensitivity. BMC Cancer 13(1), 1 (2013) 9. V. Walther, C.T. Hiley, D. Shibata, C. Swanton, P.E. Turner, C.C. Maley, Can oncology recapitulate paleontology? Lessons from species extinctions. Nat. Rev. Clin. Oncol. 12(5), 273– 285 (2015)

Uncertainty Assessment of a Hybrid Cell-Continuum Based Model for Wound Contraction Fred Vermolen

Abstract We assess the uncertainty in a hybrid cell-based, continuum-based model for wound contraction. We explore the correlations between the final contraction of a wound and the stiffness of the tissue, forcing applied by fibroblasts, plastic forces, death rate of cells, differentiation rate of cells, amount of random walk, and the chemotactic strength. Furthermore, we compute the likelihood that serious contractions occur. Although the current model is very simple, the principles can be used to unravel the most important biological mechanisms behind wound contraction.

1 Introduction Surgical interference in burn injuries generally aims at preventing complications such as hypertrophic scars and contractures, which may lead to disabilities in patient mobility in severe cases. In order to avoid these complications, it is crucially important to understand the underlying biological processes so that therapies can be improved. It is widely accepted that the differentiation mechanism from fibroblasts, which are the main cells in skin and which are responsible to regenerate skin collagen to maintain the integrity of skin, to myofibroblasts largely contributes to the severity of contracture and the development of hypertrophic scar tissue. Myofibroblasts are strong fibroblasts that pull harder on their environment than fibroblasts do. Furthermore, myofibroblasts shorten the long chains of polymeric collagen molecules. The underlying biology of contracture development [1] is very complex and at many points still unknown, and certainly too complicated to be contained in a single mathematical model. Several models for wound contraction have been developed, see [2, 3] to mention a few of them. The model by Valero et al. [2] takes into account the dependence of fibroblast-myofibroblast differentiation on

F. Vermolen () Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_21

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elongations that these cells experience. In [3] a morpho-elastic formalism to model permanent contractions is considered. Several modeling approaches exist in the literature, such as fully continuumbased models in which partial differential equations are dealt with, and small-scale models where one models colonies of cells, such as in [4–6]. Cell-based models can be distinguished into cellular-automata models and hybrid models. Hybrid models consider individual cells where cells are treated as separate entities, with partial differential equations for chemical quantities or for the resulting mechanical balance. Examples of such models are given in [7, 8]. The work in [7] describes the interplay between the development of anisotropy of tissue and the immune system. This paper is based on [8], where only the mechanical behavior of the skin is dealt with in relation with the ingress of fibroblasts, and differentiation of fibroblasts to myofibroblasts. All these models vary in complexity, where the more complex models attempt to describe the underlying biology accurately, whereas the simpler models are more tractable in the sense that their parameter space is much smaller. The poor knowledge of the parameter values needs a statistical assessment, and therefore, Monte Carlo simulations are carried out where in each simulation a sample from statistical distributions is taken. The uncertainty reflects that cellular and tissue properties vary from patient to patient as a result of lifestyle, gender, age and genetic inheritance. Such a statistical assessment allows the estimation of averages, variances, correlations and more interestingly, the computation of the likelihood that complications occur for certain wounds. The main innovation of this paper is the statistico-probabilistic assessment of the model in [8].

2 The Mathematical Model We consider a rectangular initial domain D(0) = (0, 1)2 (mm) with cells.

2.1 The Cells and Their Forces Each cell exerts pulling forces on its environment, which we obtain through integration over its boundary. Let F(t, x) be the total force exerted by all cells in the domain of computation, then we have 9

−∇ · σ = F(t, x), in D, σ · n + Ku = 0, on ∂D,

(1)

where σ represents the stress experienced in domain D. In this paper, we use simple Hooke’s Law, as well as infinite strain theory to arrive at the relation between the local displacements and the strain and stress. As boundary conditions to the above

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equation, we use the second equation in the above relation, where u = u(t, x) denotes the local displacement vector. Further K represents a spring force between the boundary of the domain and the tissue far away. The time-dependence in the displacement results from the migration and pulling by fibroblasts as well as from cell death and cell differentiation. As in [8], the force is written as a sum of a temporary and a plastic part, respectively denoted by FT (t, x) and FP (t, x), that is F(t, x) = FT (t, x) + FP (t, x), in D.

(2)

In [8], it was derived that the force exerted by a single active cell, with index i, via its boundary, is given by  FiT (t, x) =

Fc n(t, x )δ(x − x )dΓ  ,

(3)

Γi (t )

where FiT (t, x), Γi (t), Fc , n, respectively denote the total force that active cell i exerts on its environment, the boundary of cell i, intrinsic pulling force and inward normal vector over the cell boundary. The intrinsic pulling force has been assumed to be constant over each simulation. The plastic forces result from shortening the polymeric collagen molecules that is accomplished by the myofibroblasts. We assert that the magnitude of the force is related to the time of exposure of the control volume to myofibroblasts. More details can be found in [8]. The plastic forces associated with each control volume De , with boundary ∂De (D ≈ ∪e∈C D e (where C denotes the set of control volumes)) have been modelled using a phenomenological principle that the plastic force is obtained via the integration over the boundary of each control volume  FeP (τe , x) = Qe (τe )

n(x )δ(x − x )dΓ  ,

(4)

∂De

where Qe , τe , respectively, denote the intrinsic plastic force exerted by control volume De and the τe is the effective time that the control volume has contained a myofibroblast. For τe , we phenomenologically have 

t

τe (t) = 0

|De (s) ∩ DM (s)| ds, |De (s)|

(5)

where DM (t) denotes the region occupied by all myofibroblasts. The measure |.| denotes area in the current two-dimensional simulations. In R3 , this measure would denote a volume. The intrinsic forces exerted by the control volume follows from (see [8]) Qe (τe ) = Qmax (1 − e−ατe ),

(6)

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which follows a simple differential equation, motivated by chemical relaxation. Let F (t) denote the set of living cells, then

F(t, x) =

i∈F (t )

FTi (t, x) +

FPe (τe (t), x).

(7)

e∈C

2.2 The Cells and Their Migration, Death and Differentiation Despite knowing that cells migrate as a result of various other mechanisms, we take into account the mechanisms of random walk, haptotaxis, intercellular collisions and drag. Random walk represents cell diffusion in upscaled models. Haptotaxis is the migration of cells through a rigid tissue as a result of the gradient of a concentration, which we incorporate via the gradient of the TG-β growth factor secreted by the immune system. The concentration profile of TG-β is modelled through a phenomenological approximation of the solution to the heat equation. Haptotaxis should not be confused with chemotaxis, which is cell migration up to a concentration gradient in a fluid environment. Intercellular collisions are modelled using the principles outlined in [8], where Hertz’ contact model has been used. Hencewith, the migration of cell i located at position xi (t) is modelled by the following stochastic differential equation −xi dt) dxi (t) = wi zi (σcell dW(t) + βcell e−0.1t e−||xi −xW || ||xxW W −xi ||

+vc (t, xi (t))dt + vd (t, xi (t))dt.

(8)

Here wi and zi , respectively denote the binary variables indicating whether a cell is alive (viable) and whether a cell is a fibroblast (or a myofibroblast), that is ⎧ ⎨1, cell i is viable, wi = ⎩0, cell i is dead,

⎧ ⎨1, cell i is a fibroblast, zi = ⎩0, cell i is myofibroblast.

(9)

In Eq. (8), dW(t) represents a vector (2D) Wiener process for random walk, where both components are realizations from independent identical normal distributions with zero mean and variance dt. Haptotaxis is modelled as a drift term, where we phenomenologically model the gradient of the TG-β concentration, which is supposed to decrease as the location gets further away from the wound centre (located at xW ) and to decay as time proceeds. We used some fixed parameter values here. The parameters βcell and σcell , respectively, represent the drift velocity 2 = 2D parameter and random walk parameter (σcell cell , with cell diffusivity Dcell ). Further, vc represents the contribution to the velocity vector as a result of collisions with neighboring cells [8]. The velocity vector vd accounts for the contribution as a

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result of drag. This velocity follows from the movement of all points in the domain as a result of the cellular forces via vd (t, x) = du(t,x) dt . Cell death and cell differentiation are modelled as binary entirely stochastic transformations here because the history paths of individual cells is unknown. To this extent, we consider the following conditional probabilities from the commonly used exponential distributions  P(wi (t + τ ) = 0|wi (t) = 1) =

t +τ

pe−p(s−t )ds.

(10)

t

Here p represents the probability rate of cell differentiation. The back differentiation from myofibroblasts to fibroblast is neglected, hence P(wi (t + τ ) = 0|wi (τ ) = 0) = 1. Further cell death is modelled similarly with probability rate q. Cell death has also been assumed to be irreversible for obvious reasons. Hence fibroblasts may go through the following states: (wi , zi ) = (1, 1) →q (wi , zi ) = (1, 0), (wi , zi ) = (1, 1) →p,q (wi , zi ) = (0, 0), (wi , zi ) = (1, 1) →p (wi , zi ) = (0, 1) →q (wi , zi ) = (0, 0).

(11) The third transition would require the simultaneous death and differentiation, and it is less likely to occur since the likelihood is approximately pqτ 2 during τ (taking τ small), compared to the other likelihoods that occur with likelihood of O(τ ) over τ . One can use Markov Chain modeling to model the average fractions of dead cells, fibroblasts and myofibroblasts. Initially, the cells are all phenotypes as fibroblasts (wi = 1, zi = 1 for all cells at t = 0) and they are all randomly located in the computational domain outside the wound area. In the course of time, the fibroblasts migrate into the wound area, and start pulling their environment, by which the wound boundary moves towards the wound centre. As the fibroblasts differentiate to myofibroblasts, the myofibroblasts change their environment so that permanent stresses remain. This causes a permanent contraction.

3 Numerical Method and Uncertainty Assessment We use a linear triangles-based finite-element method to solve the mechanical balance. Further, to compute the cellular forces, we divide the boundaries of the cells into four segments (i.e. squares). The coordinates of the mesh points in the domain of computation are updated using x(t, X) = X + u(t, X), relating the Eulerian coordinates x to the Lagrangian coordinates X on the reference (initial) domain via the local deformations. Drag velocities for the cells are determined by mapping from displacement velocities from finite-element mesh points. The triangles from the mesh are used for the plastic forces.

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For the cell positions, we use the Euler-Maruyama time-integration method. In our simulations, we use constant values of all the parameter per sample run. Lack of knowledge about parameters requires sampling: 2 ), E ∼ N (μE , σE2 ), F ∼ N (μF , σF2 ), Q ∼ N (μQ , σQ p ∼ N (μp , σp2 ), q ∼ N (μq , σq2 ), β ∼ N (μβ , σβ2 ), σcell ∼ N (μσcell , σσ2cell ). (12)

We take N samples, which results in a Monte Carlo Error (MCE), next to errors from modelling, time-integration, rounding and from the finite-element discretization. We track the final wound area in each sample. Next to the expected value of final contraction, which is unbiasedly estimated by μA = E(A) ≈ μN A =

N 1 Aj , N

(13)

j =1

where Aj is the final area of each run. We determine the probability density and correlations between the final area A and the parameters using sampling from the earlier mentioned normal distributions. First, we estimate the MCE. The Strong Law of Large Numbers dictates that μN A → μA as N → ∞. Furthermore, from the Central Limit Theorem, it follows that √ d 2 N (μN A − μA ) → N (0, σA ), as N → ∞, where σA2 = E((A − μA )2 ) → 0⇒ Var(μN A) →

σA2 N

→

1 N 2 sA N

N

j =1 (Aj

2 2 − μN A ) =: sA as N → ∞

as N → ∞.

Hence, as N → ∞, the MCE, ε(μN A ), is given by Koehler et al. [9] ⎞1/2 ⎛ N 2

1 ⎝ sA 2⎠ N ε(μN = (Aj − μN =: εˆ (μN A ) := Var(μA ) → √ A) A ). N N j =1

(14)

This above relation is used as an estimator of the MCE. Since the definition of the MCE contains a variance over a batch of Monte Carlo simulations (say m) of the expected value (hence a set of approximations of expected value μN A with N samples, hence needing m × N samples totally), the main advantage of the above formula is that it enables the use of only one batch of Monte Carlo samples to estimate the error. If G(s) is the probability density, then we have for the probability that A ≤ A∗ for a specific A∗ : P(A ≤ A∗ ) =



A

G(s)ds. 0

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4 Computer Simulations Input values from Table 1 have been used to generate Monte Carlo samples. The reader who is interested in the evolution of the area versus time, as well as in the ingress of (myo)fibroblasts into the wound area, is referred to [8]. From the simulations, and using Eq. (14), on the normalized final area, where we set A := AA0 (A0 being the initial wound area), we obtained an average contraction N N of E(A) ≈ μN A = 0.8636, and MCE of ε(μA ) ≈ εˆ (μA ) ≈ 0.0015. From the Anderson-Darling test, we concluded that A most likely (pvalue = 0.9772) follows a normal distribution. Figure 1 shows the probability density, the cumulative probability of occurrence of a contraction reduction to an area smaller than A∗ and the average area versus the number of samples, which converges to the average final area only subject to discretization, time integration and rounding errors. It can be seen that heavy contractions, normalized areas of less than 0.85 are to occur with probability of about 0.28 given the data that we used. Furthermore, we the computed correlations between the final wound area and the input parameters have been listed in Table 2. In this table, only the significant correlations with respect to the null hypothesis H : r = 0 (no correlation for a quantity) have been shown (it was decided that there was no correlation if pvalue > 0.05). From Table 2 and in the wake of desiring minimal contraction, hence a final area closest to the initial radius, it can be concluded that it is beneficial for a patient to have large skin stiffness, small intrinsic plastic strains (forces, hence small shortening rate of polymeric molecules), small portion of chemically directed fibroblast migration towards the wound centre, and large apoptosis (death) rate of (myo)fibroblasts. In this sense, a weakly performing immune system, which decreases the amount of haptotaxis (migration directed by the concentration gradient of TG-β), could be beneficial for the prevention of contractions (though wound healing would proceed more slowly). Similar conclusions were also hinted at, though not statistically sustained, on the basis of the more complicated model by [7]. We did not investigate the possible correlation between the minimal area over time and the (myo)fibroblasts exists. Table 1 Input values for the simulations

Quantity Rcell p q Fc Qmax E ν α K βchem σcell

Mean 0.05 0.15 0.02 30 5 0.5 0.35 1 1 30 0.02

Variance 0 0.015 0.002 3 0.5 0.05 0 0 0 3 0.002

Unit mm h−1 h−1 kg/mm/h2 kg/mm/h2 kp/mm/h2 – h−1 kg/mm2 /h2 mm3 /mol/h mm/h−1/2

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Fig. 1 Left: histogram of the final area A—Middle: cumulative probability of occurrence of a contraction with area reduction to an area smaller than A —Right: Average final area versus the number of samples Table 2 Statistically significant Pearson correlation values with the final area

Quantity E Qmax r 0.404 −0.5397 pvalue < 0.0001 < 0.0001

βchem −0.1413 0.0460

q 0.1920 0.0064

5 Conclusions We formulated a Bayesian framework that is capable of estimating correlations and likelihoods from finite-element simulations that specific final contractions take place. Regarding the correlations, the patient is best of with large tissue stiffness, small plastic strains (hence small amount of shortening of polymeric chains in collagen), small portion of chemically directed migration of fibroblasts, and large death (apoptosis) rates of (myo)fibroblasts. Despite its simplicity, the model has some preliminary predictive power.

References 1. J. Tomasek, G. Gabbiani, B. Hinz, C. Chaponnier, R. Brown, Myofibroblasts and mechanoregulation of connective tissue remodelling. Nat. Rev. Mol. Cell Biol. 3, 349–363 (2002) 2. C. Valero, E. Javierre, J.M. Garcia-Aznar, M.K. Gomez-Benito, A cell regulatory mechanism involving feedback between contraction and tissue formation guides wound healing progression. PLoS One 9(3), e92774 (2014). https://doi.org/10.1371/journal.pone.0092774 3. D.C. Koppenol, F.J. Vermolen, Biomedical implications from a morphoelastic continuum model for the simulation of contracture formation in skin grafts that cover excised burns. Biomech. Model. Mechanobiol. 16(4), 1187–1206 (1988) 4. D. Drasdo, S. Höhme, A single-cell-based model for tumor growth in vitro: monolayers and spheroids. Phys. Biol. 2(3), 133–147 (2005) 5. H. Byrne, D. Drasdo, Individual-based and continuum models of growing cell populations: a comparison. J. Math. Biol. 58, 657–687 (2009) 6. C. Borau, W.J. Polacheck, R.D. Kamm, J.M. Garcia-Aznar, Probabilistic voxel-Fe model for single cell motility in 3D, In Silica Cell Tissue Science 1(2), (2014). https://doi.org/10.1186/ 2196-050X-1-2

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7. W.M. Boon, D.C. Koppenol, F.J. Vermolen, A multi-agent cell-based model for wound contraction. J. Biomech. 49(8), 1388–1401 (2016) 8. F.J. Vermolen, A. Gefen, Semi-stochastic cell–level computational modelling of cellular forces: application to contractures in burns. Biomech. Model Mechanobiol. 14(6), 1181–1195 (2009) 9. E. Koehler, E. Brown, S.J.-P.A. Haneuse, On the assessment of Monte Carlo error in simulationbased statistical analyses. Am. Stat. 63(2), 155–162 (2009)

Part VII

Structure Preserving Discretizations and High Order Finite Elements for Differential Forms

The Discrete Relations Between Fields and Potentials with High Order Whitney Forms Ana M. Alonso Rodríguez and Francesca Rapetti

Abstract When using the lower order Whitney elements on a simplicial complex, the matrices describing the external derivative, namely, the differential operators gradient, curl and divergence, are the incidence matrices between edges and vertices, faces and edges, tetrahedra and faces. For higher order Whitney elements, if one adopts degrees of freedom based on moments, the entries of these matrices are still equal to 0, 1 or −1 but they are no more incidence matrices. If one uses instead the “weights of the field on small simplices” as alternative degrees of freedom, the matrices representative of the external derivative are incidence matrices for any polynomial degree.

1 Introduction Together with the list of nodes and of their positions, the mesh data structure also contains incidence matrices (with entries 0, 1 or −1) saying which node belongs to which edge, which edge bounds which face, etc., and there is a notion of (inner) orientation of the simplices to consider. In short, an edge, a face, is not only a two-node [ni , nj ], three-node [ni , nj , nl ], subset of the set of the mesh nodes, but such a set plus an orientation of the simplex it subtends that is defined by the order of its vertices (e.g., [ni , nj ] is the edge oriented from ni to nj ). These matrices, besides containing all the information about the topology of the domain, for the lowest approximation polynomial degree, they help connecting the degrees of freedom (dofs) describing the potentials to those describing the fields. As an example, the relation E = −grad V at the continuous level becomes e = −G v

A. M. Alonso Rodríguez () Dip. di Matematica, Università degli Studi di Trento, Povo, Trento, Italy e-mail: [email protected] F. Rapetti Dep. de Mathématiques J.-A. Dieudonné, Univ. Côte d’Azur, Nice, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_22

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where G coincides with the node-to-edge incidence matrix and e (resp. v) is the vector of edge circulations (resp. values at nodes) of the electric field E (resp. the scalar electric potential V ), when nodal (resp. edge) finite elements are used to approximate V (resp. E). As pointed out in [4], nodal and edge finite element approximations, that belong to the lowest order reconstructions by the finite element families introduced in [6] for R3 and in [9] for R2 , are Whitney differential forms [10]. We wish to investigate the block-structure of the incidence matrices, when fields and potentials are approximated by high order Whitney forms [3, 8], with dofs given either by the well-known moments [6] or by the more recent weights on the small simplices [7, 8]. We do not have in mind a particular differential equation to solve but we work on the algebraic equivalent of the field/potential relation when they both are approximated by forms of higher polynomial degree. We will see that the involved matrices present a structure by blocks, each block multiplying the dofs of the potential associated to a geometrical dimension. In the following we adopt the notation of the finite element exterior calculus, see e.g. [2]. Let Pr (Rd ) and Hr (Rd ) be, respectively, the space of polynomials in n variables of degree at most r and the space of homogenous polynomials in n variables of degree equal to r. Then, Pr Λk (Rd ) = Pr Λk and Hr Λk (Rd ) = Hr Λk denote the corresponding spaces of polynomial differential forms. The space of trimmed polynomial differential forms Pr− Λk can be defined by involving the Koszul operator, κ : Λk+1 → Λk being Λk the space of k-differential forms in Rd (see, e.g. [1]), that satisfies κ(u ∧ η) = (κu) ∧ η + (−1)k u ∧ (κη) ,

u ∈ Λk ,

η ∈ Λ2 .

Moreover, κ(f u) = f κu if f is a function and κ(dxi ) = xi . These properties fully determine κ and yield κ(dxi ∧ dxj ) = xi dxj − xj dxi , κ(dxi ∧ dxj ∧ dxk ) = xi dxj ∧ dxk − xj dxi ∧ dxk + xk dxi ∧ dxj . In particular κ : Hr Λk+1 → Hr+1 Λk and, for d = 3, κu is equal to x · u (k = 0), x × u (k = 1), x u (k = 2), and 0 (k = 3), respectively. The space Pr− Λk is intermediate between Pr−1 Λk and Pr Λk as follows: Pr− Λk = Pr−1 Λk + κHr−1 Λk+1 = {u ∈ Pr Λk : κu ∈ Pr Λk−1 } . Let Ω be a bounded polyhedral domain of Rd and T a simplicial mesh of Ω. The spaces of finite element differential forms with respect to the triangulation T are denoted Pr− Λk (T ). By H Λk (Ω) we denote the Sobolev space H 1 (Ω) if k = 0, H (curl; Ω) if k = 1, H (div; Ω) if k = 2 and the space L2 (Ω) if k = 3. Then we define Pr− Λk (T ) = {u ∈ H Λk (Ω) : u|T ∈ Pr− Λk for all T ∈ T } . It is well known (see, for instance, [2]) that these finite element spaces are that of the Lagrange finite elements of degree r if k = 0, the first family of Nédélec finite elements of order r conforming in H (curl; Ω) if k = 1, the first family of Nédélec

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finite elements of order r conforming in H (div; Ω) if k = 2, and discontinuous elements of degree ≤ r − 1 if k = 3. For 0 ≤ j ≤ d, let Δj (T ) be the set of all j -dimensional subsimplices of the simplex T and Δj (T ) be the set of all j -dimensional subsimplices of the mesh T . Let us denote by Λk (T ) the space of all smooth differential k-forms on a tetrahedron T . The trace operator TrS : Λk (T ) → Λk (S), for S ∈ Δj (T ) with k ≤ j < 3, is the map defined in the following way for j = 2⎧(S = f ) for j = 1 (S = e) for j = 0 (S = ni ) 9 ⎪ if k = 0, ⎨ u|f u|e if k = 0, Trf u = nf × u if k = 1, Tre u = Trni u = u(ni ) if k = 0. ⎪ u · τ e if k = 1 ; ⎩ u · n if k = 2 ; f (v −v )×(v −v )

v −v

where nf = |(vjj −vii )×(vkk −vii )| , for f = [ni , nj , nk ], τ e = |vjj −vii | , if e = [ni , nj ], with vi the Cartesian coordinates of the node ni . In the finite element exterior calculus, we adopt spaces of differential forms u which are piecewise smooth, usually polynomials, with respect to T , i.e., the restriction u|T is smooth for each T ∈ T . Then, for f ∈ Δj (T ) with j ≥ k, the form Trf u can be multi-valued, that is, we can assign a value for each T ∈ T containing f by restricting u to T and then taking the trace on f . If all such traces coincide, we say that Trf u is single-valued. It is well-known that if Trf u is singlevalued for all f ∈ Δj (T ), k ≤ j ≤ d − 1, then u ∈ H Λk (Ω)

2 Moments and Potentials The construction of dofs based on moments for the space Pr− Λk (T ) requires the use of Pr Λk (f ), for certain f subsimplices of T . Here, Pr Λk (f ) is the space of forms obtained by restricting those of Pr Λk (Rs ) to f being s = dim(f ) ≤ d. Definition 1 The moments of a polynomial k-form u ∈ Pr− Λk (T ) with 0 ≤ k ≤ n and r ≥ 1 are the scalar quantities  Trf u ∧ η,

η ∈ Pr+k−s−1 Λs−k (f ),

(1)

f

for each f ∈ Δs (T ) and k ≤ s = dim(f ) ≤ d. The moments of a k-form u are associated to subsimplices of T of dimension equal to or greater than k. If r + k − s − 1 < 0 in (1), there are no moments associated to subsimplices of dimension s. For r = 1, the moments are associated only to k-subsimplices of T . The wedge product ∧ appearing in (1) extends the notion of exterior product to forms. It holds that u(k) ∧ v () = (−1)k v () ∧ u(k) , where u(k) stands for the k-form u. If k +  > d it is zero. In details, for d = 2, 3, we

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have u(0) ∧ v (k) = (uv)(k) for k ≤ d, u(1) ∧ v (1) = (u × v)(2) , and, for d = 3, u(1) ∧ v (2) = u(2) ∧ v (1) = (u · v)(3) . When considering moments the main tool for the identification of the matrix of the exterior derivative operator is the integration by parts formula 





Trf du ∧ η = f

Tr∂f (Trf u ∧ η) + (−1)k−1 ∂f

Trf u ∧ dη ,

u ∈ Λk (T ).

f

(2) Another important tool is the choice of a basis of Pr+k−s−1Λs−k (f ) to represent η. To this purpose, we introduce  the multi-index α = (α0 , . . . , αs ) of s + 1 integers αi ≥ 0 and weight |α| = si=1 αi . The set of multi-indices α with s + 1 components and weight r is denoted I(s + 1, r). Any (oriented) 3-simplex T = [ni . nj , nl , np ] is associated to an increasing map σT0 : {0 1 2 3} → {1 2 . . . Nv } and T = [nσ 0 (0) , . . . , nσ 0 (3)]. Any (oriented) s-simplex S ∈ Δs (T ) is associated T

T

to an increasing map σS1 : {0 1 . . . s} → {0 1 2 3} such that if σS = σS1 ◦ σT0 then S = [nσf (0) , . . . , nσf (s) ]. For each S ∈ Δs (T ) and α ∈ I(s + 1, r) we set λα = λασS0 (0) . . . λασSs (s) . Then α  Pr  Λ0 (S) = {λ = α : α ∈ I(s + 1, r )},  > 1 Pr  Λ (S) = λ dλi : α ∈ I(s + 1, r ), i ∈ {σS (1), . . . σS (s)} = > Pr  Λ2 (S) = λα dλi ∧ dλj : α ∈ I(s + 1, r  ), i, j ∈ {σS (1), . . . σS (s)}, i < j , = Pr  Λ3 (S) = λα dλi ∧ dλj ∧ dλ : α ∈ I(s + 1, r  ), i, j,  ∈ {σS (1), . . . σS (s)}, > i 1 in the simplex T = {nσ 0 (0) nσ 0 (1) nσ 0 (2) nσ 0 (3) } is the set T

T

T

T

: Lr+1 (T ) = x ∈ T : λσ 0 (i) (x) ∈ {0, T

; 1 2 r , ,..., , 1}, 0 ≤ i ≤ 3 . r +1 r +1 r +1

Let us consider the principal lattice Lr+1 (T ), the multi-index α ∈ I(4, r), and a k-subsimplex S of T . We denote by vi the (Cartesian) coordinates of the node ni in R3 . The small simplex {α, S} is the k-simplex that belongs to the small tetrahedron

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v3

v2 v0 v1

Fig. 1 Visualization (on the left) of all the small simplices associated to the principal lattice of degree r + 1 = 3 in the tetrahedron T = {n0 n1 n2 n3 }. The same set of small simplices in a fragmented visualization (on the right). A small tetrahedron in blue, a small face in green, a small edge in red, and a small nodes in brown (in black, the “twin” node of the brown one in the fragmented view)

 with barycenter at the point of coordinates 3i=0 [( 14 + αi )vσ 0 (i) ]/(r + 1), which is T parallel and 1/(r + 1)-homothetic to the (big) sub-simplex S of T . For r + 1 = 3, in Fig. 1, the blue volume is the small tetrahedron {(0, 0, 0, 2), T }, the green triangle corresponds to the small face {(0, 1, 0, 1), {n1 n2 n3 }}, the red segment indicates the small edge {(0, 1, 1, 0), {n0 n1 }} and all the nodes of L3 (T ) are small nodes. − Λk (T ), with 0 ≤ k ≤ 3 Definition 2 The weights of a polynomial k-form u ∈ Pr+1 and r ≥ 0, are the scalar quantities  u, (3) {α,S}

on the small simplices {α, S} with α ∈ I(4, r) and S ∈ Δk (T ). − Λk (T ) along all the small simplex The weights (3) of a Whitney k-form u ∈ Pr+1 {α, S} of a mesh T are unisolvent, as stated in [5, Proposition 3.14]. Since the result on unisolvence holds true also by replacing T with F ∈ Δn−1 (T ) − then TrF u ∈ Pr+1 Λk (F ) is uniquely determined by the weights on small simplices − in F . It thus follows that a locally defined u, with u|T ∈ Pr+1 Λk (T ) and singlek valued weights, is in H Λ (Ω). We thus can use the weights on the small simplices − {α, S} as dofs for the fields in the finite element space Pr+1 Λk (T ) being aware that their number is greater than the dimension of the space. These dofs have again a meaning as cochains and this relates directly the matrix describing theexterior derivative with an incidence matrix. The key point is the Stokes’ theorem C du =  − k ∂C u , where u is a (k−1)-form and C a k-chain. More precisely, if u ∈ Pr+1 Λ (T ) − k+1 then z = du ∈ Pr+1 Λ (T ) and



 {α,S}

z=

{α,S}

 du =

u= ∂{α,S}

{β,F }

 B{α,S},{β,F }

{β,F }

u

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being B the incidence matrix with as many rows as small simplices of dimension k and as many columns as small simplices of dimension k − 1. The small simplices {α, S} inherit the orientation of the simplex S so the coefficient B{α,S},{β,F } is equal to the coefficient BS,F of the incidence of face F on simplex S if β = α and zero otherwise. This is straightforward if dim(F ) > 0 but also when dim(F ) = 0, as illustrated in Fig. 1 (right). The brown dot (on the left) denotes the small node with barycentric coordinates (2, 0, 1, 0) in T . In the fragmented visualization, this small node in the notation {α, n} becomes {(2, 0, 0, 0), n2 } (the brown dot) in {(2, 0, 0, 0), T } and {(1, 0, 1, 0), n0 } (the black dot) in {(1, 0, 1, 0), T }. Acknowledgements Ana Alonso Rodríguez thanks the Laboratoire de Mathématiques J.A. Dieudonné, Université Côte d’Azur, Nice, France, where this work started.

References 1. D.N. Arnold, Spaces of finite element differential forms, in Analysis and Numerics of Partial Differential Equations. Springer INdAM Series, vol. 4 (Springer, Berlin, 2013), pp. 117–140 2. D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006) 3. A. Bossavit, Generating Whitney forms of polynomial degree one and higher. IEEE Trans. Magn. 38, 341–344 (2002) 4. A. Bossavit, Computational Electromagnetism (Academic, San Diego, CA, 1998) 5. S.H. Christiansen, F. Rapetti, On high order finite element spaces of differential forms. Math. Comput. 85, 517–548 (2016) 6. J.-C. Nédélec, Mixed finite elements in R3 . Numer. Math. 35, 315–341 (1980) 7. F. Rapetti, A. Bossavit, Geometrical localization of the degrees of freedom for Whitney elements of higher order. IEE Proc. Sci. Meas. Technol. 1, 63–66 (2007). Issue on “Computational Electromagnetism” 8. F. Rapetti, A. Bossavit, Whitney forms of higher degree. SIAM J. Numer. Anal. 47, 2369–2386 (2009) 9. P.-A. Raviart, J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975). Lecture Notes in Mathematics, vol. 606 (Springer, Berlin, 1977), pp. 292–315 10. H. Whitney, Geometric Integration Theory (Princeton University Press, Princeton, NJ, 1957)

Model Order Reduction of an Elastic Body Under Large Rigid Motion Ashish Bhatt, Jörg Fehr, and Bernard Haasdonk

Abstract A parametrized equation of motion in the absolute coordinate formulation is derived for an elastic body with large rigid motion using continuum mechanics. The resulting PDE is then discretized using linear FEM which results in a high dimensional system. Such high dimensional systems are expensive to solve especially in multi-query settings. Therefore, the system is reduced using a reduced order basis and we investigate the error introduced due to the reduction step. Simulations illustrate the efficacy of the procedure for a pendulum example.

1 Introduction Model order reduction (MOR) of nonlinear mechanical systems is a technique of reducing the computational complexity of the corresponding mathematical model by reducing associated degrees of freedom and thereby obtaining a model of reduced dimension. This reduced model can then be simulated efficiently in multi-query scenarios. The gain in simulation efficiency often comes at the cost of the low fidelity of the reduced model. It is important to certify the reduction method by estimating the error incurred to use the simulation for management decisions and engineering applications. In this paper, we derive the equation of motion of an elastic body under large rigid motion in the absolute coordinate formulation (ACF) using principles of continuum mechanics [2, 7]. For this purpose, the motion of the elastic body is decomposed into a large rigid body displacement and a small elastic displacement with respect to some fixed inertial frame. The principle of virtual work is then used to derive a

A. Bhatt () · B. Haasdonk University of Stuttgart, IANS, Stuttgart, Germany e-mail: [email protected]; [email protected] J. Fehr University of Stuttgart, ITM, Stuttgart, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_23

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second order nonlinear differential equation, with constant mass matrix, governing the motion of the body with respect to the inertial frame. Constraints to the elastic body are modeled with additional algebraic equations. The overall motion of the body is therefore governed by a differential algebraic equation (DAE) system. The floating frame of reference formulation (FFRF) is another widely used formulation in the simulation of mechanical systems [7]. In contrast to ACF, FFRF uses a detached local reference frame, referred to as the floating frame, for each elastic body in the system. The displacement of a body is then decomposed into local elastic displacement with respect to its floating frame and the global rigid body displacement with respect to the inertial reference frame of the system. As a consequence of the interaction between these local and global displacements, one therefore obtains a nonlinear equation of motion with nonlinear mass matrix. Because of the inversion at every step of time integration, this nonlinear mass matrix has practical ramifications in the simulation and reduction of the system. If one wants to avoid repeated nonlinear matrix inversions, the ACF is advantageous for the purpose of computational efficiency. This paper uses ACF instead of FFRF, see [2, 3, 6] for a comparison of the two approaches. The parametric equation obtained from spatial discretization of either one of the two aforementioned formulations is often very high dimensional and therefore it is computationally prohibitive to solve it especially in multi-query scenarios. One therefore needs a reduction strategy to simulate the system dynamics efficiently. An overview of DAE model order reduction can be found in e.g. [1]. We are interested in the reduction and estimation of the error incurred therein. For an overview of error estimators for DAEs and ODEs, see e.g. [1, 8]. In the next section, we derive the PDE governing the motion of an elastic body from the principles of continuum mechanics and derive the time continuous equations using FEM. In Sect. 3, we discuss MOR and error estimation of the resulting system. We present simulation results in Sect. 4 and conclude in Sect. 5.

2 Absolute Coordinate Formulation (ACF) Consider an elastic body with rigid and elastic degrees of freedom occupying the space Ω ⊂ Rd , d ∈ {2, 3}. We derive the equation of motion of the body with large rigid body motion and small strain along the lines of [2, 3, 6]. Let u, u0 , uf lex : + × (0, T ) → Rd be such that the total displacement u consists of the rigid body displacement u0 and the flexible (i.e. elastic) displacement uf lex of a point x ∈ Ω on the body, see Fig. 1. The rigid body displacement u0 is the sum of x-independent translational part ut : (0, T ) → Rd and x-dependent rotational part uR : + × (0, T ) → Rd , respectively. Then from vector calculus it follows u(x, t) = ut [u](t) + uR [u](x, t) + uf lex (x, t)

(1)

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Fig. 1 Elastic and rigid body displacements in an elastic body. The Elastic displacement is exaggerated for the clarity of visualization

with   uR [u](x, t) = R[u](t) − I x and uf lex (x, t) = R[u](t)uf lex  (x, t). The matrix function R : (0, T ) → Rd×d represents the rotation of the floating frame X of the body with respect to the inertial frame X and we explicitly denote the dependence of ut , uR , and R on u by enclosing the latter within square brackets e.g. y[u](·, ·). The vector uf lex  denotes the flexible displacement with respect to X . The displacement gradient ∇u then reads ∇u(x, t) = R[u](t) − I + ∇uf lex (x, t) and results in the linearized corotated strain tensor, also referred to as reduced strain tensor,  T 1  8 ∇u(x, t) + ∇u(x, t) − I E(x, t) = 2  T 1  = ∇uf lex (x, t) R[u](t) + R T [u](t)∇uf lex (x, t) 2   = sym R T [u](t)∇uf lex (x, t) . Using the principle of virtual work [5], the resultant differential algebraic equation of motion of the elastic body then reads 



8 : E : δE 8 d+ = E

ρ u¨ · δu d+ + +

+



 +

g s · δu d+ +

gt · δu d+, ∂+

(2)

C(u) = 0. Here ρ is the constant mass density of the elastic body, g s (x, t) is the source function, gt (x, t) is the traction, C denotes the accompanying constraints (e.g. pivots or joints), over dot denotes a time derivative as usual, δu denotes variation in u, ∂Ω is the boundary of the region Ω, and we have omitted space and time

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8 can be dependence in the interest of brevity of exposition. The variation δ E computed using   8 = sym R T [u]δ∇u + δR T [u]∇uf lex δE with δR T [u] =

dR T [u] δu du

and the fourth order elasticity tensor E is defined as E = λL I ⊗ I + μL I

(3)

with Lamé constants λL , μL ∈ R, identity matrix I , tensor product ⊗, and the fourth order symmetric identity tensor (I)ij kl = δik δj l + δil δj k . Here δij are the Kronecker delta functions. We remark that Eq. (3) is only valid for hyperelastic anisotropic materials and additional constitutive equations are needed for more complex materials. We now discretize Eq. (2) to derive the semi-discretized system of equations. To this end, substituting the finite element ansatz u(x, t) ≈ uh (x, t) = N (x)q(t), where N (x) ∈ Rd×n is the finite element shape function matrix and q ∈ Rn is the unknown solution vector, into Eq. (1), we obtain the following discretized system M q¨ + R h (q)KR h T (q)q f lex + f nl (q) = f s + f t , Bq = 0. with  ρN T N d+,

M= +

 R h (q)KR h T (q)q f lex =

+

sym(∇uf lex ) : E : sym(δ∇uf lex ) d+,

 fs =

 T

+

N g s (x, t) d+, f t =

+

N T g t (x, t) d+,

(4)

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where M is the constant mass matrix, K is the constant stiffness matrix, f s and f t are source and traction force vectors, respectively, ⎡ ⎢ R h (q) = ⎢ ⎣

⎤

R[uh ] ..

.

⎥ ⎥ ∈ Rn×n , ⎦ R[uh ]

f nl is a nonlinear function of q, and B ∈ Rm×n is the constraint matrix resulting from the linear constraint function C of Eq. (2). Typically m  n and the matrices M and K depend on the system parameters ρ, λL , and μL . The vector q f lex is the flexible part of the solution vector q. For details of the derivation of Eq. (4) from Eq. (2), see [2, 3] for instance. Now that we have introduced the finite element discretization, let us return briefly to Eq. (1) at this point. There is an inherent implicitness in this equation as both rigid and elastic displacements depend on the total displacement and vice-versa. This mutualism in ACF is a source of ambiguity in the solution of the resulting governing Eq. (2). To extract the displacement of the rigid body motion u0 from the total displacement u, one can use a combination of the rigid body displacements of the nodes in the finite element mesh of the body and their gradients as discussed in [2, 3]. We will use rigid displacements of two maximally distant nodes of the mesh to determine the rigid body motion.

3 MOR of Eq. (4) Numerically solving the parametrized high dimensional system (4) is often prohibitively computationally expensive. Therefore, a model of reduced dimension is desirable. Reducing a DAE is particularly challenging as the reduced system does not preserve the constraints in general. In this exposition, we will use the first order formulation of the DAE for the purpose of MOR. To this end, let us rewrite Eq. (4) as a first order DAE with Lagrange multiplier λ ∈ Rm as q˙ = M −1 p, p˙ = −R h KR h T q f lex − B T λ + f (q), Bq = 0, Bp = 0, q(0) = q 0 , p(0) = p 0

(5)

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Here f = f s + f t − f nl (q) and q 0 , p0 are the initial conditions. We can further rewrite Eq. (5) in a more compact form as a system of order 2n y˙ = g(y, λ), By = 0,

(6)

y(0) = y 0 , T

where y = [q T , p T ] ∈ R2n , 

 B 0 B= , 0 B and the function g denotes the right hand side of the differential part of Eq. (5). In this paper, we use a projection based model order reduction technique, see [4] for an overview of the reduced basis methods. Model order reduction methods rely on the assumption that a high dimensional solution manifold can be well approximated by a manifold of much lower dimension. A basis which projects the high dimensional manifold onto the low dimensional one is referred to as a reduced (order) basis. For given reduced bases V , W ∈ R2n×k , with 2n 3 k, such that y ≈ y r = V z and W T V = I ∈ Rk×k , Eq. (6) can be reduced to the DAE, for an unknown vector z ∈ Rk , ˜ z˙ = W T g(V z, λ), BV z = 0,

(7)

z(0) = z0 . with z0 = W T y 0 and λ˜ ∈ Rm being the Lagrange multiplier. This surrogate model of reduced dimension k can then be simulated with an efficient DAE solver, e.g. leapfrog or shake-rattle algorithm. Note, however, that the reduced complexity still is dependent on the full dimension 2n as a reconstruction step is required for R h evaluations. This is typical of POD approaches that do not make use of some hyperreduction strategy, e.g. DEIM. This reduced complexity, however, comes at the cost of low fidelity. The error thus incurred is detrimental to the cause of MOR, having a bound on the error therefore improves reliability of the reduced model. Let us now derive an equation for approximating error in the solution of full system Eq. (6) due to the simulation of the alternative reduced order model Eq. (7). To this end, let us define the error in the solution y to be e(t) = y(t) − y r (t) so that e˙ (t) = g(y(t)) − V W T g(y r (t)), e(0) = y(0) − V W T y(0).

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In [8], the following error estimator was derived e(t)G ≤ Δ(t), for all t ∈ [0, T ]

(8)

with  Δ(t) =

t

α(s)e

t s

β(τ )dτ

ds + e

t 0

β(τ )dτ

e(0)G

0

where α(t) = (I − V W T )g(y r (t))G and β(t) = LG [g](y r (t)). Here  · G is some appropriate norm and LG [g](y(t)) is the local logarithmic Lipschitz constant defined by LG [g](y) = sup x∈Rd

x − y, g(x) − g(y)G . x − y2G

(9)

As an alternative to the estimator in Eq. (8), we propose to approximate the time derivative of the equation with a Taylor expansion i.e. e˙ (t) = y˙ − V z˙ = g(y) − V W T g(V z) 1 = ∇g(V z)e(t) + (I − V W T )g(V z) + ∇ 2 g(ξ )e(t) ⊗ e(t), 2

(10)

where ξ ∈ B(V z, e) and ⊗ denotes the tensor product. Both Eqs. (8) and (10) assume that the solution satisfies the constraint By = 0 and therefore lies on the manifold defined by the constraint. While the former requires evaluations of an expensive logarithmic norm, the latter requires solving a high dimensional ODE.

4 Pendulum Example Consider a thin 2D elastic beam of slenderness ratio 10 made of steel which is free to rotate about a fixed end. We discretize the beam with an irregular triangular mesh with 93 nodes each with two (x and y) flexible degrees of freedom. The beam is subjected to gravity and traction and is set into motion from an initial angle of 45 degrees from the vertical with zero initial velocity. By simulating the system with the leapfrog integrator, we produce a reduced basis using weighted proper orthogonal decomposition, see e.g. Prop. 2.90 in [4], with weighted norm   K 0 G= 0 M

A. Bhatt et al.

pivot disp. error

max disp. error

276 10 -6

1 0.5 0 0

0.1

0.2

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

10 -10

2 1 0 0

time

Fig. 2 Maximum absolute error in the beam nodal displacements (top) and absolute error in the pivot displacement (bottom)

from a training set of density parameters ρ ∈ [7854, 7855, 7857, 7858], test it on the parameter ρ = 7856, and obtain the maximum absolute error (max(q − q r )) as shown in Fig. 2 for final time t = 0.5. Although the error starts out very small, one can see a growing trend in it as the time increases. We plot the displacement in the fixed point of the beam in the lower part of the same figure and observe a much smaller error in the reproduction of the displacement of the fixed point.

5 Conclusions We have derived and discretized using FEM the equation of motion of a constrained elastic body with large rigid motion in absolute coordinate formulation. The resulting parametrized DAE is then reduced with a POD basis. An ODE is derived whose solution bounds the error due to MOR. The results are illustrated with a beam example which demonstrate the efficacy of the reduction technique. MOR and error estimation of nonlinear DAEs require further investigation. This work can be extended to MOR of coupled elastic bodies. Acknowledgements The authors gratefully acknowledge the support of DFG grants FE1583/21 and HA5821/5-1. The authors are also thankful to Patrick Buchfink and Dennis Grunert for constructive discussions.

References 1. P. Benner, T. Stykel, Model Order Reduction for Differential-Algebraic Equations: A Survey. Surveys in Differential-Algebraic Equations IV (Springer, Berlin, 2017), pp. 107–160 2. J. Gerstmayr, J. Ambrósio, Component mode synthesis with constant mass and stiffness matrices applied to flexible multibody systems. Int. J. Numer. Methods Eng. 73(11), 1518–1546 (2008)

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3. J. Gerstmayr, J. Schöberl, A 3D finite element method for flexible multibody systems. Multibody Syst. Dyn. 15(4), 305–320 (2006) 4. B. Haasdonk, Reduced basis methods for parametrized PDEs – a tutorial introduction for stationary and instationary problems, in Model Reduction and Approximation: Theory and Algorithms, ed. by P. Benner, A. Cohen, M. Ohlberger, K. Willcox (SIAM, Philadelphia, 2017), pp. 65–136 5. C. Lanczos, The Variational Principles of Mechanics (Courier Corporation, North Chelmsford, 2012) 6. A. Pechstein, D. Reischl, J. Gerstmayr, A generalized component mode synthesis approach for flexible multibody systems with a constant mass matrix. J. Comput. Nonlinear Dyn. 8(1), 011019 (2013) 7. A.A. Shabana, Dynamics of Multibody Systems (Cambridge University Press, Cambridge, 2013) 8. D. Wirtz, D. Sorensen, B. Haasdonk, A posteriori error estimation for DEIM reduced nonlinear dynamical systems. SIAM J. Sci. Comput. 36(2), A311–A338 (2014)

On Surface Area and Length Preserving Flows of Closed Curves on a Given Surface Miroslav Koláˇr, Michal Beneš, and Daniel Ševˇcoviˇc

Abstract In this paper we investigate two non-local geometric geodesic curvature driven flows of closed curves preserving either their enclosed surface area or their total length on a given two-dimensional surface. The method is based on projection of evolved curves on a surface to the underlying plane. For such a projected flow we construct the normal velocity and the external nonlocal force. The evolving family of curves is parametrized by a solution to the fully nonlinear parabolic equation for which we derive a flowing finite volume approximation numerical scheme. Finally, we present various computational examples of evolution of the surface area and length preserving flows of surface curves. We furthermore analyse the experimental order of convergence. It turns out that the numerical scheme is of the second order of convergence.

1 Introduction In this article we discuss a motion of closed and nonselfintersecting curves Gt , t ≥ 0, on a given two dimensional surface M ⊂ R3 . We suppose that the surface M is represented by the graph of a given smooth function ϕ : R2 → R and the curve Gt is evolved in the outer normal direction by the following nonlocal geometric evolution

M. Koláˇr () · M. Beneš Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic e-mail: [email protected]; [email protected] D. Ševˇcoviˇc Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská Dolina, Bratislava, Slovakia e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_24

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equation: VG = −KG + F ,

(1)

where Gt |t =0 = Gini ⊂ M is the initial condition—a C 1 smooth Jordan curve, VG is the normal velocity, KG is the geodesic curvature of the curve Gt , and F is the nonlocal force term. We present results of the surface area preserving geodesic curvature flow with the force FA , and the length preserving geodesic curvature flow with the force FL . The surface area preserving flow has been analysed in the recent paper [1]. In this paper we furthermore investigate the length preserving flow of surface curves. We compare both types of preserving flows. Recall that the length L(Gt ) and the enclosed surface area A(Gt ) of a surface closed curve Gt satisfy the following identities:  d L(Gt ) = KG VG dS, dt Gt

 d A(Gt ) = VG dS, dt Gt

(see e.g. [1]). With help of these identities the nonlocal forces given by  1 FA = KG dS, L(Gt ) Gt

 G FL =  t Gt

KG2 dS KG dS

,

represent the surface area and length preserving flows, respectively. The constrained motion driven by (1) is a generalization of the geometric motion in the plane, which is broadly discussed in the literature (see, e.g., [2–6] for the area-preserving flow or, e.g., [7] for the length-preserving flow). In general, the physical context of moving interfaces driven by the curvature is also discussed in [8, 9] within the context of the Allen-Cahn equation [10, 11] or within the context of the recrystallization effects (see [12]).

2 Parametric Description and Projection to the Plane In accordance with [1, 13], the geometric motion law (1) is treated by means of the vertical projection Γt of a surface curve Gt to the plane, i.e., Gt = {(X, ϕ(X)T : X ∈ Γt )}. Here Γt denotes the time-dependent closed projected planar curve moving in the normal direction (see Fig. 1). Then, Γt is described by the position vector X = X(u, t), u ∈ [0, 1], where u is a parameter from a fixed interval, and X is required to be 1-periodic in u.

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Fig. 1 An example of a curve Gt with an outer normal vector N on a given surface M = graph(ϕ) and its projection Γt to the underlying plane R2 (see [1])

Similarly to [1, 14, 15], one can derive a system of governing equations for the parametrization X(u, t) of the curve Γt , provided that Gt evolves on the surface M 1  2 1+|∇ϕ|2 vΓ where in the normal direction by the velocity VG . Then VG = 1+(∇ϕ·t 2 Γ) vΓ is the normal velocity of the projected curve Γt (see [14]). We assume the parametrization X is oriented counter-clockwise, and that the periodic boundary conditions for X at u = 0 and u = 1 are imposed, i.e., X|u=0 = X|u=1 and ∂u X|u=0 = ∂u X|u=1 . Then, the unit tangential vector tΓ , the outer unit normal vector nΓ , and the curvature κΓ of Γt are expressed in terms of X as the following tΓ =

∂u X , |∂u X|

nΓ =

∂u X⊥ , |∂u X|

κΓ = −

1 ∂ |∂u X| ∂u



∂u X |∂u X|

 · nΓ .

Here a · b denotes the Euclidean inner product of vectors a and b. For a curve Gt on the surface M, its geodesic curvature can be expressed in terms of properties of Γt as follows (see [1, 14]):

KG =

tTΓ ∇ 2 ϕ tΓ (∇ϕ (1+|∇ϕ|2 )1/2 (1 + (∇ϕ · tΓ )2 )3/2

(1 + |∇ϕ|2 )1/2 κΓ −

· nΓ )

.

Having this geometrical framework, one can construct a geometric equation for the normal velocity vΓ of Γt as vΓ = β(X, nΓ , κΓ ), where β is the normal component of the velocity of the planar curve Γt , i.e., β = ∂t X · nΓ . For technical details on the derivation of the following system of equations, we refer the reader to, e.g., [1]. The curve Gt evolves according to the motion law (1) provided the parametrization X(u, t) of the projected curve Γt satisfies the following system of nonlinear parabolic equations: 1 ∂ ∂t X = a |∂u X| ∂u



∂u X |∂u X|



  ∂u X⊥ , + b + cF |∂u X|

(2)

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subject to the initial condition X|t =0 = Xini , and tTΓ ∇ 2 ϕ tΓ (∇ϕ · nΓ ) 1 , , b = 1 + (∇ϕ · tΓ )2 (1 + (∇ϕ · tΓ )2 )(1 + |∇ϕ|2 ) 1  1 + (∇ϕ · tΓ )2 2 c= . 1 + |∇ϕ|2

a=

3 Numerical Solution In our approach the projected planar curve Γt is replaced by a piece-wise linear curve, and for spatial discretization, the technique of flowing finite volume method is used. The principle of the method can be found in, e.g., [1, 2, 16]. The method was successfully applied in, e.g., dislocation dynamics [17], image processing [18] or computational geometry [2]. The method is based on positioning of discrete nodes Xi = X(ui , t) for i = 0, 1, . . . , M, along the curve Γt . Then, linear segments connecting the neighboring nodes represent the finite volumes. We denote the length of a finite volume as di = |Xi − Xi−1 | for i = 1, 2, . . . , M, where X0 = XM . Additionally, we denote ϕi = ϕ(Xi ), and Di = |(Xi , ϕi ) − (Xi−1 , ϕi−1 )| as the length of the segment of the discretized curve Gt . The approximation of the unit tangent a normal vectors is as follows: tj =

Xj +1 − Xj −1 , dj +1 + dj

nj =

⊥ X⊥ j +1 − Xj −1

dj +1 + dj

,

and the discrete geodesic curvature is calculated as:

Ki =

tTi ∇ 2 ϕi ti (∇ϕi (1+|∇ϕi |2 )1/2 (1 + (∇ϕi · ti )2 )3/2

(1 + |∇ϕi |2 )1/2 κi −

· ni )

.

(3)

Finally, the semidiscrete scheme for solving (2) then reads as follows:   (X⊥ − X⊥ dXi di+1 + di Xi − Xi−1 Xi+1 − Xi i−1 ) =ai , − + (bi + ci F ) i+1 dt 2 di+1 di 2 (4) tTj ∇ 2 ϕj tj (∇ϕj · nj ) 1 aj = , bj = , 1 + (∇ϕj · tj )2 (1 + (ϕj · tj )2 )(1 + |∇ϕj |2 ) 1  1 + (∇ϕj · tj )2 2 cj = , 1 + |∇ϕj |2

(5)

(6)

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satisfying the initial condition Xi (0) = Xini (ui ) for i = 1, 2, . . . , M and F = FA in the case of the surface area-preserving flow and F = FL in the case of lengthpreserving flow (see [1]). The terms FA and FL are given by 1

FA = M

M

j =1 Dj j =1

Dj +1 + Dj , Kj 2

M FL =

2 Dj+1 +Dj j =1 Kj 2 M Dj+1 +Dj j =1 Kj 2

.

4 Computational Experiments We present our qualitative and quantitative results of computational studies for the surface area-preserving and length-preserving flows of closed curves evolving on a surface driven by (1). Both problems are treated by the numerical scheme (4)–(6) for parametric equation (2). In the following examples we demonstrate how solutions of (1) evolve in time and converges towards stationary curves. For the quantitative analysis, we measure the experimental orders of convergence (EOC) for our numerical scheme. We perform evaluation of EOC in such a way that the conserved quantities—the surface area A(Gt ) enclosed by the curve Gt and the length L(Gt ) of the curve Gt serve as testing parameters for computations of EOCs. In the case of the surface area-preserving flow, we evaluate differences given by the area at the initial time A(Gini ), and the areas A(GTi ) at given data output times Ti , i = 1, . . . , N, i.e., ei = |A(GTi ) − A(Gini )|. For the length-preserving flow, the differences between the initial length L(Gini ) and lengths L(GTi ) were measured for same time levels Ti , i.e., ei = |L(Gini ) − L(GTi )|. Considering a mesh with M segments, the following maximum and discrete L1 (with time stepping Δtk ) norms of errors depending on the number of finite volumes M are evaluated as follows: errormax (M) =

max

k=1,2,...,N

N 1 errorL1 (M) = ek . TN

ek ,

k=1

The order of convergence of the scheme (4)–(6) between two meshes with M1 and M2 volumes is estimated as  EOC =

ln

errorI (M1 ) errorI (M2 )



ln

M2 M1



 ,

I ∈ {max, L1 }.

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1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0.5

0

-0.5

1 -1

0.5 -0.5

0

0 0.5

-0.5 1 -1

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

-1 -1

-0.5

0

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1.5 1 0.5 0 -0.5 -1

-1.5 -1

-0.5 0 0.5 1 1.5

1 0 0.5 -1 -0.5

-1.5 -1.5

-1

-0.5

0

0.5

1

1.5

Fig. 2 Examples of the length-preserving flow of the surface curve Gt on two different surfaces M and its projection Γt to plane. The EOCs are shown in Table 1

In the following computational experiments shown in Figs. 2 and 3, we investigate the length an surface area preserving flows driven by (1) on the surface M given by a graph of the function ϕ(x, y) = x 2 − y 4 (top) and ϕ(x, y) = sin(πy) (bottom). In both sets of examples, we chose a dumbbell shaped initial curve and its rotation by 90◦ given by the parametrization Xini (u) = (sin(2πu), −(sin(2πu)2 + 0.1) cos(2πu)), u ∈ [0, 1]. We also computed EOCs for the length (Table 1) and surface area preserving flow (Table 2). In both experiments the EOC is approximately 2 which indicates that the numerical scheme is of the second order of experimental convergence.

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1

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0.5

0

-0.5

1 -1

0.5 -0.5

0

0

-0.5

0.5

-1

1-1

-1

-0.5

0

-0.5

0

1

0.5

1

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0.5

0

-0.5

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1 -1 -1

1

0.5

Fig. 3 Examples of the surface area-preserving flow of the surface curve Gt on two different surfaces M and its projection Γt to plane. The EOCs are shown in Table 2

Table 1 EOCs for the length-preserving flow depicted in Fig. 2

M errormax EOC errorL1 The surface M with ϕ(x, y) = x 2 − y 4 100 2.45246 ·10−3 – 2.45246 ·10−3 −4 200 6.06811 ·10 2.0149 5.84099 ·10−4 −4 300 2.69402 ·10 2.0026 2.59340 ·10−4 −4 400 1.51633 ·10 1.9978 1.45975 ·10−4 −5 500 9.71800 ·10 1.9937 9.35550 ·10−5 The surface M with ϕ(x, y) = sin(πy) 100 4.53052 ·10−3 – 4.30824 ·10−3 −3 200 1.15305 ·10 1.9742 1.09608 ·10−3 −4 300 5.15109 ·10 1.9873 4.89453 ·10−4 −4 400 2.90710 ·10 1.9885 2.76130 ·10−4 −4 500 1.86633 ·10 1.9861 1.77203 ·10−4

EOC – 2.0143 2.0024 1.9977 1.9936 – 1.9747 1.9884 1.9897 1.9878

286 Table 2 EOCs for the surface area preserving flow depicted in Fig. 3

M. Koláˇr et al. M errormax EOC errorL1 The surface M with ϕ(x, y) = x 2 − y 4 100 3.7837 ·10−4 – 3.8695 ·10−4 200 9.4747 ·10−5 1.9976 9.6895 ·10−5 300 4.2245 ·10−5 1.9920 4.3195 ·10−5 400 2.3870 ·10−5 1.9843 2.4400 ·10−5 500 1.5365 ·10−5 1.9741 1.5701 ·10−5 The surface M with ϕ(x, y) = sin(πy) 100 4.82926 ·10−3 – 4.63840·10−3 200 1.23896 ·10−3 1.9626 1.19223·10−3 300 5.51443 ·10−4 1.9964 5.30688·10−4 400 3.10376 ·10−4 1.9978 2.98703·10−4 500 1.98734 ·10−4 1.9979 1.91262·10−4

EOC – 1.9976 1.9925 1.9852 1.9757 – 1.9599 1.9962 1.9977 1.9978

5 Conclusions We studied the length and surface area preserving non-local geometric flows driven by the geodesic curvature and external force. We applied a projection method for a flow of surface curves into the underlying plane. We presented a formula for the normal velocity of a projected flow and we proposed a numerical discretization scheme. The scheme is based on the flowing finite volume method resulting in a semi-discrete scheme which can be solved by the method of lines. We presented results of computation of the length and surface area preserving flows. We also performed quantitative analysis of the experimental order of convergence of the numerical method showing the second order of convergence. Acknowledgements Miroslav Koláˇr and Michal Beneš were partly supported by the project No. 14-36566G of the Czech Science Foundation and by the project No. OHK4-001/17 201719 of the Student Grant Agency of the Czech Technical University in Prague. The third author was supported by the VEGA grant 1/0062/18.

References 1. M. Koláˇr, M. Beneš, D. Ševˇcoviˇc, Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete Contin. Dyn. Syst. - Ser. B 22(10), 367–3689 (2017) 2. M. Koláˇr, M. Beneš, D. Ševˇcoviˇc, Computational analysis of the conserved curvature driven flow for open curves in the plane. Math. Comput. Simul. 126, 1–13 (2016) 3. I.C. Dolcetta, S.F. Vita, R. March, Area preserving curve shortening flows: from phase separation to image processing, Interfaces Free Bound. 4, 325–343 (2002) 4. M. Gage, On an area-preserving evolution equation for plane curves. Contemp. Math. 51, 51–62 (1986) 5. C. Kublik, S. Esedo¯glu, J.A. Fessler, Algorithms for area preserving flows. SIAM J. Sci. Comput. 33, 2382–2401 (2011)

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6. J. McCoy, The surface area preserving mean curvature flow. Asian J. Math. 7, 7–30 (2003) 7. D. Ševˇcoviˇc, S. Yazaki, On a gradient flow of plane curves minimizing the anisoperimetric ratio, IAENG Int. J. Appl. Math. 43(3), 160–171 (2013) 8. J. Rubinstein, P. Sternberg, Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48, 249–264 (1992) 9. M. Beneš, S. Yazaki, M. Kimura, Computational studies of non-local anisotropic Allen-Cahn equation. Math. Bohem. 136, 429–437 (2011) 10. J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. III. Nucleation of a twocomponent incompressible fluid. J. Chem. Phys. 31, 688–699 (1959) 11. S. Allen, J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979) 12. I.V. Markov, Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth, and Epitaxy, 2nd edn. (World Scientific Publishing Company, Springer, 2004) 13. K. Deckelnick, Parametric mean curvature evolution with a Dirichlet boundary condition. J. Reine Angew. Math. 459, 37–60 (1995) 14. K. Mikula, D. Ševˇcoviˇc, Computational and qualitative aspects of evolution of curves driven by curvature and external force. Comput. Vis. Sci. 6, 211–225 (2004) 15. K. Mikula, D. Ševˇcoviˇc, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force. Math. Methods Appl. Sci. 27, 1545–1565 (2004) 16. D. Ševˇcoviˇc, K. Mikula, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math. 61, 1473–1501 (2001) 17. M. Beneš, J. Kratochvíl, J. Kˇrišt’an, V. Minárik, P. Pauš, A parametric simulation method for discrete dislocation dynamics. Eur. Phys. J. ST 177, 177–192 (2009) 18. M. Beneš, M. Kimura, P. Pauš, D. Ševˇcoviˇc, T. Tsujikawa, S. Yazaki, Application of a curvature adjusted method in image segmentation. Bull. Inst. Math. Acad. Sinica (N. S.) 3, 509–523 (2008)

Derivation of Higher-Order Terms in FFT-Based Numerical Homogenization Felix Dietrich, Dennis Merkert, and Bernd Simeon

Abstract In this paper, we first introduce the reader to the Basic Scheme of Moulinec and Suquet in the setting of quasi-static linear elasticity, which takes advantage of the fast Fourier transform on homogenized microstructures to accelerate otherwise time-consuming computations. By means of an asymptotic expansion, a hierarchy of linear problems is then derived, whose solutions are looked at in detail. It is highlighted how these generalized homogenization problems depend on each other. We extend the Basic Scheme to fit this new problem class and give some numerical results for the first two problem orders.

1 Introduction Numerical homogenization deals with the efficient computation of macroscopic quantities, the so-called effective properties, by solving microstructural problems in representative volume elements (RVEs). Based on the assumption of a periodic microstructure, efficient FFT-based algorithms [2, 7, 8] can be applied for this purpose. They have recently been shown to be very competitive, taking advantage of imaging data, i.e. pixels or voxels, as computational mesh. We study an extension of this methodology that includes higher-order derivatives of the macroscopic quantities, with the aim of attaining higher accuracy for the microscopic solutions and the effective properties in this way. This idea has been introduced by Boutin [1], and we discuss here the algorithmic treatment in a unified framework. For the state-of-the-art in FFT-based numerical homogenization, we mention the work on the variational scheme based on the Hashin-Shtrikman energy principle [2], the polarization scheme [7], and the extension to non-linear problems, such as elasto-plasticity [10] or elasto-viscoplasticity [3].

F. Dietrich () · D. Merkert · B. Simeon Technische Universität Kaiserslautern, Kaiserslautern, Germany e-mail: [email protected]; [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_25

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The paper is organized as follows. We start with a compact summary of the so-called Basic Scheme by Moulinec and Suquet [8] and its CG-formulation by Vondˇrejc [12]. Then we derive a class of generalized homogenization problems [11] that include higher-order terms and show how the Basic Scheme can be easily extended for this case. The paper closes with numerical examples and comparisons.

2 FFT-Based Homogenization We examine the problem of periodic quasi-static linear elasticity in a representative 5 d volume element. The d-dimensional torus Td Rd /Zd ∼ = − 1 , 1 is chosen as 2 2

reference domain to enforce the periodicity. For x ∈ Td , the microstructure is (in a weak sense) characterized by the strain-displacement equation (x) =

  T  1 ∇u(x) + ∇u(x) , 2

(1)

a constitutive equation in form of Hooke’s law   σ (x) = C(x)∶ (x) + E ,

(2)

with ∶ being the double-dot product, and the balance of momentum ∇ · σ (x) = 0 ,

(3)

where we assume that no external forces are applied. For a given stiffness  d×d×d×d distribution C ∈ L∞ Td with major and minor symmetries and a sym

prescribed symmetric macroscopic strain tensor E ∈ Rd×d sym , the problem has a unique weak solution for the displacement u ∈ H 1 (Td )d , with strain and stress  d×d  , such that the mean value Td u(x) dx is equal to tensors (u), σ (u) ∈ L2 Td sym

zero; see [5]. By introducing a regular isotropic reference tensor C 0 ∈ Rd×d×d×d , whose sym 0 0 entries are given in the case d = 3 for the Lamé parameters λ ∈ R and μ ∈ R\{0} as   Cij0 kl = λ0 δij δkl + μ0 δik δj l + δil δj k ,

i, j, k, l ∈ {1, 2, 3},

with δij being the Kronecker-Delta, we can rewrite (2) and (3) as   ∇ · C 0 ∶(x) + τ (x) = 0

(4)

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291

  with the polarization term τ (x) C(x) − C 0 ∶(x) + C(x)∶E. The solution of (4) is given by the periodic Lippmann-Schwinger equation [8]   (x) = − Γ 0 ∗ τ (x)

5

6 − Γˆ 0 (ξ )∶τˆ (ξ ) exp(iξ · x) ,

(5)

ξ =0 d ξ ∈(2πZ)

where the Fourier coefficients of the Green strain operator Γ 0 are given as Γˆij0 kl (ξ ) =

1 λ 0 + μ0 ξ i ξ j ξ k ξ l , 7 72 (δki ξl ξj +δli ξk ξj +δkj ξl ξi +δlj ξk ξi )− 0 0 7 7 μ (λ + 2μ0 ) 7ξ 74 4μ0 7ξ 7

for frequencies ξ = 0. While (5) can directly be used for an iterative solution scheme called Basic Scheme, it is preferred to bring all terms containing the strain  to the left-hand side, resulting in the equation      Id + Γ 0 (x) ∗ C(x) − C 0 ∶(x) = −Γ 0 (x) ∗ C(x)∶E .

(6)

The advantage of this formulation is that Krylov subspace methods such as the CG method are directly applicable; see [12]. The resulting CG-version of the Basic Scheme then reads as follows. Although different convergence criteria 7 7 are possible, we will stick to a simple Cauchy criterion of the form 7n+1 − n 7L2 / 0 L2 < TOL.

3 General Homogenization Problem of Order α Our goal is to extend the problem of quasi-static linear elasticity such that not only macroscopic strains but also macroscopic strain gradients or even higherorder derivatives can be included. This can be achieved by a scale separation as presented in [1]. A characteristic length, denoted by L and  respectively in the following, is to be associated with both the macro- and the microscale. We define the macroscopic variable Y x/L and the microscopic variable yx/, which allow for the displacement to be formally written as an asymptotic series expansion in the scale ratio κ/L such that   u(Y, y) = L u0 (Y, y) + κu1 (Y, y) + κ 2 u2 (Y, y) + κ 3 u3 (Y, y) + . . . .

(7)

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In the following, we will usually drop the dependencies on the spatial variables for the sake of better readability. However, if a quantity might only depend on either the microscopic or the macroscopic variable alone, we will denote this explicitly.  By splitting the nabla operator ∇ = L1 ∇Y + κ1 ∇y and by defining symmetric macroscopic and microscopic gradients eY (ui ) :=

1 (∇Y ⊗ ui + ui ⊗ ∇Y ) 2

and ey (ui ) :=

 1 ∇y ⊗ ui + ui ⊗ ∇y 2

for i = 0, 1, . . . , accordingly, the series expressions for the strain  and the stress σ can be derived. Inserting (7) into the strain-displacement equation (1) gives  = eY (u0 ) + κ −1 ey (u0 ) + κeY (u1 ) + ey (u1 ) + . . .

(8)

and after an application of Hooke’s law (2) we end up with σ = C : eY (u0 ) + κ −1 C : ey (u0 ) + κC : eY (u1 ) + C : ey (u1 ) + . . . .

(9)

If we furthermore insert (9) into the balance of momentum (3), we get 5 6 5 6 0 = ∇Y · C : eY (u0 ) + κ −1 ∇y · C : eY (u0 ) 5 6 5 6 + κ −1 ∇Y · C : ey (u0 ) + κ −2 ∇y · C : ey (u0 ) + . . . .

(10)

Each term ui appears in four different addends, of which one consists only of purely macroscopic derivatives, one only of purely microscopic derivatives and the remaining two consist of mixed derivatives. We introduce the notation 5 6 P 0 (ui ) := ∇Y · C : eY (ui ) , 5 6 5 6 P −1 (ui ) := ∇Y · C : ey (ui ) + ∇y · C : eY (ui ) , 5 6 P −2 (ui ) := ∇y · C : ey (ui ) . Rearranging the terms of (10) with respect to the exponent of κ leads to the expression 6 6 5 5 0 = κ −2 P −2 (u0 ) + κ −1 P −2 (u1 ) + P −1 (u0 ) 5 6 + κ 0 P −2 (u2 ) + P −1 (u1 ) + P 0 (u0 ) + . . . , where each bracket has to vanish for the left-hand side to be zero. This structure allows us to solve for the term ui successively in a hierarchical manner.

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Algorithm 1 Basic scheme (CG-version) 1: INIT:   2: 0 (x) = −Γ 0 (x) ∗ C(x)∶E , ∀x ∈ Td 3: ITERATION:   4: τn (x) = C(x) − C 0 ∶n (x), ∀x ∈ Td 5: 6: 7: 8: 9: 10:

τˆn = F (τn )  d ηˆ n (ξ ) = −Γˆ 0 (ξ )∶τˆn (ξ ), ∀ξ ∈ 2πZ \ {0} ηˆ n (0) = 0 ηn = F −1 (ηˆ n ) n+1 (x) = n (x) − ηn (x), ∀x ∈ Td Check convergence criterion

// Fourier Transform

// Inverse Fourier Transform

  The first problem 0 = P −2 (u0 ) = ∇y · C : ey (u0 ) is trivially solved by a purely macroscopic displacement u0 (Y, y) = U (Y ). The second problem takes the form 0 = P −2 (u1 ) + P −1 (u0 ) ? ?  @  @ 5 6 = ∇y · C : ey (u1 ) + ∇Y · C : ey U (Y ) + ∇y · C : eY U (Y ) 5 6 5 6 = ∇y · C : ey (u1 ) + ∇y · C : E(Y ) , which coincides with the classical problem presented in Sect. 2. Its solution can be computed with Algorithm 1. All the higher-order problems have essentially the same structure. We restrict ourselves to the second order problem 0 = P −2 (u0 ) + P −1 (u1 ) + P 0 (u2 ) 5 6 5 6 = ∇y · C : ey (u2 ) + ∇Y · C : ey (u1 ) + 5 6 5  6 ∇y · C : eY (u1 ) + ∇Y · C : eY U (Y ) , but the following idea applies to the remaining higher-order problems as well. The displacement term u1 (Y, y) depends linearly on the macroscopic strain E(Y ). Therefore, we use a separation of variables to make the ansatz u1 (Y, y) = X1 (y)∶E(Y ) with X1 (y) ∈ H 1 (Td )d×d×d being a third-order tensor depending solely on the microscopic variable, which has to be determined beforehand. The above problem then reads  5 6 5 6 0 = ∇y · C : ey (u2 ) + ∇Y · C : ey X1 (y)∶E(Y )   6 5 6 5 . + ∇y · C : eY X1 (y)∶E(Y ) + ∇Y · C : eY U (Y )

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After rearranging the terms, one can define the polarization p2 and the body force g2 accordingly (see Definition 1 for the exact formulas) such that the problem is reduced to the equation 5 6 0 = C∶ey (u2 ) + p2 + g2 ,

(11)

a general form also taken by the remaining higher-order problems [1, 11]. Thus, we define the generalized homogenization problem of order α, for α = 1, 2, . . . , as follows; see also [11]. Definition 1 For Y ∈ Ω fixed and y ∈ Td , the generalized homogenization problem of order α takes the form   ∇y · C∶α (uα ) + pα + gα = 0 , where the polarization ⎧ ⎪ ⎨C∶E(Y ) , for α = 1, ?  T @ pα = 1 ⎪ 2 C∶ Xα−1 (y) · ∇ α−1 E(Y ) + Xα−1 (y) · ∇ α−1 E(Y ) , for α ≥ 2, ⎩ and the body force gα = ∇Y · hα with ⎧ ⎪ ⎪ 0 ⎪ ⎨ 5   6  hα = C∶ ey X1 (y)∶E(Y ) + eY U (Y ) 5  ⎪   ⎪ ⎪ ⎩C∶ e X (y) · ∇ α−2 E(Y ) + e X y

α−1

Y

α−2 (y) · ∇

, for α = 1, α−3 E(Y )

6

, for α = 2, , for α ≥ 3,

are order-dependent terms. It has a unique weak solution uα ∈ H 1 (Td )d assuming the displacements have a mean value of zero [11, 12]. The solution can be computed with a slight variation of Algorithm 1 presented at the end of Sect. 2. The only part that has to be adapted is the initialization in Line 2, whereas the iteration loop remains unchanged. We define the quantity θα which has a closed expression in terms of its Fourier coefficients for non-zero frequencies ξ [11, eq. (38)] that reads θˆα (ξ ) =

? @     i 2 (ξ ) · ξ − g ˆ (ξ ) ⊗ ξ + ξ ⊗ g ˆ (ξ ) ξ  ξ ⊗ ξ g ˆ . α α α ξ 4

(12)

The then reads the same as before with the initialization 0 = −Γ 0 ∗  algorithm  pα + θα , ∀x ∈ Td instead.

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4 Numerical Results We use Hashin’s structure as a benchmark problem for our numerical computations; see [4]. It consists of a coated circular inclusion in a matrix material and each material Ωi is assumed to be isotropic. The Young’s moduli and Poisson’s ratios are denoted by Ei and νi , accordingly. In our tests, the radii are set to r1 = 0.25 and r2 = 0.4. If not mentioned otherwise, Young’s moduli have the values E1 = 100 GPa for the core material, E2 = 1000 GPa for the coating and a resulting E3 = 453.685 GPa for the matrix material, following the formulas found in [6]. Poisson’s ratio is chosen to be ν = 0.3 for all materials. A tolerance of 10−6 was used for the following computations. Figure 1 shows the structure in more detail. Figure 2 shows how the Basic Scheme—shortened as FFTH for FFT-based Homogenization—and its CG-version behave if the number of grid points gets larger. The standard algorithm needs much more iterations than the CG-version, especially for second order problems. It is important to note that the number of

y2

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Fig. 3 Number of iterations and computation time plotted against the contrast of core and coating materials

iterations is essentially independent of the grid size, although the Basic Scheme needs noticeably more iterations for second order problems on smaller grids. The computation time for second order problems is in both algorithms noticeably higher than for first order problems. Due to the hierarchical structure of the problems, at least a factor of four was to be expected (three first order problems plus the second order problem itself). The computation of the polarization and body force terms result in additional overhead. For the plots shown in Fig. 3, we kept E1 at a value of 100 GPa and changed the Young’s modulus E2 of the coating material. The computations were performed on a grid with 642 points. In addition to the time gap between first and second order problems already shown before, we can see here that the number of iterations for the Basic Scheme surpasses 104 iterations already for contrasts smaller than 10−3 or greater than 103, whereas the CG-version can still handle these problems within a few hundred iterations. For the most part, its computation time is smaller as well.

5 Conclusion Starting from an FFT-based scheme and its CG-version, we have presented the generalization to higher-order derivatives and a comparison of the schemes for different orders in terms of number of iterations and computation time. We are still working on an extensive quantitative analysis of the implications on the effective properties. This should be combined with a multiscale simulation using higher-order terms (FE-FFT coupling) [9] in order to obtain a meaningful assessment of the pros and cons of this approach. Acknowledgements The collaboration with H. Andrä, M. Kabel and M. Schneider, Fraunhofer ITWM Kaiserslautern, is gratefully acknowledged.

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References 1. C. Boutin, Microstructural effects in elastic composites. Int. J. Solids Struct. 33, 1023–1051 (1996) 2. S. Brisard, L. Dormieux, FFT-based methods for the mechanics of composites: A general variational framework. Comput. Mater. Sci. 49, 663–671 (2010) 3. P. Eisenlohr et al., A spectral method solution to crystal elasto-viscoplasticity at finite strains. Int. J. Plast. 46, 37–53 (2013) 4. Z. Hashin, The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150 (1962) 5. M. Kabel, H. Andrä, Fast numerical computation of precise bounds of effective elastic moduli, Berichte des Fraunhofer ITWM, 224 (2013) 6. M. Kabel, D. Merkert, M. Schneider, Use of composite voxels in FFT-based homogenization. Comput. Methods Appl. Mech. Eng. 294, 168–188 (2015) 7. V. Monchiet, G. Bonnet, A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast. Int. J. Numer. Methods Eng. 89, 1419– 1436 (2012) 8. H. Moulinec, P. Suquet, A fast numerical method for computing the linear and nonlinear properties of composites. Comptes rendus de l’Académie des Sci. Série II, Mécanique, physique, chimie, astronomie 318, 1417–1423 (1994) 9. J. Spahn et al., A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Comput. Methods Appl. Mech. Eng. 268, 871–883 (2014) 10. P. Suquet, N. Lahellec, Elasto-plasticity of heterogeneous materials at different scales, Procedia IUTAM 10, 247–262 (2014) 11. T. Tran, V. Monchiet, G. Bonnet, A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media. Int. J. Solids Struct. 49, 783–792 (2012) 12. J. Vondˇrejc, J. Zeman, I. Marek, An FFT-based Galerkin method for homogenization of periodic media. Comput. Math. Appl. 68, 156–173 (2014)

Part VIII

Monge-Ampère Solvers with Applications to Illumination Optics

A Least-Squares Method for a Monge-Ampère Equation with Non-quadratic Cost Function Applied to Optical Design N. K. Yadav, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman

Abstract Freeform optical surfaces can transfer a given light distribution of the source into a desired distribution at the target. Freeform optical design problems can be formulated as a Monge-Ampère type differential equation with transport boundary condition, using properties of geometrical optics, conservation of energy, and the theory of optimal mass transport. We present a least-squares method to compute freeform lens surfaces corresponding to a non-quadratic cost function. The numerical algorithm is capable to compute both convex and concave surfaces.

1 Introduction The optical design problem with one or two freeform surfaces can be formulated as an inverse problem: compute an optical system consisting of freeform reflectors and/or lenses that converts a given light distribution of the source into a desired target light distribution. A partial differential equation of Monge-Ampère (MA) type with transport boundary condition can be derived using conservation of energy and the laws of geometrical optics [1–3]. For certain optical systems [1, 4, 5], this problem can be interpreted as an optimal mass transport (OMT) problem with a quadratic cost function, related to the optical path length, and we refer to the corresponding

N. K. Yadav () · J. H. M. ten Thije Boonkkamp Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands e-mail: [email protected]; [email protected] W. L. IJzerman Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands Philips Lighting, Philips Research, Eindhoven, The Netherlands e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_26

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differential equation as the standard MA equation. This MA equation can be solved using a least-squares (LS) method [1, 4]. The goal of this paper is to compute the freeform surfaces of a lens with parallel incoming and outgoing light rays, which corresponds to a non-quadratic cost function. In this case, the optical system is represented by a more complicated MA type equation than the standard equation [1, 4, 5]. In this article we give a brief overview of an extended least-squares (ELS) method to compute the freefrom surfaces of a lens characterized by a non-quadratic cost function. The ELS method is a two-stage procedure like the LS method: first we determine an optimal map by minimizing three functionals iteratively, then we compute the freefrom surfaces from the mapping. In the first stage, there are two nonlinear minimization steps, which can be performed pointwise, but in the third step two coupled elliptic PDEs have to be solved, for the detailed algorithm see [5]. Uniqueness of the mapping is guaranteed by constructing a c-convex/concave pair of freeform surfaces, which relates to the OMT problem. We have organized our paper as follows. In Sect. 2 we give a general formulation of the freeform design problems as well as the specific model for a freefrom lens. A short overview of the proposed solution method is presented in Sect. 3. We apply the numerical method to two test problems in Sect. 4, and the concluding remarks are given in Sect. 5.

2 Mathematical Formulation In this section we present the mathematical formulation in Cartesian coordinates. We consider optical systems for illumination purposes. Clearly, the wave length of light is much smaller than the size of these systems, therefore the laws of geometrical optics are applicable. Let the source and target domains be represented by the compact sets S and T , respectively, and f (x), g(y) ≥ 0 are the emittance/illuminance functions. The key idea for the design of such an optical system is to find a mapping y = m(x) : S → T that satisfies the energy conservation relation 

 A

f (x) dx =

m(A)

g(y) dy,

(1)

for each A ⊂ S, after a change of variables the conservation law becomes f (x) = g(m(x))| det(Dm(x))|, ∀x ∈ S,

(2)

where Dm is the Jacobian of m. The accompanying boundary condition m(∂S) = ∂T ,

(3)

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states that all light from the source S must be transferred to the target T . Using the laws of geometrical optics we can derive a generalized mathematical structure for optical systems consisting of one or two freeform surfaces u1 (x) + u2 (y) = c(x, y), with y = m(x),

(4)

where u1 (x), u2 (y) represent the geometry of the optical system, and c(x, y) is a distance function, depending on the optical path length, and is known as the cost function in mass transport theory [6]. OMT originates from the following problem: how to fill a hole with a heap of sand from another location while minimizing the transportation cost [6]. In optical design problems we do not consider a hole and heap of sand, but instead a light source with an emittance and a target with a desired illuminance. It was shown that several optical design problems can be viewed as an OMT problem [7]. Freeform Lens The geometrical structure of a lens with two freefrom surfaces is shown schematically in Fig. 1. Let (x1 , x2 , z) ∈ R3 denote the Cartesian coordinates with z the coordinate along the horizontal/optical axis and x = (x1 , x2 ) ∈ R2 the coordinates in the plane z = 0, denoted by α1 . Let the source S emit parallel light rays propagating in the positive z-direction with the emittance f (x) [lm/m2], x ∈ S. The light rays are refracted at the lens surfaces L1 and L2 , and leave the lens as a parallel bundle, and reach the target T at a distance  > 0 from the plane α1 . The desired illuminance is g(y) [lm/m2], y ∈ T where y ≡ (y1 , y2 ) ∈ R2 are the Cartesian coordinates of the target plane α2 . The location of the lens surfaces is given by functions z ≡ u1 (x), x ∈ S and w ≡  − z = u2 (y), y ∈ T . Applying the geometrical properties of the lens and the law of refraction, we obtain relation (4) for the location of the lens surfaces with the following cost

Fig. 1 Sketch of a freeform lens

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function c(x, y) =  −

2 n β − β 2 − (n2 − 1)|x − y|2 , n2 − 1 n2 − 1

(5)

where β = L− is the reduced optical path length (L is the total optical path length) and n is the refractive index of the lens. The optical mapping is given by m(x) = x − -

β∇u1 (x) n2

+ (n2 − 1)|∇u1 |2

,

(6)

defining the coordinates on the target T of a ray with initial position x ∈ S. By substituting this expression in relation (2) we can obtain a MA type equation representing the freefrom surface u1 of the lens. Next, we assume that both lens surfaces are c-convex/concave to obtain a unique optical system [6]. The lens surfaces u1 and u2 are c-convex if u1 (x) = max{c(x, y) − u2 (y)} ∀ x ∈ S,

(7a)

u2 (y) = max{c(x, y) − u1 (x)} ∀ y ∈ T ,

(7b)

y∈T

x∈S

and, likewise c-concave if we replace max by min. It has been proved that for a continuously differentiable function c ∈ C 1 (S × T ), the mapping y = m(x) is implicitly given by the relation ∇u1 (x) = ∇x c(x, m(x)),

(8)

under the condition that the matrix C = Dxy c = (cxi yj ), (i, j = 1, 2) is invertible [6], in other words, we could also have derived the expression for m from (5) and (8). We can verify from relation (5) that the matrix C is symmetric negative definite, consequently det(C) > 0 and tr(C) < 0. Since the cost function c is continuously differentiable, we obtain from relation (8) CDm(x) = D2 u1 (x) − Dxx c(x, m(x)) = P ,

(9)

where D2 u1 is the Hessian of u1 . The matrix P = D2 u1 (x) − Dxx c(x, m(x)) is negative semi-definite for a c-concave pair (u1 , u2 ) and positive semi-definite for a c-convex pair (u1 , u2 ). In the following, we discuss the convex case, thus we require the matrix P to be positive semi-definite. Substituting Dm from the above equation into the energy conservation law (2), we obtain det(P (x)) f (x) = , det(C(x, m(x))) g(m(x))

∀ x ∈ S.

(10)

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The positive semi-definiteness of the matrix P guarantees c-convexity of the lens surfaces u1 , u2 . Because det(C) > 0 and f, g ≥ 0, it is obvious that det(P ) ≥ 0. So, the only requirement is tr(P ) ≥ 0 for convex optical surfaces. Next, we outline the ELS algorithm. The method presented here is based on [1]. Compared to [1] we deal with a non-quadratic cost function that results in the presence of C in (9).

3 The Extended Least-Squares Algorithm First, we calculate the mapping m using the ELS method as follows: we enforce the equality CDm = P by minimizing the functional JI (m, P ) =

1 2

 ||CDm − P ||2F dx.

S

(11)

The norm used in this functional is the Frobenius norm. Next, we address the boundary condition (3) by minimizing the functional JB (m, b) =

1 2

A ∂S

|m − b|2 ds,

(12)

where |.| denotes the 2 -norm for vectors, and b is a vector function mapping ∂S to ∂T . We combine the functionals JI for the interior and JB for the boundary domain by taking a weighted average J (m, P , b) = αJI (m, P ) + (1 − α)JB (m, b).

(13)

The parameter α (0 < α < 1) controls the weight of the first functional compared to the second functional. The minimizer gives us the mapping m which is implicitly related to the surface function u1 as shown in relation (8). We minimize the functionals (11)–(13) iteratively starting with an adequate initial guess m0 . In this article, we give a brief description of the minimization of P and m; for more details see [1]. Minimizing Procedure for P We assume m fixed and minimize JI (m, P ) over all matrices that satisfy (10). Since the integrand of JI (m, P ) does not depend on derivatives of P , the minimization procedure can be performed pointwise. Thus, we minimize ||CD − P ||F for each grid point x ∈ S, where D is the central difference approximation of Dm. This gives rise to the following quadratic minimization problem minimize HS (p11 , p22 , p12 ) ≡

1 ||QS − P ||2F , 2

(14a)

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subject to det(P ) =

f det(C), g

(14b)

tr(P ) = p11 + p22 ≥ 0,

(14c)

where Q = CD and QS = (Q + QT )/2. This problem can be solved using a Lagrangian multiplier λ, and the possible minimizer has to satisfy the following algebraic system p11 + λp22 = q11 ,

(15a)

λp11 + p22 = q22 ,

(15b)

(1 − λ)p12 = qS ,

(15c)

2 = p11 p22 − p12

f det(C). g

(15d)

The system (15a)–(15c) is linear in p11 , p22 and p12 , and is regular if λ = ±1, for detailed solution see [5]. Minimizing Procedure for m Here, we assume b and P are fixed and minimize J (m, P , b). The minimizer is obtained from δJ (m, P , b)[η] = 0,

∀η ∈ [C 2 (S)]2 ,

(16)

where δ represents the first variation of J with respect to m in the direction η. This gives the following coupled elliptic equations with Robin boundary conditions ∇ · (C T CDm) = ∇ · (C T P ),

x ∈ S,

(1 − α)m + α(C T C∇m) · nˆ = (1 − α)b + αC · P n, ˆ

(17a) x ∈ ∂S.

(17b)

We discretize the elliptic equations using the finite volume method, for the detailed solution see [5]. Calculation of the Freeform Surfaces We compute the first lens surface u1 (x) from relation (8) in the least-squares sense, i.e.,  1 u1 (x) = argminφ I (φ), I (φ) = |∇φ(x) − ∇x c(x, m(x))|2 dx, (18) 2 S and this gives the following Poisson problem with Neumann boundary condition: Δu1 = ∇ · ∇x c(x, m), ˆ ∇u1 · nˆ = ∇x c · n,

x ∈ S, x ∈ ∂S.

(19a) (19b)

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The second lens surface is calculated from (4)–(5), after substituting the converged mapping m(x) and the first lens surface u1 (x).

4 Numerical Examples We tested our algorithm to design both convex and concave optical surfaces. In this article, we present two test problems for convex lens surfaces: in the first test problem we map a square with uniform emittance into a circle with uniform illuminance, in the second test problem we map a uniform square into the illuminance corresponding to the projection of a picture on a screen. The numerical results are verified using Monte-Carlo ray tracing [4]. In the first test problem, the source domain is given by S = [−1, 1]×[−1, B 1] with emittance f (x1 , x2 ) = 1/4 and the target domain by T = {(y1 , y2 ) ∈ R2 B ||y||2 ≤ 1}. The target plane is at a distance  = 40, the refractive index n = 1.5 and β = 3π. The target T is illuminated by a parallel beam of light rays with uniform illuminance g(y1 , y2 ) = 1/π. We discretize the source domain S uniformly by 200 × 200 grid with 1000 points on the boundary. The resulting mapping for α = 0.65, after 200 iterations is shown in Fig. 2a and the convergence history of the algorithm is shown in Fig. 2b. The square converged into a perfect circle see Fig. 2b. Numerical results provide evidence that α = 0.65 is the best choice to balance the residuals JI and JB . Note that the values of JI and JB stall at a value of approximately 10−6 , probably due to discretization errors.

Fig. 2 Square-to-circle problem: the mapping and convergence history. (a) The mapping after 200 iterations. (b) Convergence history of the functional JI and JB

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Fig. 3 Square-to-picture problem: the picture showing costumes of Indian classical dance Bharatanatyam. The original is shown on the left and the ray trace result on the right on a 800×800 grid. Courtesy Wikipedia

Next, we apply our algorithm to map a uniform emittance of a square into the picture shown in Fig. 3. The emittance and the parameters are the same as defined in the previous example. The desired target illumination g(y1 , y2 ) is given by the grayscale test image shown in Fig. 3. Because the target distribution contains many details, e.g., the pattern of costumes and jewellery, it provides a challenging test for our algorithm. Note that the picture is converted into grayscales and contains some black regions, which results in g(y1 , y2 ) = 0 for some (y1 , y2 ) ∈ T . This would give division by 0 in the least-squares algorithm. Therefore, we increased the illuminance to 10% of the maximum value if it is below this threshold. We discretized the source S on a 500 × 500 grid, with 1000 boundary points. The target illuminance for ten million rays is plotted in Fig. 3. The output images is very close to the corresponding original image, although the image is slightly blurred, but even complex details can be identified.

5 Concluding Remarks The standard freeform design problem is equivalent to a MA equation with transport boundary condition, corresponding to a quadratic cost. In this contribution we presented an extended least-squares (ELS) method to compute freeform lens surfaces from a modified MA equation, corresponding to a non-quadratic cost function.

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The ability to simulate an optical system with non-quadratic cost function for both convex and concave cases, makes our method a valuable addition to existing methods. The ELS algorithm works efficiently, and can compute the freeform surfaces for complex targets distributions representing detailed pictures. In future work we would like to apply the algorithm to more complex cost functions, e.g., point light sources and far-field problems.

References 1. C.R. Prins, R. Beltman, J.H.M. ten Thije Boonkkamp, W.L. IJzerman, T.W. Tukker, A leastsquares method for optimal transport using the Monge-Ampère equation. SIAM J. Sci. Comput. 37(6), B640–B660 (2015) 2. V. Oliker, Differential equations for design of a freeform single lens with prescribed irradiance properties. Opt. Eng. 53(3), 031302 (2013) 3. K. Brix, Y. Hafizogullari, A. Platen, Designing illumination lenses and mirrors by the numerical solution of Monge-Ampère equations. J. Opt. Soc. Am. A 32(11), 2227–2236 (2015) 4. N.K. Yadav, J.H.M. ten Thije Boonkkamp, W.L. IJzerman. A least-squares method for the design of two-reflector optical system. J. Comput. Appl. Math. (2017, manuscript submitted) 5. N.K. Yadav, J.H.M. ten Thije Boonkkamp, W.L. IJzerman, A Monge-Ampère problem with nonquadratic cost function to compute freeform lens surfaces. J. Sci. Comput. (2018, manuscript submitted) 6. C. Villani, Topics in Optimal Transportation, vol. 58 (American Mathematical Society, Providence, RI, 2003) 7. V. Oliker, Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport. Arch. Ration. Mech. Anal. 201(3), 1013–1045 (2011)

Solving Inverse Illumination Problems with Liouville’s Equation Bart S. van Lith, Jan H. M. ten Thije Boonkamp, and Wilbert L. IJzerman

Abstract We aim to solve inverse problems in illumination optics by means of optimal control theory. This is done by first formulating geometric optics in terms of Liouville’s equation, which governs the evolution of light distributions on phase space. Choosing a metric that measures how close one distribution is to another, the formal Lagrange method can be applied. We show that this approach has great potential by a simple numerical example of an ideal lens.

1 Introduction Illumination optics deals with designing optics for lighting systems. Since illumination generally involves large objects relative to the wavelength of light, it is usually described in terms of geometric optics [1]. In the illumination setting, we consider some optical axis denoted z and hypothetical screens perpendicular to it, given by z = const. The ray position on the screen q changes as the screen is moved depending on the angle the ray makes with the screen. The momentum p is related to the angle between the ray and the optical axis γ by |p| = n sin γ , with n the refractive index [2]. As it turns out, the evolution of a ray can be described by a

B. S. van Lith () · J. H. M. ten Thije Boonkamp Eindhoven University of Technology, Eindhoven, The Netherlands e-mail: [email protected] W. L. IJzerman Eindhoven University of Technology, Eindhoven, The Netherlands Philips Lighting, High Tech Campus 7, Eindhoven, The Netherlands © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_27

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Hamiltonian system, given by ∂h dq = , dz ∂p

(1a)

dp ∂h =− , dz ∂q

(1b)

where 2 h(q, p) = −σ n(q)2 − |p|2 ,

(1c)

with σ ∈ {−1, 0, 1} describing backward, perpendicular or forward moving rays, respectively. We will assume for simplicity that all rays we discuss have σ = 1. The collection of all positions and momenta is called phase space, denoted P. Only in the special case that the refractive index is constant will phase space be the product of position space Q and momentum space P . It is well known that Hamiltonian flows preserve volume in phase space [3]. For the remainder of this work, we consider two-dimensional optics, so that position q and momentum p are both scalars. In the absence of diffusion and attenuation, the energy carried by a light ray is constant. Furthermore, illumination usually deals with static lighting situations. If we define some power density on phase space, call it ρ, the power carried by a light ray is given by ρ dy, with dy = dq dp the phase space volume element associated with the ray. Since volume in phase space is preserved by the Hamiltonian flow, we see that ρ is also constant along a ray, i.e.,     ρ z + Δz, q(z + Δz), p(z + Δz) = ρ z, q(z), p(z) .

(2)

This relation holds even when dealing with optical interfaces where the ray momentum changes discontinuously, e.g., lenses and mirrors. Discontinuous transformations on phase space, meaning Snell’s law and the law of specular reflection, are in fact symplectic [4]. Hence, one way of computing the energy distribution on some target screen is by finding ray positions on the target screen given their initial position. This process is known as ray tracing, see for instance [5]. When everything is sufficiently smooth, we may subtract the right-hand side of (2), divide by Δz and pass to the limit as Δz → 0. Together with Hamilton’s equations (1), this yields Liouville’s equation, dρ ∂ρ ∂h ∂ρ ∂h ∂ρ = + · − · = 0. dz ∂z ∂p ∂q ∂q ∂p

(3)

Using the fact that the corresponding velocity field is divergence-free, we can write Liouville’s equation in conservative form, so that ρz + ∇ · (ρS∇h) = 0,

(4a)

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where ∇ is the gradient operator on phase space and S is the symplectic matrix 

 0 1 S= . −1 0

(4b)

Hence, another way of computing the target light distribution is by solving Liouville’s equation. Clearly, some special techniques have to be applied around optical interfaces to deal with their effects, see [6]. Snell’s law can be incorporated by taking one-sided limits of (2) towards the interface, leading to     ρ ζ (q)+ , q, p = ρ ζ (q)− , q, S(p) ,

(5)

where S denotes the action of Snell’s law. Here, the one-sided limits in z are indicated with superscript pluses and minuses. The rule (5) will also be abbreviated as ρ + = ρ − . In the same work, the authors have also shown that solving (3) can indeed provide some computational advantages over ray tracing.

2 Optics as Open-Loop Controllers Due to space constraints, we only give here a brief outline. More details can be found in [7] and future publications. A typical goal in the design of optical systems is to achieve a certain output distribution given an input. In terms of lighting, think for instance of the uniform illumination of a wall using one or several LEDs. Stated in these terms, it is perhaps a small step to think of an optic as an open-loop controller acting on the light distribution as a whole. The controller is open-loop, since we usually cannot adjust the optic after fabrication. Now that we have identified optics with open-loop controllers, we can immediately apply the machinery of optimal control theory to illumination optics [8]. Suppose we are given an input distribution ρ0 at z = 0 and some desired output distribution ρ ! at z = Z. We wish to solve the inverse problem and find an optic that transforms the input distribution to an output that closely matches the desired state, for instance as measured by the L2 -norm on phase space. Thus, we wish to minimise   2 1 ρ(Z) − ρ ! dy, 2 P

where Z is the total length of the optic. For simplicity, we consider only a single refractive surface z = ζ (q) as the optic. The surface is, however, completely freeform. As our control parameter, we choose the gradient of the surface, rather than the surface position, since Snell’s law depends on the local surface normal.

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Thus, we set u = ζ  (q) and we find the q-component of the local unit normal as u ν(u) = √ . 1 + u2

(6)

After finding the optimal u, we have to integrate it once more. We assume that the endpoints of the surface are fixed, hence we can find ζ by solving Poisson’s equation, i.e., ζ  = u ,

(7)

where ζ is subject to Dirichlet boundary conditions. Snell’s law itself is enforced by using (5). To find the optimal surface, we employ the formal Lagrange multiplier method, which calls for the assembly of the Lagrangian. The Lagrangian, just as in more common optimisation settings, is the original objective function where the constraints are added with a Lagrange multiplier. In this case, both Liouville’s equation (3) and Snell’s law (5) need to be enforced, each with their own Lagrange multiplier. Thus, the Lagrangian is given by  L[ρ, u, ϕ, μ] = 12



2 ρ(Z) − ρ ! dy +

P

 Z −



 u2 dq +

α 2

β 2

Q



 ϕ ρz + ∇ · (ρS∇h) dz dy −

P 0

(u )2 dq

Q



(8) μ(ρ + − ρ − )dy,

P

where the last two integrals represents the constraints. In (8) we have introduced the regularisations, measured by α > 0 and β > 0, while ϕ and μ are the Lagrange multipliers. Note the order of integration in the fourth term, which can be viewed as two integrals, one over the region z < ζ and one over the region z > ζ . The variable ϕ is also known as the costate. The first-order necessary conditions for the optimal surface are now given by a stationary point of the Lagrangian. These can therefore be found by applying the calculus of variations [9].

2.1 First-Order Necessary Conditions For brevity’s sake, we only show here the result of the application of the calculus of variations. A more detailed presentation can be found in the aforementioned thesis [7] and will also be given elsewhere. The optimality system is found by demanding that the first variation of the Lagrangian (8) vanishes. It consists of two hyperbolic

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PDEs and an extra constraint equation, given by ρz + ∇ · (ρS∇h) = 0,

(9a)

ϕz + (S∇h) · ∇ϕ = 0,  ∂ρ −  dp = 0, αu − βu + μ ∂u

(9b) (9c)

P

where and μ = ϕ − = ϕ + . Furthermore, Snell’s law (5) should hold across the interface. This system is completed by an initial condition for ρ and the terminal condition for the costate given by ϕ(Z) = ρ(Z) − ρ ! .

(10)

Furthermore, a suitable set of boundary conditions for ρ and a complementary set of boundary conditions for ϕ have to be specified. The optimality system (9) constitutes a two-point boundary value problem, which are as a rule pretty difficult to solve. Therefore, an iterative approach is needed. Typically, this involves dropping one of the three condition of (9) and iteratively improving until all three conditions are satisfied. We take here the approach where the extra constraint (9c) is dropped, while the state and costate satisfy Liouville’s equation. This tacitly introduces a reduced functional j [u] = L[Rρ (u), u, Rϕ (u), Rμ (u)] with Rρ the solution operator for ρ, etc. The variation is given by  δL j  [u] = δu dq, (11) δu Q

L is the L2 -gradient of L with respect to u and it is given by the left-hand where δδu side of (9c). Consequently, the control input u can be updated using a gradient descent technique. In our implementation, we actually use a Newton type descent method. Thus, the solution strategy is relatively straightforward: we make an initial guess L of the control u, compute the solutions ρ and ϕ by solving (9a)–(9b), compute δδu and update u accordingly. This process is repeated until some specified convergence criterion is met. The exit condition can be specified in terms of the L2 -norm of the gradient or by testing whether the updates of L are sufficiently small. Here, we use the latter option, so that we stop when

B B B Lk+1 − Lk B B < , B B B Lk

(12)

with  = 10−4 and Lk the value of the Lagrangian at iteration k. By the definition L of the first variation, this exit criterion implies that δδu is of order ε.

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3 Results We show here a relatively simple example of our optimisation procedure: an ideal lens. An ideal lens rotates the initial phase space distribution by π2 in going from one focal plane to the other. Our approach requires a fixed initial position, hence our result will only approximate an ideal lens for the particular input we choose. However, for a given input ρ0 , we set ρ ! (q, p) = ρ0 (p, −q). The initial condition we consider is given by a Gaußian, i.e.,     1 q2 p2 ρ0 (q, p) = exp − 2 exp − 2 , 2πσq σp 2σq 2σp

(13)

2 1 and σp = 10 , see Fig. 1 . where σq = 10 One side of the lens is taken to be a flat plane, see Fig. 1. As an initial guess for the freeform surface, we use a spherical lens with focal length f = 3 units, resulting in a total length of the optic of Z = 6 units. The edges of the freeform are 2  fixed with ζ (−1) = ζ (1) = f , leading to q 2 + ζ (q) − z0 = R 2 , with z0 such that ζ (±1) = f . The lens material is chosen to have index n2 = 1.6, resulting in R = 1.8 to find the correct focal point. Hence, the initial control input is given by

u0 (q) = − -

q R2 − q 2

.

(14)

The ambient refractive index is set to n1 = 1, corresponding roughly to air. Liouville’s equation is solved by means of a discontinuous Galerkin method on a 20 × 20 regular Cartesian mesh. The lens surface is solved on a onedimensional mesh spanning [−1, 1] with 20 elements by a continuous Galerkin method. Furthermore, we set a maximum momentum of |p| ≤ 34 to speed up computations somewhat.

Fig. 1 Sketch of the initial situation: a spherical lens with a focal distance f . Several light rays are drawn originating from a point source on the optical axis

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Figure 2 shows the initial condition ρ0 and the initial output, the output distribution resulting from the spherical lens. The desired output is simply the input rotated by π2 . Any deviation from this output is due to optical aberrations. Notably, there is a significant amount of stretching in the q-direction and squeezing in the p-direction, while mirror symmetry across the lines q = 0 and p = 0 is lost as well. The end result after optimisation is shown in Figs. 3 and 4. Specifically, Fig. 3 shows the target distribution and the final output distribution. By eye, it is hard to judge which distribution appears closer to the target, the initial output or the optimised one. However, the reduction of the L2 -distance is almost 80%. The optimised lens surface itself is shown in Fig. 4. The immediate feature that catches the eye is increased curvature near the centre. Closer to the edges, the curvature seems less pronounced.

Fig. 2 Initial condition (left) and the initial output of the optic (right)

Fig. 3 Target distribution (left) and the output distribution after optimisation (right)

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Fig. 4 Initial guess for the lens surface (black) and lens surface after optimisation (blue)

4 Conclusion In this short contribution we have shown a new approach to the optimisation of optical systems, especially aimed at illumination optics. Our approach consists of solving Liouville’s equation to compute the light distribution in phase space, rather than ray tracing techniques. In other work, we have shown that this idea has merits of itself. It also allows the formulation of an optimal control problem, where the optic is considered an open-loop controller acting on the light distribution. The formulation in terms of Liouville’s equation allows for a formal Lagrange approach, where Liouville’s equation and Snell’s law are enforced through Lagrange multipliers. Note that it is, in principle, quite easy to add additional constraints, be they equality or inequality constraints. For instance, in some cases the curvature of the lens has to be below a certain threshold due to manufacturing constraints. Or perhaps the lens has to fit in a certain space, setting limits to the thickness. Such things can be dealt with by adding more Lagrange multipliers in the case of equality constraints, or Karush-Kuhn-Tucker multipliers in the case of inequality constraints. Our simple example is the approximation of an ideal lens. In particular, we optimise the lens surface to rotate the input distribution by π2 . Compared to the initial guess for the lens surface, a spherical lens, we achieve an almost 80% reduction in the L2 -distance between the output and desired distributions. Our approach has great potential for application, as it provides a relatively straightforward optimisation technique. Acknowledgements B.S. van Lith wishes to thank J.H.M. ten Thije Boonkkamp for presenting the work in Norway in his stead.

References 1. J. Chaves, Introduction to Non-imaging Optics (CRC Press, Boca Raton, 2008) 2. K.B. Wolf, Geometric Optics on Phase Space (Springer, Berlin, 2004) 3. V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, Berlin, 1978) 4. A.J. Dragt, J.M. Finn, Lie series and invariant functions for analytic symplectic maps. J. Math. Phys. 17(12), 2215–2227 (1976)

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5. A.S. Glassner, An Introduction to Ray Tracing (Academic, London, 1991) 6. B.S. van Lith, J.H.M. ten Thije Boonkkamp, W.L. IJzerman, T.W. Tukker, A novel scheme for Liouville’s equation with a discontinuous Hamiltonian and applications to geometrical optics. J. Sci. Comput. 68, 739–771 (2016) 7. B.S. van Lith, Principles of computational illumination optics. PhD thesis, TU/e (2017) 8. F. Tröltzsch, J. Sprekels, Optimal Control of Partial Differential Equations: Theory, Methods and Applications (American Mathematical Society, Providence, RI, 2010) 9. M. Giaquinta, S. Hildebrandt, Calculus of Variations I (Springer, Berlin, 2004)

Part IX

Mixed and Nonsmooth Methods in Numerical Solid Mechanics

Strong vs. Weak Symmetry in Stress-Based Mixed Finite Element Methods for Linear Elasticity Bernhard Kober and Gerhard Starke

Abstract Based on the Hellinger-Reissner principle, accurate stress approximations can be computed directly in suitable H (div)-like finite element spaces treating conservation of momentum and the symmetry of the stress tensor as constraints. Two stress finite element spaces of polynomial degree 2 which were proposed in this context will be compared and relations between the two will be established. The first approach uses Raviart-Thomas spaces of next-to-lowest degree and is therefore H (div)-conforming but produces only weakly symmetric stresses. The stresses obtained from the second approach satisfy symmetry exactly but are nonconforming with respect to H (div). It is shown how the latter finite element space can be derived by augmenting the componentwise next-to-lowest Raviart-Thomas space with suitable bubbles. However, the convergence order of the resulting stress approximation is reduced from two to one as will be confirmed by numerical results. Finally, the weak stress symmetry property of the first approach is discussed in more detail and a post-processing procedure for the construction of stresses which are element-wise symmetric on average is proposed.

1 Introduction The Hellinger-Reissner principle formulates the variational problem of linear elasticity in terms of stresses alone with conservation of momentum and stress symmetry as constraints. A number of finite element combinations were proposed for this approach starting with [1], see [2] and [4] for a recent overview of the stateof-the-art. Two particularly appealing stress finite elements which are of lowest possible (to achieve quadratic convergence) polynomial degree 2 are investigated in this contribution. The first one is due to Boffi et al. [3] and uses RaviartThomas spaces of next-to-lowest degree for the stress approximation which is only

B. Kober () · G. Starke Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_28

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weakly symmetric. The second one is due to Gopalakrishnan and Guzmán [7] and produces exactly symmetric stresses but is nonconforming with respect to H (div). In contrast to the derivation in [7] where symmetric basis functions for the stress approximation are used, we will enforce (exact) symmetry by Lagrange multipliers here. To this end, a nonsymmetric version of their nonconforming stress space is introduced and shown to be obtained from augmenting the componentwise next-tolowest order Raviart-Thomas space with suitable bubbles (four per element in the two-dimensional case).

2 The Hellinger-Reissner Principle and Stress Finite Element Spaces We consider the reference configuration Ω ⊂ Rd (d = 2, 3) of a linear elastic object with boundary ∂Ω = Γ = ΓN ∪ ΓD (ΓN and ΓD disjoint and nonempty), a volume load f : Ω → Rd and some prescribed displacement uD : ΓD → Rd and surface forces g : ΓN → Rd on the separate boundary parts. The HellingerReissner Principle can then be obtained by considering the optimality conditions for minimizing the elastic energy in terms of stress J (σ ) :=

1 2



 Ω

 Aσ : σ dx −

 (σ · n) · uD ds ΓD

constrained by div σ = −f, as σ = 0 in Ω and σ · n = g on ΓN . Here σ : Ω → Rdxd denotes the stress tensor, n denotes the outer unit normal, as denotes the antisymmetric part of a matrix, div denotes the divergence (applied row-wise) and A denotes the compliance tensor given by Aσ :=

  1 λ (tr σ )I σ− 2μ dλ + 2μ

(1)

with the material dependent Lamé constants λ and μ. In the incompressible limit (λ → ∞) Aσ is reduced to the deviatoric (trace-free) part of σ (and thus remains bounded). The problem is then to find σ ∈ Σ with σ · n = g on ΓN , u ∈ U and θ ∈ Θ satisfying 

Aσ , τ

 0

  + (u, div τ )0 + θ , as τ 0 = (uD , τ · n)0,ΓD ,   (div σ , v)0 = − f, v 0 ,   as σ , γ 0 = 0 ,

(2)

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for all τ ∈ Σ with τ · n = 0 on ΓN , v ∈ U and γ ∈ Θ. Here, (·, ·)0 and (·, ·)0,ΓD denote the L2 inner products on Ω and ΓD . The used function spaces are defined as follows Σ := H (Ω, div)d = {τ ∈ L2 (Ω)d : div v ∈ L2 (Ω)}d , U := L2 (Ω)d , Θ := L2 (Ω)d×d,as = {γ ∈ L2 (Ω)d×d : γ + γ T = 0}, where the Lagrange multiplier enforcing the divergence constraint can be interpreted as the displacement and the one enforcing the symmetry constraint as the antisymmetric part of the gradient of displacement, i.e. the rotations. The saddle point problem (2) is wellposed (see [3]). Using finite dimensional (sub)spaces one obtains the corresponding mixed finite element method: Find σ h ∈ Σ h with σ h · n = g on ΓN , uh ∈ U h and θ h ∈ Θ h satisfying     Aσ h , τ h 0 + (uh , div τ h )0 + θ h , as τ h 0 = (uD , τ h · n)0,ΓD ,   (div σ h , vh )0 = − f, vh 0 ,   as σ h , γ h 0 = 0,

(3)

for all τ h ∈ Σ h with τ h · n = 0 on ΓN , vh ∈ U h and γ h ∈ Θ h . For the work with finite element spaces we denote by Th a simplicial mesh of Ω with the set of sides Sh and by Pk (T ) the polynomials of degree k or less on a simplex T ∈ Th . The first finite element combination which we consider was suggested in [3] and features next-to-lowest degree Raviart-Thomas elements: ˜ : p ∈ Pk (T )d, p˜ ∈ Pk (T )} RTk (T ) := {p(x) + xp(x) Σ h := RT1 (Th )d = {τ h ∈ Σ : τ h |T ∈ RT1 (T )d ∀ T ∈ Th } U h := DP1 (Th )d = {vh ∈ U : vh |T ∈ P1 (T )d ∀ T ∈ Th } Θ h := P1 (Th )d×d,as = {γ h ∈ Θ ∩ C(Ω)d×d : γ h |T ∈ P1 (T )d×d ∀ T ∈ Th } The second combination as suggested in [7] uses strongly symmetric stress approximations and thus the second Lagrange multiplier is no longer needed. U h remains the same but Σ h is replaced by: Σ Sh := {τ h ∈ L2 (Ω)2×2 : as τ h = 0, τ h |T ∈ P2 (T )2×2 for all T ∈ Th ,   τ h · nS S , p1 0,S = 0 for all p1 ∈ P1 (S)2 for all S ∈ Sh }.

(4)

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Where  · S denotes the jump over the side S. Since Σ Sh ⊂ Σ, this combination yields a non-conforming method and requires the use of an   element-wise divergence operator divh . The problem is then to find σ h ∈ Σ Sh with σ h · nS − gS , p1 0,S = 0 for S ⊂ ΓN and uh ∈ U h satisfying 

Aσ h , τ h

 0

+ (uh , divh τ h )0 = (uD , τ h · n)0,ΓD ,   (divh σ h , vh )0 = − f, vh 0

(5)

  for all τ h ∈ Σ Sh with τ h · nS , p1 0,S = 0 for S ⊂ ΓN and vh ∈ U h . Both methods are proven to be wellposed and convergent. For the first one standard results are used after verifying an inf-sup condition of the form inf

sup

vh ∈U h τ h ∈Σ h γ h ∈Θ h

(div τ h , vh )0 + (as τ h , γ h )0 ≥ β, τ h Σ (vh U + γ h Θ )

(6)

and for the second one a direct analysis of the resulting linear system is necessary since the standard techniques are not applicable due to the non-conformity. Both stress spaces have good approximation properties but the non-conformity in the second method only allows an a-priori error estimate of order O(h), while the first one retains the optimal convergence order O(h2 ).

3 The H (div)-Nonconforming Space by Augmenting Raviart-Thomas Elements Our aim is now to describe a method which uses Lagrange multipliers to find a solution in Σ Sh , without having to construct symmetric basis functions. We therefore remove the symmetry requirement from our ansatz space: 2 d×d Σ∪ : τ h |T ∈ P2 (T )d×d for all T ∈ Th , h := {τ h ∈ L (Ω)   τ h · nS S , p1 0,S = 0 for all p1 ∈ P1 (S)d for all S ∈ Sh }. d×d,as and propose Moreover we extend the space of rotations to Θ ∪ h := DP2 (Th ) the following: ∪ Theorem 1 The mapping as : Σ ∪ h → Θ h is surjective.

We will give the idea of the proof after having a closer look at Σ ∪ h in the next subsection, but first we want to state the following corollary. Theorem 2 If (5) has a unique solution, then the following problem has a unique solution and the stress and displacement approximations coincide: Find σ h ∈ Σ ∪ h

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  with σ h · nS − gS , p1 0,S = 0 for S ⊂ ΓN , uh ∈ U h and θ h ∈ Θ ∪ h satisfying 

Aσ h , τ h

 0

  + (uh , divh τ h )0 + θ h , as τ h 0 = (uD , τ h · n)0,ΓD ,   (divh σ h , vh )0 = − f, vh 0 ,   as σ h , γ h 0 = 0,

(7)

  for all τ h ∈ Σ ∪ h with τ h · nS , p1 0,S = 0 for S ⊂ ΓN , vh ∈ U h and γ h ∈ Θ h . Proof Suppose (σ h , uh , θ h ) is solution of (7), then σ h ∈ Σ Sh because of as σ h ∈ S ∪ Θ∪ h . Testing with τ h ∈ Σ h ⊂ Σ h immediately gives that (σ h , uh ) is also solution of (5). We now argue that, assuming the wellposedness of (5), the problem (7) with homogeneous right hand side (f, g, uD = 0) possesses only the zero solution. This is due to Theorem 1. The result then follows because (7) is a square system. & % Just like Θ ∪ h could be interpreted (locally) as an extension of Θ h , the nonconformΔ ing space Σ ∪ h can be described as an enrichment of Σ h by an augmenting space Σ h consisting curls of certain bubble functions on each simplex.

3.1 The Augmenting Space Σ Δ h The structure of Σ Δ h is very similar for d = 2 and d = 3, so for lack of space we will describe it in two dimensions only. We will denote by λTi , i = 1, 2, 3 the T  barycentric coordinates on the triangle T and by ∇ ⊥ := ∂2 −∂1 the orthogonal gradient (or 2D-curl). First we need to define the needed bubble functions. For each triangle T ∈ Th let bT (x) := λT1 (x)λT2 (x)λT3 (x) (extended by zero outside T ) be the cubic element-bubble-function on T and for each edge Si ∈ ∂T let bT ,Si (x) := λTi (x)λTi+1 (x)(λTi (x) − λTi+1 (x)) (extended by zero outside T ) be the cubic edge-bubble-function of Si on T . Note that while bT vanishes on ∂T and thus is globally continuous, bT ,Si vanishes only on Sj with j = i and thus is not globally continuous. Taking the curls of these functions we obtain the desired extension. We introduce the space of bubble-curls by BC(Th ) := span{∇ ⊥ bT , ∇ ⊥ bT ,S1 , ∇ ⊥ bT ,S2 , ∇ ⊥ bT ,S3 }T ∈Th , where ∇ ⊥ is to be understood elementwise. Finally the augmenting space is defined as 2 ΣΔ h := BC(Th )

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and we can obtain the following relationship: Theorem 3 Δ Σ∪ h = Σh ⊕ Σh ∪ ∪ Proof Σ h ⊕ Σ Δ h ⊂ Σ h : Σ h ⊂ Σ h is trivial since τ h · nS S = 0 for τ h ∈ Σ. For

τh ∈ ΣΔ h we have to verify the jump condition 

τ h · nS S , p1

 0,S

= 0 ∀ p1 ∈ P1 (S)2 , ∀ S ∈ Sh .

The simple identity nS · ∇ ⊥ = tS · ∇ suggests that we are to consider the tangential derivatives of the bubble functions. Those derivatives vanish on all edges except for one edge for each edge-bubble-function. However for the curl of an edge-bubblefunction restricted to the nonzero edge we have nSi · ∇ ⊥ bT ,Si = tSi · ∇(λi λi+1 (λi − λi+1 )) = (λi+1 (λi − λi+1 ) − λi (λi − λi+1 ) + 2λi λi+1 )(tSi · ∇λi ) = (−6λ2i + 6λi − 1)(tSi · ∇λi ), which is zero when tested against linear functions on Si . Δ Σ∪ h ⊂ Σ h ⊕ Σ h : On a single triangle we have dim RT1 (T ) = 8 and

dim BC(T ) = 4. Since ∇ ⊥ bT is zero at the center of mass of T and has vanishing normal component on ∂T we have ∇ ⊥ bT ∈ RT1 (T ). The same holds true for each ∇ ⊥ bT ,Si since its normal component is a quadratic function on Si . Therefore dim RT1 (T ) ⊕ BC(T ) = 12 = dim P2 (T )2 . & % Using this representation of Σ ∪ h one can now prove Theorem 1 by combining a smart choice of basis functions of RT1 (Th ) and BC(Th ) to obtain a basis of Θ ∪ h. Due to lack of space we will omit the details here and proceed with the presentation of our numerical results.

4 A Numerical Comparison We tested both methods featuring Lagrange multipliers against the Cook’s membrane problem, as well as against a regularized problem, both for a nearly incompressible (λ = 49,999) and a compressible material (λ = 2). μ is always set to 1. The shape of Ω as well as the boundary conditions are summarized in  T Fig. 1 with Fc = 0 0.0625 and F(x2 ) = 4Fc max{0, 1 − 25|x2 − 0.52|}. The volume load f was set to zero in both cases.

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Fig. 1 Cooks membrane on the left and the regularized domain on the right

Due to the lack of an analytical solution we need an alternative for assessing the quality of our approximations. The least-squares-functional Jls (σ h , uR h ) :=

1 1 2 2 Aσ Sh − ε(uR h )0 + as σ h 0 , 2 2

R R with σ Sh := σ h − as σ h and ε(uR h ) := ∇uh − as ∇uh is equivalent to the approximation error (see [5, Theorem 3.1]). The displacement reconstruction uR h ∈ H 1 (Ω) necessary for the evaluation of Jls (σ h , uR ) is constructed following ideas h from [6] and [8]: uR h is obtained by first solving the following problem locally on every triangle T ∈ Th :

      ∇ u˜ R + ph , vh 0 = Aσ h + θ h , ∇vh 0 h , ∇vh 0     u˜ R = uh , qh 0 h , qh 0

∀ vh ∈ P2 (T )2 , (8) ∀ qh ∈ P0 (T )2

and subsequently averaging the values of u˜ R h on the vertices and edge midpoints to 2 ⊂ H 1 (Ω). retrieve uR ∈ P (T ) 2 h h The results for the Cook’s membrane problem are summarized in Table 1, where l := log2 (h0 / h) is the level of refinement, σ 1h is the solution of (3), σ 2h is the solution of (7), n1dof and n2dof are the respective sizes of the linear systems and nt is the number of triangles. In Table 2 the results for the regularized problem are displayed with the same notation. Plotting the square root of the functional against the level of refinement yields the graphs in Fig. 2. Our tests confirm the predicted convergence behaviour for the regularized problem. In case of the Cook’s membrane problem both methods achieve only an order of convergence of approximately 0.7, which is due to the fact, that the solution of the continuous problem is less regular.

nt

44 176 704 2816 11264 45056 180224

l

0 1 2 3 4 5 6

717 2885 11577 46385 185697 743105 2973057

n1dof

Table 1 Results for cook’s membrane

1300 5240 21040 84320 337600 1351040 5405440

n2dof

λ = 49,999 7.51712e−06 2.42195e−06 8.87635e−07 3.60592e−07 1.56550e−07 7.05565e−08 3.24436e−08

Jls (σ 1h , uR h)

λ = 49,999 1.23805e−05 5.14683e−06 2.04684e−06 8.23787e−07 3.43616e−07 1.48839e−07 6.64434e−08

Jls (σ 2h , uR h)

λ=2

7.15095e−06 2.08703e−06 6.68831e−07 2.32852e−07 8.65879e−08 3.37006e−08 1.35153e−08

Jls (σ 1h , uR h)

λ=2

1.09624e−05 3.63318e−06 1.20104e−06 4.11980e−07 1.47983e−07 5.55008e−08 2.15629e−08

Jls (σ 2h , uR h)

330 B. Kober and G. Starke

nt

49 196 784 3136 12544 50176 200704

l

0 1 2 3 4 5 6

816 3248 12963 51797 207081 828113 3312033

n1dof

Table 2 Results for the regularized problem

1464 5868 23496 94032 376224 1505088 6020736

n2dof 2.96362e−06 8.86412e−07 8.67680e−08 7.66000e−09 6.27767e−10 5.04845e−11 4.11928e−12

λ = 49,999

Jls (σ 1h , uR h)

4.14240e−06 1.34541e−06 4.76183e−07 1.64916e−07 5.01953e−08 1.38663e−08 3.63725e−09

λ = 49,999

Jls (σ 2h , uR h)

3.57976e−06 1.11751e−06 1.08365e−07 9.41993e−09 7.67974e−10 6.16212e−11 5.02207e−12

λ=2

Jls (σ 1h , uR h)

3.85810e−06 1.22544e−06 3.33871e−07 9.13749e−08 2.38911e−08 6.06266e−09 1.52063e−09

λ=2

Jls (σ 2h , uR h)

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-2

10

-2

10 -3 10 -3

10 -4 10

10 -4

-5

10 -6

0

1

2

3

4

5

6

10

-7

0

1

2

3

4

5

6

Fig. 2 Cook’s membrane on the left and the regularized problem on the right

5 Postprocessing for Elementwise Symmetry on Average Now we sketch a possible way for the construction of a modified stress approximation σ Sh with the property that as σ Sh vanishes on average for all T ∈ Th . In order to keep the equilibration property (second equation in (3)) unaffected, we compute a correction in Σ⊥ h = {σ h ∈ Σ h : div σ h = 0, σ h · n = 0 on ΓN } B = {∇ ⊥ χ h : χ h ∈ HΓ1N (Ω)2 , χ h BT ∈ P2 (T )2 } =: ∇ ⊥ Ξ h , where Ξ h is the standard conforming piecewise quadratic finite element space with zero boundary conditions on ΓN . The constraint on the anti-symmetric part of the modified stress approximation σ Sh = σ h + ∇ ⊥ χ h then reads ⎛

⎞ 0 1 ⎠ 0 = ⎝as σ Sh , −1 0

⎛



L2 (T )

⎞ 0 1 ⎠ = ⎝as σ h , −1 0 

− (div χ h , 1)L2 (T )

L2 (T )

for all T ∈ Th . A minimal correction χ ⊥ h ∈ Ξ h is constructed such that ∇ ⊥ χ h 2 → min! subject to the constraints ⎛  ⎞ 0 1 ⎠ (div χ h , 1)L2 (T ) = ⎝as σ h , −1 0

for all T ∈ Th

(9)

L2 (T )

among all χ h ∈ Ξ h . The inf-sup stability of the P2 − P0 element (in 2D) (cf. [4, Sect. 8.4.3]) and the coercivity in HΓ1N lead to the well-posedness of (9).

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References 1. D.N. Arnold, F. Brezzi, J. Douglas, PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1, 347–367 (1984) 2. D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155 (2006) 3. D. Boffi, F. Brezzi, M. Fortin, Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8, 95–121 (2009) 4. D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications (Springer, Heidelberg, 2013) 5. Z. Cai, G. Starke, Least squares methods for linear elasticity. SIAM J. Numer. Anal. 42, 826–842 (2004) 6. A. Ern, M. Vohralík, Polynomial-degree-robust a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, mixed discretizations. SIAM J. Numer. Anal. 53, 1058–1081 (2015) 7. J. Gopalakrishnan, J. Guzmán, Symmetric nonconforming mixed finite elements for linear elasticity. SIAM J. Numer. Anal. 49, 1504–1520 (2011) 8. R. Stenberg, Postprocessing schemes for some mixed finite elements. Math. Model. Numer. Anal. 25, 151–167 (1991)

Evolution of Load-Bearing Structures with Phase Field Modeling Ingo Muench

Abstract We suggest an algorithm to generate the topology of load-bearing structures with help of a phase field model. The objective function homogenizes equivalent stress within the isotropic elastic material. However, local inhomogeneities in the stress field, e.g., at concentrated loads, do not distract the convergence of the algorithm. Beside a certain threshold in the equivalent stress field, the desired filling level of the design space is the main parameter of our objective function. The phase field parameter describes the density and stiffness of the substance in a closed interval. An Allen-Cahn equation regulates the phase transition, which is not conserving the mass of the system. The model evolves continuous regions of voids or dense material, whereas voids retain an infinitesimal residual stiffness, which is a million times smaller than the stiffness of the dense material. The evolution of structures is discussed by numerical examples.

1 Introduction The topology and shape of load-bearing structures conduct their mechanical performance. Therefore, numerical optimization methods have become of great interest in the corresponding branch of mechanics. Since Bendsøe and Kikuchi [3], Bendsøe [2] proposed material distribution concepts they have inspired many methods. Review articles on different approaches are given in [5, 8, 10, 11]. Let us introduce a continuous variable ϕ to specify the topology of evolving structures. This phase field variable defines voids by ϕ → −1, substance by ϕ → +1, and is coupled to the density of the material such that intermediate states 0 ≤ ρ/ρ0 ≤ 1 occur. From a technical point of view the intermediate density of material between voids (ρ/ρ0 = 0) and substance (ρ/ρ0 = 1) is unrequested. On the other

I. Muench () Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_29

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hand, such diffuse transition zones are well-known from bio-mechanical processes like the formation of bones. Similarly, diffuse transition zones in the phase field model play the role of sensors and effectors to evolve the structure. Phase interfaces remain mechanically sensitive and enable the substance phase to grow into voids, and vice versa. In contrast, voids are insensitive to mechanical strain. Whereas level set models move boundaries between substance and voids with help of sensitivity analysis [1, 11, 12], the phase field approach generates driving forces in the transition zone as result of an explicit sensitivity function. From our point of view, a sensitivity function is computationally cheap compared to sensitivity analysis. Additionally, it allows for parameters evaluated by integrals over the design domain. The outline of the paper is as follows: In Sect. 2 we give the ideas of the model and explain the governing equations. Some aspects of numerical implementation are also discussed. Then, the performance of the algorithm is demonstrated in Sect. 3. Finally, we conclude in Sect. 4.

2 Stress Based Evolution Model Let us introduce some operators for vectors a and tensors A by Grad[a] = ai,j ei ⊗ ej , Grad[A] = Aij,k ei ⊗ ej ⊗ ek , Div a = ai,i , Div A = Aij,j ei , tr[A] = Aii , devA = A − 1/3 tr[A] 11 , where e1 , e2 , e3 denote Euclidean basis vectors, and 11 is the identity. To keep the computational cost of the algorithm as low as possible, our objective function F does not base on gradients of equivalent stress, which is given by ( σV =

3 tr[(devσ )2 ] . 2

(1)

We expect that the inhomogeneity of the equivalent stress field σV (ϕ) can be measured with help of a certain value σ¯ V within the design space B. This motivates the objective function F (σV (ϕ), σ¯ V ). A solution with σV ≡ σ¯ V in the substance, and σV ≡ 0 in voids would be the perfect result of our algorithm. However, this thought teaches us, that we actually seek for homogeneous equivalent stress in the substance only. In regions of phase transition the homogeneity of the equivalent stress must be relinquished. Further, concentrated forces inevitably yield inhomogeneous equivalent stress σV (ϕ). Thus, the objective function F (σV (ϕ), σ¯ V ) must generally accept inhomogeneities in the equivalent stress, which is also found as statement in [6, 9].

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Of course, the value of σ¯ V (ϕ) must adapt to the desired filling level  (ϕ + 1) dV Kˆ = B  B dV

(2)

within the design space. Compared to higher filling levels, low filling levels Kˆ require higher σ¯ V (ϕ) in the evolving structure for equivalent load. Thus, we denote ˆ ϕ) and consider: σ¯ V (K,  F =

ˆ ϕ) − σV (ϕ) σ¯ V (K, ϕ dV → min w.r.t. ϕ . ˆ ϕ) σ¯ V (K, B ) *+ , ˆ ϕ) γ (K,

(3)

ˆ ϕ) measures the inhomogeneity of the equivalent stress concerning The term γ (K, σ¯ V . Latter is a threshold for the growth or disaggregation of substance. Derivation of Eq. (3) towards its minimizer requests ∂F =0 ∂ϕ

 ⇔

∂γ ϕ + γ dV = 0 . B ∂ϕ

(4)

Since the phase field variable ϕ couples to the density of material by ρϕ f (ϕ) ρ0 ,

f (ϕ)

eα ϕ , eα ϕ + 1

(5)

we can interpret γ as pressure to adjust the density of substance in B. The continuous function f (ϕ) has lower and upper limits eα ϕ = 0, ϕ→−∞ eα ϕ + 1 lim

lim

ϕ→+∞

eα ϕ = 1, eα ϕ + 1

(6)

and increases monotonously between its limits. The first and second order derivative of f (ϕ) with respect to ϕ reads 

f (ϕ) =

α eα ϕ (eα ϕ + 1)2

,



f (ϕ) =

  α2 eα ϕ 1 − eα ϕ (eα ϕ + 1)3

.

(7)

Similarly to the density of the substance, the elasticity matrix Cϕ is coupled to the phase field variable Cϕ f (ϕ) C0 ,

(8)

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where C0 is the elasticity matrix of the employed material. The filling level in the design domain is given by 

 ρϕ dV f (ϕ) dV = B . ρ dV B 0 B dV

K(ϕ) B

(9)

Our variational approach considers the Gibbs free energy Ψ (ε, ϕ, Grad[ϕ]) = Ψmech (ε , ϕ) + Ψwell (ϕ) + Ψgrad (Grad[ϕ]) ,

(10)

with inner mechanical energy, double well potential, and gradient energy:   1 Ψmech ε, ϕ = f (ϕ) ε : C0 : ε , 2

Ψwell(ϕ) = ϕ 6 − ϕ 4 − ϕ 2 + 1 ,

Ψgrad (Grad[ϕ]) =

7 72 1 Lc 7Grad[ϕ]7 . 2

(11)

The parameter Lc regulates the thickness of the phase transition. The total energy Π of the model penalizes the objective function (3) as additional energy by factor cγ   ˆ ϕ) ϕ dV Π(u, ϕ, Grad[ϕ]) = B Ψ (ε(u), ϕ, Grad[ϕ]) dV + B cγ γ (K,   (12) − B ρϕ b · u dV − ∂ B (t · u + y ϕ) dA . The term ρϕ b accounts for body forces, e.g., the weight of substance. Similarly, the ˆ ϕ) ϕ can be interpreted as external work for the injection or extraction term cγ γ (K, of substance to evolve a structure with best possible homogeneous equivalent stress. Variation of the total energy Π with respect to its minimizers u and ϕ requires  ∂Ψ ∂Ψ ∂Ψ : δε + δϕ + · Grad[δϕ] dV ∂ϕ ∂(Grad[ϕ]) B ∂ε  + (cγ δγ ϕ + cγ γ δϕ − f (ϕ) ρ0 b · δu − f  (ϕ) ρ0 δϕ b · u) dV

  δΠ =

 −

B

!

∂B

(t · δu + y δϕ) dA = 0 ,

∂Ψ ∈ R3×3 , σ ∂ε

η

∂Ψ ∈ R, ∂ϕ

ξ

∂Ψ ∈ R3 . ∂Grad[ϕ]

(13)

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Applying the divergence theorem in Eq. (13) to terms with σ and ξ yields  δΠ = 

B B

(−Div σ − f (ϕ) ρ0 b) · δu dV + (η − Div ξ − f  (ϕ) ρ0 b · u + cγ γ ) δϕ + cγ δγ ϕ dV  +

!

∂B

(σ · n − t) · δu + (ξ · n − y) δϕ dA = 0 .

(14)

Allen and Cahn [4] suggest to relax the balance equation (14) by  δΠ(u, ϕ) = −

B

β ϕ˙ δϕ dV ,

β > 0.

(15)

Then, the Euler equations of the model read Div σ + f (ϕ) ρ0 b = 0 , η − Div ξ − f  (ϕ) ρ0 b · u + cγ γ = −β ϕ˙ .

(16)

We solve the above equations in the weak form within a nonlinear finite element framework and the backward Euler time integration scheme. The median σ¯ V in Eq. (3) adapts to the desired filling level Kˆ as follows. First, the equivalent stress of all integration points nGP at position xGP is stored in the array fi = σV (xGP ) ,

i = 1 . . . nGP .

(17)

Then, the components of fi are sorted in ascending order fi → fj ,

fj +1 ≥ fj ,

i, j = 1 . . . nGP .

(18)

With help of the index k we define with 2 n + 1 components of fj : k = int

?

@ j =k+n 

1 − Kˆ · nGP , σ¯ V  j =k−n

fj . 2n+ 1

(19)

Random noise in σ¯ V during the evolution process is reduced to an acceptable level with n = 0.001 nGP . The update of σ¯ V is performed after the converged time step. For more details we refer to [7].

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3 Numerical Examples 3.1 Plate with Top-Load We consider an initially filled rectangular region of length L = 100, and height H = 50 as design space B with thickness d = 1. The isotropic elastic material has the elastic modulus E = 1000, and Poisson’s ratio ν = 0.3. Plane stress is assumed for the evolving structure with 10% desired filling level of B. A line load q = 0.6 of length l = 4 is at the top of the plate. We use 200 × 100 biquadratic finite element for the mesh, and the numerical parameters Lc = 2 and cγ = 10. The evolution of the structure is shown in Fig. 1. First of all, a double arch bridge evolves before the algorithm reduces the final state to a triangle of beams. Then, except at the bar end of beams the equivalent stress becomes homogeneous, see Fig. 2. Increasing the

a)

b) design space

L

2L

c)

d)

e)

f)

Fig. 1 Plate with top-load reducing 95% mass: initial (a), intermediate (b)–(e), and final (d) structure for cγ = 10

Evolution of Load-Bearing Structures with Phase Field Modeling

a)

b)

c)

d)

0.0

0.2

0.4

0.6

0.8

341

1.0

Fig. 2 Plate with top-load: equivalent stress in the initial (a), intermediate (b)–(c), and the final (d) structure for cγ = 10

a)

b)

Fig. 3 Plate with top-load for cγ = 50: intermediate (a) and final (b) structure

nucleation factor up to cγ = 50 yields more complex intermediate states, however, the final topology is similar to the solution with cγ = 10, see Fig. 3.

3.2 Beam with Top-Load This example increases the span with of the load-bearing structure to a rectangular design space with L = 600, H = 50, and d = 1 such that 1200 × 100 biquadratic

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a)

b)

c)

d)

e)

f)

g)

Fig. 4 (a) Initial state with 100% substance in the design space. (b) Start of evolution process with trajectories indicated by colored lines. (c) Progressive evolution reduces the number of trajectories. (d) The structure simplifies and yellow trajectories disappear. (e) Further degradation of beams (blue trajectories) yields significant bending of remaining beams. (f) Substance accumulates to beams in regions of significant bending highlighted by the dashed orange circle. (g) Final structure with 15% substance in the design space. Deformations exaggerated by factor 50

Fig. 5 Increased span width (800 × 50) of the design space yields a twofold topology: an overall truss like structure with a substructure marked by dashed orange lines

finite elements are used. Material, loading, and parameters as given in Sect. 3.1. The evolution process of the structure is explained in Fig. 4. It is the classical solution with trajectories. However, further increase of the span will lead to a topology, which is well known in civil engineering, since it uses a substructure for the central beam of an overall truss like structure, see Fig. 5.

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4 Conclusions A numerical model for the evolution of load-bearing structures with almost homogeneous equivalent stress is presented. The objective function enters the variational approach as penalty function. The corresponding penalty factor cγ has great impact to evolving structures. Our observations are promising, since the evolving structures confirm classical results as well as meaningful alternatives.

References 1. G. Allaire, F. Jouve, A. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393 (2004) 2. M. Bendsøe, Optimal shape design as a material distribution problem. Struct. Optim. 1(4), 193–202 (1989) 3. M. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988) 4. J. Cahn, S. Allen, A microscopic theory for domain wall motion and its experimental verification in fe-al alloy domain growth kinetics. J. de Physique Colloques 38(C7), 51–54 (1977) 5. J. Deaton, R. Grandhi, A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49, 1–38 (2014) 6. C. Le, J. Norato, T. Bruns, C. Ha, D. Tortorelli, Stress-based topology optimization for continua. Struct. Multidiscip. Optim. 41(4), 605–620 (2010) 7. I. Münch, Ch. Gierden, W. Wagner, A phase field model for stress based evolution of loadbearing structures. Int. J. Numer. Methods Eng., 1–21 (2018). https://doi.org/10.1002/nme. 5909 8. D. Munk, G. Vio, G. Steven, Topology and shape optimization methods using evolutionary algorithms: a review. Struct. Multidiscip. Optim. 52, 613–631 (2015) 9. G. Rozvany, On design-dependent constraints and singular topologies. Struct. Multidiscip. Optim. 21(2), 164–172 (2001) 10. O. Sigmund, K. Maute, Topology optimization approaches. Struct. Multidiscip. Optim. 48, 1031–1055 (2013) 11. N. van Dijk, K. Maute, M. Langelaar, F. van Keulen, Level-set methods for structural topology optimization: a review. Struct. Multidiscip. Optim. 48(3), 437–472 (2013) 12. M. Wang, X. Wang, D. Guo, A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192, 227–246 (2003)

An Accelerated Newton Method for Nonlinear Materials in Structure Mechanics and Fluid Mechanics Thomas Richter and Carolin Mehlmann

Abstract We analyze a modified Newton method that was first introduced by Turek and coworkers. The basic idea of the acceleration technique is to split the Jacobian A (x) into a “good part” A1 (x) and into a troublesome part A2 (x). This second part is adaptively damped if the convergence rate is bad and fully taken into account close to the solution, such that the solver is a blend between a Picard iteration and the full Newton scheme. We will provide first steps in the analysis of this technique and discuss the effects that accelerate the convergence.

1 Introduction On the ENUMATH 2015 conference in Ankara, Turek and coworkers [4] presented an accelerated Newton scheme, the operator related damped Newton method, for the solution of complex non-Newtonian flow problems. The basic idea is to split the Jacobian A(x) into a “good part” A1 (x) that stabilizes the solution and into the “bad part” A2 (x). This splitting is mostly ad hoc. The Jacobian is replaced by Aδ (x) = A1 (x) + δA2 (x) and the parameter δ ∈ [0, 1] is adaptively tuned to the recent convergence history. Since the first notion of the method it has been used by different authors in several application problems like granulate flow [5], fracture propagation [7] and also by ourselves in a study on a viscous-plastic rheology used in describing the dynamics of the sea ice layer on the arctic and antarctic ocean [6]. In most of these applications, the partitioning of the Jacobian matrix A(x) is ad hoc. We discuss, as example, the Navier-Stokes equations: the Newton linearization of the convective term N(v)(φ) := (v · ∇v, φ) in a search direction w is given by (v · ∇w, φ) + (w · ∇v, φ). Here, a possible choice for the “good part” would by (v · ∇w, φ) due to definiteness, the remaining part (w · ∇v, φ) would be left due T. Richter () · C. Mehlmann University of Magdeburg, Institute for Analysis and Numerics, Magdeburg, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_30

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to its undetermined sign that could possibly cause problems. The optimal choice is less obvious for complex problems like fracture mechanics [7]. A second problem of the operator related damped Newton approach is the proper control of the parameter δ used for partial damping. In [5] the author computationally tested various techniques for adaptively controlling this parameter and derived a formula that performs very well in different applications [6, 7]. There is however no rigorous proof for the robustness of this strategy that relates the damping parameter to the convergence rate of the Newton scheme. The relevant mechanism of the operator related damped Newton scheme is not understood. Several applications show that a very small deviation of δ from 1 already shows a substantial effect. Aim of this article is to give better inside to the operator related Newton scheme. In Sect. 2 we start by presenting some results from an application of this Newton strategy to a viscous-plastic flow model taken from [6]. Next in Sect. 3 we give first steps for an analysis of this Newton method. Finally, we conclude.

2 An Application of the Operator Related Newton Scheme to a Viscous-Plastic Flow Problem The results presented in this section are mostly taken from [6] and have also been presented on the 2017 Enumath conference.

2.1 A Brief Description of a Viscous-Plastic Sea Ice Model A common and widely used sea ice model was introduced by Hibler [2] and is based on the following modeling assumptions: 1. The ice layer on the ocean is described as a two dimensional vector field denoting the velocity v : Ω → R2 , by the scalar ice height H : Ω → R+ and the scalar ice concentration A : Ω → R ∩ [0, 1] that indicates whether a certain area is covered by stiff ice (A = 1) or by open water (A = 0). H and A are advected with the ice velocity v. 2. Ice mass is conserved by linearized conservation laws ρice ∂t v − div σ = ρice f,

(1)

where f describes forcing by ocean and atmosphere current as well as all gravitational or rotational effects (here these effects are simplified and we refer to Hibler [2] and [6] for a complete description of the model. 3. Ice is considered as a viscous-plastic material following a normal flow. The  yield stress (the ice strength P ) is given by P = P ! H exp − 20(1 − A) , where

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P ! = 2.75 · 104 N/m3 is a constant. Below this critical value a linear viscous behavior is assumed. The stresses consist with an elliptical (eccentricity e = 2) yield curve giving the stress-strain relationship σ = 2e−2 ζ ˙  + ζ tr(˙ )I − 1/2P I with  = 12 (∇v + ∇vT ), ˙  = ˙ − 12 tr(˙ )I and the viscosity ζ . 4. In [6] we have derived the following formulation to express the viscosity ζ =

P , 2Δ(˙ )

1  2 Δ(˙ ) = Δ2min + 2e−2 ˙  : ˙  + tr(˙ )2 ,

(2)

The constant Δmin = 2 · 10−9 s determines the critical strain for a transition to the viscous case. Δ(˙ ) contains a smooth transition between viscous and plasti‘c behavior to ensure differentiability. Besides problems of data uncertainty, size and variability of geometry and parameters the great challenge in sea ice simulations is the nonlinearity coming from the plastic material behavior. The viscosities given by (2) can greatly vary. For ice 13 2 at rest, their maximum is bound by P ! Δ−1 min ≈ 10 Ns/m while for typical strain rates of fast moving ice (usually still less than 50 cm/s) the viscosities can go down to values as small as 102 Ns/m2 . Currently there exists no satisfying nonlinear solver for accurately solving this nonlinear sea ice models. The most advanced and reliable techniques are based on a Newton method with various globalization techniques, see Lemieux, Losch et al. [3].

2.2 Linearization by Newton We only discuss the linearization of the critical stress terms. The complete derivation is given in [6]. The stresses A(v)(φ) = (σ , ∇φ) can be written as ⎞

⎛

√ −1  ⎟ ⎜ 1 P τ (v), τ (φ) ⎟ A(v)(φ) := ⎜  1 ⎠ , τ (φ) = 2e ˙ (φ) + √2 tr(˙ (φ))I. ⎝ 2 Δ2min + τ (v) : τ (v) Ω

(3) This allows for a simple expression of the Jacobian as ⎞

⎛ ⎜ A (v)(w, φ) = ⎜ ⎝ )

⎞

⎛

⎟ ⎜ P (τ (v) : τ (φ))τ (w), τ (v) ⎟ ⎟ ⎟ ⎜ 1 ⎠ − ⎝  3 ⎠ . 2 2 2 2 Δmin + τ (v) : τ (v) Δmin + τ (v) : τ (v) Ω Ω *+ , ) *+ , P τ (w), τ (φ)

=:A1 (v)(w,φ)

=:A2 (v)(w,φ)

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Theorem 1 (Jacobian of the Sea Ice Problem) Let τ (·) be continuous with a Korn’s like inequality c1 wH 1 (Ω) ≤ τ (w) ≤ c2 wH 1 (Ω) . Then, the Jacobian is positive definite satisfying the estimates 9

Δ2min c1 min Ω Δ2

C

 ζ w2H 1 (Ω) ≤ A (v)(w, w) ≤ c2  ζ w2H 1 (Ω)

c1  ζ w2H 1 (Ω) ≤ A1 (v)(w, w) ≤ c2  ζ w2H 1 (Ω) 9 C Δ2min  −c1 min 1 − ζ w2H 1 (Ω) ≤ A2 (v)(w, w) ≤ 0. Ω Δ2 Proof The proof is given in [6]. From the analysis of the Jacobian we derive the following consequences: as A (v) is positive definite, a properly damped Newton scheme should converge globally. The Jacobian’s lower bound can get very small for high strain rates (such that Δmin /Δ ≈ Δmin = 2 · 10−9 ). The first part of the Jacobian A1 (v) is positive definite and bounded with constants that are properly scaled. This explains the choice as “good term”. We add arguments in Sect. 3. Finally, the second part A2 (v) is negative semidefinite and the reason for the critical lower bound in the full Jacobian. Hence, this part is selected for damping.

2.3 The Operator Related Damped Newton Method for the Simulation of the Sea Ice Dynamics We apply the operator related damped Newton scheme to the partitioning Aδ (v) = A1 (v) + δA2 (v) using the adaptive control introduced in [5]. In Fig. 1 we show the progression of the Newton scheme with and without control of the parameter δ.

3

δ (l) reduction rate relative residuum

1.4

d (l) = 1 reduction rate relative residuum

mesh size 2 km 2.5

mesh size 2 km

1.2

2

1

1.5

0.8 0.6

1

0.4

0.5

0.2

0 0

50

100 Newton iteration

150

200

0

0

10

20

30 40 Newton iteration

50

60

Fig. 1 Sea ice simulation on a mesh of 2 km mesh size (which is considered as very accurate within the community). Left: simulation with standard Newton with line-search. No convergence is observed. Right: the operator related damped Newton gives convergence

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In addition, both settings are completed with a standard line search scheme. While standard Newton does not give convergence, that operator related damping helps. We observe, that the damping parameter δ is never decreased to values below δ ≈ 0.98. Nevertheless, the effect is immense. See [6] for further details.

2.4 Related Results for Various Viscosity Models The stress formulation of the sea ice problem (3) can be easily altered to fit various other applications. The p-Laplacian could be interpreted by choosing the tensor τ (v) = ∇v and by relaxing the exponent to p ∈ (1, ∞)    p−2 2 A(v)(φ) =  2 + τ (v) · τ (v) τ (v) · τ (φ) dx, Ω

where  > 0 is a regularization parameter. For the Jacobian we get the same results as indicated in Theorem 1. The limit case p = 1 is the parabolic minimal surface problem that also falls into our class of possible formulations. Power-law fluid models like the Carreau fluid can be represented by using the tensor τ (v) = ˙  (v) and the variational formulation ⎛ ⎞  − ν ν 0 ∞ ⎝ν∞ + ⎠ A(v)(φ) =   1−n τ (v) : τ (φ) dx 2 Ω 1 + λ τ (v) : τ (v) 2 with limiting viscosities ν0 and ν∞ , the power-index n and the relaxation time λ. For all these cases we can show similar bounds to Theorem 1. The complete Jacobian— in all these cases—is positive definite with a lower bound that is possibly close to zero and that depends on the regularization parameter  and the limiting viscosities ν0 , ν∞ . The derivative of the linear part is positive definite with robust bounds, while the remaining part (the derivative with respect to the viscosity) is negative semidefinite. This allows a clear splitting of the Jacobian into an easy part A1 and into a destabilizing part A2 .

3 First Analysis of the Operator Related Newton Scheme The effect of the operator related Newton scheme is an acceleration of the convergence. It is not strictly a globalization method as we apply it to problems with positive definite Jacobian that already allow for global convergence. The problem in numerical realization is the identification of optimal line search parameters. To start the exposition we give a variant of a well-known result concerning globalization of Newton schemes (compare Lemma 3.5 in [1]).

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Theorem 2 Let f : D ⊂ Rn → Rn with f ∈ C 1 (D) and Lipschitz continuous Jacobian f  = f1 + f2 (Lipschitz constant being L > 0). For δ ∈ [0, 1] define fδ := f1 + δf2 such that fδ (x) ≥ γδ x,

γδ > 0.

Then, let δ ∈ [0, 1] and fδ := f1 + δf2 the following estimate holds for the operator related damped Newton iterate xω = x − ωfδ (x)−1 f (x)  f (xω ) ≤ 1 − ω

+ (1 − δ)ωf2 (x)fδ (x)−1  +

 Lω2  −1 f (x) f (x) f (x). 2γδ δ

Proof Let x ∈ D be the current iterate. We define an intermediate newton step ?  xs := x − sfδ (x)−1 f (x) ⇒ f (xω ) = I −

ω

0

@ f  (xs )fδ (x)−1 f (x) ds f (x)

Insertion of ±fδ (x) and estimation gives the result. We first discuss the standard case δ = 1 corresponding to the full Newton scheme. The convergence rate ρ and the optimal damping parameter ωopt with a resulting optimal bound for the convergence rate is given by ρ(ω) = 1 − ω +

Lω2 2γ

⇒

ωopt =

γ L

⇒

ρ(ωopt ) = 1 −

γ , 2L

(4)

where γ := γ1 is the lower bound for the full Jacobian from Theorem 1. The sea ice problem yields γ ≈ 10−14 (this value is taken from simulations and is actually less worse than the estimate γ ∼ Δ2min = 4 · 10−18 might suggest). This explains the very slow convergence of standard Newton schemes. Furthermore, with ωopt being defined by (4) it is difficult to numerically identify a correct parameter ω in this tiny range. To understand the effect of the operator related damping we give bounds for the Jacobian Aδ comparable to those indicated in Theorem 1. Combining A1 (v) and A2 (v) directly gives C⎫ Δ2min ⎬  ζ w2H 1 (Ω) c1 max 1 − δ, δ min ⎩ Ω Δ2 ⎭ ⎧ ⎨

9

≤ Aδ (v)(w, w) ≤ c2  ζ w2H 1 (Ω) such that the lower bound is shifted to γδ ≥ 1 − δ. Compared to γ ∼ 10−14 even very small deviations of δ from 1 result in a substantial improvement of this lower bound. In light of (4) the improved bound would give a prediction of ωopt,δ and

Analysis of an Accelerated Newton Solver

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ρ(ωopt,δ ) shows a great improvement ωopt,δ =

γδ L

⇒

ρ(ωopt,δ ) = 1 −

γδ . 2L

However, this result is only part of the truth, as the complete estimate given in Theorem 2 for δ < 1 includes the term (1 − δ)ωf2 (x)fδ (x)−1  which is skipped here and that counteracts the acceleration by an improved bound for γδ > 0. A complete analysis cannot be performed without further insight to the differential operators.

3.1 Numerical Analysis of the Operator Related Newton Method To validate our findings and the role of the damping parameter δ we analyze as simpler problem (compared to the sea ice problem) the limit-case of the pLaplacian, given by the variational formulation   A(u, φ) = ζ (u)∇u, ∇φ Ω ,

− 1  ζ (u) := ε + |∇u|2 2 .

This is a variant of the minimal surface problem which is obtained for ε = 1. The Jacobian is given by     A (u)(w, φ) = ζ (u)∇w, ∇φ Ω − ζ (u)3 (∇u · ∇w)∇u, ∇φ Ω ) *+ ,) *+ , =:A1 (u)(w,φ)

=:A2 (u)(w,φ)

and it directly suggests a splitting. It is easy to check that the partially damped Jacobian Aδ (u)(w, φ) := A1 (u)(w, φ) + δA2 (u)(w, φ) is positive definite for all δ ∈ [0, 1] and allows for the bounds :

 ; 2 2 c1 max 1 − δ, min ε ζ (u)  ζ ∇w2 Aδ (u)(w, w) ≤ c2  ζ ∇w2 . Ω

Given ε  1 even small deviations of δ from 1 have a large effect. We numerically study this problem for ε2 = 10−4 using a standard finite element discretization with piecewise linear finite elements on a quadrilateral mesh of the domain Ω = (0, 1)2 that consists of 64 × 64 elements. We study the homogeneous problem with Dirichlet u(x, y) = sin(πxy) + sin(π(2x + y)) on (0) ∂Ω. The initial solution uh is set to zero within the domain and it complies with the Dirichlet data on the boundary. This set of parameters is sufficiently difficult to pose great challenges to a standard Newton scheme that—by theory— should globally converge, as the Jacobian is positive definite. It is however very

352

T. Richter and C. Mehlmann 1.14 1.12 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92

damping parameter ω 0

0.02

0.04

0.06

0.08

0.1

Fig. 2 Convergence of the standard Newton scheme in the very first step. Variation of the relative residual change for different linesearch parameters. Convergence is obtained for only a very small range of parameters 1.4

δ=1

1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5

δ = 0.99

1.2 1 0.8 δ = 0.9

0.6

δ= 0 δ = 0.7

0.4 0

0.2

damping parameter ω δ = 0.3 0.4 0.6 0.8

1

δ = 0.99

δ= 1

δ = 0.3, δ = 0

δ = 0.9

δ = 0.7

damping parameter ω 0

0.2

0.4

0.6

0.8

1

Fig. 3 Convergence of the operator related damped Newton scheme for different damping parameters δ. Each figure shows the relative change of the residual after one Newton step for different linesearch parameters

difficult to numerically find suitable line search parameters ω. In Fig. 2 we show the convergence rates that would result from different choices of the damping parameter ω in the very first step of the Newton iteration. Here, convergence is only surely obtained for ω ∈ (0, 0.01). Finally Fig. 3 shows the effect of controlling the parameter δ ∈ [0, 1] for a damping of the negative semidefinite part A2 (u). We show the results for the very first Newton step and after some iterations when the initial residual was already reduced by two orders of magnitude. These results show two effects of the operator related damped Newton scheme: first, the optimal convergence rates are strongly improved by choosing a proper value for the damping parameter δ. Second—and even more important—a good choice of δ < 1 enlarges the interval where a suitable linesearch parameter ω might be found. This eases the construction of numerically robust solvers. In the first step, given δ = 0, convergence is given for all ω ∈ (0, 1] instead of ω ∈ (0, 0.01) in the undamped case.

Analysis of an Accelerated Newton Solver

353

4 Conclusion We conclude with two findings: the operator related damped Newton scheme that has been introduced by Mandal, Ouazzi and Turek [4] is highly effective. In different applications it has proven to successfully accelerate Newton solvers. The effects are partially due to an enlarging of the range of linesearch parameters that give convergence. This allows for efficient numerical schemes with few residual evaluations during the linesearch procedure and good convergence rates. Second by correct damping we can improve the convergence rates. A rigorous analysis of the operator related Newton scheme is still open. In particular a robust adaptive procedure for the choice of the damping parameter that leads to a convergent solver is missing. Acknowledgements The authors acknowledge the financial support by the Deutsche Forschungsgemeinschaft (314838170), GRK 2297 MathCoRe, the Federal Ministry of Education and Research of Germany (05M16NMA) and the German Federal Environmental Foundation.

References 1. P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Computational Mathematics. vol. 35 (Springer, Berlin, 2011) 2. W.D. Hibler, A dynamic thermodynamic sea ice model. J. Phys. Oceanogr. 9, 815–846 (1979) 3. M. Losch, A. Fuchs, J.F. Lemieux, A. Vanselow, A parallel Jacobian-free Newton-Krylov solver for a coupled sea ice-ocean model. J. Comput. Phys. 257, 901–911 (2014) 4. S. Mandal, A. Ouazzi, S. Turek, Modified Newton Solver for Yield Stress Fluids. Proceedings of the ENUMATH 2015 (Springer, Berlin, 2016) 5. S. Mandal, Efficient FEM solver for quasi-Newtonian problems with application to granular materials. Dissertation, Technical University Dortmund (2016) 6. C. Mehlmann, T. Richter, A modified global Newton solver for viscous-plastic sea ice models. Ocean Model. 116, 96–107 (2017) 7. T. Wick, Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation. Comput. Methods. Appl. Mech. Eng. 325, 577–611 (2017)

Part X

A Posteriori Error Estimation, Adaptivity and Approximation

Best Error Localizations for Piecewise Polynomial Approximation of Gradients, Functions and Functionals Andreas Veeser

Abstract We consider the approximation of (generalized) functions with continuous piecewise polynomials or with piecewise polynomials that are allowed to be discontinuous. Best error localization then means that the best error in the whole domain is equivalent to an appropriate accumulation of best errors in small domains, e.g., in mesh elements. We review and compare such best error localizations in the three cases of the Sobolev-Hilbert triplet (H01 , L2 , H −1 ).

1 Introduction to Best Error Localizations Let V be a normed function space over some domain Ω ⊂ Rd and denote by  · Ω its norm. We consider the approximation of any v ∈ V by functions from a space S that is built with the help of a mesh. The best possible error is then given by infs∈S v − sΩ . Our interest here is to describe the best error in the following manner:

p ∀v ∈ V inf v − sΩ ≈ inf v − ppωi (1) s∈S

i∈I

p∈S|ωi

where the power p ∈ [1, ∞) is related to the employed norm, {ωi }i∈I is a finite covering of subdomains, ideally the mesh elements, arising from the underlying mesh,  · ωi is the restriction of the norm  · Ω to ωi , and S|ωi is the restriction of the approximation space S to ωi . Notice that the quantities of the right-hand side are local best errors, thus counterparts of the global best error on the left-hand side. We therefore refer to such an equivalence as a best error localization.

A. Veeser () Università degli Studi di Milano, Dipartimento di Matematica F. Enriques, Milano, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_31

357

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A. Veeser

A best error localization is a basic result about the approximation problem of the above type and in particular useful in the following contexts: – Error bounds: It suffices to bound the local best errors in terms of some given regularity of v beyond  · ωi . The resultant bound for the global error will be automatically in terms of local meshsize and local regularity. Notice that one is completely free in the amount of extra regularity, since v in (1) is arbitrary. – Parallelization: The fact that the local best errors are independent of each other allows for the approximate parallel computation of the global best error. – Adaptive mesh refinement: In adaptive mesh refinement, one iteratively applies local refinements in order to construct meshes that are well-tuned for a given function v. The local best errors as in (1) can be invoked in fast tree approximation [1] in order to choose suitable local refinements in the case when v is known. Tree approximation can be employed for coarsening in the adaptive solution of partial differential equations. The rest of this article is organized as follows. Section 2 introduces the choices for V ,  · Ω , and S that will be analyzed and compared in Sects. 3, 4, and 5. To complete the discussion, Sect. 6 presents conclusions for the case, where we approximate simultaneously in all previously considered norms.

2 Sobolev-Hilbert Triplet, Meshes, and Piecewise Polynomials Let Ω ⊂ Rd , d ∈ N, be a domain (an open, connected, and bounded set) with polyhedral boundary ∂Ω. In what follows, we shall approximate elements of the Sobolev spaces  2

L (Ω) with

1

v0;Ω :=

|v|

2

2

,

Ω

H01 (Ω) := {v ∈ H 1 (Ω) | v|∂Ω = 0} with H −1 (Ω) := H01 (Ω)∗

with

f −1;Ω :=

|v|1;Ω := ∇v0;Ω , f, ϕ , ϕ∈H 1 (Ω)\{0} |ϕ|1;Ω sup

0

which satisfy H01 (Ω) ⊂ L2 (Ω) ⊂ H −1 (Ω) and form a Hilbert triplet. The following notions will be useful in connection with our approximants and their best error localizations. An n-simplex C in Rd with n ∈ {1, . . . , d} is the convex hull of n + 1 points in d R that do not lie on an (n − 1)-dimensional plane, while a 0-simplex is a point. The set of vertices of C is indicated by V(C).

Best Error Localizations in Piecewise Polynomial Approximation

359

Let M be a simplicial, face-to-face mesh of 4 Ω. More precisely, M is a finite collection of d-simplices in Rd such that Ω = K∈M K, the intersection K1 ∩ K2 of any K1 , K2 ∈ M is either empty or an n-simplex with n ≤ d and V(K1 ∩ K2 ) = V(K1 ) ∩ V(K2 ), and the intersection K ∩ ∂Ω for each K ∈ M is either empty or an n-simplex with n ≤ d − 1. Apart from elements, the following subdomains will be used in the best error localizations below. We let V denote the set of all vertices of M and associate with each vertex z ∈ V the star 3 ωz := {K ∈ M : K 6 z}. Furthermore, F stands for the set of all internal (d − 1)-faces of M. Given such a face F ∈ F , let K1 , K2 ∈ M be the two elements containing F and set ωF := K1 ∪ K2 , which we call, a bit improper, the pair associated with F . The following three properties of a mesh M will be important for the validity or equivalence constants of the best error localizations below. We say that M has face-connected stars whenever, for any K1 , K2 ∈ M sharing a vertex z ∈ V, there exists a path (Ki )ni=1 of elements such that K1 = K1 , Kn =  K2 , Ki ⊂ ωz and Ki ∩ Ki+1 ∈ F for all i = 1, . . . , n − 1. Observe that our assumptions on Ω and M allow that Ω is on both sides of the boundary at some places. As a consequence, face-connectedness of stars may not be verified; cf. [10, Fig. 1]. Given an element K ∈ M, we denote by hK its diameter and by ρK the diameter of its largest inscribed ball. The shape coefficient of M is then σM := max hK /ρK . K∈M

A condition that is somewhat weaker than shape regularity is local quasiuniformity. Here we shall quantify it with the help of the following two numbers: μM := max

max |K  |/|K| and μ˜ M := max #{K ∈ M | K 6 z}.

K∈M K  ∩K=∅

z∈V

We conclude this section by introducing the approximants. Given  ∈ N0 , we let P denote the set of all polynomials of total degree ≤  and write S0 (M) := {s ∈ L2 (Ω) | ∀K ∈ M s|K ∈ P }. for the space of discontinuous piecewise polynomials over M. We have S0 (M) ⊂ L2 (Ω) ⊂ H −1 (Ω),

but S0 (M) ⊂ H01 (Ω).

360

A. Veeser

In order to measure the approximation error in spite of the nonconformity of S0 (M) with respect to H01 (Ω), we extend the H01 -seminorm as follows: ⎛ |v|1;Ω

:= ⎝



⎞1 2

|∇v|

2⎠

(2)

,

K∈M K

where we suppress the dependence on the mesh M for the sake of notational simplicity. The largest H01 (Ω)-conforming subspace of S0 (M) is S1 (M) := S0 (M) ∩ H01 (Ω) = {s ∈ S0 (M) | ∀F ∈ F the trace s|F is well-defined, s|∂Ω = 0}, which coincides with the space {s ∈ S0 (M) | s ∈ C 0 (Ω), s|∂Ω = 0} of continuous piecewise polynomials whenever M has face-connected stars. Notice the convention that k in Sk (M) stands for the number of weak, and not classical, derivatives. In the following sections, we fix the domain Ω and the mesh M and omit them in the notation for function spaces. Finally, if ω ⊂ Ω is a subdomain made of elements from M and k ∈ {0, 1}, we write Sk |ω := {s|ω | s ∈ Sk } for the restriction of Sk to ω.

3 Approximating Gradients Both spaces Sk , k ∈ {0, 1}, are popular choices in Galerkin methods for the solution of linear elliptic partial differential equations of second order. Quasi-optimality results like Céa’s lemma and nonconforming counterparts [11] ensure that the Galerkin error in H01 is bounded in terms of the best approximation error in H01 . Theorem 1 (Element-Element Localization) If the mesh M has face-connected stars, then, for any v ∈ H01 , ⎛ inf |v − s|1;Ω

s∈S0

=⎝

K∈M

⎞1 2

inf |v

p∈P

− p|21;K ⎠

≈ inf |v − s|1;Ω . s∈S1

The hidden constant in  is bounded in terms of d, , and σM . Proof The perfect localization of the best error with discontinuous piecewise polynomials by an equality is an immediate consequence of the extension (2) and the fact that local approximants in S0 are uncoupled. Further, we have infs∈S 0 |v − 

s|1;Ω ≤ infs∈S 1 |v − s|1;Ω because of S1 ⊂ S0 . The converse inequality with 

Best Error Localizations in Piecewise Polynomial Approximation

361

a multiplicative factor is covered by Veeser [10, Corollary 1]. The key issue is to find suitable values of a continuous approximant in shared nodes. To this end, one can use Scott-Zhang interpolation [5] or average nodal values of local best approximations. In both cases, the values of the target function v on one element affect the approximant in other elements. The error incurred by this information transport is controlled with the help of the face-connectedness of stars. Shape regularity arises through the scaling of derivatives. & % Let us discuss several aspects of Theorem 1 briefly. First, we see that, although of different dimension, S0 and S1 offer essentially the same approximation power. In particular, their decay rates for shape regular refinements always coincide. This hinges on the critical direction , for which the algebraic inclusion S0 ∩ H01 ⊂ S1 is a necessary condition. The fact that Theorem 1 localizes to elements may be also described by saying that the best error with S1 is quasi-optimal with respect to its shape functions P . This is a positive assessment of requiring continuity and zero boundary values. The localization to elements is advantageous for adaptive tree approximation in [1], because the local best errors are then not only simple but also monotone local error functionals; cf. [10, §4.2]. Finally, let us illustrate the application of Theorem 1 to error bounds. Given 1 < r ≤  + 1, elementwise application of the Bramble-Hilbert lemma gives ⎛ inf |v − s|1;Ω ≤ inf |v − s|1;Ω  ⎝

s∈S0

s∈S1

⎞1 2

2(r−1)

hK

|v|2r;K ⎠ ,

K∈M

where | · |r;K is the H r (K)-seminorm. Notice that the extra regularity on the righthand side is broken. Thus, if v happens to be in S1 , the right-hand side is 0, instead of being strictly positive or ∞ for unbroken regularity with 1 < r < 32 or r ≥ 32 , respectively. Furthermore, error bounds with broken regularity are crucial when Ω is a polyhedral surface; cf. [3].

4 Approximating Functions The approximation of a function in L2 is perhaps the most elementary approximation of functions in a Hilbert space and therefore of interest by its own. In particular, [10, Theorem 3] reduces the gradient approximation of the preceding section to the uncoupled approximation of the d partial derivatives in L2 . Moreover, discretizations basing on the standard weak formulation of parabolic problems require the L2 -approximation of the initial value and also here the spaces Sk , k ∈ {0, 1} are popular choices. This example is also related to the approximation in the so-called reaction-diffusion norm, where L2 -approximation appears as a limiting case.

362

A. Veeser

Theorem 2 (Element-Pair Localization) If M has face-connected stars, then, for any v ∈ L2 , ⎛ ⎝

K∈M

⎞1 2

inf v

p∈P

− p20;K ⎠

= inf v − s0;Ω s∈S0

⎛ ≤ inf v − s0;Ω ≈ ⎝ s∈S1

F ∈F

⎞1 2

inf v − s20;ωF ⎠ ,

s∈S1 |ωF

where the hidden constants are bounded in terms of d, , μM , and μ˜ M . Proof The perfect localization of the best error with S0 as well as the inequalities infs∈S 0 v − s20;Ω ≤ infs∈S 1 v − s20;Ω and 



F ∈F

inf v − s20;ωF  inf v − s20;Ω

s∈S1 |ωF

s∈S1

are again straight-forward; the hidden constant non depends on d. The missing, 2 trivial inequality infs∈S 1 v − s20;Ω  inf v − s is [9, 1 F ∈F s∈S |ωF 0;ωF  Theorem 5.1]. The proof is similar to the corresponding step in the proof of Theorem 1, but with several differences. One difference is the definition of the shared nodal values, which, in the spirit of Scott-Zhang interpolation, can be chosen by means of dual basis functions associated to elements instead of faces. Another difference regards the control of the information transport. Here one employs a path of pairs instead of elements, whose existence also results from the faceconnectedness of stars. Shape regularity does not arise as here derivatives are not involved. & % Let us compare Theorem 2 with Theorem 1. We first observe that a truly discontinuous piecewise polynomial is in L2 but not in S1 and so S0 ∩ L2 ⊂ S1 does not hold. Consequently, ≤ in Theorem 2 is strict in certain cases and element localization is not possible for the best error with S1 . This does not preclude the application of tree approximation [1], but leads to complications. In particular, one has to employ local best error on so-called minimal pairs that are only weakly monotone; cf. [9, §8.2] Let us see how the different approximation powers of S0 and S1 reflect in the error bounds. Given 0 < r ≤  + 1 and writing hF for the diameter of ωF , pairwise application of the fractional Bramble-Hilbert lemma in [4] implies for the continuous case ⎛ inf v − s0;Ω  ⎝

s∈S1

F ∈F

⎞1 2

2 ⎠ , h2r F |v|r;ωF

Best Error Localizations in Piecewise Polynomial Approximation

363

where, in contrast to the discontinuous case, regularity across element boundaries is required. This is however only relevant for low regularity. In fact, if r > 12 , one can replace in the bound the pairs by elements. This follows from the comparison of local best errors on pairs and on elements in [9, §6], which is also of interest in the context of mesh-dependent norms and for the choice of the penalty parameter in discontinuous Galerkin methods.

5 Approximating Functionals The explicit approximation of functionals may appear unusual. However, for the weak formulation of the Poisson problem −Δu = f in Ω, u = 0 on ∂Ω, the natural space for f is H −1 and therefore, generally speaking, its approximation is of interest. Similarly, for the standard weak formulation of a parabolic equation, the time derivative of the solution is in general, for a given time, only in H −1 and its discrete counterpart is often from one of the spaces Sk , k ∈ {0, 1}. Theorem 3 (Star-Star Localization) For any f ∈ H −1 , ⎛ ⎝

z∈V

⎞1 2

inf f − s2−1;ωz ⎠ ≈ inf f − s−1;Ω

s∈S0 |ωz

s∈S0

⎛ ≤ inf f − s−1;Ω ≈ ⎝ s∈S1

z∈V

⎞1 2

inf f

s∈S1 |ωz

− s2−1;ωz ⎠

,

where the hidden constants are bounded in terms of d, , and σM . Proof In addition to infs∈S 0 f − s−1;Ω ≤ infs∈S 1 f − s−1;Ω , the inequalities 



z∈V

inf f − s2−1;ωz  inf f − s2−1;Ω ,

s∈Sk |ωz

s∈Sk

k ∈ {0, 1},

(3)

are straight-forward. See [8] for the nontrivial converses of (3). Their proofs present a new ingredient: a so-called localization operator, which already implicitly appeared in a posteriori error analysis. Apart from that, shared nodal values are defined with the help of dual basis functions with respect to locally supported weights. The control of the information transport is done with the help of a ‘trivial path’ consisting of two stars, whose existence follows from the definition of stars. Shape regularity arises because negative and so dual derivatives are involved. & % A striking difference between Theorem 3 and its precursors is that the best error with discontinuous piecewise polynomials is not localized to elements. This follows immediately by considering a functional that is supported in a (d − 1)-face. Further,

364

A. Veeser

approximating the Dirac measure in a vertex in H −1 (Ω), [9] shows that the best error with both spaces Sk , k ∈ {0, 1}, also does not localize to pairs. Accordingly, local best errors for H −1 -tree approximation will involve minimal rings as in [2]. As in the previous cases, let us conclude with associated error bounds. Given −1 < r ≤  + 1 and writing hz for the diameter of ωz , starwise application of Friedrichs inequalities and the Bramble-Hilbert lemma yields

inf f − s−1;Ω ≤ inf f − s−1;Ω

s∈S0

s∈S1

⎞1 ⎛ 2

2(r+1) 2 ⎠ ⎝  hz |f |r;ωz , z∈V

where the difference between continuous and discontinuous case is not visible anymore in the bound.

6 Approximating Simultaneously The quasi-optimality results in [6, 7] for parabolic semidiscretizations involve, e.g., the norm of L2 (0, T ; H01) ∩ H 1 (0, T ; H −1 ). This gives rise to the following question: Does there exist an approximant in, say, S1 with local near best errors simultaneously in all three spaces H01 ⊂ L2 ⊂ H −1 ? Since stability is necessary for being near best, a candidate for such an approximant is the one invoked to prove the respective nontrivial inequality of Theorem 3. In [8], we show that this candidate is viable. Theorem 4 (Simultaneous Locally Near Best Interpolation) Assume that M has face-connected stars. There is an interpolation operator Π onto S1 such that, for any v ∈ H01 ,

|v − Πv|21;Ω  inf |v − p|21;K , K∈M

v − Πv20;Ω 

F ∈F

v

− Πv2−1;Ω



p∈P

inf

v − sF 20;ωF ,

inf

v − sz 2−1;ωz ,

sF ∈S1 |ωF

z∈V

sz ∈S1 |ωz

where the hidden constants can be bounded in terms of d, , and σM . In contrast to the previous theorems, Theorem 4 has been given using an interpolation operator. We could have formulated also the previous theorems in this manner and the present one as localization of the best error with respect to 

| · |21;Ω +  · 20;Ω +  · 2−1;Ω

1 2

,

where the three local contributions also become uncoupled.

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365

Acknowledgements The support by the GNCS, part of the Italian INdAM, is gratefully acknowledged.

References 1. P. Binev, R. DeVore, Fast computation in adaptive tree approximation. Numer. Math. 97(2), 193–217 (2004) 2. P. Binev, W. Dahmen, R. DeVore, Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004) 3. F. Camacho, A. Demlow, L2 and pointwise a posteriori error estimates for fem for elliptic pdes on surfaces. IMA J. Numer. Anal. 35(3), 1199–1227 (2015) 4. T. Dupont, R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34(150), 441–463 (1980) 5. L.R. Scott, S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990) 6. F. Tantardini, A. Veeser, The L2 -projection and quasi-optimality of Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 54(1), 317–340 (2016) 7. F. Tantardini, A. Veeser, Quasi-optimality constants for parabolic galerkin approximation in space, in Numerical Mathematics and Advanced Applications ENUMATH 2015, ed. by B. Karasözen et al. (Springer International Publishing, Cham, 2016), pp. 105–113 8. F. Tantardini, A. Veeser, R. Verfürth, H −1 -approximation with piecewise polynomials (in preparation) 9. F. Tantardini, A. Veeser, R. Verfürth, Robust localization of the best error with finite elements in the reaction-diffusion norm. Constr. Approx. 42(2), 313–347 (2015) 10. A. Veeser, Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. 16(3), 723–750 (2016) 11. A. Veeser, P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. III – DG and other interior penalty methods. SIAM J. Numer. Anal. (accepted for publication)

Adaptive Discontinuous Galerkin Methods for Flow in Porous Media Birane Kane, Robert Klöfkorn, and Andreas Dedner

Abstract We present an adaptive Discontinuous Galerkin discretization for the solution of porous media flow problems. The considered flows are immiscible and incompressible. The fully adaptive approach implemented allows for refinement and coarsening in both the element size, the polynomial degree and the time step size.

1 Introduction The strong heterogeneity and anisotropy of many geological systems entails various challenges for the modeling and the numerical simulation of many environmental problems such as groundwater flow and petroleum engineering. In order to fathom the inherent geological complexity of such problems, we require locally conservative discretization methods such as Discontinuous Galerkin (DG) methods and adaptive strategies allowing to gain efficiently an accurate approximation of the true solution. This paper extends our previous work [5, 6] focusing on h-adaptive and hpadaptive schemes for 2d and 3d two-phase flow problems with strong heterogeneity, discontinuous capillary pressure functions and gravity effects. We consider here adaptive approaches allowing for refinement/coarsening in both the element size, the polynomial degree and the time step size. This allows thereby to refine the mesh when the solution is estimated to be rough and increase the local polynomial

B. Kane () Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Stuttgart, Germany e-mail: [email protected] R. Klöfkorn International Research Institute of Stavanger, Stavanger, Norway e-mail: [email protected] A. Dedner Mathematics Institute, University of Warwick, Coventry, UK e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_32

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degree when the solution is estimated to be smooth. It also grants more flexibility regarding the time step size without hindering the convergence of the method. To our knowledge, this is the first time the concept of adaptivity is incorporated to such an extent in the DG discretization of fully-implicit fully-coupled two-phase flow problems. The milestone contribution of [7] limits to a 2d flow decoupled formulation with continuous capillary-pressure functions and h-adaptivity. An aposteriori estimator for homogeneous two-phase flow problems was presented by Vohralik et al. in [10], it has not yet been applied to DG methods. The rest of this paper is organized as follows. In the next section, we describe the two-phase flow model. The DG discretization is introduced in Sect. 3. The adaptive strategy is outlined in Sect. 4. Numerical examples are provided in Sect. 5. Concluding remarks are provided in the last section.

2 Governing Equations We consider an open and bounded domain Ω ⊂ Rd , d ∈ {1, 2, 3} and the time interval J = (0, T ), T > 0. The two-phase flow problem considered is a system of equations with two unknowns pw and sn , −∇ · (λt K∇pw + λc K∇sn − (ρw λw + ρn λn )Kg) = qw + qn , φ

∂sn − ∇ · (λn K(∇pw − ρn g)) − ∇ · (λc K∇sn ) = qn . ∂t

(1)

Here, K is the permeability of the porous medium, ρα is the phase density, qα is a source/sink term and g is the constant gravitational vector, φ > 0 is the porosity and λt = λw + λn denotes the total mobility. The phase mobilities are λα = kμrαα , α ∈ {w, n}, where μα is the phase viscosity and krα is the relative permeability of phase α. The relative permeabilities are functions that depend on the phase saturation in nonlinear fashion (i.e. krα = krα (sα )). For example, in the Brooks-Corey model [2], 2+3θ

2+θ

krw (sw,e ) = sw,eθ , krn (sn,e ) = (sn,e )2 (1 − (1 − sn,e ) θ ), the effective saturation s −sα,r sα,e is sα,e = 1−sαw,r −s , ∀α ∈ {w, n}. Here, sα,r , α ∈ {w, n} are the phase n,r residual saturations. The parameter θ ∈ [0.2, 3.0] is a result of the inhomogeneity of the medium. The capillary pressure pc = pc (sw,e ) is a function of the phase −1/θ saturation pc (sw,e ) = pd sw,e where pd ≥ 0 is the constant entry pressure and λc (sn ) = λn (sn )pc (sn ). In order to have a complete system we add boundary and initial conditions. Thus, we assume that the boundary of the system is divided into disjoint sets, ∂Ω = ΓD ∪ ΓN . We define the total inflow Jt = Jw + Jn as the sum of the phases inflow on the Neumann boundary ΓN .

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3 Discretization Let Th = {E} be a family of non-degenerate, quasi-uniform, possibly nonconforming partitions of Ω consisting of Nh elements of maximum diameter h. Let Γ h be the union of the open sets that coincide with internal interfaces of elements of Th . Dirichlet and Neumann boundary interfaces are collected in the set ΓDh and ΓNh . Let e denote an interface in Γ h shared by two elements E− and E+ of Th ; we associate with e a unit normal vector νe directed from E− to E+ . We also denote by |e| the measure of e. The discontinuous finite element space is Dr (Th ) = {v ∈ L2 (Ω) : v|E ∈ Pr (E) ∀E ∈ Th }, where Pr (E) denotes Qr (resp. Pr ) the space of polynomial functions of degree at most r ≥ 1 on E (resp. the space of polynomial functions of total degree r ≥ 1 on E). We approximate the pressure and the saturation by discontinuous polynomials of total degrees rp and rs respectively. For any function q ∈ Dr (Th ), we define the jump operator  ·  and the average operator {·} over the interface e: ∀e ∈ Γ h , q := qE− − qE+ , {q} := 12 qE− + 12 qE+ , and ∀e ∈ ∂Ω, q = {q} := qE− . The derivation of the semi-discrete DG formulation is standard (see [7]). First, we multiply each equation of (1) by a test function and integrate over each element, then we apply Green formula to obtain the semi-discrete weak DG formulation. Hence, the aforementioned formulation consists in finding the continuous in time approximations Pw,h (·, t) ∈ Drp (Th ), Sn,h (·, t) ∈ Drs (Th ) such that: Bh (Pw,h , ϕ; Sn,h ) = lh (ϕ)

∀ϕ ∈ Drp (Th ), ∀t ∈ J ,

(Φ∂t Sn,h , ψ) + ch (Pw,h , ψ; Sn,h ) + dh (Sn,h , ψ) = rh (ψ)

∀ψ ∈ Drs (Th ), ∀t ∈ J .

(2) The bilinear form Bh in the total fluid conservation equation of (2) is: Bh (Pw,h , ϕ; Sn,h ) = Bbulk,h + Bcons,h + Bsym,h + Bst ab,h.

(3)

The first term Bbulk,h := Bbulk,h (Pw,h , ϕ; Sn,h ) of (3) is the volume term: Bbulk,h =

 E∈Th

E

(λt K∇Pw,h + λc K∇Sn,h ) · ∇ϕ −

(ρn λn + ρw λw )Kg · ∇ϕ.

E∈Th

The second term Bcons,h := Bcons,h (Pw,h , ϕ; Sn,h ) is the consistency term: Bcons,h = −

e∈Γ h ∪ΓDh

+

e∈Γ h ∪ΓDh

 e

 e

{λt K∇Pw,h }ω · νe ϕ −

e∈Γ h ∪ΓDh

{(ρn λn + ρw λw )Kg)}ω · νe ϕ.

 e

{λc K∇Sn,h }ω · νe ϕ

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The weighted average operator {·}ω [5] is used to handle the strong heterogeneity. The term Bsym,h := Bsym,h (Pw,h , ϕ; Sn,h ) is the symmetry term. Depending on  we get different methods, SIPG ( = −1), NIPG ( = 1), IIPG ( = 0):

Bsym,h =



e∈Γ h ∪ΓDh

e

{λt K∇ϕ}ω Pw,h  + 



e∈Γ h ∪ΓDh

The stability term is Bst ab,h := Bst ab,h(Pw,h , ϕ) =

e

{λc K∇ϕ}ω Sn,h .



p



e Pw,h ϕ. e∈Γ h ∪ΓDh rp (rp +d−1)|e| p , σp ≥ γe = σp min(|E − |,|E+ |)

γe

0. Following [1], the penalty formulation is: The right hand side of the total fluid conservation equation of (2) is a linear form including the boundary conditions and the source terms. The second equation of (2) is the discrete weak formulation of the nonwetting-phase conservation equation. For more details see [5]. The interface conditions due to the capillarypressure heterogeneity (see e.g., [9]) are enforced weakly through the penalties on interelement jumps of the saturation similarly to [1]. In the following, we implement both a first and second order implicit time discretization method for (2). This implicit space time discretization leads to a fully coupled nonlinear system to which the NewtonRapshon iterative scheme is applied.

4 Adaptivity The choice between h-adaptivity and p-adaptivity depends heavily on the value of r ηE a smoothness indicator ςE = r−1 where ηE , E ∈ Th is a given error indicator and ηE

r−1 ηE is the same indicator evaluated for the L2 projection of the solution into a lower order polynomial space. The derivation of this L2 projection is quite straightforward due to the hierarchical aspect of the modal DG bases implemented. The smoothness indicator ςE allows to refine the mesh when the solution is estimated to be rough and increase the polynomial degree when the solution is estimated to be smooth. In the sequel, we implement an explicit estimator originally designed for instationnary convection-diffusion problems. A thorough analysis is available in [8]. Applying the estimator to the nonwetting phase conservation equation of (1) yields: 2 7 7 1 2 = hE 7R 72 ηE vol L2 (E) + 2 2 rs e∈Γ h

+

e∈∂E∩∂Ω





7 he 7 7Re 72 2 + 2 L (e) r

37 7 7 he 7 7Re 72 2 + r 7Re 72 2 2 L (e) 1 L (e) r he

7 r3 7 7Re 72 2 1 L (e) he



 .

(4)

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Here Rvol is the interior residual indicating how accurate the discretized solution satisfies the original PDE at every interior point of the domain, Rvol = qn − φ

∂Sn,h + ∇ · [λhn K(∇Pw,h − ρn g)] + ∇ · [λhc K∇Sn,h ]. ∂t

The term Re1 is the numerical zero order inter-element (resp. Dirichlet boundary condition residual) depending on the jump of the discrete solution at the elements boundaries (resp. at the Dirichlet boundary), hence reflecting the regularity of the DG approximation (resp. the accuracy of the approximation on the Dirichlet boundary), 9 {γes λhc δKν }Snh  if e ∈ Γ h Re1 = s h γe λc δKν (sD − Sn,h ) if e ∈ Γ D rs (rs +d−1)|e| where δKν = νeT Kνe and γes = σs min(|E , σs ≥ 0. − |,|E+ |) The term Re2 is the first order numerical inter-element residual (resp. Neumann boundary condition residual) depending on the jump of numerical approximation of the normal flux at the elements boundaries (resp. at the Neumann boundary). It also allows to assess the regularity of the DG approximation (resp. the accuracy of the approximation on the Neumann boundary),

Re2

⎧ ⎨λh K(∇Pw,h − ρn g) + λh K∇Sn,h  · νe if e ∈ Γ h n  c  = ⎩Jn + λn K(∇Pw,h − ρn g) + λhc K∇Sn,h · νe if e ∈ Γ N .

Following [3], we choose the temporal error indicator at time step i as (ηti ime )2 =

7

7 7 i − S i−1 )72 7∇(Sn,h n,h 7 2

E∈Th

L (E)

.

(5)

The use of heuristic error indicators requires a maximum level of allowed hrefinement maxlevel to be specified to avoid overly aggressive refinement. In Algorithm 1, whenever an element is selected for h-refinement it is also selected for p-coarsening in order to reduce the oscillations in the vicinity of the front of the propagation. Algorithm 2 describes the time step adaptation process. Depending on the value of the time indicator ηt ime , we increase, decrease or maintain the time step size. The parameters timef act− , timef act+ , ttolmax and ttolmin in Algorithms 1 and 2 are user-chosen constants.

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Algorithm 1 r be given 1: Let ηE 2: for all E ∈ Th do 3: hE = diam(E), rE = poldeg(E) r > stol then 4: if ηE 5: if ςE > threshold then 6: if maxlevel > level(E) then hE 7: hnew E := 2 8: else 9: rEnew = rE − 1 10: end if 11: else 12: rEnew = rE + 1 13: end if r < 0.1 × stol then 14: else if ηE 15: if ςE < threshold then 16: rEnew = rE + 1 17: else 18: hnew E := 2hE 19: end if 20: end if 21: end for

Algorithm 2 1: 2: 3: 4: 5: 6: 7: 8:

i Let ηtime and Δti be given i if ηtime > ttolmax then Δti+1 = timef act− × Δti i else if ηtime < ttolmin then Δti+1 = timef act+ × Δti else Δti+1 = Δti end if

5 Numerical Simulations In this section we present some numerical tests for the adaptive DG scheme. All test cases are implemented with the Interior Penalty methods combined with second order Adams-Moulton time discretization. In order to ensure second order accuracy, we employ a central differencing of the mobility for internal interfaces thus following a similar approach to that of [4]. A container is filled with sand and saturated with water with density ρw = 1000 kg/m3 and viscosity μw = 1 × 10−3 kg/m s. The DNAPL considered in the experiment is Tetrachloroethylene with density ρn = 1460 kg/m3 and viscosity μn = 9 × 10−4 kg/m s. Brooks-Corey’s constitutive relations are used for the capillary pressure and the relative permeabilities. Initial conditions where the domain is fully saturated with water and hydrostatic pressure distribution are 0 = (0.65 − y) · 9810, s 0 = 0). Discretization of the system is considered (i.e. pw n

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performed by Interior Penalty DG methods with a fully implicit and fully coupled approach. All test cases in this section include gravitational forces and capillary pressure effects.

5.1 2d Two-Phase Flow We consider here a two-dimensional DNAPL infiltration problem. Detailed boundary conditions are specified in Table 1 and Fig. 1. The material properties of the problem are detailed in Table 2. The bottom of the reservoir is impermeable for both phases. Homogeneous Dirichlet conditions for the saturation sn are prescribed at the left and right boundaries. A flux of Jn = −5.137 × 10−5 m s−1 of the DNAPL is infiltrated into the domain from the top. The permeability tensor K is 

 10−10 −5−11 K= m2 . −5−11 10−10 The initial mesh consists of 600 quadrilateral elements. We choose an initial time step of size Δt = 1 s, we set timef act− = 0.5, and timef act+ = 2. The final time is T = 2000 s. We consider a Newton solver tolerance newtT ol = 10−6 and a linear solver tolerance linabstol = 5 × 10−7 . Table 1 Boundary conditions

Fig. 1 2d domain

ΓI N ΓN ΓS ΓE ∪ ΓW

Jn = −5.137 × 10−5 , Jw = 0 Jn = 0.00, Jw = 0.00 Jw = 0, Jn = 0.00 pw = (0.65 − y) · 9810, sn = 0

374 Table 2 Parameters

B. Kane et al. Φ [-] 0.40

Swr [-] 0.12

Snr [-] 0.00

θ [-] 2.70

pd [Pa] 755

Fig. 2 Saturation contours Table 3 Estimates for the 2d case

Avg ηsp Final ηsp Avg ηtime Final ηtime max Δt [s] min Δt [s]

5.23 × 10−3 5.83 × 10−3 0.66 0.79 32 1

Figure 2 and Table 3 show the numerical results for the IIPG scheme with constant polynomial order rs = rp = 2. The non-wetting front propagates in the dominating direction of the anisotropy. There is also a steady increase of the time step size until it reaches a plateau with Δt = 32 s. As expected, we witness a heavy refinement in the vicinity of the front, this helps to limit the undershoots to negligible values.

5.2 3d Two-Phase Flow In this section, we focus on 3d test cases. A flux of 0.25 kg s−1 m−2 of the DNAPL is infiltrated from the top into a domain of depth of 1 m. For the anisotropic case,

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we keep the parameters of Table 2 and we set the absolute permeability tensor K as in [11]. Figure 3 and Table 4 provide the details of the heterogeneous case. ⎛

⎞ 10−10 0 −5−11 ⎜ ⎟ K = ⎝ 0 10−10 5−11 ⎠ m2 −5−11 5−11 −5−11 Anisotropic Case The initial mesh consists of 8 × 8 × 8 hexahedral elements. We choose an initial time step of size Δt = 0.5 s, we set timef act− = 0.75, and timef act+ = 1.5. The final time is T = 1500 s. The first row of Fig. 4 illustrates the evolution of the nonwetting saturation for the anisotropic case, the effects of the hp-adaptive algorithm are reflected in the mesh distribution showing an intense refinement and lower polynomial degree in the parts of the domain where the value of the indicator is above the threshold value. We notice a drastic improvement of the front shape when h and hp-adaptive methods

Fig. 3 3d domain Table 4 3d-heterogeneous case parameters

Φ [-] k [m2 ] Swr [-] Snr [-] θ [-] pd [Pa]

Ω1 0.39 6.64 × 10−16 0.1 0.00 2.0 5000

Ω2 0.39 6.64 × 10−15 0.1 0.00 2.0 5000

Ω\Ω1 ∩ Ω\Ω2 0.39 5.621 × 10−11 0.098 0.00 2.49 1323.95

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Fig. 4 First row: anisotropic 3d-problem, from left to right, contour plot of saturation after 1500 s of injection, mesh distribution, polynomial degree distribution. Second row: heterogeneous 3d-problem, from left to right, mesh distribution after 1650 s of injection, polynomial degree distribution, saturation profile along the slice y=0.45, polynomial degree distribution along the slice y=0.45 Table 5 Estimates for the 3d-anisotropic case

Avg ηsp Final ηsp Avg ηtime Final ηtime max Δt [s] min Δt [s]

0.096 0.087 1.67 1.77 19.23 0.5

are used. The time step size steadily increases until it reaches Δt = 19.23 s. Table 5 shows the numerical results for the fully adaptive IIPG scheme. Heterogeneous Case The second row of Fig. 4 shows the evolution of the nonwetting saturation for the heterogeneous case. It is worth noting the variation of time step size, as we set timef act− = 0.75, and timef act+ = 1.5 in Algorithm 2. The time step size decreases drastically once the front reaches the top edge of the lens and it usually increases again after some kind of equilibrium is reached. Moreover, we notice as expected, an intense refinement and lower polynomial degree around the lens. Table 6 provides the details of the numerical results for the fully adaptive IIPG scheme.

Adaptive Discontinuous Galerkin Methods for Flow in Porous Media Table 6 Estimates for the 3d-heterogeneous case

377 Avg ηsp Final ηsp Avg ηtime Final ηtime max Δt [s] min Δt [s]

0.041 0.04 0.1 0.42 17.086 1

6 Conclusion and Outlook In this work, we have introduced the first fully adaptive discontinuous Galerkin scheme for incompressible, immiscible two-phase flow in anisotropic and strongly heterogeneous porous media with gravity forces and capillary effects. We considered as test cases anisotropic and heterogeneous DNAPL infiltration in an initially water saturated reservoir. The oscillations appearing in the vicinity of the front of the propagation are reduced with the local mesh refinement and the decrease of the local polynomial order. Acknowledgements Birane Kane would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. Robert Klöfkorn acknowledges the Research Council of Norway and the industry partners; ConocoPhillips Skandinavia AS, Aker BP ASA, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, ENGIE E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS, DEA Norge AS of The National IOR Centre of Norway for support. The authors would like to thank the reviewers for helpful comments to improve this work.

References 1. P. Bastian, A fully-coupled discontinuous galerkin method for two-phase flow in porous media with discontinuous capillary pressure. Comput. Geosci. 18(5), 779–796 (2014) 2. R.H. Brooks, A.T. Corey, Hydraulic properties of porous media and their relation to drainage design. Trans. ASAE 7(1), 26–0028 (1964) 3. D.A. Di Pietro, M. Vohralík, A review of recent advances in discretization methods, a posteriori error analysis, and adaptive algorithms for numerical modeling in geosciences. Oil Gas Sci. Technol. - Revue dIFP Energies nouvelles 69(4), 701–729 (2014) 4. Y. Epshteyn, B. Rivière, Fully implicit discontinuous finite element methods for two-phase flow. Appl. Numer. Math. 57(4), 383–401 (2007) 5. B. Kane, Using dune-fem for adaptive higher order discontinuous galerkin methods for strongly heterogenous two-phase flow in porous media. Arch. Numer. Softw. 5:1 (2017). 6. B. Kane, R. Klöfkorn, C. Gersbacher, hp–adaptive discontinuous galerkin methods for porous media flow, in International Conference on Finite Volumes for Complex Applications (Springer, Cham, 2017), pp. 447–456. 7. W. Klieber, B. Rivière, Adaptive simulations of two-phase flow by discontinuous galerkin methods, Comput. Methods Appl. Mech. Eng. 196(1), 404–419 (2006)

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8. S. Sun, M.F. Wheeler, L2 (H 1 ) norm a posteriori error estimation for discontinuous galerkin approximations of reactive transport problems. J. Sci. Comput. 22(1), 501–530 (2005) 9. C. Van Duijn, J. Molenaar, M. De Neef, The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media. Transp. Porous Media 21(1), 71–93 (1995) 10. M. Vohralík, M.F. Wheeler, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows. Comput. Geosci. 17(5), 789–812 (2013) 11. M. Wolff, Y. Cao, B. Flemisch, R. Helmig, B. Wohlmuth, Multi-point flux approximation lmethod in 3d: numerical convergence and application to two-phase flow through porous media, in Simulation of Flow in Porous Media: Applications in Energy and Environment. Radon Series on Computational and Applied Mathematics, vol. 12 (De Gruyter, Berlin, 2013), pp. 39–80

An Adaptive E-Scheme for Conservation Laws Ebise A. Abdi, Christian V. Hansen, and H. Joachim Schroll

Abstract An adaptive E-scheme for possibly degenerate, viscous conservation laws is presented. The scheme makes use of both given and numerical diffusion to establish the E-property. In the degenerate case it reduces to local Lax–Friedrichs. Both explicit and time-implicit E-schemes are monotone and TVD. Numerical experiments demonstrate the robustness and improved accuracy of the adaptive scheme.

1 Introduction Numerical methods for hyperbolic conservation laws ut + f (u)x = 0 require artificial viscosity to be stable. Below we show that enough viscosity leads to Eschemes [5]. Tadmor [7] showed that E-schemes are entropy-stable. In this paper, we outline the construction of an adaptive scheme for possibly degenerate viscous conservation laws ut + f (u)x = (dux )x ,

d = d(x) ≥ 0

based on the observation that enough effective (natural and artificial) diffusion 2D = 2(d + Δx) ≥ |f  |Δx

(1)

leads to E-schemes. While it is well known that monotone schemes [2] are Eschemes, we prove in [1] that E-schemes are monotone. It is also well-known

E. A. Abdi School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia C. V. Hansen · H. J. Schroll () Department of Mathematics and Computer Science, University of Southern Denmark, Odense M, Denmark e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_33

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that monotone schemes are L1 -contracting and total variation diminishing (TVD). By a compactness argument a subsequence converges to a weak solution, see [4, Chap. 15]. By satisfying (1) locally, our adaptive viscosity scheme is E-scheme; it is monotone, TVD and entropy-stable. Monotonicity would be difficult to verify directly as the local diffusion coefficient depends on the solution via the characteristic speed f  (u). Numerical experiments with scalar conservation laws and the Navier–Stokes system confirm that the adaptive E-scheme is stable but more accurate than traditional schemes designed for hyperbolic problems and augmented by the given, natural diffusion.

2 Enough Diffusion Makes E-Schemes As a starting point let us consider the classical Lax–Friedrichs scheme in conservation form un+1 = unj − j

 Δt  F (unj , unj+1 ) − F (unj−1 , unj ) = H(un )j Δx

(2)

 uj +1 − uj 1 f (uj ) + f (uj +1 ) − Dj +1/2 2 Δx

(3)

with numerical flux F (uj , uj +1 ) =

and numerical diffusion 2 = Δx/Δt ≥ f  ∞ augmented by the natural, given diffusion d Dj +1/2 = dj +1/2 + Δx . Expanding both fluxes f (v) and f (w) around u between v ≤ u ≤ w, we find  1 1 1 f (w) + f (v) = f (u) + f  (ξ )(w − u) − f  (η)(u − v) , 2 2 2 with some ξ ∈ (u, w) and η ∈ (v, u). When the diffusion in relation to the step size is large enough 2D ≥ f  ∞ Δx, the numerical flux (3) cannot exceed the given, natural flux F (v, w) =

 1 D (w − v) ≤ f (u) . f (w) + f (v) − 2 Δx

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Osher [5] defined E-schemes as schemes in conservation form (2) with so-called E-fluxes   sign(w − v) F (v, w) − f (v + s(w − v)) ≤ 0 ,

∀s ∈ [0, 1] .

(4)

Lemma 3.1 in [1] states that locally enough diffusion 2Dj +1/2 ≥ |f  |j +1/2 Δx ,

|f  |j +1/2 := max |f  (uj + s(uj +1 − uj ))| 0≤s≤1

(5)

guarantees E-fluxes and E-schemes. The augmented, classical Lax–Friedrichs scheme satisfies this condition by maximizing the characteristic speed  2DjLF +1/2 = 2dj +1/2 + f ∞ Δx .

The local version, that is the Rusanov-scheme [6], adapts its artificial viscosity to the local characteristic speed 2DjR+1/2 = 2dj +1/2 + |f  |j +1/2 Δx . In this contribution we suggest to exploit any available natural diffusion to avoid unnecessary overdamping. Sufficient for (5) actually is 2Dj +1/2 = max(2dj +1/2 , |f  |j +1/2 Δx) .

(6)

Note that the adaptive viscosity scheme (2), (3) with effective diffusion (6) is by construction an E-scheme and as such it is entropy-stable. Next we show the scheme is still monotone and hence TVD. Obviously the adaptive diffusion coefficient (6) is smallest of all three.

3 E-Schemes Are Monotone The explicit scheme (2) is monotone in the sense of Crandall and Majda [2] if H is a non-decreasing function in all its arguments. As already pointed out by Osher [5] monotone schemes are E-schemes. In turn, consistent E-schemes with Δt CFL condition Δx ∂v F − ∂w F ∞ ≤ 1 are monotone, see [1, Lemma 4.1 and 4.2]. Consider for example the case uj < uj +1 . By the E-property (4)

F (uj , uj +1 ) ≤

⎧ ⎪ ⎨f (uj ) = F (uj , uj ) ⎪ ⎩f (uj +1 ) = F (uj +1 , uj +1 )

.

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Hence, F (v, w) is non-decreasing in v and non-increasing in w. The diagonal element in DH(u) is 1 −Δt/Δx(∂v F −∂w F ) and hence the CFL condition controls monotonicity of H(u)j with respect to uj . This shows that E-schemes cannot extend the class of monotone schemes and thus are limited to first order of accuracy. However, as monotone schemes, E-schemes are L1 -contracting and TVD. One can also show that implicit time stepping with E-schemes is monotone and TVD, see [1] for details.

4 Less Artificial Diffusion Is More Resolution To compare the adaptive viscosity scheme vs local Lax–Friedrichs consider Burgers’ equation ut +

  1  2 u = d(x)ux x , x 2

x ∈ (−6, 6]

augmented with degenerate diffusion   d(x) = max 0.025 sin(πx/6), 0 . Boundary conditions are periodic, and the initial data is the piecewise constant characteristic function χ[−5,−3]∪[1,3]. On an equidistant mesh Δx = 0.05 with CFLnumber 0.9 and at time t = 3.0, approximations obtained by local Lax–Friedrichs and the adaptive E-scheme are shown in Fig. 1. It is obvious that for negative x and without natural diffusion both schemes coincide. On the right half line, however, the local Lax–Friedrich’s numerical diffusion (dashed red) grows with f  (u) = u and, on top, is almost as big as the given diffusion (dashed green). The adaptive E-scheme in contrast requires only very little artificial diffusion. Accordingly, the approximation of the latter scheme (solid blue) appears sharper especially at its peak near x = 4. Let’s consider next the scalar conservation law in two variables   ut + ∇ · f (u) = ∇ · d(x)∇u ,

f (u) =

1  2 4 T u ,u 2

(7)

on the domain (x, y) ∈ (−12, 12] × (−6, 6]. The degenerate diffusion depends on x alone     d(x) = max 1/2 + sin π(x + 6)/12 , 0 /32 .     Initial data u(0, x, y) = sin (x − 6)π/6 sin (y + 3)π/6 and boundary conditions are periodic. The mesh size in both directions is Δx = Δy = 0.1. In a

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1 u adaptive E u

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dimensional splitting approach as described in [4, Chap. 18], the one dimensional schemes are applied coordinate-by-coordinate; one step in x-direction followed by one step in y-direction. By this approach the CFL limit is 0.5. The local Lax–Friedrichs scheme applies artificial diffusion to stabilize the scheme disregarding the given diffusion. As a consequence this schemes’ effective diffusion is locally larger than what would be necessary for stability, leading to overdamping. Figure 2 displays the intersection along y = 2 for both the local Lax– Friedrichs and the adaptive E-scheme. While the adaptive E-scheme on the given

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mesh sharply resolves the minimum near x = 4, local Lax–Friedrichs can hardly reproduce the feature.

5 Application to Isentropic Navier–Stokes Equations Most relevant are adaptive viscosity E-schemes for incomplete and degenerate parabolic systems like the compressible Navier–Stokes equations [3]. In the isentropic case without thermodynamic effects, the equations consist of conservation of mass and momentum ρt + ∇ · (ρu) = 0 , ρDu/Dt + ∇p = ∇ · σ . Here ρ and ρu denote mass- and momentum densities respectively. D/Dt is the material derivative  and p the pressure. The Cauchy stress tensor is given by σ = λ(∇ · u)I + μ ∇u + (∇u)T . The system is closed by an equation of state, for example the gamma-law p = ρ γ , γ = 1.4. Figure 3 shows the adaptive E-scheme when applied to the one-dimensional system 





 ρ ρu

ρu ρu2 + p

+ t

 =

x

 0 (2μ + λ)ux

(8) x

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on the interval (−6, 6] and with given diffusion   2μ + λ = max 0.1 sin((x + 3)π/6), 0 . The initial density is ρ(0, x) = 1.0 + 0.5 sin(xπ/6) with zero initial momentum and periodic boundary conditions. The mesh size Δx is 0.1 with CFL number 0.9. Due to zero artificial diffusion around the origin, the approximation by the adaptive E-scheme appears much sharper than the local Lax–Friedrichs scheme that suffers from unnecessary numerical diffusion and overdamping. In the last experiment, the new scheme is applied to the two-dimensional system ⎛

⎛ ⎛ ⎞ ⎞ ⎞ ρ ρu ρv ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ ρu ⎠ + ⎝ ρu2 + p ⎠ + ⎝ ρuv ⎠ = ρv ρuv ρv 2 + p t x y ⎛ ⎞ ⎞ ⎛ 0 0 ⎜ ⎟ ⎟ ⎜ μuy ⎝ (2μ + λ)ux + (μ + λ)vy ⎠ + ⎝ ⎠ . μvx (μ + λ)ux + (2μ + λ)vy x

(9)

y

The dynamic viscosity is given by   μ = max 0.05 sin((x − 3)π/6) sin((y − 3)π/6), 0 and the bulk viscosity λ = −2/3μ follows from Stokes’ hypothesis. Initially, the flow is at rest and the density is a perturbed sine wave ρ(0, x, y) = 1 + 0.2 sin((x − 6)π/3) sin((y + 3)π/3) with ρ(0, x, y) = 0.5 inside the rectangle [−2, 2] × [−1, 1]. The mesh size in both x and y-direction is 0.05 and the CFL number is 0.4. The density at later time is depicted in Fig. 4. Despite low and locally degenerate artificial viscosities, the scheme is stable and produces approximations free of spurious oscillations. The artificial viscosity is adapted componentwise. As there is no natural diffusion in the continuity equation, the numerical viscosity for ρ is strictly positive. Local and adaptive viscosities are identical in their first components. For the momentum, however, the adaptive scheme makes use of the given viscosity. Where enough natural viscosity is available the artificial viscosity degenerates to zero. The difference in density between the adaptive E-scheme and local Lax–Friedrichs is depicted in Fig. 5. Again we observe large local differences due to less artificial diffusion and higher resolution in the adaptive E-scheme on the finite mesh. Recall that the difference is due to artificial diffusion which decays in the limit of mesh-size to zero.

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Fig. 4 Density in Navier–Stokes (9) at t = 2

Fig. 5 Difference in density ρadaptive E − ρlocal LF at t = 2

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6 Concluding Remarks Enough diffusion, both given and artificial, leads to E-schemes. E-schemes are monotone, L1 -contracting, and TVD. As monotone schemes, they are limited to first order of accuracy. For possibly degenerate, viscous conservation laws however, adaptive viscosity E-schemes are more accurate than traditional monotone schemes augmented by given diffusion; without compromising stability! Therefore, this class of methods is attractive as robust, basic method in high resolution finite volume and discontinuous Galerkin schemes. Acknowledgements The authors acknowledge funding by the Ethiopian Department of Education, and the Norwegian Agency for Development Cooperation.

References 1. E.A. Abdi, C.V. Hansen, H.J. Schroll, An adaptive viscosity E-scheme for balance laws. Preprint SDU (2017) 2. M.G. Crandall, A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comput. 34(149), 1–21 (1980) 3. H.O. Kreiss, J. Lorenz, Initial–Boundary Value Problems and the Navier–Stokes Equations (Academic, Bosten, 1989) 4. R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Basel, 1992) 5. S. Osher, Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21, 217–235 (1984) 6. V.V. Rusanov, Calculation of interaction of nonsteady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279 (1961) 7. E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43, 369–381 (1984)

Adaptive Filtered Schemes for First Order Hamilton-Jacobi Equations Maurizio Falcone, Giulio Paolucci, and Silvia Tozza

Abstract In this paper we consider a class of “filtered” schemes for some first order time dependent Hamilton-Jacobi equations. A typical feature of a filtered scheme is that at the node xj the scheme is obtained as a mixture of a high-order scheme and a monotone scheme according to a filter function F . The mixture is usually governed by F and by a fixed parameter ε = ε(Δt, Δx) > 0 which goes to 0 as (Δt, Δx) is going to 0 and does not depend on n. Here we improve the standard filtered scheme introducing an adaptive and automatic choice of the parameter ε = εn (Δt, Δx) at every iteration. To this end, we use a smoothness indicator in order to select the regions where we can compute the regularity threshold εn . The numerical tests presented confirms the effectiveness of the adaptive scheme.

1 Introduction The accurate numerical solution of Hamilton-Jacobi (HJ) equations is a challenging topic of growing importance in many fields of application, e.g. control theory, KAM theory, image processing and material science. Due to the lack of regularity of viscosity solutions, this issue is delicate and the construction of high-order methods is usually rather difficult (e.g. ENO, WENO). In recent years a general approach to

All the authors are members of the INdAM Research group GNCS. M. Falcone · G. Paolucci () Department of Mathematics, Sapienza University of Rome, Rome, Italy e-mail: [email protected]; [email protected] S. Tozza Istituto Nazionale di Alta Matematica/Department of Mathematics, Sapienza University of Rome, Rome, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_34

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the construction of high-order methods using filters has been proposed in [7] and further developed in [8]. We consider a class of “filtered” schemes for the scalar evolutive HamiltonJacobi equation in one dimension 9

vt + H (vx ) = 0, v(0, x) = v0 (x),

(t, x) ∈ [0, T ] × R, x ∈ R,

(1)

where the hamiltonian H and the initial data v0 are Lipschitz continuous functions. It is well known that with these assumptions we have an existence and uniqueness result for the viscosity solution [3]. Our aim is to present a rather simple way to construct convergent schemes to the viscosity solution v of (1) with the property to be of high-order in the region of regularity. A typical feature of a filtered scheme S F is that at the node xj it is a mixture of a high-order scheme S A and a monotone scheme S M according to a filter function F . The scheme is written as   A n S (u )j − S M (un )j F n M n , j ∈ Z, (2) un+1 ≡ S (u ) := S (u ) + εΔtF j j j Δt where ε = εΔt,Δx > 0 is a parameter going to 0 as (Δt, Δx) is going to 0 and does not depend on n. Filtered schemes are high-order accurate where the solution is smooth, monotone otherwise, and this feature is crucial to prove a convergence result for viscosity solutions as in [1]. In this work we improve the filtered scheme (2) introducing an adaptive and automatic choice of the parameter ε = εn at every iteration. To this end, we use a smoothness indicator in order to select the regions where we can compute the regularity threshold εn . Our smoothness indicators are based on the ideas of Jiang and Peng [5], but other indicators with similar properties can be used. In Sect. 2 we define our new filtered scheme, explaining how we choose the smoothness indicators. Section 3 is focussed on the filter function and the tuning of the parameter εn , we state there the convergence result. In Sect. 4 we present some numerical tests, comparing our scheme with the state-of-the-art methods. Finally, Sect. 5 contains some conclusions and future perspectives.

2 A New Adaptive Filtered Scheme Starting from the ideas of [1] on filtered schemes, we introduce a procedure to compute the regularity threshold ε in an automatic way, in order to exploit the local regularity of the solution. Let us begin defining a uniform grid in space xj := j Δx, j ∈ Z, and in time tn := t0 + nΔt, n ∈ [0, N], with (N − 1)Δt < T ≤ NΔt. Then,

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we introduce our Adaptive Filtered (AF) scheme as follows:  un+1 j

=S

AF

(u )j := S (u n

M

n

)j + φjn εn ΔtF

S A (un )j − S M (un )j εn Δt

 ,

(3)

where un+1 := u(tn+1 , xj ), S M and S A are respectively the monotone and the highj order scheme, F is the filter function, εn is the switching parameter at time tn and φjn is the smoothness indicator function at the node xj and time tn . Note that if εn ≡ εΔx, with ε > 0 and φjn ≡ 1, we get the Standard Filtered (SF) scheme of [1]. As a first big step towards the construction of our scheme, we have to define the smoothness indicator function φ. A good choice is 9 φjn

=

φ(ωjn )

:=

1 if the solution un is regular in Ij , 0 if Ij contains a point of singularity,

(4)

where Ij = (xj −1 , xj +1 ) and ωjn is the smoothness indicator at the node xj depending on the values of the approximate solution un . The function φ will be also used to compute εn , in order to exploit the local regularity of the solution. Next, we give more details on ωjn and briefly review the theory of smoothness indicators which have been used for the construction of the weighted essentially non-oscillatory (WENO) schemes for (1) (see [5] for details). These indicators are βk = βk (un )j :=

r 

l=2

xj

xj−1

 2 (l) Δx 2l−3 Pk (x) dx,

(5)

for k = 0, . . . , r − 1, where Pk is the Lagrange polynomial of degree r interpolating the values of un on the stencil Sj +k = {xj +k−r , . . . , xj +k }. Before going on with the construction of φ, let us state a fundamental result on the behavior of the indicators (5).   Proposition 1 Assume f ∈ C r+1 Ω \ {xs } where Ω is a neighborhood of a singular point xs , and f  (xs− ) = f  (xs+ ). Then, for k = 0, . . . , r − 1 and j ∈ Z, the followings are true: ◦

1. If xs ∈ Ω \ S j +k ◦

⇒ βk (f ) = O(Δx 2 );

2. If xs ∈ S j +k ⇒ βk (f ) = O(1), > ◦ = where Sj +k = xj −r+k , . . . , xj +k and S j +k = (xj −r+k , xj +k ). At this point we define as in [5] αk± := 1/(βk± + Δx 2 ) , at a fixed time t n and node xj , with βk− defined as in (5) and βk+ analogously with integral between xj and xj +1 . We focus on the information in the stencil {xj −1 , xj , xj +1 } defining ω− := α1− /(α0− + α1− ) and ω+ := α0+ /(α0+ + α1+ ) which quantify the regularity on (xj −1 , xj ] and [xj , xj +1 ), respectively. 2

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Then, taking ωj := min{ω− , ω+ } and using Proposition 1, we can prove that 9 O(Δx 4 ), if xj −1 < xs < xj +1 , ωj = 1 (6) + O(Δx), otherwise. 2 Due to lack of space, we refer to [4] for the proofs of these results. Finally, we need to define a function φ such that φ = 0 if ωj is close to 0 and φ = 1, otherwise. The simplest choice is φ(ω) = χ{ω≥M} , with M a small positive constant (e.g. M = 0.1), but of course other definitions of φ are possible (see [4] for more details). Note that to construct φ it is enough to use the indicators (5) with r = 2, requiring only five points to inspect the regularity in Ij .

3 Construction of the Scheme and Convergence Once we have defined the smoothness indicators, we are able to complete the construction of our scheme. First, let us specify some requirements on the schemes S M and S A . Let us introduce the assumptions on S M . 1. The scheme can be written in differenced form ≡ S M (un )j := unj − Δt hM (D − unj , D + unj ) un+1 j un

−unj

for a function hM (p− , p+ ), with D ± unj := ± j±1 Δx 2. hM is a Lipschitz continuous function 3. (Consistency) ∀v, hM (v, v) = H (v) 4. (Monotonicity) for any functions u, v, u ≤ v ⇒

S M (u) ≤ S M (v).

Our assumptions for S A are the following: 1. The scheme can be written in differenced form = S A (un )j := unj − ΔthA (D k,− uj , . . . , D − unj , D + unj , . . . , D k,+ unj ), un+1 j un

−un

j for some function hA (p− , p+ ) (in short), with D k,± unj := ± j±k kΔx A 2. h is a Lipschitz continuous function 3. (High-order consistency) Fixed k ≥ 2 order of the scheme, then for all l = 1, . . . , k and for all functions v ∈ C l+1 , there exists a constant CA,l ≥ 0 such that

B B B v(t + Δt, x) − S A (v(t, ·))(x) B B B EA (v)(t, x) := B B B B Δt   ≤ CA,l Δt l ||∂tl+1 v||∞ + Δx l ||∂xl+1v||∞ .

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Note that no assumptions on the stability of the high-order scheme are made. This is because filtered schemes are able to stabilize a possibly unstable high-order scheme. In our approach the filter function F must satisfy: – F (x) ≈ x for |x| ≤ 1 so that if |S A − S M | ≤ Δtεn and φjn = 1 ⇒ S AF ≈ S A , – F (x) = 0 for |x| > 1 so that if |S A − S M | > Δtεn or φjn = 0 ⇒ S AF = S M . It is clear that several choices for F are possible. In all our tests presented in this paper, we used the simple discontinuous function of [1], that is 9 F (x) =

x if |x| ≤ 1, 0 otherwise,

(7)

but more regular functions with similar properties can be easily found. See [4] for more details on that point. The last step is to show how to compute the switching parameter εn , which allows for the adaptivity of our scheme. The idea is the following: if we want the scheme (3) to switch to the high-order scheme when some regularity is detected, we have to choose εn such that B B B B B S A (v n ) − S M (v n ) B B hA (·) − hM (·) B j jB B B B for (Δt, Δx) → 0, B B=B B ≤ 1, B B B B εn Δt εn   in the region of regularity Rn at time tn , that is Rn = xj : φ(ωjn ) = 1 . Therefore, computing directly by Taylor expansion and using the consistency properties of the schemes S A and S M , we can obtain the inequality B B   B Δx B Δt 2 B M M 2 2 B vxx ∂p+ hj − ∂p− hj + Hp (vx ) + O(Δt ) + O(Δx )B . ε ≥B B 2 B 2 n

Finally, we use a numerical approximation of the lower bound on the right hand side of the previous inequality to obtain the following formula for εn : B  B   5 6 B ε = max K BH D unj − H D unj − λ H (D + unj ) − H (D − unj ) B xj ∈Rn 6 5 + hM (D unj , D + unj ) − hM (D unj , D − unj ) 6BB 5 M + n n M − n n B − h (D uj , D uj ) − h (D uj , D uj ) B , n

with K > 12 , λ :=

Δt Δx

and D unj :=

unj+1 −unj−1 . 2Δx

(8)

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Before stating our main convergence result, let us give a definition of εmonotonicity and a useful proposition. Definition 2 (ε-Monotonicity) A numerical scheme S is ε-monotone if for any functions u, v, u ≤ v ⇒ S(u) ≤ S(v) + CεΔt, where C is a constant and ε tend to 0 as Δ = (Δt, Δx) tends to 0. Proposition 3 Let un be the solution obtained by the scheme (3)–(8) and assume that v0 and H are Lipschitz continuous functions. Assume also that λ = Δt/Δx = constant. Then, εn is well defined and un satisfies, for any i and j , the discrete Lipschitz estimate |uni − unj | Δx

≤L

for some constant L > 0, for 0 ≤ n ≤ T /Δt. Moreover, there exists a constant C > 0 such that εn ≤ CΔx. It is clear that our scheme is ε-monotone by construction. Finally, we conclude this section giving our convergence result for the AF scheme. We refer to [4] for the proofs of these results. Theorem 4 Let the assumptions on S M and S A be satisfied. Assume that v0 and H are Lipschitz continuous functions, un+1 is computed by (3)–(8), with K > 1/2 and j λ = constant. Let us denote by vjn := v(t n , xj ) the values of the viscosity solution on the nodes of the grid. Then, 1. the AF scheme (3) satisfies Crandall-Lions estimate [2] √ ||un − v n ||∞ ≤ C1 Δx,

∀ n = 0, . . . , N,

for some constant C1 > 0 independent of Δx. 2. (First order convergence for regular solutions) Moreover, if v ∈ C 2 ([0, T ] × R), then ||un − v n ||∞ ≤ C2 Δx,

∀ n = 0, . . . , N,

for some constant C2 > 0 independent of Δx. 3. (High-order local consistency) Let k ≥ 2 be the order of the scheme S A . If v ∈ C l+1 in some neighborhood of a point (t, x) ∈ [0, T ]×R, then for 1 ≤ l ≤ k, EAF (v n )j = EA (v n )j = O(Δx l ) + O(Δt l ) for t n − t, xj − x, Δt, Δx sufficiently small.

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4 Numerical Tests In this section we present some one-dimensional examples designed to show the properties of our scheme, stated in Theorem 4. Our goal is also to compare the performance of our scheme S AF with those of the S F scheme of [1] and of the WENO scheme of second/third order of [5]. For each test we specify the monotone and high-order schemes composing the filtered scheme and compute the errors and orders in L∞ and L1 norm. Example 1: Transport Equation In order to test the capability of our scheme to handle both regular and singular regions, let us consider the transport equation with constant velocity and initial condition ⎧ ⎪ ⎨ 1 − |x + 3| if − 4 ≤ x ≤ −2, v0 (x) = (1 − x 2 )4 if − 1 ≤ x ≤ 1, ⎪ ⎩0 otherwise, in the domain (t, x) ∈ [0, 1.8] × [−4.5, 4.5]. This problem models the transport of a function composed by two peaks, the first with three points of singularity whereas the second is more regular. For this test we have used as S M the Central Upwind scheme [6] with the numerical hamiltonian hM (p− , p+ ) :=

5 6 1 − + − − + − + − a H (p ) − a H (p ) − a a (p − p ) , a+ − a−

(9)

where a + = max{Hp (p− ), Hp (p+ ), 0} and a − = min{Hp (p− ), Hp (p+ ), 0} and as S A the Lax-Wendroff-Richtmyer (LWR) hamiltonian  −

+

h (p , p ) := H A

 ?    @ Δt p− + p+ + − − H p −H p . 2 2Δx

(10)

Looking at Fig. 1 we can see that, although the filtered scheme allows the formation of little oscillations before the peaks, it seems to get the kinks of the singularities

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Table 1 Test 1. Errors and orders in L∞ and L1 norms

Nx 100 200 400 800

Nt 24 48 96 192

Monotone L∞ err 1.03E−1 7.95E−2 5.83E−2 4.28E−2

Nx 100 200 400 800

Nt 24 48 96 192

AF-LWR L∞ err 6.34E−2 4.29E−2 3.03E−2 1.86E−2

0.58 0.5 0.7

L1 err 5.63E−2 1.90E−2 6.39E−3 2.17E−3

Nx 100 200 400 800

Nt 24 48 96 192

WENO 2/3 L∞ err ord 1.21E−1 7.87E−2 0.62 4.84E−2 0.7 2.95E−2 0.72

L1 err 1.93E−1 5.57E−2 1.33E−2 3.57E−3

ord 0.38 0.45 0.45 ord

L1 err 1.47E−2 7.58E−2 3.85E−2 1.94E−2

ord 0.95 0.98 0.99 ord 1.57 1.57 1.56 ord 1.8 2.07 1.89

better than both the monotone and WENO schemes, which smooth the corners. Moreover, we can observe a very good resolution of the regular part. From Table 1 we note that our scheme is of high-order at least in L1 norm and presents lower errors than the WENO scheme in both norms. Example 2: Burgers’ Equation Let us consider the Burgers’ equation for HJ with a Lipschitz continuous initial condition 9

vt (t, x) + 12 vx (t, x)2 = 0, v0 (x) = max{1 − x 2 , 0}.

(t, x) ∈ (0, 1) × (−2, 2),

In this test we focus on the comparison between the filtered schemes S F and S AF , using the scheme (9) as monotone and the following Heun-Centered scheme (HC) 9

± n u∗ = un − ΔthA ∗ (D u ) 1 n 1 ∗ n+1 A ± ∗ u = 2 u + 2 u − Δt 2 h∗ (D u ),

(11)

 + − − , p+ ) := H p +p with hA (p , as high-order scheme. For S F we take ε = 5Δx ∗ 2 as suggested in [1]. In Fig. 2, the adaptive tuning of the parameter εn reduces the oscillations caused by the unstable Heun-Centered scheme for S F , although S F seems to have a better resolution of the singularity (at least for Δx = 0.1). This behavior is confirmed also by Table 2, where S F has better errors in L∞ for the first refinements of the grid, but clearly loses with respect to S AF in the L1 norm, in both

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Fig. 2 Test 2. Plots of the exact solution (solid line) and the approximated solution (+) obtained by the SF-HC scheme with ε = 5Δx (left), Filtered AF-HC (center) and WENO (right), with Δx = 0.1 Table 2 Test 2. Errors and orders in L∞ and L1 norms

Nx 40 80 160 320

Nt 32 64 128 256

SF-HC L∞ err 2.38E−2 1.42E−2 9.24E−3 4.93E−3

Nx 40 80 160 320

Nt 32 64 128 256

AF-HC L∞ err 4.49E−2 1.69E−2 9.23E−3 4.70E−3

Nt 32 64 128 256

WENO 2/3 L∞ err 1.31E−2 1.33E−2 6.43E−3 3.21E−3

Nx 40 80 160 320

ord 0.75 0.62 0.91 ord 1.41 0.87 0.97 ord −0.02 1.05 1.00

L1 err 1.93E−2 4.88E−3 7.93E−4 2.39E−4 L1 err 1.87E−2 2.01E−4 4.28E−4 1.02E−4 L1 err 1.26E−2 2.63E−3 4.37E−4 7.30E−5

ord 1.99 2.62 1.73 ord 3.21 2.23 2.07 ord 2.26 2.59 2.58

errors and orders. With respect to the WENO scheme, the AF-HC scheme produces always comparable performances, some times also better (see e.g. the first order in both norms or the second and third errors in L1 ).

5 Conclusions We have presented a rather simple way to construct convergent schemes, which are of high-order in the regions of regularity for the solution. The filter method is able to stabilize an otherwise unstable (high-order) scheme, still preserving its accuracy. The novelty here is the adaptive and automatic choice of the parameter εn which improves the filtered scheme in [1]. The computation of εn , although more

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expensive, is still affordable in low dimensions. The adaptive scheme is able to reduce the oscillations which may appear choosing a constant ε and, as shown by the numerical tests, gives always better results. Finally, we note that the adaptive filtered scheme, with a wise choice for the high-order scheme, has results close to the WENO scheme in terms of errors but seems to have a better accuracy on the kinks.

References 1. O. Bokanowski, M. Falcone, S. Sahu, An efficient filtered scheme for some first order HamiltonJacobi-Bellman equations. SIAM J. Sci. Comput. 38(1), A171–A195 (2016) 2. M.G. Crandall, P.L. Lions, Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43(167), 1–19 (1984) 3. M. Falcone, R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and HamiltonJacobi Equations (SIAM, Philadelphia, 2014) 4. M. Falcone, G. Paolucci, S. Tozza, Convergence of adaptive filtered schemes for first order evolutive Hamilton-Jacobi equations (submitted) 5. G. Jiang, D.-P. Peng, Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000) 6. A. Kurganov, S. Noelle, G. Petrova, Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23(3), 707–740 (2001) 7. P.L. Lions, P. Souganidis, Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton–Jacobi equations. Numer. Math. 69, 441–470 (1995) 8. A.M. Oberman, T. Salvador, Filtered schemes for Hamilton-Jacobi equations: a simple construction of convergent accurate difference schemes. J. Comput. Phys. 284, 367–388 (2015)

Goal-Oriented a Posteriori Error Estimates in Nearly Incompressible Linear Elasticity Dustin Kumor and Andreas Rademacher

Abstract In this article, we consider linear elastic problems, where Poisson’s ratio is close to 0.5 leading to nearly incompressible material behavior. The use of standard linear or d-linear finite elements involves locking phenomena in the considered problem type. One way to overcome this difficulties is given by selective reduced integration. However, the discrete problem differs from the continuous one using this approach. This fact has especially to be taken into account, when deriving a posteriori error estimates. Here, we present goal-oriented estimates based on the dual weighted residual method using only the primal residual due to the linear problem considered. The major challenge is given by the construction of an appropriate numerical approximation of the error identity. Numerical results substantiate the accuracy of the presented estimator and the efficiency of the adaptive method based on it.

1 Introduction Materials with nearly incompressible behavior are often found in practice. Consequently, there is a great interest in efficient and reliable simulation techniques for this problem class. One approach suitable to them among many others is given by finite elements with selective reduced integration, see, for instance, [7, Chapter 4]. In this setting assuming linear elastic material behavior the continuous and discrete bilinear forms are different due to the stabilization in the discrete bilinear form. Hence, we have no Galerkin orthogonality anymore, which complicates among other things the derivation of a posteriori error estimators. Nonetheless, one is interested in obtaining such estimators to check the accuracy of the discretization and to construct efficient adaptive methods.

D. Kumor · A. Rademacher () Technische Universität Dortmund, Institute of Applied Mathematics, Dortmund, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_35

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The article at hand focuses on the derivation of a posteriori error estimates in a linear quantity of interest. To this end, we utilize the dual weighted residual (DWR) method, see, e.g., [1, 2] for an overview on it. The usual proceeding, which we also follow here, is to derive an error identity in the residuals at first. Since this identity involves unknown analytic values in general, it has to be approximated numerically by suitable methods. However, the derivation of the error identity rely on Galerkin orthogonality arguments. Because of missing Galerkin orthogonality in our setting, we have to improve the arguments. In this context, the work [13] is very useful, where similar settings are considered to measure the error induced by inexact solving algorithms like iterative solvers. One drawback of the DWR method is that the accuracy of the numerical approximation techniques can only be shown under very strict regularity assumptions. However, the numerical results discussed herein substantiate the accuracy of the presented error estimator and the efficiency of the adaptive method based on it. In [10], a modification of the DWR method to improve the stability of the adaptive algorithm based on it is discussed. Alternative approaches to gain a posteriori estimates in linear quantities of interests for linear problems are discussed, for instance, in [11, 12]. We refer also to the survey article [6]. Moreover, the (quasi) optimal convergence of adaptive algorithms based on this approach is shown in [3, 5, 9]. The article is organized as follows: Sect. 2 is devoted to the introduction of the continuous problem setting. A finite element discretization based on selective reduced integration is shortly described in the following section. The derivation of the error estimator is in the focus of Sect. 4. In the first step, an error identity is proven and then numerically approximated. A numerical example is considered in Sect. 5 substantiating the theoretical findings. Finally, we draw some conclusions and present a short outlook on further tasks.

2 Continuous Problem Setting We introduce the problem of linear elasticity for nearly incompressible materials in this section. To simplify the presentation, we consider a two-dimensional problem by applying the plain strain assumption. To this end, let Ω ⊂ R2 be a polyhedral domain. Its boundary Γ := ∂Ω is subdivided into two parts ΓD and ΓN with |ΓD | > 0. The semilinear form A is given by an additive composition of a bilinear form a and a linear form l with A(ψ)(ϕ) = a(ψ, ϕ) − l(ϕ),   a(ψ, ϕ) = C : D ψ, D ϕ 0 ,     l(ϕ) = f, ϕ 0 + g, γ (ϕ) Γ , N : ;   B B V = ψ ∈ H 1 Ω, R2 B γ (ψ) = 0 on ΓD .

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The continuous problem consists in finding u ∈ V with A(u)(ϕ) = 0 for all ϕ ∈ V . Here, the standard L2 (Ω) scalar product is denoted by (·, ·)0 and (·, ·)ΓN is the L2 scalar product on the Neumann boundary ΓN . The symmetric gradient is given by D, the tensor double contraction by   : and the trace by γ . The linear  form l is defined by the body force f ∈ L2 Ω, R2 and the traction force g ∈ L2 ΓN , R2 . The fourth order elasticity tensor C is determined by the shear modulus μ > 0 and the bulk modulus K > 0 for linear isotropic material behavior. It is more convenient to use the elasticity modulus E > 0 and Poisson’s ratio ν ∈ [0, 0.5). Here, it is noteworthy that K → ∞ for ν → 0.5, where ν = 0.5 corresponds to incompressible material behavior. It is well known that for standard finite element methods the constant in the a priori error estimates goes to infinity for ν → 0.5. I.e. the finite element error can be and usually is significantly larger than the interpolation error. This situation is called volumetric or Poisson’s locking. In the next section, we discuss a finite element approach, which circumvents this difficulty.

3 Finite Element Discretization Using Selective Reduced Integration At the end of the last section, we have outlined the complex of problems arising by nearly incompressible material behavior. Here, we shortly introduce a finite element approach, whose error constants do not depend on Poisson’s ratio ν. The basic idea lies in the different treatment of the volumetric and deviatoric part of the stress. Introducing the mechanical pressure p, the volumetric part of the stress is given by σ vol = −pI . The deviatoric part is defined by 1 trace(σ )I = (C : D u)dev . 3   We define the operator P : H 1 Ω, R2 → L2 (Ω) by P (ϕ) = −K div(ϕ). Using this notation, we can write the bilinear form a in the following way: σ dev := σ −

    a(ψ, ϕ) = C : (D ψ)dev , (D ϕ)dev + P (ψ), P (ϕ) K −1 0

    with the weighted L2 scalar product (·, ·)K −1 = K −1 ·, · = ·, K −1 · . 0 0 The basic idea is to use standard bilinear elements for the displacement u but a piecewise constant approximation for the mechanical pressure p. Let Th be an admissible triangulation of Ω using quadrilaterals, Kˆ the reference element, FK :   2 ˆ ˆ K → K the connected transformation for K ∈ Th , and Q1 K, R the space of vector valued functions on the reference element Kˆ with bilinear polynomials

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in each component. The discrete trial and test space for the displacement is then defined by 9 C B   B 2 ˆ R ∀K ∈ Th . Vh := ψh ∈ V BB ψh|K ◦ FK ∈ Q1 K, Furthermore, let C

9

B   B Mh := ξh ∈ L (Ω) BB ξh|K ◦ FK ∈ P0 Kˆ 2

∀K ∈ Th

  with the space of constant scalar functions P0 Kˆ . We now define the discrete   operator Ph : H 1 Ω, R2 → Mh by 

Ph (ϕ), ζh

 K −1

  = −K div(ϕ), ζh K −1

∀ζh ∈ Mh .

Setting Ah (ψ)(ϕ) = ah (ψ, ϕ) − l(ϕ),     ah (ψ, ϕ) = C : (D ψ)dev , (D ϕ)dev + Ph (ψ), Ph (ϕ) K −1 , 0

we end up with the problem to find uh ∈ Vh with Ah (uh )(ϕh ) = 0 for all ϕ ∈ Vh . The connection to selective reduced integration consists in the realization of Ph by applying a one point Gaussian quadrature rule only on the volumetric part, cf. [7, Chapter 4] for more details.

4 A Posteriori Error Estimation In this section, we derive an a posteriori error estimate with respect to the discretization error measured in a linear quantity of interest J : V → R. The derivation is separated into two steps. At first, we proof an analytic error identity. In the second step, a numerical approximation method is proposed.

4.1 Error Identity A key ingredient of the error identity is the dual solution z, which represents the influence of a certain point in space on the error measured in the quantity of interest. The continuous dual problem is to find z ∈ V , which fulfills a(ϕ, z) = J (ϕ)

∀ϕ ∈ V .

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The discrete dual problem consists in finding zh ∈ Vh such that ah (ϕh , zh ) = J (ϕh )

∀ϕh ∈ Vh .

Due to the symmetry of a and ah the dual problems coincided with the original or primal problems with another right hand side. Using the continuous dual solution, we obtain the following error identity: Theorem 1 It holds J (u) − J (uh ) = ρh (uh ) (z − zh ) + Δρ (uh ) (z − zh ) + Δρ (uh ) (zh ) + ρh (uh ) (zh )

(1)

with the primal residual ρh (uh ) (·) = −Ah (uh ) (·) = l(·) − ah (uh , ·) and the difference in the residuals   Δρ (uh ) (·) = Ah (uh ) (·) − A (uh ) (·) = Ph (uh ) − K div (uh ) , div (·) 0 . Proof We use the definition of the dual solution z as well as A(u)(z) = 0 to calculate J (u) − J (uh ) = a (u − uh , z) = −A (uh ) (z) + Ah (uh ) (z) − Ah (uh ) (z) = ρh (uh ) (z) + Δρ (uh ) (z) = ρh (uh ) (z − zh ) + Δρ (uh ) (z − zh ) + Δρ (uh ) (zh ) + ρh (uh ) (zh ) . The definition of A and Ah as well as of the discrete operator Ph leads to     Δρ (uh ) (ϕ) = Ah (uh ) ϕ − A (uh ) ϕ       = Ph (uh ) , Ph ϕ − P , P ϕ (u ) h K −1 K −1    = Ph (uh ) − P (uh ) , P ϕ K −1    = Ph (uh ) − K div (uh ) , div ϕ 0

for arbitrary ϕ ∈ V , which completes the proof.

& %

4.2 Numerical Approximation of the Error Identity The error identity (1) includes the analytic dual solution z. Hence, it cannot be evaluated numerically. There exist different approaches to approximate such error identities, see [1, Section 4.1] for an overview. Here, we work with higher order

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interpolation or reconstruction techniques. To this end, we assume that the mesh Th has patch structure. We define the patchwise biquadratic interpolation operator (2) to obtain a higher order approximation of the dual solution. However, we I2h also have to approximate P (z), which corresponds to the mechanical pressure. Because Ph (zh ) is piecewise constant, we use an average type approximation here. I.e. we take the meanvalue over all adjacent mesh elements of a node as approximation in this node and interpolate this values using bilinear basis functions. The corresponding operator is denoted by IA . Finally, we set   (2) ηh := ρh (uh ) I2h zh − zh , ηa := Δρ (uh ) (IA zh − zh ) , ηn := Δρ (uh ) (zh ) + ρh (uh ) (zh ) , and end up with the following error estimator: J (u) − J (uh ) ≈ η = ηh + ηa + ηn Here, ηh +ηa measure the discretization error and ηn the numerical one, for instance, arising from the numerical solution of the linear system of equations. We also (2) consider the estimator η, ˜ in which we only use I2h . The last step is to localize the error estimator η on the single mesh cells. There exists several approaches in literature, cf., e.g., to [14] for an overview. We use the filtering technique going back to [4] here, which especially circumvents the evaluation of the strong form and of terms on edges.

5 Numerical Results We consider an example on an L-shaped domain, Ω = (−0.5, 0.5)2\[0, 0.5]2, ΓD = {0} × [0, 0.5] ∪ [0, 0.5] × {0}, and ΓN = ∂Ω\ΓD . Following [8], a nonsmooth analytic solution u with div(u) = 0 is constructed, which specifies f and g. Furthermore,  E = 2.5 and ν = 0.49999. The quantity of interest is  we set given by J (ϕ) = Ω χB ϕ1 + ϕ2 dx with a smoothed indicator function χB with respect to B = (−0.5, −0.25) × (0.25, 0.5). In Fig. 1a the mesh generated in the adaptive algorithm based on η is depicted, where we find strong refinements against the reentrant corner. The mesh based on η, ˜ cf. Fig. 1b leads to a similar mesh. In Fig. 2a, the convergence of the adaptive algorithms based on η and η˜ as well as the results using uniform refinement are compared. As expected, the order of convergence is reduced for uniform refinement due to the nonsmooth solution. The optimal order is regained for the adaptive algorithm based on η as well as for the one based on η. ˜ Furthermore, we depict the absolute value of the single contributions of η in Fig. 2a. We clearly see that ηh and

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Fig. 1 Adaptive meshes for different reconstruction techniques. (a) Adaptive mesh based on η in step 12 of the adaptive algorithm. (b) Adaptive mesh based on η˜ in step 12 of the adaptive algorithm

Fig. 2 Convergence of the different methods and accuracy of the error estimator. (a) Convergence of the adaptive method and contribution of the single parts of η. (b) Effectivity indices for uniform and adaptive refinement

ηa have to be taken into account. Moreover, also the numerical error measured by ηn becomes relevant for fine meshes because of the bad conditioning of the discrete problem, which is well known for this finite element we plot   approach. Finally, the absolute value of the effectivity index Ieff = η/ J (u) − J (uh ) in Fig. 2b for uniform and adaptive refinement. It approaches to one for the adaptive refinement and goes to a constant around 0.4 for uniform refinement, which has to be expected. The deviation in the last iteration is due to the dominating numerical error. However, we do not show the effectivity index with respect to η˜ because it is in the range of 103 . All in all, we find that η is an accurate estimator and that the adaptive method based on it is as efficient as expected.

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6 Conclusion and Outlook In this article, we have considered a finite element method based on selective reduced integration for the problem of nearly incompressible linear elasticity and have derived an error estimator in a user defined linear quantity of interest, which consists in the usual residual of the DWR method plus some additional terms arising from the fact that the continuous and discrete bilinear forms differ. Furthermore, a special numerical approximation approach is developed. Numerical results show the necessity of the additional terms and substantiate the accuracy of the estimator and the efficiency of the adaptive method based on it. We are currently extending the results presented in this article to the nonlinear case in a general framework and additionally consider the error, which emerge from the use of two different discrete problem formulations. Acknowledgements The authors gratefully acknowledge the financial support by the German Research Foundation (DFG) within the subproject A5 of the transregional collaborative research centre (Transregio) 73 “Sheet-Bulk-Metal-Forming”.

References 1. W. Bangerth, R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics, ETH Zürich (Birkhäuser, Basel, 2003) 2. R. Becker, R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001) 3. R. Becker, E. Estecahandy, D. Trujillo, Weighted marking for goal-oriented adaptive finite element methods. SIAM J. Numer. Anal. 49, 2451–2469 (2011) 4. M. Braack, A. Ern, A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1, 221–238 (2003) 5. M. Feischl, D. Praetorius, K.G. Van der Zee, An abstract analysis of optimal goal-oriented adaptivity. SIAM J. Numer. Anal. 54, 1423–1448 (2016) 6. M.B. Giles, E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145–236 (2002) 7. T.J.R. Hughes, The Finite Element Method. Linear Static and Dynamic Finite Element Analysis (Dover Publications, Mineola, 2000) 8. H. Melzer, R. Rannacher, Spannungskonzentration in Eckpunkten der Kirchhoffschen Platte. Bauingenieur 55, 181–184 (1980) 9. M.S. Mommer, R. Stevenson, A goal-oriented adaptive finite element method with convergence rates. SIAM J. Numer. Anal. 47, 861–886 (2009) 10. R.H. Nochetto, A. Veeser, M. Verani, A safeguarded dual weighted residual method. IMA J. Numer. Anal. 29, 126–140 (2009) 11. M. Paraschivoiu, J. Peraire, A.T. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations. Comput. Methods Appl. Mech. Eng. 150, 289– 312 (1997) 12. S. Prudhomme, J.T. Oden, On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Eng. 176, 313–331 (1999) 13. R. Rannacher, J. Vihharev, Adaptive finite element analysis of nonlinear problems: balancing of discretization and iteration errors. J. Numer. Math. 21, 23–61 (2013) 14. T. Richter, T. Wick, Variational localizations of the dual-weighted residual estimator. J. Comput. Appl. Math. 279, 192–208 (2015)

Nitsche’s Method for the Obstacle Problem of Clamped Kirchhoff Plates Tom Gustafsson, Rolf Stenberg, and Juha Videman

Abstract The theory behind Nitsche’s method for approximating the obstacle problem of clamped Kirchhoff plates is reviewed. A priori estimates and residual-based a posteriori error estimators are presented for the related conforming stabilised finite element method and the latter are used for adaptive refinement in a numerical experiment.

1 Introduction Nitsche’s method is widely used for the numerical approximation of contact problems, cf. [5, 6, 8, 9] and all the references therein. It was first proposed as a non-standard treatment of boundary conditions [23] and as such is related to later discovered discontinuous Galerkin methods. Over 20 years ago (cf. [25]), using a Lagrange multiplier to impose (weakly) the Dirichlet boundary condition on the Poisson equation, we observed that the Lagrange multiplier can be eliminated, element by element, from the stabilised finite element formulation, leading to an optimally conditioned, symmetric and positive definite system corresponding to a method by Nitsche which was, at that time, largely forgotten. There are, however, fundamental issues with Nitsche’s formulation since its analysis requires an additional smoothness assumption and the a posteriori estimates are based on a so-called saturation assumption, cf. [1, 10, 20]. Recently, by going back to the interpretation of Nitsche’s method as a stabilised formulation, recalling our analysis for the Stokes problem [26] and using the techniques from [13, 15],

T. Gustafsson · R. Stenberg () Department of Mathematics and Systems Analysis, Aalto University, Aalto, Finland e-mail: [email protected]; [email protected] J. Videman CAMGSD/Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_36

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we were able to give a complete error analysis (both a priori and a posteriori) for the stabilised/Nitsche’s method when applied to the membrane obstacle problem [17]. We emphasise that the stabilised form is only needed for the analysis. For the practical implementation, Nitsche’s formulation is preferable. In this note, we will review our latest results on conforming stabilised finite element methods for the plate obstacle problem, cf. [19]. Our method is based on a saddle point formulation with the contact force appearing as an additional unknown (Lagrange multiplier). We present an a priori estimate with minimal regularity assumptions and introduce the Nitsche’s formulation with Lagrange multiplier providing an approximation for the contact force and the unknown contact domain. We will also present an a posteriori error estimator and use it for adaptive refinement in a numerical experiment. Numerical approximation of fourth-order variational inequalities has been previously studied, e.g., in [2–4, 16, 24] but to our knowledge Nitsche’s or stabilised methods using conforming C 1 -continuous elements were for the first time proposed and rigorously analysed in [19].

2 Problem Statement Let Ω ⊂ R2 denote a polygonal domain occupied by (the mid-surface of) a thin plate of thickness d whose deformation is governed by the Kirchhoff–Love theory. Assume that the vertical displacement u of the plate, resulting from an applied load f ∈ L2 (Ω), is constrained by a rigid obstacle g ∈ H 2 (Ω) and suppose, for simplicity, that the plate is clamped at all edges. Letting ε(v) = 12 (∇v + ∇v T ) denote the infinitesimal strain tensor and K(u) = −ε(∇u) the curvature, the bending moment M is defined by   Ed 3 ν M(u) = tr (K(u)) I , K(u) + 12(1 + ν) 1−ν where E and ν are the Young’s modulus and the Poisson ratio (see, e.g., [12]). Defining the bilinear and linear forms a and l by   a(w, v) = M(w) : K(v) dx, l(v) = f v dx, Ω

Ω

the solution to the clamped plate obstacle problem can be characterised as @ ? 1 a(v, v) − l(v) , u = argminv∈K 2 where K = { v ∈ H02 (Ω) : v ≥ g in Ω } or, equivalently, as the solution to the variational inequality: Find u ∈ K such that a(u, v − u) ≥ l(v − u) ∀ v ∈ K .

(1)

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Remark 1 The existence of a unique solution to problem (1) follows from standard 3 (Ω) ∩ theory, see [21]. For smooth data, the solution has been shown to be in Hloc 2 3 C (Ω), in convex domains in H (Ω), cf. [7, 14], and the smoothness threshold seems to be H 7/2−ε (Ω), ε > 0, see Example 1 in [3]. Nitsche’s method for approximating the plate obstacle problem can be regarded as a stabilised finite element method for the Lagrange multiplier formulation of problem (1) with the stabilisation term arising from its strong form. Associating a Lagrange multiplier λ to the constraint v ≥ g, the strong form reads as follows:

λ ≥ 0,

u − g ≥ 0,

A(u) − λ = f

⎫ ⎬

λ(u − g) = 0

⎭

u = 0 and

∂u =0 ∂n

in Ω,

on ∂Ω,

(2)

(3)

where A(u) =

Ed 3 Δ2 u . 12(1 − ν 2 )

The Lagrange multiplier λ corresponds to a reaction force exerted on the plate by the obstacle and it belongs to the space Λ = {μ ∈ Q : v, μ ≥ 0 ∀v ∈ V s.t. v ≥ 0 a.e. in Ω}, where V = H02 (Ω), Q = H −2 (Ω) = [H02 (Ω)] and ·, · : V × Q → R denotes the duality pairing. Let us define a bilinear form B : (V × Q) × (V × Q) → R and a linear form L : V × Q → R through B(w, ξ ; v, μ) = a(w, v) − v, ξ  − w, μ, L(v, μ) = (f, v) − g, μ. Problem (2)–(3) can now be written as the following variational inequality: Find (u, λ) ∈ V × Λ such that B(u, λ; v, μ − λ) ≤ L(v, μ − λ)

∀(v, μ) ∈ V × Λ.

The bilinear form B is continuous and stable (cf. [19]) with respect to the norm 1/2  |||(w, ξ )||| = w22 + ξ 2−2 ,

(4)

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where ·2 and ·−2 are the usual norms in H 2 (Ω) and H −2 (Ω). The saddle-point formulation (4) and formulation (1) are equivalent, cf. [11].

3 Stabilised Finite Element Method Let Ch be a conforming shape-regular triangulation of Ω into triangles K and let Vh ⊂ V and Qh ⊂ Q be finite element subspaces. Defining Λh = {μh ∈ Qh : μh ≥ 0 in Ω} ⊂ Λ and introducing stabilised bilinear and linear forms Bh and Lh by Bh (w, ξ ; v, μ) = B(w, ξ ; v, μ) − α Lh (v, μ) = L(v, μ) − α

h4K (A(w) − ξ, A(v) − μ)K ,

K∈Ch

h4K (f, A(v) − μ)K ,

K∈Ch

where α > 0 is a stabilisation parameter, the stabilised finite element method becomes: Find (uh , λh ) ∈ Vh × Λh such that Bh (uh , λh ; vh , μh − λh ) ≤ Lh (vh , μh − λh )

∀(vh , μh ) ∈ Vh × Λh .

(5)

The stabilised formulation is consistent and, assuming that α ∈ (0, CI ) where CI > 0 is the constant from the inverse inequality CI

h4K A(wh )20,K ≤ a(wh , wh )

∀wh ∈ Vh ,

K∈Ch

it is stable which leads to a quasi-optimal a priori error estimate (see [19]): |||(u − uh , λ − λh )|||  inf

vh ∈Vh , μh ∈Λh

  |||(u − vh , λ − μh )||| + u − g, μh  + osc(f ).

Above osc(f ) denotes the data oscillation defined by osc(f )2 =

h2K f − fh 0,K ,

K∈Ch

with fh ∈ Vh standing for the L2 -projection of f .

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4 Nitsche’s Method Assume that the finite element spaces consist of a C 1 -element for the displacement field (in our example, the Argyris element) coupled with a piecewise polynomial and discontinuous approximation of the Lagrange multiplier. The Lagrange multiplier can thus be eliminated elementwise from the stabilised formulation (5) leading to the Nitsche’s method: Find uh ∈ Vh such that ah (uh , vh ; uh ) = lh (vh ; uh )

∀vh ∈ Vh ,

(6)

where ah (uh , vh ; wh ) = a(uh , vh ) +





1

u ,v α H4 h h

  − uh , A(vh ) Ω

C (wh

lh (vh ; wh ) = (f, vh ) +



  − f, vh Ω

1

α H4

ΩC (wh )

  4 − αH A(u ), A(v ) h h ) 

g, vh

C (wh )

  − A(uh ), vh Ω

ΩC (wh )

C (wh )

Ω\ΩC (wh )

,

  − g, A(vh ) Ω

C (wh )

− (αH4 f, A(vh ))Ω\ΩC (wh ) .

Above, H ∈ L2 (Ω) is defined as H|K = hK , ∀K ∈ Ch , and the contact set ΩC (wh ) = {(x, y) ∈ Ω : F (wh ) > 0}, where F (wh ) =

 1  g − wh + αH4 (A(wh ) − f ) , 4 + αH

w+ = max(w, 0),

is the reaction force (Lagrange multiplier), is approximated iteratively using the previous displacement field to linearise problem (6). Based on an a posteriori error analysis of the stabilised method, the local error estimator used in an adaptive refinement strategy is defined as 2 2 EK = ηK +

1 2 ηE + ((uh − g)+ , λh )K + (g − uh )+ 22,K , 2 E⊂K

where E are the (interior) edges of K ∈ Ch and 2 ηK = h4K A(uh ) − λh − f 20,K ,

2 ηE = h3E Vn (uh )20,E + hE Mnn (uh )20,E ,

with Vn (uh ) and Mnn (uh ) denoting the jumps over interior edges of the Kirchhoff shear force and the normal moment, see [18, 19] for more details.

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5 Numerical Results We will consider the problem from [4, Example 7.4] where the domain Ω is given by Ω = (−0.5, 0.5)2 \ [0, 0.5]2 and the obstacle and load function are g(x) = − sin(2π(x + 0.5)(y + 0.5)) sin(4π(x − 0.5)(y − 0.5)) − 0.35 ⎧ 2 2 ⎪ ⎨ 500 e(x+0.25) +(y+0.25) , f (x) = 0, ⎪  3/2 ⎩ 1000 0.5 + (x − 0.25)2 + (y + 0.25)2 ,

x ≤ 0, y > 0 x ≤ 0, y ≤ 0 x > 0, y ≤ 0

The stabilisation parameter is chosen as α = 10−5 , and the marking and adaptive refinement strategies are as in [19]. For this problem the contact set is onedimensional. The discrete solution and the discrete contact set are visualised in Fig. 1. The adaptive meshes and the total error are given in Fig. 2. Note that due to the re-entrant corner, the exact solution belongs to H 2.54(Ω) (cf. [22]) which corresponds to the convergence rate N −0.27 obtained with uniform refinement, with N denoting the number of degrees of freedom, and that the adaptive meshing strategy recovers the optimal rate of convergence N −2 for fifth order elements.

Fig. 1 The discrete solution uh and the discrete contact set, ΩC (uh ), after two adaptive mesh refinements

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Fig. 2 A sequence of adaptively refined meshes and the total error as a function of the number of degrees of freedom. The upper-left panel depicts the initial mesh

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Acknowledgements The authors are grateful for the financial support from the Portuguese Science Foundation (FCOMP-01-0124-FEDER-029408), Tekes (Decision number 3305/31/2015), the Finnish Academy of Science and Letters, and the Finnish Cultural Foundation.

References 1. R. Becker, P. Hansbo, R. Stenberg, A finite element method for domain decomposition with non-matching grids. ESAIM: Math. Model. Numer. Anal. 37, 209–225 (2003) 2. S. Brenner, L.-Y. Sung, Y. Zhang, Finite element methods for the displacement obstacle problem of clamped plates. Math. Comput. 81, 1247–1262 (2012) 3. S. Brenner, L.-Y. Sung, H. Zhang, Y. Zhang, A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates. J. Comput. Appl. Math. 254, 31– 42 (2013) 4. S. Brenner, J. Gedicke, L.-Y. Sung, Y. Zhang, An a posteriori analysis of C 0 interior penalty methods for the obstacle problem of clamped Kirchhoff plates. SIAM J. Numer. Anal. 55, 87–108 (2017) 5. E. Burman, A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM J. Numer. Anal. 50, 1959–1981 (2012) 6. E. Burman, P. Hansbo, M.G. Larson, The penalty-free Nitsche’s method and non-conforming finite elements for the Signorini problem. SIAM J. Numer. Anal 55, 2523–2539 (2017) 7. L.A. Caffarelli, A. Friedman, The obstacle problem for the biharmonic operator. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6, 151–183 (1979) 8. F. Chouly, P. Hild, A Nitsche-based method for unilateral contact problems: numerical analysis. SIAM J. Numer. Anal. 51, 1295–1307 (2013) 9. F. Chouly, P. Hild, Y. Renard, Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: theory and numerical experiments. Math. Comput. 84, 1089– 1112 (2015) 10. F. Chouly, M. Fabre, P. Hild, J. Pousin, Y. Renard, Residual-based a posteriori error estimation for contact problems approximated by Nitsche’s method. IMA J. Numer. Anal. 38, 921–954 (2018). https://doi.org/10.1093/imanum/drx024 11. I. Ekeland, R. Temam, Convex Analysis and Variational Problems (SIAM, Philadelphia, 1999) 12. K. Feng, Z.-C. Shi, Mathematical Theory of Elastic Structures (Springer/Science Press, Berlin/Beijing, 1996) 13. L.P. Franca, R. Stenberg, Error analysis of Galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal. 28, 1680–1697 (1991) 14. J. Frehse, Zum Differenzierbarkeitsproblem bei Variationsungleichungen höherer Ordnung. Abh. Math. Sem. Univ. Hamburg 36, 140–149 (1971) 15. T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79, 2169–2189 (2010) 16. T. Gudi, K. Porwal, A C 0 interior penalty method for a fourth-order variational inequality of the second kind. Numer. Methods Partial Differ. Equ. 32, 36–59 (2016) 17. T. Gustafsson, R. Stenberg, J. Videman, Mixed and stabilized finite element methods for the obstacle problem. SIAM J. Numer. Anal. 55, 2718–2744 (2017) 18. T. Gustafsson, R. Stenberg, J. Videman, A posteriori estimates for conforming Kirchhoff plate elements. arXiv preprint: 1707.08396 19. T. Gustafsson, R. Stenberg, J. Videman, A stabilized finite element method for the plate obstacle problem. arXiv preprint: 1711.04166 20. M. Juntunen, R. Stenberg, Nitsche’s method for general boundary conditions. Math. Comput. 78, 1353–1374 (2009) 21. J.L. Lions, G. Stampacchia, Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)

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22. H. Melzer, R. Rannacher, Spannungskonzentrationen in Eckpunkten der vertikal belasteten Kirchhoffschen Platte. Bauingenieur 55, 181–189 (1980) 23. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Univ. Hamburg 36, 9–15 (1970/1971) 24. R. Scholz, Mixed finite element approximation of a fourth order variational inequality by the penalty method. Numer. Funct. Anal. Optim. 9, 233–247 (1987) 25. R. Stenberg, On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63, 139–148 (1995) 26. R. Stenberg, J. Videman, On the error analysis of stabilized finite element methods for the Stokes problem. SIAM J. Numer. Anal. 53, 2626–2633 (2015)

Part XI

Noncommutative Stochastic Differential Equations: Analysis and Simulation

Stochastic B-Series and Order Conditions for Exponential Integrators Alemayehu Adugna Arara, Kristian Debrabant, and Anne Kværnø

Abstract We discuss stochastic differential equations with a stiff linear part and their approximation by stochastic exponential Runge–Kutta integrators. Representing the exact and approximate solutions using B-series and rooted trees, we derive the order conditions for stochastic exponential Runge–Kutta integrators. The resulting general order theory covers both Itô and Stratonovich integration.

1 Introduction The idea of expressing the exact and numerical solutions of different blends of differential equations in terms of B-series and rooted trees has been an indispensable tool ever since John Butcher introduced the idea in 1963 [4]. Naturally then, such series have also been derived for stochastic differential equations (SDEs) by several authors, see e.g. [6] for an overview. In this paper, the focus is on d-dimensional SDEs of the form   M

  dX(t) = AX(t) + g0 X(t) dt + gm (X(t)) ! dWm (t),

X(0) = x0 ,

(1)

m=1

A. A. Arara Hawassa University, Department of Mathematics, Hawassa, Ethiopia K. Debrabant University of Southern Denmark, Department of Mathematics and Computer Science, Odense M, Denmark e-mail: [email protected] A. Kværnø () Norwegian University of Science and Technology - NTNU, Department of Mathematical Sciences, Trondheim, Norway e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_37

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or in integral form 

t

X(t) = e x0 + tA

e

(t −s)A

g0 (X(s))ds+

0

M 

t

e(t −s)Agm (X(s))!dWm (s),

(2)

m=1 0

in which case the linear term AX(t), A ∈ Rd×d constant will be treated with particular care by the use of exponential Runge–Kutta integrators, see e.g. [1, 5, 10] and references therein. The integrals w. r. t. the components of the M-dimensional Wiener process W (t) can be interpreted e. g. as an Itô or a Stratonovich integral. The coefficients gm : Rd → Rd are assumed to be sufficiently differentiable and to satisfy a Lipschitz and a linear growth condition. For Stratonovich SDEs, we  g require in addition that the coefficients gm are differentiable and that also the gm m satisfy a Lipschitz and a linear growth condition. In the following, we will denote dt = dW0 (t). For the numerical solution of (1) we consider a general class of ν-stage stochastic exponential Runge–Kutta integrators: Hi = eci hA Yn +

M ν

(m)

Zij (A) · gm (Hj ),

i = 1, . . . , ν,

(3a)

m=0 j =1

Yn+1 = ehA Yn +

ν M

zi(m) (A) · gm (Hi ),

(3b)

m=0 i=1

where typically the coefficients Zij(m) and zi(m) are random variables depending on the stepsize h, the matrix A and the Wiener processes, and ci are real coefficients, i = 1, . . . , ν. For vanishing A the method (3) reduces to a standard stochastic Runge–Kutta method. For an example of a 2-stage stochastic exponential Runge– Kutta method, see Example 11 below. Although convergence and order results of specific stochastic exponential methods proposed in the literature are given, see for instance [1, 5, 10], there is to our knowledge up to now no general order and convergence theory for stochastic exponential Runge–Kutta methods. In this paper, such a theory is provided. The theory is derived based on a combination of the ideas of stochastic B-series and rooted trees developed in [6], and the similar ideas for deterministic exponential Runge–Kutta methods, as derived in [2, 8].

2 Some Notation, Definitions and Preliminary Results on Stochastic B-Series In Sect. 3 we will develop B-series for the exact solution of the stochastic differential equation (1) and stochastic exponential Runge–Kutta integrators of the form (3). For this, we will use the following definitions of the trees associated to (1), their corresponding elementary differentials and associated B-series.

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Definition 1 (Trees) The set of M + 2-colored, rooted trees T = {∅} ∪ T0 ∪ T1 ∪ · · · ∪ TM ∪ TA is recursively defined as follows: 1. The graph •m = [∅]m with only one vertex of color m belongs to Tm , and •A = [∅]A with only one vertex of color A belongs to TA , 2. Let τ = [τ1 , τ2 , . . . , τκ ]m be the tree formed by joining the subtrees τ1 , τ2 , . . . , τκ each by a single branch to a common root of color m and τ = [τ1 ]A be the tree formed by joining the subtree τ1 to a root of color A. If τ1 , τ2 , . . . , τκ ∈ T \ {∅}, then τ = [τ1 , τ2 , . . . , τκ ]m ∈ Tm and [τ1 ]A ∈ TA , for m = 0, . . . , M. Definition 2 (Elementary Differential) For a tree τ ∈ T the elementary differential is a mapping F (τ ) : Rd → Rd defined recursively by 1. F (∅)(x0 ) = x0 , 2. F (•m )(x0 ) = gm (x0 ), F (•A )(x0 ) = Ax0 , 3. If τ1 , τ2 , . . . , τκ ∈ T \ {∅}, then F ([τ1 ]A )(x0 ) = AF (τ1 )(x0 ) and (κ) (x0 )(F (τ1 )(x0 ), . . . , F (τκ )(x0 )) F ([τ1 , τ2 , . . . , τκ ]m )(x0 ) = gm

for m = 0, . . . , M. Now we give the definition of B-series. Definition 3 (B-Series) A (stochastic) B-series is a formal series of the form B(φ, x0 ; h) =

α(τ ) · φ(τ )(h) · F (τ )(x0 ),

τ ∈T

where φ(τ )(h) is a random variable satisfying φ(∅) ≡ 1 and φ(τ )(0) = 0 for all τ ∈ T \ {∅}, and α : T → Q is given by α(∅) = 1,

α(•m ) = 1,

α([τ1 , . . . , τκ ]m ) =

α(•A ) = 1,

κ . 1 α(τk ), r1 !r2 ! · · · rl !

α([τ1 ]A ) = α(τ1 ),

k=1

where r1 , r2 , . . . , rl count equal trees among τ1 , τ2 , . . . , τκ , and m = 0, . . . , M. Next we give an important lemma to derive B-series for the exact and numerical solutions. It states that if Y (h) can be expressed as a B-series, then f (Y (h)) can also be expressed as a B-series where the sum is taken over trees with a root of color f and subtrees in T .

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Lemma 4 If Y (h) = B(φ, x0 ; h) is some B-series and f ∈ C ∞ (Rd , Rd ), then f (Y (h)) can be written as a formal series of the form f (Y (h)) =

β(u) · ψφ (u)(h) · G(u)(x0 )

(4)

u∈Uf

where Uf is a set of trees derived from T , by (i) [∅]f ∈ U , and u = [τ1 , τ2 , . . . , τκ ]f ∈ Uf , (ii) G([∅]f )(x0 ) = f (x0 ) and G([τ1 , τ2 , . . . , τκ ]f )(x0 ) = f (κ) (x0 )(F (τ1 )(x0 ), . . . , F (τκ )(x0 )), < (iii) β([∅]f ) = 1 and β([τ1 , . . . , τκ ]f ) = r1 !r21!···rl ! κk=1 α(τk ), with r1 , r2 , . . . , rl counting equal trees among τ1 , τ2 , . . . , τκ , < (iv) ψφ ([∅]f ) ≡ 1 and ψφ ([τ1 , τ2 , . . . , τκ ]f )(h) = κk=1 φ(τk )(h), for all τ1 , τ2 , . . . , τκ ∈ T \ {∅} and κ = 1, 2, . . . . & %

Proof The proof of this lemma is given in [6]. Applying Lemma 4 to the functions gm on the right hand side of (1) gives gm (B(φ, x0 ; h)) =

 α(τ ) · φm (τ )(h) · F (τ )(x0 ),

(5)

τ ∈Tm

where ⎧ ⎨1  (τ )(h) = 1 and

q [τˆ ]A

+ ,) * = [. . . [[τˆ ]A ]A . . .]A for τˆ ∈ T \ TA .

Proof Write the exact solution X(h) of (1) at t = h as a B-series B(ϕ, x0 ; h). Substituting X(h) = B(ϕ, x0 ; h) in (2) and using (5) gives B(ϕ, x0 ; h) = ehA x0 +

M 

m=0 0

h

e(h−s)A

τˆ ∈Tm

 α(τˆ ) · ϕm (τˆ )(s) · F (τˆ )(x0 ) ! dWm (s).

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Inserting the series representation ehA x0 =

B(ϕ, x0 ; h) = x0 +

∞

hq q=1

+

M

α(τˆ )

m=0 τˆ ∈Tm

q!

∞ 

h 0

q=0

∞

q=0

hq Aq q! x0

yields

Aq x0  (h − s)q  q ϕm (τˆ )(s) ! dWm (s) · A F (τˆ )(x0 ) . q!

(8)

q

Note that any tree τ ∈ T can be rewritten as τ = [τˆ ]A for q = 0, 1, . . ., with τˆ ∈ T \ TA , that means τˆ = ∅ or τˆ = [τ1 , . . . , τκ ]m for an m ∈ {1, . . . , M}. It holds q q q that F ([τˆ ]A ) = Aq F (τˆ ) and α([τˆ ]A ) = α(τˆ ). Especially, for τ = [∅]A it holds that q q α(τ ) = 1 and F (τ )(x0 ) = A F (∅) = A x0 . Using (6) and the linear independence of the elementary differentials finishes the proof. & % Example 1 Let τ = , where the colors black, white and red correspond to the deterministic function g0 , the stochastic function g1 and an application of the matrix A, respectively. Then α(τ ) = 1, F (τ )(x0 ) = g0 (g1 , g1 (Ax0 , g0 ))(x0 ) and  h s ϕ(τ )(h) = 0 W1 (s) 0 s12 ! dW1 (s1 ) ds. Note also that e.g. τ = ∈ / T since it is impossible for node to have more than one branch. Next we derive the B-series representation for one step of the stochastic exponential Runge–Kutta integrator (3). Theorem 9 Assume that the coefficients Zij(m) (A) and zi(m) (A) can be expressed as power series of the form Zij(m) (A) =

∞

(m,q)

Zij

Aq

zi(m) (A) =

and

q=0

∞

(m,q)

zi

Aq ,

(9)

q=0

for i, j = 1, . . . , ν, and m = 0, . . . , M. Then the stage values Hi and the numerical solution Yn+1 defined by (3) can be written as B-series Hi = B(Φi , Yn ; h), i = 1, . . . , ν, and Yn+1 = B(Φ, Yn ; h) with the following recurrence relations for the functions Φi (τ )(h) and Φ(τ )(h), Φi (∅) = Φ(∅) ≡ 1,

q

Φi ([∅]A )(h) =

q Φi ([[τ1 , . . . , τκ ]m ]A )(h)

=

ν

(ci h)q , q! (m,q) Zij

j =1 q

Φ([[τ1 , . . . , τκ ]m ]A )(h) =

ν

i=1

q

Φ([∅]A )(h) =

κ .

Φj (τk )(h),

k=1 (m,q)

zi

κ . k=1

Φi (τk )(h),

hq , q!

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for τ1 , . . . , τκ ∈ T , κ = 1, 2, . . ., q = 0, 1, . . . and m = 0, . . . , M, where τj = ∅ for j = 1, . . . , κ if κ > 1. Proof Write the stage values Hi and the approximation Yn+1 to the exact solution as B-series: Hi = B(Φi , Yn ; h),

i = 1, . . . , ν

Yn+1 = B(Φ, Yn ; h).

and

(10)

Substituting (5) into (3) and using (9) we get Hi =

∞

(ci h)q

q!

q=0

=

∞

(ci h)q

q!

q=0

Aq Yn +

M ν

Zij(m) (A)

m=0 j =1

Aq Yn +

ν M



α(τ ) · Φj (τ )(h) · F (τ )(Yn )

τ ∈Tm

α(τ )

m=0 j =1 τ ∈Tm

∞

(m,q)

Zij

Φj (τ )(h) · Aq F (τ )(Yn ),

q=0

and similarly Yn+1 =

∞

hq q=0

q!

Aq Yn +

M ν

m=0 i=1 τ ∈Tm

α(τ )

∞

(m,q)

zi

Φi (τ )(h) · Aq F (τ )(Yn ).

q=0

Now using (10) and the linear independence of the elementary differentials yields the assertion. & % Remark 10 When A vanishes, all elementary differentials corresponding to trees in TA are zero, and the above B-series theory agrees with the B-series theory developed in [6]. In the deterministic case, with gm = 0 for m ≥ 1, the elementary differentials corresponding to trees in Tm , m ≥ 1, vanish, and our results agree with those given in [2, 8]. We conclude this article with an example on how to apply the theorems of this section to decide the order of a given stochastic exponential integrator. Example 11 We will apply Theorems 6, 7, and 9 to the 2-stage stochastic exponential time-differencing Runge–Kutta (SETDRK) method for M = 1 given by

Yn+1

√ H1 = Yn , H2 = Yn + hg1 (H1 ),  tn+1 hA = e Yn + e(tn+1 −s)A ds · g0 (H1 )  +

tn tn+1 tn

1 +√ h



e(tn+1 −s)A ! dW1 (s) · g1 (H1 ) tn+1 tn

  e(tn+1 −s)A W1 (s) ! dW1 (s) · −g1 (H1 ) + g1 (H2 ) ,

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where tn+1 = tn + h. Using the expansion (for the manipulation of stochastic integrals, see e. g. [7]) 



h

e

(h−s)A

h

! dW1 (s) =

0





0 h

+ 0

h

1 ! dW1 (s)A + 0

(h − s) ! dW1 (s)A1

0

(h − s)2 2

! dW1 (s)A2 + . . .

∗ ∗ ∗ = I(1) A0 + I(10) A1 + I(100) A2 + . . .

hs s ∗ = 0 0 1 . . . 0 n−1 !dWm1 (sn ) · · · ! dWmn (s1 ), and the similar where I(m 1 ...mn ) h ∗ A0 + I ∗ 1 expansion 0 e(h−s)A W1 (s) ! dW1 (s) = I(11) (110) A + . . . we obtain z1(0) (1)

 

z1 =

z2(1)

h

= 0

h 0

e(h−s)A ds = hA0 +

h2 1 h3 2 A + A + ..., 2 6

W1 (s) e(h−s)A (1 − √ ) ! dW1 (s) h

∗ ∗ I(11) I(110) ∗ = (I1∗ − √ )A0 + (I(10) − √ )A1 + . . . , h h  h ∗ ∗ I(11) I(110) W1 (s) = e(h−s)A √ ! dW1 (s) = √ A0 + √ A1 + . . . . h h h 0 (0)

We also have (with colors as in Example 1) z2 = 0, Φ1 ( ) = Φ2 ( ) = Φ1 ( ) = √ Φ2 ( ) = Φ1 ( ) = Φ1 ( ) = Φ2 ( ) = 0 and Φ2 ( ) = h, resulting in the weight functions given in Table 1. While the weight functions for the exact solution and the numerical approximation of the order 1.5 trees do not coincide, their expectation values coincide in case of Itô integrals but not for Stratonovich integrals (when τ = ). Thus, by Theorem 7 the above method has mean square order 1 in the Itô case but only 0.5 in the Stratonovich case.

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Table 1 Trees, corresponding order and weight functions for Example 11

τ ρ(τ ) ϕ(τ )(h) ∗ 0.5 I(1) 1 h h ∗ I(11) ∗ 1.5 I(10)

∗ ∗ hI(1) − I(01) ∗ I(01) ∗ I(01) h 2 W1 (s) dW1 (s) 0 ∗ I(111)

Φ(τ )(h) (1,0) (1,0) ∗ z1 + z2 = I(1) (0,0) (0,0) z1 + z2 =h h (1,0) (1,0) ∗ z1 Φ1 ( ) + z2 Φ2 ( ) = I(11) (0,0)

z1

(0,0)

Φ2 ( ) = 0 (1,1) (1,1) ∗ z1 Φ1 (∅) + z2 Φ2 (∅) = I(10) (1,0) (1,0) z1 Φ1 ( ) + z2 Φ2 ( ) = 0 (1,0) (1,0) z1 Φ1 ( ) + z2 Φ2 ( ) = 0 √ ∗ (1,0) (1,0) z1 Φ21 ( ) + z2 Φ22 ( ) = hI(11) (1,0)

z1

Φ1 ( ) + z2

(1,0)

Φ1 ( ) + z2

Φ2 ( ) = 0

References 1. S. Becker, A. Jentzen, P.E. Kloeden, An exponential Wagner–Platen type scheme for SPDEs. SIAM J. Numer. Anal. 54(4), 2389–2425 (2016) 2. H. Berland, B. Owren, B. Skaflestad, B-series and order conditions for exponential integrators. SIAM J. Numer. Anal. 43(4), 1715–1727 (2005) 3. K. Burrage, P.M. Burrage, Order conditions of stochastic Runge–Kutta methods by B-series. SIAM J. Numer. Anal. 38(5), 1626–1646 (2000) 4. J.C. Butcher, Coefficients for the study of Runge-Kutta integration processes. J. Aust. Math. Soc. 3, 185–201 (1963) 5. D. Cohen, S. Larsson, M. Sigg, A trigonometric method for the linear stochastic wave equation. SIAM J. Numer. Anal. 51(1), 204–222 (2013) 6. K. Debrabant, A. Kværnø, B-series analysis of stochastic Runge–Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J. Numer. Anal. 47(1), 181–203 (2008) 7. K. Debrabant, A. Kværnø, Stochastic Taylor expansions: weight functions of B-series expressed as multiple integrals. Stoch. Anal. Appl. 28(2), 293–302 (2010) 8. M. Hochbruck, A. Ostermann, Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43(3), 1069–1090 (2005) 9. G.N. Milstein, Numerical Integration of Stochastic Differential Equations. Mathematics and Its Applications, vol. 313 (Kluwer Academic Publishers Group, Dordrecht, 1995). Translated and revised from the 1988 Russian original 10. A. Tambue, J.M.T. Ngnotchouye, Weak convergence for a stochastic exponential integrator and finite element discretization of stochastic partial differential equation with multiplicative & additive noise. Appl. Numer. Math. 108, 57–86 (2016)

What Is a Post-Lie Algebra and Why Is It Useful in Geometric Integration Charles Curry, Kurusch Ebrahimi-Fard, and Hans Munthe-Kaas

Abstract We explain the notion of a post-Lie algebra and outline its role in the theory of Lie group integrators.

1 Introduction In recent years classical numerical integration methods have been extended beyond applications in Euclidean space onto manifolds. In particular, the theory of Lie group methods [15] has been developed rapidly. In this respect Butcher’s B-series [14] have been generalized to Lie–Butcher series [19, 20]. Brouder’s work [2] initiated the unfolding of rich algebro-geometric aspects of the former, where Hopf and pre-Lie algebras on non-planar rooted trees play a central role [5, 18]. Lie– Butcher series underwent similar developments replacing non-planar trees by planar ones [16, 21]. Correspondingly, pre-Lie algebras are to B-series what post-Lie algebras are to Lie-Butcher series [9, 12]. In this note we explore the notion of a post-Lie algebra and outline its importance to integration methods.

C. Curry () · K. Ebrahimi-Fard Norwegian University of Science and Technology (NTNU), Institutt for matematiske fag, Trondheim, Norway e-mail: [email protected]; [email protected] H. Munthe-Kaas University of Bergen, Department of Mathematics, Bergen, Norway e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_38

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2 Post-Lie Algebra and Examples We begin by giving the definition of a post-Lie algebra followed by a proposition describing the central result. The three subsequent examples illustrate the value of such algebras, in particular to Lie group integration methods. Definition 1 A post-Lie algebra (g, [·, ·], 7) consists of a Lie algebra (g, [·, ·]) and a binary product 7 : g ⊗ g → g such that, for all elements x, y, z ∈ g the following relations hold x 7 [y, z] = [x 7 y, z] + [y, x 7 z],

(1)

[x, y] 7 z = a7 (x, y, z) − a7 (y, x, z),

(2)

where the associator a7 (x, y, z) := x 7 (y 7 z) − (x 7 y) 7 z. Post-Lie algebras first appear in the work of Vallette [22] and were independently described in [21]. Comparing these references the reader will quickly see how different they are in terms of aim and scope, which hints at the broad mathematical importance of this structure. Proposition 2 Let (g, [·, ·], 7) be a post-Lie algebra. For x, y ∈ g the bracket x, y := x 7 y − y 7 x + [x, y]

(3)

satisfies the Jacobi identity. The resulting Lie algebra is denoted (g, ·, ·). Corollary 3 A post-Lie algebra with an abelian Lie algebra (g, [·, ·] = 0, 7) reduces to a left pre-Lie algebra, i.e., for all elements x, y, z ∈ g we have a7 (x, y, z) = a7 (y, x, z).

(4)

Example 4 Let X (M) be the space of vector fields on a manifold M, equipped with a linear connection. The covariant derivative ∇X Y of Y in the direction of X defines an R-linear, non-associative binary product X 7 Y on X (M). The torsion T , a skewsymmetric tensor field of type (1, 2), is defined by T (X, Y ) := X 7 Y − Y 7 X − X, Y ,

(5)

where the bracket ·, · on the right is the Jacobi bracket of vector fields. The torsion admits a covariant differential ∇T , a tensor field of type (1, 3). Recall the definition of the curvature tensor R, a tensor field of type (1, 3) given by R(X, Y )Z = X 7 (Y 7 Z) − Y 7 (X 7 Z) − X, Y  7 Z. In the case that the connection is flat and has constant torsion, i.e., R = 0 = ∇T , we have that (X (M), −T (·, ·), 7) defines a post-Lie algebra. Indeed, the first Bianchi

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identity shows that −T (·, ·) obeys the Jacobi identity; as T is skew-symmetric it therefore defines a Lie bracket. Moreover, flatness is equivalent to (2) as can be seen by inserting (5) into the statement R = 0, whilst (1) follows from the definition of the covariant differential of T : 0 = ∇T (Y, Z; X) = X 7 T (Y, Z) − T (Y, X 7 Z) − T (X 7 Y, Z). The formalism of post-Lie algebras assists greatly in understanding the interplay between covariant derivatives and integral curves of vector fields, which is central to the study of numerical analysis on manifolds. Example 5 We now consider planar rooted trees with left grafting. Recall that a rooted tree is made out of vertices and non-intersecting oriented edges, such that all but one vertex have exactly one outgoing line and an arbitrary number of incoming lines. The root is the only vertex with no outgoing line and is drawn on bottom of the tree, whereas the leaves are the only vertices without any incoming lines. A planar rooted tree is a rooted tree with an embedding in the plane, that is, the order of the branches is fixed. We denote the set of planar rooted trees by OT. OT =



 , , ,

, ,

,

,

,... .

,

The left grafting of two trees τ1 7 τ2 is the sum of all trees resulting from attaching the root of τ1 via a new edge successively to all the nodes of the tree τ2 from the left. 7

=

+

+

.

(6)

Left grafting means that the tree τ1 , when grafted to a vertex of τ2 becomes the leftmost branch of this vertex. We consider now the free Lie algebra L(OT) generated by planar rooted trees. In [16] is was shown that L(OT) together with left grafting defines a post-Lie algebra. In fact, it is the free post-Lie algebra PostLie( ) on one generator [16]. Ignoring planarity, that is, considering non-planar rooted trees, turns left grafting into grafting, which is a pre-Lie product on rooted tree satisfying (4) [17]. The space spanned by non-planar rooted trees together with grafting defines the free pre-Lie algebra PreLie( ) on one generator [3]. Example 6 Another rather different example of a post-Lie algebra comes from projections on the algebra Mn (K) of n × n matrices with entries in the base field K. More precisely, we consider linear projections involved in classical matrix factorization schemes, such as LU , QR and Cholesky [6, 7]. Let π+∗ be such a projection on Mn (K), where ∗ = LU , QR, Ch. It turns out that both π+∗ and π−∗ := id − π+∗ satisfy the Lie algebra identity [π±∗ M, π±∗ N] + π±∗ [M, N] = π±∗ ([π±∗ M, N] + [M, π±∗ N]),

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for all M, N ∈ Mn (K). In [1] it was shown that M 7 N := −[π−∗ M, N] defines a post-Lie algebra with respect to the Lie algebra defined on Mn (K). Corollary 3 is more subtle in this context as it reflects upon the difference between classical and modified classical Yang–Baxter equation [7, 8, 11].

3 Post-Lie Algebras and Lie Group Integration We now consider post-Lie algebras as they appear in numerical Lie group integration. Recall the standard formulation of Lie group integrators [15], where differential equations on a homogeneous space M are formulated using a left action · : G × M → M of a Lie group G of isometries on M, with Lie algebra g. An infinitesimal action · : g × M → T M arises from differentiation, B ∂ BB V · p := exp(tV ) · p. ∂t Bt =0 In this setting any ordinary differential equation on M can be written as y  (t) = f (y(t)) · y(t),

(7)

where f : M → g. For instance ODEs on the 2-sphere S 2 ( SO(3)/SO(2) can be expressed using the infinitesimal action of so(3). Embedding S 2 ⊂ R3 realizes the action and infinitesimal action as matrix-vector multiplications, where SO(3) is the space of orthogonal matrices, and so(3) the skew-symmetric matrices. To obtain a description of the solution of (7), we begin by giving a post-Lie algebra structure to gM , the set of (smooth) functions from M to g. For f, g ∈ gM , we let [f, g](p) := [f (p), g(p)]g and B ∂ BB g(exp(tf (p)) · p). (f 7 g)(p) := ∂t Bt =0 The flow map of (7) admits a Lie series expansion, where the terms are differential operators of arbitrary order, which live in the enveloping algebra of the Lie algebra generated by the infinitesimal action of f . Recall that for a Lie algebra (g, [·, ·]), the enveloping algebra is an associative algebra (U (g), ·) such that g ⊂ U (g) and [x, y] = x · y − y · x in U (g). As a Lie algebra g with product 7, the enveloping algebra of (g, [·, ·], 7) is U (g) together with an extension of 7 onto U (g) defined such that for all x ∈ g and y, z ∈ U (g) x 7 (y · z) = (x 7 y) · z + y · (x 7 z) (x · y) 7 z = a7 (x, y, z).

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Many of these operations are readily computable in practice. Recall that g has a second Lie algebra structure g¯ associated to the bracket ·, ·, reflecting the Jacobi bracket of the vector fields on M generated by the infinitesimal action of g. As  a vector space, its enveloping algebra U (¯g), ∗ is isomorphic to U (g). The Lie series solution of (7) is essentially the exponential in U (¯g), which in contrast to the operations of U (g) is in general difficult to compute. We are lead to the following: Basic Aim The fundamental problem of numerical  Lie group integration is the ∗ in U (¯ approximation of the exponential exp g ), ∗ in terms of the operations of   U (g), ·, 7 , where (g, [·, ·], 7) is the free Post-Lie algebra over a single generator. Remark 7 One may wonder why we use post-Lie algebras, which require flatness and constant torsion, and not structures corresponding to constant curvature and zero torsion such as the Levi-Civita connection on a Riemannian symmetric space. The key is that the extension of 7 onto U (g) allows for a nice algebraic representation of parallel transport, requiring flatness of the connection 7. Indeed, the basic assumption is that 7 extends to the enveloping algebra such that x 7 (y 7 z) = (x ∗ y) 7 z. From this follows x, y 7 z = (x ∗ y − y ∗ x) 7 z = x 7 (y 7 z) − y 7 (x 7 z), and hence R(x, y, z) = 0. For any connection 7, the corresponding parallel transport of g is g + tf 7 g +

t3 t2 f 7 (f 7 g) + f 7 (f 7 (f 7 g)) + · · · . 2 3!

If the basic assumption above holds, this reduces to the formula exp∗ (tf ) 7 g. Recall that the free Post-Lie algebra over a single generator is the post-Lie algebra of planar rooted trees postLie({ }) given in Example 5. Freeness essentially means that it is a universal model for any post-Lie algebra generated by a single element, and in particular the post-Lie algebra generated by the infinitesimal action of a function f ∈ gM on M. For instance, if we decide that represents the element f ∈ gM , then there is a unique post-Lie morphism F : postLie({ }) → gM such that F ( ) = f . Moreover, we then have, e.g., F ( ) = F ( 7 ) = F ( ) 7 F ( ) = f 7 f, or F ([ , ]) = [f, f 7 f ], and so on. This F is called the elementary differential map, associating planar rooted trees and commutators of these with vector fields on M. Hence, all concrete computations in gM involving the operations 7 and [·, ·] can be lifted to symbolic computations in the free post-Lie algebra postLie({ }).

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Revisiting our basic aim, we require a description of U (postLie({ })), which is given as the linear combination of all ordered forests (OF) over the alphabet of planar rooted trees, including the empty forest I,   OF = I, , , , , , , , , ,..., , ,... . So an element a ∈ U (postLie({ })) could be, for instance, of the following form a = 3I + 4.5 − 2

+3

+6

+7

−2

··· .

To be more precise, U (postLie({ })) consists of all finite linear combinations of this  kind, while infinite combinations such as the exponential live in U postLie({ }) and are obtained  by an inverse limit construction [12]. Elements in the space U postLie({ }) we call Lie–Butcher (LB) series. Note that all computations on such infinite series are done by evaluating the series on something finite in U (postLie({ })). Indeed, we consider U := U postLie({ }) as the (linear) dual   space of U := U postLie({ }) , with a bilinear pairing ·, · : U × U → R defined such that OF is an orthonormal basis, i.e. for ω, ω ∈ OF we have ω, ω  = 1 if ω = ω , and zero if ω = ω . Two important subclasses of LB-series are   gLB := α ∈ U (postLie({ }) : α, I = 0, α, ω ω  = 0 ∀ω, ω ∈ OF\{I}   GLB := α ∈ U (postLie({ }) : α, ω ω  = α, ωα, ω  ∀ω, ω ∈ OF ,







where


denotes the usual shuffle product of words, e.g., a
I = I
a = a, ab
cd = a(b
cd) + c(ab
d).

Here elements in gLB are called infinitesimal characters, representing vector fields on M and elements in GLB are characters, representing flows (diffeomorphisms) on M. GLB forms a group under composition called the Lie–Butcher group. A natural question is how does an element γ ∈ GLB represents a flow on M? The elementary differential map sends γ to the (formal1) differential operator, i.e., F (γ ) =

γ , ωF (ω) ∈ U (g)M .

ω∈OF

The flow map Ψγ : M → M is such that the differential operator F (γ ) computes the Taylor expansion of a function φ ∈ C ∞ (M, R) along the flow Ψγ : F (γ )[φ] = φ ◦ Ψγ . 1 Neglecting

convergence of infinite series at this point.

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Recall from Proposition 2 that any post-Lie algebra comes with two Lie algebras g and g. Hence there are two enveloping algebras U (g) and U (g), with two different associative products. It turns out that U (g) and U (g) are isomorphic as Hopf algebras [8, 9], such that the product of the latter can be represented in U (g). For LBseries, the resulting two associative products in U (g) are called the concatenation product and Grossman–Larson (GL) product [13]. Indeed, we have ω · ω = ωω (sticking words together). The GL product is somewhat more involved, i.e., for α, β ∈ gLB we have α ∗ β = α · β + α 7 β, see [9] for the general formula. Interpreted as operations on vector fields on M, the GL product represents the standard composition (Lie product) of vector fields as differential operators, while the concatenation represents frozen composition, for α, β : M → U (g) we have α · β(p) = α(p) · β(p). The two associative products on U (g) yield two exponential mappings exp· , exp∗ between gLB and GLB obtained from these, 1 1 1 1 exp· (α) = I+α+ α·α+ α·α·α+· · · , exp∗ (α) = I+α+ α∗α+ α∗α∗α+· · · . 2 6 2 6 Both send vector fields on M to flows on M. However, it turns out that the Grossman–Larson exponential exp∗ sends a vector field to its exact solution flow, while the concatenation exponential exp· sends a vector field to the exponential Euler flow, y1 = exp(hf (y0 )) · y0 .

(8)

All the basic Lie group integration methods can be formulated and analysed directly   in the space of LB-series U postLie({ }) with its two associative products and the lifted post-Lie operation. The Lie-Euler method which moves in successive timesteps along the exponential Euler flow is one such example. A slightly more intricate example is Example 8 (Lie Midpoint Integrator) On the manifold M a step of the Lie midpoint rule with time step h for (7), is given as K = hf (exp(K/2) · y0 ) y1 = exp(K) · y0 .   In U postLie({ }) , the same integrator → Φ : gLB → GLB is given as: K = exp· (K/2) 7 (h ) ∈ gLB Φ = exp· (K) ∈ GLB .

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We conclude by commenting that the two exponentials are related exactly by a map χ : g → g, called the post-Lie Magnus expansion [8, 10, 11], such that exp· (f ) = exp∗ (χ(f )), f ∈ g. The series χ(f) corresponds  to the backward error analysis related to the Lie–Euler method. In U postLie({ }) we find χ( ) = −

1 1 1 1 + [ , ]+ + 2 12 3 12

−

1 12

+ ···

This should be compared with the expansion β on page 184 in [16]. Chapoton and Patras studied the equality between these exponentials in the context of the free pre-Lie algebra [4]. Acknowledgements The research on this paper was partially supported by the Norwegian Research Council (project 231632).

References 1. C. Bai, L. Guo, X. Ni, Nonabelian generalized Lax pairs, the classical Yang–Baxter equation and PostLie algebras. Commun. Math. Phys. 297(2), 553–596 (2010) 2. Ch. Brouder, Runge-Kutta methods and renormalization. Eur. Phys. J. C12, 512–534 (2000) 3. F. Chapoton, M. Livernet,Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. 2001, 395–408 (2001) 4. F. Chapoton, F. Patras, Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula. Int. J. Algebra Comput. 23(4), 853–861 (2013) 5. Ph. Chartier, E. Hairer, G. Vilmart, Algebraic structures of B-series. Found. Comput. Math. 10, 407–427 (2010) 6. M.T. Chu, L.K. Norris, Isospectral flows and abstract matrix factorizations. SIAM J. Numer. Anal. 25, 1383–1391 (1988) 7. P. Deift, L.C. Li, C. Tomei, Matrix factorizations and integrable systems. Commun. Pure Appl. Math. 42(4), 443–521 (1989) 8. K. Ebrahimi-Fard, I. Mencattini, Post-Lie algebras, factorization theorems and Isospectral flows, arXiv:1711.02694 9. K. Ebrahimi-Fard, A. Lundervold, H.Z. Munthe-Kaas, On the Lie enveloping algebra of a postLie algebra. J. Lie Theory 25(4), 1139–1165 (2015) 10. K. Ebrahimi-Fard, A. Lundervold, I. Mencattini, H.Z. Munthe-Kaas, Post-Lie algebras and Isospectral flows. SIGMA 25(11), 093 (2015) 11. K. Ebrahimi-Fard, I. Mencattini, H.Z. Munthe-Kaas,Post-Lie algebras and factorization theorems. J. Geom. Phys. 119, 19–33 (2017) 12. G. Fløystad, H. Munthe-Kaas, Pre- and Post-Lie Algebras: The Algebro-Geometric View. Abel Symposium Series (Springer, Berlin, 2018) 13. R. Grossman, R. Larson, Hopf algebraic structures of families of trees. J. Algebra 26, 184–210 (1989) 14. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31 (Springer, Berlin, 2002) 15. A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett, A. Zanna, Lie-group methods. Acta Numer. 9, 215–365 (2000)

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16. A. Lundervold, H.Z. Munthe-Kaas, On post-Lie algebras, Lie–Butcher series and moving frames. Found. Comput. Math. 13(4), 583–613 (2013) 17. D. Manchon, A short survey on pre-lie algebras, in Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory, ed. by A. Carey. E. Schrödinger Institut Lectures in Mathematics and Physics (European Mathematical Society, Helsinki, 2011) 18. R. McLachlan, K. Modin, H. Munthe-Kaas, O. Verdier, Butcher series: a story of rooted trees and numerical methods for evolution equations. Asia Pac. Math. Newsl. 7, 1–11 (2017) 19. H. Munthe-Kaas, Lie–Butcher theory for Runge–Kutta methods. BIT Numer. Math. 35, 572– 587 (1995) 20. H. Munthe-Kaas, Runge–Kutta methods on Lie groups. BIT Numer. Math. 38(1), 92–111 (1998) 21. H. Munthe-Kaas, W. Wright, On the Hopf Algebraic Structure of Lie Group Integrators. Found. Comput. Math. 8(2), 227–257 (2008) 22. B. Vallette, Homology of generalized partition posets. J. Pure Appl. Algebra 208(2), 699–725 (2007)

On Non-commutative Stochastic Exponentials Charles Curry, Kurusch Ebrahimi-Fard, and Frédéric Patras

Abstract Using non-commutative shuffle algebra, we outline how the Magnus expansion allows to define explicit stochastic exponentials for matrix-valued continuous semimartingales and Stratonovich integrals.

1 Introduction Non-commutative stochastic exponentials occur in a large variety of forms and applications. Conceptually the simplest examples arise from linear matrix-valued semimartingale stochastic differential equations (SDEs). See for instance [11, 12, 15]. These are a fundamental object in the study of flows of SDEs, as the derivative of a stochastic flow with respect to the initial condition obeys such an equation. Explicit solutions of linear SDEs in Rn can also be derived using matrix-valued stochastic exponentials. The reader is referred to Protter’s book [20] for details and background. Magnus’ classical expansion [17] defines a Lie series which permits the expression of the solution of a linear non-autonomous matrix-valued ordinary differential equation in terms of the usual matrix exponential. See references [5, 19, 21] for more details and applications. This result has an abstract formulation in terms of non-commutative shuffle algebras and can be generalised to pre-Lie algebras [8]. We employ the aforementioned abstract approach to Magnus’ result in the context of Stratonovich integration to explicit stochastic exponentials for matrixvalued semimartingales in terms of the matrix exponential.

C. Curry · K. Ebrahimi-Fard () Norwegian University of Science and Technology (NTNU), Institutt for matematiske fag, Trondheim, Norway e-mail: [email protected]; [email protected] F. Patras Univ. Côte d’Azur, CNRS, UMR 7351, Nice Cedex, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_39

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2 Stochastic Exponentials Let xt be a semimartingale on a filtered probability space (Ω, F , (Ft )t ≥0 , P ), assumed to satisfy the usual hypotheses, and presume that x0 = 0. Following Protter [20] the classical (Doléans–Dade) stochastic exponential yt = E(x)t exists and solves the linear stochastic integral equation 

t

yt = 1 +

(1)

ys− dxs , 0

corresponding to the linear SDE dyt = yt dxt with initial value y0 = 1. It is  (n−1)  known [14] that E(x) = 1 + n>0 x (n) , where x (0) := 1 and x (n) := x− dx are iterated stochastic integrals. Doléans–Dade’s stochastic exponential admits the explicit representation [20] .     1 1 yt = exp xt − [x, x]t 1 + Δxs exp − Δxs + Δxs2 . 2 2 s≤t

(2)

In the sequel we will elaborate on algebraic structures related to Stratonovich integration that allow the generalization of (2) to matrix-valued continuous semimartingales X. The more complicated case of general non-commutative semimartingales and Itô integration will be addressed in future work. Non-commutativity arises naturally from the consideration of matrix-valued stochastic integrals. The simplest such instance comprises so-called right- and left stochastic integrals of Itô type [20]  (H 8 X)t :=



t

Hs− dXs ,

t

(X ≺ H )t :=

0

dXs Hs−

(3)

0

where Hs and Xs are a n × n-matrices of semimartingales, and Hs− = lim Ht . t 0, depending only on the regularity of M but not on h, u − uh h ⎡

B B⎤  B B Bah (u, wh ) − f (x)wh (x)dx BB ⎥ ⎢ B ⎥ ⎢ Ω ≤ C ⎢ inf u − vh h + sup ⎥. ⎦ ⎣vh ∈Vh w  h h wh ∈Vh \{0}

(8)

2.3 Two-Point Flux Approximation Finite Volumes on Cartesian Meshes A second example of non-conforming scheme is given by the “two-point flux approximation” (TPFA) finite volume scheme [5]. The TPFA scheme is widely used in petroleum engineering: constant values are considered in control volumes over which a discrete mass balance of the various components is established. Let M be a rectangular mesh of a rectangle Ω ⊂ R2 . In addition to the notations K, σ and x σ defined in Sect. 2.2, we introduce the following (see Fig. 2): – for each K ∈ M, x K is the intersection of the bisectors of the edges of K (since K is a rectangle, x K is also the centre of mass of K) and FK is the set of edges of K, – V is the set of vertices of the mesh and, for K ∈ M, VK is the set of vertices of K, – for each K ∈ M and each s ∈ VK , VK,s is the rectangle defined by x σ , s, x σ  and x K , where σ and σ  are the edges of K touching s, – uK (resp. uσ ) represents an approximate value of the unknown u at x K (resp. x σ ). Fig. 2 Notation for a rectangular mesh

σ

uσ

s

L uK

VK,s xK

K

σ uσ xσ

nK,σ

xL

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The idea of finite volume schemes consists in finding approximate values FK,σ  of the exact fluxes − σ ∇u · nK,σ dγ (x) (nK,σ is the normal to σ outward K), and in writing the following discrete flux balance in each cell ∀K ∈ M ,

 FK,σ =

f (x)dx,

(9)

K

σ ∈FK

and the flux conservativity across each interior edge: ∀σ ∈ Fint common face of K and L , FK,σ + FL,σ = 0.

(10)

Relation (9) simply mimicks the Stokes formula applied to the continuous problem (1): 

 ∇u · nK,σ dγ (x) = f (x)dx. − σ ∈FK

σ

K

The TPFA finite volume scheme consists in substituting, in the previous equations, FK,σ = −|σ |

uσ − uK . dist(x σ , x K )

(11)

The boundary condition is imposed by setting uσ = 0 if σ ⊂ ∂Ω.

(12)

There is no clear way to see the TPFA scheme method as a non-conforming finite element method. However, it can be recast into a variational form. Consider a family ((vK )K∈M , (vσ )σ ∈F ) such that vσ = 0 if σ ⊂ ∂Ω. Multiplying (9) by vK and summing over K ∈ M yields

K∈M σ ∈FK



|σ | (vσ − vK )(uσ − uK ) = vK f (x)dx. dist(x σ , x K ) K

(13)

K∈M

The study of the TPFA scheme was performed in [5] using finite volume techniques, and the following results were obtained: if the size of the mesh tends to 0 while the ratio height/width of each cell remains bounded by above and below, then an error estimate holds, which depends on the regularity of u. We present in the next section a way of merging this scheme with conforming or non-conforming finite elements and many other methods, and to extend the second Strang lemma to the approximate solution.

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3 The Gradient Discretisation Method We introduce a finite dimensional space XD,0 (“D” for “discretisation”, and the 0 to indicate that, in some way, this space accounts for the homogeneous boundary condition in (1)), as well as two linear operators ΠD and ∇D , which respectively reconstruct, from the discrete unknowns, a function on Ω and its “gradient”, such that ΠD : XD,0 → L2 (Ω)

and

∇D : XD,0 → L2 (Ω)d .

It is assumed that ∇D · L2 (Ω)d is a norm on XD,0 (which is natural in this case where we consider homogeneous boundary conditions). Using the objects (XD,0 , ΠD , ∇D ), it is now possible to design a unified convergence analysis for a number of conforming and non conforming methods, and to extend the second Strang lemma for this framework. Let us give the choice of (XD,0 , ΠD , ∇D ) for the three classical methods that were mentioned in the previous section. 1. For conforming P1 finite elements, – the set XD,0 consists in vectors vD of values at the vertices of the mesh, with zero at vertices on ∂Ω, – the reconstructed function ΠD vD ∈ C(Ω) is the piecewise linear function on the mesh that takes these values at the vertices, – the approximate gradient is simply defined by ∇D vD = ∇(ΠD vD ). 2. For non-conforming P1 elements, – the set XD,0 consists in vectors vD of values at the centres of mass of the edges, with zero for edges on ∂Ω, – the reconstructed function ΠD vD is the piecewise linear function on the mesh which takes these values at these centres of mass, – the approximate gradient is ∇D vD = ∇M (ΠD vD ), that is the broken gradient defined in (6). 3. For the TPFA scheme, – the set XD,0 is the space of real families vD = ((vK )K∈M , (vσ )σ ∈F ) satisfying the boundary conditions (12), – the reconstructed function ΠD vD is the piecewise constant function equal to vK on the cell K, – the approximate gradient ∇D vD is the piecewise constant function equal to ∇K,s vD on VK,s , for any cell K and any vertex s ∈ VK , where, if σ and vσ −vK σ  are the edges of K sharing the vertex s, ∇K,s vD = dist(x nK,σ + σ ,x K ) vσ  −vK dist(x σ  ,x K )

nK,σ  .

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Using (XD,0 , ΠD , ∇D ) as defined in the three examples, schemes (2), (7) and (13) presented in the previous section can be rewritten as Find uD ∈ XD,0 such that, for all  vD ∈ XD,0 ,  ∇D uD (x) · ∇D vD (x)dx = f (x)ΠD vD (x)dx. Ω

(14)

Ω

Except in the conforming case, the discrete operators ΠD , ∇D are not expected to satisfy an exact discrete Stokes formula, only an approximate one. We measure the resulting defect of conformity of the method by the quantity WD (ϕ) such that, for any sufficiently regular function ϕ : Ω → Rd ,

WD (ϕ) =

B B B B B (ϕ(x) · ∇D vD (x) + divϕ(x)ΠD vD (x))dx B B B Ω

sup

∇D vD L2 (Ω)d

vD ∈XD,0 \{0}

.

(15)

Define the “best interpolation error” by SD (u) :=

min

wD ∈XD,0

  ΠD wD − uL2 (Ω) + ∇D wD − ∇uL2 (Ω)d .

Define CD , the norm of the linear operator ΠD , by CD =

max

vD ∈XD,0 \{0}

ΠD vD L2 (Ω) /∇D vD L2 (Ω)d ,

which corresponds to the constant in a discrete Poincaré inequality. It is then possible to prove that the solution uD of (14) exists and is unique, and satisfies 1 (SD (u) + WD (∇u)) 2 ≤ ΠD uD − uL2 (Ω) + ∇D uD − ∇uL2 (Ω)d ≤ (CD + 2)(SD (u) + WD (∇u)),

(16)

where u is the unique solution of (1) (see [4, Theorem 2.29]). This result first includes the error estimate (3) of the conforming methods, since in this case WD (∇u) = 0 and the right hand side of (3) is controlled by SD (u). It also contains the error estimate (8): indeed, the second term of the right hand side of (8) is identical to WD (∇u) and the first term is controlled by SD (u) with the above definitions for these quantities. It finally provides an error estimate for the TPFA method, which extends the one previously established in [5, Theorem 9.4] in the case where u ∈ H 2 (Ω) ∩ H01 (Ω).

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If a sequence (XDm ,0 , ΠDm , ∇Dm )m∈N is selected such that (CDm )m∈N remains bounded, then (16) shows that, as m → ∞, ΠDm uDm → u in L2 (Ω) and ∇Dm uDm → ∇u in L2 (Ω)d if and only if SDm (u) → 0 and WDm (∇u) → 0. These properties are satisfied by the three classical methods detailed in this paper, and by all the methods listed in the introduction.

4 Conclusions The advantage of the unified framework presented here is that the common results apply to the numerous methods included in this framework. It also applies to non-linear problems, in which case the proofs rely on compactness and discrete functional analysis arguments rather than error estimates.

References 1. P. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978) 2. P. Ciarlet, The Finite Element Method, Part I, ed. by P.G. Ciarlet, J.-L. Lions. Handbook of Numerical Analysis, vol. III (North-Holland, Amsterdam, 1991) 3. D.A. Di Pietro, J. Droniou, G. Manzini, Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes. J. Comput. Phys. 355, 397–425 (2018). https://doi.org/10.1016/j.jcp.2017. 11.018 4. J. Droniou, R. Eymard, T. Gallouët, C. Guichard, R. Herbin, The Gradient Discretisation Method. Mathématiques et Applications (Springer, Berlin, 2017) 5. R. Eymard, T. Gallouët, R. Herbin, Finite volume methods, in Techniques of Scientific Computing, Part III, ed. by P.G. Ciarlet, J.-L. Lions. Handbook of Numerical Analysis, vol. VII (North-Holland, Amsterdam, 2000), pp. 713–1020 6. R. Eymard, C. Guichard, R. Herbin, Small-stencil 3d schemes for diffusive flows in porous media. M2AN 46, 265–290 (2012) 7. G. Strang, Variational crimes in the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Proceedings of a Symposium, University of Maryland, Baltimore, MD, 1972) (Academic, New York, 1972), pp. 689–710

Quasi-Optimal Nonconforming Methods for Second-Order Problems on Domains with Non-Lipschitz Boundary Andreas Veeser

and Pietro Zanotti

Abstract We introduce new nonconforming finite element methods for elliptic problems of second order. In contrast to previous work, we consider mixed boundary conditions and the domain does not have to lie on one side of its boundary. Each method is quasi-optimal in a piecewise energy norm, thanks to the discretization of the load functional with a moment-preserving smoothing operator.

1 Introduction We review and extend the construction of the quasi-optimal Crouzeix-Raviartlike (CR-like) and Discontinuous Galerkin (DG) methods proposed in [7, 8]. To introduce the setting and the issues addressed here, consider the problem − Δu = f

in Ω,

u = 0 on ΓD ,

∂n u = 0

on ΓN ,

(1)

on a polygonal domain Ω ⊆ R2 , where ΓD and ΓN form a partition of the boundary ∂Ω. Given a triangulation M of Ω, with edges F , let CR indicate the space of Crouzeix-Raviart functions [2], vanishing in the midpoints of the edges along ΓD . We approximate the weak solution u of (1) by  U ∈ CR

such that ∀σ ∈ CR Ω

∇M U · ∇M σ = f, Eσ 

(2)

A. Veeser · P. Zanotti () Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_41

461

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where ∇M denotes the piecewise gradient on M and E : CR → H 1 (Ω) is a linear operator preserving the integral mean on the edges not lying on ΓD . The abstract theory in [6, 7] then provides the quasi-optimality ∇M (u − U )L2 (Ω) ≤ EL(CR,H 1 (Ω)) infs∈CR ∇M (u − s)L2 (Ω) which distinguishes (2) from other discretizations of (1) with the space CR. We construct the operator E combining a bubble smoother with an averaging operator into the space S11 of continuous piecewise affine functions. If ∂Ω is not Lipschitz because Ω does not lie on one side of ∂Ω, the stability of the latter (and so that one of E) is a delicate issue. We attain this property through a careful definition of the edges and Lagrange nodes of M, ensuring that the nodal basis functions of S11 have face-connected support in the sense of [5]. Thus, our analysis applies under mild assumptions on Ω and covers, for instance, the limit case of a re-entrant corner, i.e. {(x1 , x2 ) ∈ R2 | |x1 |, |x2 | < 1 and (x2 = 0 or x1 < 0)}.

(3)

The rest of this article is organized as follows. In Sects. 2 and 3 we detail the technical preliminaries to Sect. 4, where we review the construction of DG and CRlike methods of arbitrary fixed order for the Poisson problem (1) in Rd as well as a first-order CR method for linear elasticity.

2 Face-Connected Meshes of Non-Lipschitz Domains Let C be a n-simplex in Rd , n ∈ {0, . . . , d}. We indicate by L1 (C) and FC the vertices and (n − 1)-dimensional faces of C, while hC and ρC stand for the diameter of C and the diameter of the largest n-dimensional ball in C. The set Pp (C), p ∈ N0 , consists of all polynomials on C with total degree ≤ p and is uniquely determined by the evaluation at the Lagrange nodes Lp (C). We consider simplicial, face-to-face meshes M of an open, bounded and connected set Ω4⊆ Rd . More precisely, M is a finite collection of d-simplices such that Ω = K∈M K, the intersection of K1 , K2 ∈ M is either empty or an n-simplex with n ≤ d and L1 (K1 ∩ K2 ) = L1 (K1 ) ∩ L1 (K2 ), and the intersection K ∩ ∂Ω, K ∈ M, is either empty or an n-simplex with n ≤ d − 1. The shape coefficient of M is γM := max{hK /ρK | K ∈ M} and we write C∗,γM for a generic function, not necessarily the same at each occurrence, increasing in γM and possibly depending on other parameters ∗. We use the subscript M to indicate the broken version of a differential operator, acting on some broken Sobolev space H k (M) := {v ∈ L2 (Ω) | ∀K ∈ M v|K ∈ H k (K)},

k ∈ {1, 2}.

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Our assumptions on M prescribe that Ω is polyhedron, but do not rule out domains like (3). Hence, it may happen that the intersection of two simplices K1 , K2 ∈ M is a common (d − 1)-simplex lying on ∂Ω. In this case, it is not convenient to consider K1 ∩ K2 a single face of M, as for non-boundary faces. We therefore define the faces of M as the quotient set = > F := (Γ, K) | K ∈ M, Γ ∈ FK / ∼ (Γ1 ,K1 ) ∼ (Γ2 , K2 )

:⇐⇒

with

Γ1 = Γ2 and Γ1  ∂Ω.

(4)

2 Notice that, given a face F = [(Γ,  K)] ∈ F and a function w ∈ L (Γ ), we can 2 write w ∈ L (F ) and set F w := Γ w. We say that F = [(Γ, K)] is a boundary face and write F ∈ F b whenever Γ ⊆ ∂Ω, which immediately yields #F = 1. Hence, we define a unit normal n and the average and jump of v ∈ H 1 (M) on F by

n|F := nK ,

{{v}}|F (x) := v|K (x),

v|F (x) := v|K (x),

x ∈ Γ,

(5)

where nK is the outer unit normal of K. In order to formulate mixed boundary b ∪ F b into conditions like (1), we assume that we are given a partition F b =: FD N Dirichlet and Neumann boundary faces. We define also the interior faces F i := F \ F b . Each class F ∈ F i consists of two pairs (Γ1 , K1 ) and (Γ2 , K2 ) with Γ1 = Γ2 = K1 ∩ K2 and we set n|F := nK1 ,

{{v}}|F (x) := (v|K1 + v|K2 )(x)/2,

v|F (x) := (v|K1 − v|K2 )(x)

for x ∈ K1 ∩ K2 and v ∈ H 1 (M). In what follows, the dependencies of normal and jump on the ordering of K1 and K2 will be insignificant. We thus have {{·}}|F ,  · |F : H 1 (M) → L2 (F ) for all F ∈ F . If we apply these operators componentwise to vector-valued fields, then {{∇v}}|F · nF and ∇v|F · nF are also in L2 (F ) and represent the average and jump of the normal derivative of v ∈ H 2 (M) on F . We will be interested in approximating functions from the Sobolev space b HD1 (Ω) := {v ∈ H 1 (Ω) | ∀F ∈ FD

v|F = 0}

by ‘finite’ functions from (subspaces of) the broken polynomial space Sp0 := {s ∈ H 1 (M) | ∀K ∈ M v|K ∈ Pp (K)},

p ∈ N0 .

The intersection of HD1 (Ω) and Sp0 coincides with the conforming space b Sp1 := {s ∈ Sp0 | ∀F ∈ F i ∪ FD

s|F = 0} = HD1 (Ω) ∩ Sp0 .

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It is useful to have a nodal basis of Sp1 , p ≥ 1, at hand. For this purpose, we introduce the Lagrange nodes of degree p of M, similarly to (4), by Lp := {(ν, K) | K ∈ M, ν ∈ Lp (K)}/ ≈ where (ν1 , K1 ) ≈ (ν2 , K2 ) if and only if ν1 = ν2 =: ν and there are (Kj )nj=0 ⊆ M such that K0 = K1 , Kn = K2 and we have ν ∈ Γj := Kj ∩ Kj +1 with [(Γj , Kj )] ∈ F i for all j = 0, . . . , n − 1.

(6)

With this definition, the evaluation s(z), z ∈ Lp , is well-defined for all s ∈ Sp1 , although Sp1 may fail to be a subspace of C 0 (Ω). Given z ∈ Lp and F ∈ F , we write z ∈ F if there are K ∈ M, Γ ∈ FK and ν ∈ Lp (K) ∩ Γ such that z = [(ν, K)] and F = [(Γ, K)]. Removing from Lp all nodes lying on Dirichlet faces, we define b Lp := {z ∈ Lp | ∀F ∈ FD

z∈ / F}

and the evaluation at Lp uniquely determines every function s ∈ Sp1 . p For all nodes z ∈ Lp , we construct Φz ∈ H 1 (Ω) ∩ Sp0 by setting p

p

Φz|K = Φν,K

if z = [(ν, K)]

and

p

Φz|K = 0

otherwise

p

where Φν,K is the nodal basis function of Pp (K) associated with the evaluation at p p ν. The set {Φz | z ∈ Lp } is a basis of Sp1 , because Φz (z ) = δzz for all z ∈ Lp . p Each function Φz is supported on the star ωz := {K ∈ M | z = [(ν, K)] for some ν ∈ Lp (K)} which is face-connected in view of (6), cf. [5].

3 Full Stability and Simplified Averaging Let V be a Hilbert space with scalar product a and denote by ·, · the pairing of the dual V  and V . Given a symmetric elliptic problem in the form given  ∈ V  , find u ∈ V such that ∀v ∈ V

a(u, v) = , v ,

(7)

we aim at approximating the solution u of (7) in a finite-dimensional space S with, possibly, S  V . For this purpose, let b be a nondegenerate bilinear form on S and

Quasi-Optimal Non-conforming Methods on Non-Lipschitz Domains

465

E : S → V be a linear operator. We define a nonconforming method M : V  → S via the following discrete problem for U := M: find U ∈ S such that ∀σ ∈ S

b(U, σ ) = , Eσ  .

(8)

We use the abbreviation M = (S, b, E), because this triplet determines M only up to some slight ambiguities, see [6, Remark 2.2]. Since E often maps possibly rough functions into more regular ones, we call it smoothing operator or, shortly, smoother. To assess the quality of the approximation of u by U , we assume √ that a extends to a scalar product 8 a on V + S, inducing the norm  ·  := 8 a (·, ·). Then, the nondegeneracy of b and dim(S) < +∞ ensure that E is bounded and there is a constant Cstab ≥ 0 such that ∀ ∈ V 

U  ≤ Cstab V  .

(9)

This full stability is a discrete counterpart of the identity u = V  resulting from (7) and our use of full refers to the fact that no regularity beyond  ∈ V  is involved. Full stability is necessary for quasi-optimality, cf. (13) below. Moreover, [6, Theorem 4.7] states that the use of a smoother E as in the right-hand side of (8) is necessary for (9) and relates the best value of the constant Cstab to an operator norm of E. In the remaining part of this section, we consider V = HD1 (Ω) and S ⊆ Sp0 with p ∈ N. This case covers the setting of the introduction and is closely related to all examples in the next section. A possible strategy to map functions from Sp0 into HD1 (Ω) is to approximate them by the so-called averaging operator. We define a simplified variant as follows: for all nodes z ∈ Lp , we fix a representative (νz , Kz ) of z, i.e. z = [(νz , Kz )], and set Ap σ :=



z∈Lp

p

(10)

σ|Kz (νz )Φz .

Ap is an admissible smoothing operator in the setting at hand, although its definition does not rely on the concrete form of problem (7). Lemma 1 below extends [8, Lemma 3.3] under the assumptions of Sect. 2 and clarifies in which sense Ap σ approximates σ . We are interested in this information because the simplified averaging is a building block of the nonconforming methods in the next section. Our proof exploits the face-connectedness of the stars ωz , z ∈ Lp , thus motivating (4) and (6). Lemma 1 (Approximation by Simplified Averaging) Fix σ ∈ Sp0 and z = [(ν, K)] ∈ Lp with p ∈ N. If ν ∈ / ∂K, then σ|K (ν) = Ap σ (ν), else we have |σ|K (ν) − Ap σ (ν)| ≤ Cd,p



b ,F 6z |F | F ∈F i ∪FD

− 12

σ L2 (F ) .

(11)

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Proof If ν ∈ / ∂K, we have Kz = K in (10), because K is the unique d-simplex in M containing ν. If, instead, ν ∈ ∂K, we claim that |σ|K (ν) − Ap σ (ν)| ≤



b ,F 6z |σ |F |(ν). F ∈F i ∪FD

(12)

Indeed, assume first that z ∈ Lp , so that Ap σ (ν) = σ|Kz (ν) for some Kz ∈ M with (ν, K) ≈ (ν, Kz ). Connect K to Kz through a path of simplices as in (6), with common faces (Fj )n−1 ⊆ F i . Since z ∈ Fj for all j = 0, . . . , n − 1, we j =0  infer |σ|K (ν) − σ|Kz (ν)| ≤ j =0,...,n−1 |σ |Fj (ν)| which yields (12). Next, if b such z ∈ / Lp , we have Ap σ (z) = 0 and there is a face Fz = [(Γz , Kz )] ∈ FD that z ∈ F . Connecting K to K as above and using (5), we derive |σ (ν)| ≤ z z |K  |σ  (ν)|+|σ  (ν)| which, again, provides (12). The final estimate j =0,...,n−1 |Fj |Fz then follows from (12) and an inverse inequality in Pp (F ).

4 Quasi-Optimality by Moment-Preserving Smoothing Let M = (S, b, E) be a nonconforming method for (7), with discrete problem (8). The best error in S, namely infs∈S u − s, provides a benchmark for the error u − U  of this method. Hence, we say that M is quasi-optimal in the norm  · , with quasi-optimality constant Cqopt ≥ 1, if we have ∀u ∈ V

u − U  ≤ Cqopt infs∈S u − s

(13)

and Cqopt is the best such constant. Quasi-optimality requires full stability (9) and the identity u = U whenever u ∈ V ∩ S, or equivalently, (14) below. Remarkably, this possibly mild consistency condition actually characterizes the quasi-optimality of fully stable methods, according to [6, Theorems 4.14 and 4.19]. Theorem 2 (Quasi-Optimality) A nonconforming method M = (S, b, E) is quasi-optimal in the norm  ·  induced by 8 a if and only if ∀u ∈ V ∩ S, σ ∈ S

b(u, σ ) = a(u, Eσ ).

(14)

In this case, denoting by Π the 8 a -orthogonal projection onto S, we have Cqopt = 2 1 + ρ , where ρ ≥ 0 is the smallest constant such that ∀u ∈ V , σ ∈ S

b(Πu, σ ) − a(u, Eσ ) ≤ ρu − Πu supsˆ∈S,ˆs =1 b(ˆs , σ ).

This theorem reduces to the classical Céa’s Lemma provided M is a conforming Galerkin method, i.e. S ⊆ V , b = a|S×S and E = IdS . The following examples

Quasi-Optimal Non-conforming Methods on Non-Lipschitz Domains

467

show that quasi-optimality can be achieved also in the genuine nonconforming case S  V. DG Methods for the Poisson Problem We first review a DG method from [8, Section 3.2] under the assumptions of Sect. 2. Inspired by Rivière et al. [4], we consider the following setting for the Poisson problem (1): V = HD1 (Ω),

8 aIP (v1 , v2 ) := (v1 , v2 )M + (v1 , ηh−1 v2 )Σ

S = Sp0 ,

bIP (s, σ ) := (s, σ )M − ({{∇s}} · n, σ )Σ + (s, {{∇σ }} · n + ηh−1 σ )Σ where p ∈ N, η > 0 is a parameter, h|F denotes the diameter of F , as well as  (v1 , v2 )M := Ω ∇M v1 · ∇M v2 and (w1 , w2 )Σ := b F ∈F i ∪FD F w1 w2 . We aIP and observe that 8 aIP|V ×V provides a denote by  · IP the norm induced by 8 weak formulation of (1). All functions are intended up to an additive constant in b = ∅. Only minor modifications are needed if b is the the pure Neumann case FD IP symmetric bilinear form in [1]. In view of [7, Remark 3.6], the operator Ap does not fulfill (14) in this setting, but we can achieve this property by employing also a moment-preserving smoother. To see this, we associate to any F ∈ F i ∪ FNb and K ∈ M the bubble functions < < 1 . Writing P ΦF := z∈L1 (F ) Φz1 ∈ Sd1 and ΦK := z∈L1 (K) Φz1 ∈ Sd+1 −1 = {0}, 2 2 we introduce the weighted L -projections QF : L (F ) → Pp−1 (F ) and QK : L2 (K) → Pp−2 (K) by 





(QF v)qΦF = F

vq

 (QK v)rΦK =

and

F

vr

K

K

for all q ∈ Pp−1 (F ) and r ∈ Pp−2 (K), respectively. Then, the bubble smoother Bp : H 1 (M) → HD1 (Ω) given by Bp σ := BpF σ + BpM (σ − BpF σ ) with BpF σ :=



F ∈F i ∪FNb

z∈Lp−1 (F )



 p−1 QF {{v}} (z)Φz ΦF ,

BpM σ :=

(QK v)ΦK

K∈M

and the convention Φz0 = 1 for p = 1 is such that ∀s, σ ∈ Sp0

(s, Bp σ )M = (s, σ )M − ({{∇s}} · n, σ )Σ .

(15)

Finally, since Bp typically generates an oscillatory function, we stabilize with Ap . We thus end up with the method MIP := (Sp0 , bIP , Ep ), where Ep σ := Ap σ + Bp (σ − Ap σ ).

(16)

Theorem 3 (Quasi-Optimality of MIP )2 For all η > 0, the method MIP is quasi-

optimal in the norm  · IP and Cqopt ≤

1 + Cd,p,γM η−1 .

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Proof The first part of Theorem 2 ensures that MIP is quasi-optimal, because the combination of (15) and (16) provides (14). Next, denoting by Π the 8 aIP -orthogonal projection onto Sp0 , we have, for u ∈ HD1 (Ω) and σ ∈ Sp0 , bIP (Πu, σ ) − (u, Ep σ )M = (Πu − u, Ep σ − σ )M + (Πu − u, {{∇σ }} · n)Σ . 1

We obtain the bound |(Πu − u, {{∇σ }} · n)Σ | ≤ Cd,p,γM η− 2 u − ΠuIP σ IP using a scaled trace inequality. Moreover, the expansion (σ − Ap σ )|K =  p ν∈Lp (K) (σ|K (ν) − Ap σ (ν))Φν,K , Lemma 1 and the scaling of the nodal basis  of Pp (K) yield ∇(σ − Ep σ )L2 (K) ≤ Cd,p,γM F ∩K=∅ h−1/2 σ L2 (F ) , b . Then, summing over all simplices of M, it follows where F varies in F i ∪ FD ∇M (σ − Ep σ )L2 (Ω) ≤ Cd,p,γM η−1/2 σ IP . Finally, we insert these inequalities in the identity above and use the coercivity of bIP . Thus, we derive that the constant ρ in Theorem 2 satisfies ρ ≤ Cd,p,γM η−1/2 and conclude using the formula Cqopt = 1 + ρ 2 . CR-Like Methods for the Poisson Problem Next, we review the methods in [7, Sections 3.2–3.3]. Introducing the space : ;  b CRp := s ∈ Sp0 | ∀F ∈ F i ∪ FD , q ∈ Pp−1 (F ) sq = 0 , F

with p ∈ N, we propose the following CR-like setting for (1), cf. [2]: V = HD1 (Ω),

S ⊆ CRp ,

8 aCR (v1 , v2 ) := (v1 , v2 )M ,

bCR := 8 aCR|S×S .

Since S ⊆ Sp0 , the method MCR := (S, bCR , Ep ) is well-defined. MCR is a conforming Galerkin method whenever S ⊆ Sp1 , since Ep|Sp1 = Ap|Sp1 = IdSp1 . More generally, the following counterpart of Theorem 3 holds. Theorem 4 (Quasi-Optimality of MCR ) For all S ⊆ CRp , the method MCR is quasi-optimal in the norm ∇M · L2 (Ω) and Cqopt ≤ Cd,p,γM . Proof Proceeding as in the proof of Theorem 3, one can see that MCR is quasioptimal and ρ ≤ Id − Ep L(CRp ,H 1 (M)) ≤ Cd,p,γM , cf. [7, Theorem 3.10]. A Crouzeix-Raviart Method for Linear Elasticity Finally, we review the method in [8, Section 3.3] for the linear elasticity problem − div(σ (u)) = f

in Ω,

u = 0 on ΓD ,

σ (u)n = 0

on ΓN ,

(17)

where σ (u) := 2με(u) + λtr(ε(u))I and we use standard notations, cf. [3]. For all η > 0, the restriction to HD1 (Ω)d of the bilinear form  8 aHL(v1 , v2 ) :=

η 2μεM (v1 ) : εM (v2 ) + λdivM (v1 )divM (v2 ) + (v1 , v2 )Σ h Ω

Quasi-Optimal Non-conforming Methods on Non-Lipschitz Domains

469

provides a weak formulation of (17). Moreover, inspired by Hansbo and Larson [3], we set V = HD1 (Ω)d ,

S = CRd1 ,

8 a := 8 aHL ,

bHL := 8 aHL|S×S

b = ∅, and denote by  · HL the norm induced by 8 aHL . In the pure traction case FD 0 functions are intended up to rigid motions. Since CR1 ⊆ S1 , we define the method MHL := (CRd1 , bHL , E1 ), with the convention that E1 σ indicates the application of the smoother E1 in (16) to each component of σ . The next result can be proven similarly to Theorem 3, cf. [8, Theorem 3.12].

Theorem 5 (Quasi-Optimality of MHL )2 For all η > 0, the method MIP is quasioptimal in the norm  · HL and Cqopt ≤

1 + Cd,γM (2μ + λ)η−1 .

Although this bound of the quasi-optimality constant of MHL is not uniform in the nearly-incompressible limit λ → +∞, some robustness properties of MHL follow by comparison with the method in [3] by P. Hansbo and Larson; see [8, Lemma 3.15] for further details. Acknowledgements The support by the GNCS, part of the Italian INdAM, is gratefully acknowledged.

References 1. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982) 2. P. Ciarlet Jr., C.F. Dunkl, S. Sauter, A family of Crouzeix-Raviart finite elements in 3D, arXiv:1703.03224v1 [math.NA] 3. P. Hansbo, M.G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. M2AN Math. Model. Numer. Anal. 37, 63–72 (2003) 4. B. Rivière, M.F. Wheeler, V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39, 902–931 (2001) 5. A. Veeser, Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. 16(3), 723–750 (2016) 6. A. Veeser, P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. I – Abstract theory, SIAM J. Numer. Anal. 56(3), 1621–1642 (2018). https://doi.org/10.1137/ 17M1116362 7. A. Veeser, P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. II – Overconsistency and classical nonconforming elements, arXiv:1710.03447 [math.NA] 8. A. Veeser, P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. III – DG and other interior penalty methods, SIAM J. Numer. Anal. (accepted for publication)

Convergence of Adaptive Finite Element Methods with Error-Dominated Oscillation Christian Kreuzer and Andreas Veeser

Abstract Recently, we devised an approach to a posteriori error analysis, which clarifies the role of oscillation and where oscillation is bounded in terms of the current approximation error. Basing upon this approach, we derive plain convergence of adaptive linear finite elements approximating the Poisson problem. The result covers arbritray H −1 -data and characterizes convergent marking strategies.

1 Introduction Adaptive finite element methods (AFEMs) are well-established and efficient tools for solving boundary value problems. A common form is the iteration of SOLVE

→

ESTIMATE

→

MARK

→

REFINE,

(1)

i.e. solve for the current finite element solution, estimate its error by means of socalled a posteriori indicators, and use this information in a marking strategy to make refinement decisions. The resultant problem-dependent adaptation of the mesh often significantly improves efficiency, compared to classical uniform refinement. There are theoretical results corroborating this practical observation; see the surveys [3, 9] and [5], which even obtains a nonlinear quasi-optimality result with respect to an augmented energy norm error. Almost all results about the convergence behavior of AFEMs invoke extra regularity for the data which does not fit to the proven convergence rate in the light

C. Kreuzer () Technische Universität Dortmund, Fakultät für Mathematik, Dortmund, Germany e-mail: [email protected] A. Veeser Università degli Studi di Milano, Dipartimento di Matematica F. Enriques, Milano, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_42

471

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of [2, 6]. To exemplify and explain this, let us consider the approximation of the weak solution of the Poisson problem −Δu = f in Ω ⊂ Rd ,

u = 0 on ∂Ω

(2)

by a realization of (1) with linear finite elements. Here one usually requires the regularity f ∈ L2 (Ω) beyond H −1 (Ω), irrespective of the actual regularity of the target function u. One reason for requiring f ∈ L2 (Ω) lies in a posteriori error analysis. Aiming at computability, local H −1 -norms are replaced with L2 -norms scaled by the local meshsize hT , leading to terms like the classical oscillation indicator hT minfT ∈R f − fT L2 (T ) , which needs f ∈ L2 (Ω) to be defined. Note however that such a term is not yet really computable and may cause serious overestimation; cf. [7]. Another reason lies in many convergence analyses themselves. They often rely on the fact that the scaling factors of the aforementioned L2 -norms strictly reduce under refinement and, e.g., in the case of oscillation terms, this strict reduction hinges on extra regularity of f . It is worth mentioning that the second reason does not apply to the derivations of plain convergence in [8, 10]. In [7] we present a new approach to a posteriori analysis covering arbitrary f ∈ H −1 (Ω) and prove in particular ∇(u − UT )Ω  ET + δT  ∇(u − UT )Ω , where UT is the linear finite element solution over the mesh T , the estimator ET is computable in a similar sense as UT , while the computation of the oscillation δT requires additional information on f . Here we prove plain convergence of adaptive linear finite elements for (2) by combining the a posteriori analysis [7] and the convergence analyses [8, 10]. The result forgoes extra regularity and so convergence rates are precluded. Section 2 presents the adaptive finite method in detail, along with an account of [7], and Sect. 3 is concerned with the convergence proof, thereby characterizing convergent marking strategies. Section 4 then concludes with a brief discussion of a few convergent marking strategies.

2 Adaptive Algorithm with Abstract Marking Strategy In this section we introduce adaptive finite element methods for approximating the weak solution u ∈ H01 (Ω) of (2) for some fixed but arbitrary load f ∈ H −1 (Ω).

(3)

Here H01 (Ω) is the subspace of Sobolev functions in H 1 (Ω) with vanishing trace and H −1 (Ω) denotes its topological dual space.

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Bisection and Conforming Linear Finite Elements Let T0 be a suitable simplicial, face-to-face (conforming) mesh of Ω and denote by T the family of its refinements using iterative or recursive bisection; cf. the discussion in [9] and the references therein. In what follows, ‘’ stands for ‘≤ C’, where the generic constant C may depend on the shape coefficient of T0 . For example, for any simplex arising in T, we have that its shape coefficient  1. Let T ∈ T be a mesh. We denote the set of its vertices by VT and the set of its faces by FT . The associated space of continuous piecewise affine functions is   ¯ | V|T ∈ P1 for all T ∈ T ⊂ H 1 (Ω). V(T ) := V ∈ C 0 (Ω) Its nodal basis {ΦT,z }z∈VT is given by ΦT,z ∈ V(T )

such that ΦT,z (y) = δzy for all z, y ∈ VT .

4 Note supp ΦT,z = {T ∈ T : T 6 z} =: ωT,z . The finite element functions ˚ V(T ) := {V ∈ V(T ) | ∀z ∈ VT ∩ ∂Ω V (z) = 0} ⊂ H01 (Ω) are conforming and the associated Galerkin approximation UT of (2) is characterized by  V(T ) UT ∈ ˚

such that ∀V ∈ ˚ V(T ) Ω

∇UT · ∇V dx = f, V ,

(4)

where ·, · is the duality pairing between H −1 (Ω) and H01 (Ω). In order to be able to compute UT , we suppose that ∀T ∈ T, z ∈ VT ∩ Ω

f, ΦT,z  is available.

(5)

A Posteriori Error Analysis with Error-Dominated Oscillation We outline the approach to a posteriori error analysis in [7], which prepares the ground for formulating our convergence theorem. To this end, we fix the mesh T and drop it as subscript in the following discussion. Further, given a subdomain ω ⊂ Ω, we choose ∇ · ω := ∇ · L2 (ω) as norm on H01 (ω) and denote by  · −1;ω its dual norm on H −1 (ω). We then have the global relationship ∇(u − U )Ω = f + ΔU −1;Ω between error and residual as well as, locally, ∀z ∈ V

f + ΔU −1;ωz ≤ ∇(u − U )ωz .

(6)

Exploiting Galerkin orthogonality and the fact that {Φz }z∈V provides a partition of unity, one derives that, for any v ∈ H01 (Ω) and V ∈ ˚ V(T ), f + ΔU, v = f + ΔU, v − V  

z∈V

f + ΔU −1,ωz ∇vωz ;

(7)

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see, e.g., [4, 7]. Applying the Cauchy-Schwarz inequality on the sum and recalling (6), one obtains the error localization ∇(u − U )2Ω 

z∈V

f + ΔU 2−1;ωz .

In general, upper bounds for the local residual norms f + ΔU −1;ωz , as for the global one f + Δu−1;Ω , cannot be computed by a finite number of operations from the information in (5). The reason lies in the possibly infinite-dimensional nature of f ; see [7]. One therefore may isolate this difficulty by inserting a finite-dimensional approximation of f , e.g., some piecewise constant function f¯. Unfortunately, the arising terms f − f¯−1;ωz , or the larger classical oscillation terms h(f − f¯)ωz , may overestimate the error by far on a given mesh, see [7], and even lead to worse convergence rates, see [4]. To overcome these drawbacks, we have developed in [7] a projection operator P : H −1 (Ω) → D(T ) with the following properties. The evaluation Pf is computable under (5) and the range



 = cT v dx + cF v ds D(T ) :=  ∈ H −1 (Ω) | , v = T ∈T

T

F ∈F

F

for all v ∈ H01 (Ω) with cT , cF ∈ R for T ∈ T , F ∈ F

>

contains ΔU ∈ H −1 (Ω). Moreover, P is locally stable and invariant in that, for any  ∈ H −1 (Ω), z ∈ V, and D ∈ D(T ), we have P−1;ωz  −1;ωz ,

(8a)

as well as  = P in H −1 (ωz )

whenever  = D in H −1 (ωz ).

(8b)

Consequently, we have P(ΔU ) = ΔU and the local splittings f + ΔU −1;ωz  Pf + ΔU −1;ωz + f − Pf −1;ωz ,

(9)

where Pf + ΔU −1;ωz is quantifiable under (5), but the computation or bounding of f − Pf −1;ωz requires information beyond. To sum up, we formulate the following theorem. Theorem 1 (A Posteriori Bounds) For any mesh T ∈ T, we have ∇(u − UT )2Ω 

z∈VT

PT f + ΔUT 2−1;ωT,z + f − PT f 2−1;ωT,z

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as well as, for any z ∈ VT , PT f + ΔUT 2−1;ωT,z + f − PT f 2−1;ωT,z  ∇(u − UT )2ωT,z . Adaptive Algorithm The bounds in Theorem 1 are vertex-wise, while bisection is applied element-wise. Therefore, we assume that there are element indicators ET (T ), T ∈ T , such that PT f + ΔUT 2−1;ωT,z 

ET (T )2

(10a)

T ⊂ωT,z

and

ET (T )2 

z∈T ∩VT

PT f + ΔUT 2−1;ωT,z .

(10b)

Such local equivalences hold for various indicators resembling the ones from the standard residual estimator, the hierarchical estimator, or estimators based upon discrete local problems; cf. [7]. In order to achieve a similar grouping for the oscillation, we introduce, for any D T ∈ T , its minimal ring ω(T ) := {ωT  (T ) : T  ∈ T, T  6 T }, where ωT (T ) := ∪{T  ∈ T : T  ∩ T = ∅}. Then δ(T ) := f − PT f −1;ω(T )

(11)

does not depend on the surrounding mesh, and we can derive the inequalities f − PT f 2−1;ωT,z 

T ⊂ωT,z

δ(T )2 ,

δ(T )2 

z∈T ∩VT

f − PT f 2−1;ωT,z . (12)

Algorithm (AFEM) Starting from T0 and k = 0, compute (Vk )k , (Uk )k , (Tk )k iteratively as follows, where step 3 will be specified further below. 1. Set Vk := ˚ V(Tk ) and let Uk ∈ Vk be the solution of (4) with T = Tk . 2. Compute the indicators of estimator and oscillation in (10) and (11), respectively, and write Ek as an abbreviation for ETk . 3. Choose a subset Mk ⊂ Tk with the help of the values of the indicators {Ek (T )}T ∈Tk and {δ(T )}T ∈Tk . 4. Let Tk+1 be the smallest refinement of Tk in T such that Mk ∩ Tk+1 = ∅. 5. Increment k and go to step 1.

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3 Convergence and Marking Strategy The above adaptive method generalizes global uniform refinement, which is the special case corresponding to Mk = Tk for all k. The new key feature is that certain elements may not be refined anymore. In other words: it may happen that T ∗ := {T | ∃m ∈ N0 ∀k ≥ m T ∈ Tk } = ∅ and so

V∞ :=

3

Vk = H01 (Ω).

k

It is then the task of step 3, the marking strategy, to preclude u ∈ H01 (Ω) \ V∞ , which is equivalent to non-convergence. Let us first derive a necessary condition for the marking strategy. If limk→∞ ∇(u − Uk )Ω = 0, then, for any element T ∈ T ∗ , the local lower bounds in (10), (11), and Theorem 1 imply limk→∞ Ek (T ) + δ(T ) = 0 for the associated indicators. Hence it is necessary for convergence that the marking strategy ensures ∀T ∈ T ∗

δ(T ) = 0 and lim Ek (T ) = 0, k→∞

(13)

which specifies [8, (5.1)]. It turns out that this condition is also sufficient for convergence, as it complements the following property of our refinement procedure. Let hk denote the meshsize function of4 Tk given by hk |T = |T |1/d for all T ∈ Tk and let χk the characteristic function of {T ∈ Tk : T ∈ Tk+1 }. Then [9, Lemma 9] shows lim hk χk L∞ (Ω) = 0,

k→∞

(14)

which generalizes maxΩ hk → 0 for non-adaptive, uniform refinement. Theorem 2 (Adaptive Convergence) Let f ∈ H −1 (Ω) be arbitrary and assume that the indicators of the estimator satisfy (10). Then the approximate solutions (Uk )k of the AFEM converge to the exact solution u if and only if the marking strategy ensures (13). Proof We adopt the proof of [9, Theorem 8], which essentially follows [10]. Using the Lax-Milgram theorem, let U∞ ∈ V∞ be such that  Ω

∇U∞ · ∇V dx = f, V  for all V ∈ V∞ .

Thanks to [9, Lemma 7], we have lim ∇(U∞ − Uk )Ω = 0

k→∞

(15)

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and it remains to show that U∞ = u or, as an alternative, that its residual vanishes:  ∇U∞ · ∇v dx for all v ∈ H01 (Ω). (16) 0 = R∞ , v := f, v − Ω

Here we can take the test functions only from C0∞ (Ω), because C0∞ (Ω) is a dense subset of H01 (Ω). Consequently, the convergence (15) shows that (16) follows from ∀ϕ ∈ C0∞ (Ω)

lim Rk , ϕ = 0,

k→∞

(17)

where Rk := f + ΔUk ∈ H −1 (Ω) is the residual of Uk . In order to verify this, let ϕ ∈ C0∞ (Ω) and k,  ∈ N0 with k ≥  and introduce the set T∗ := T ∩ T ∗ . The inclusion V ⊂ Vk , the abstract error bound (7), the local equivalences (9), as well as the upper bounds in (10) and (11) imply ∗ , Rk , ϕ = Rk , ϕ − I ϕ  S,k + S,k

(18)

where I denotes Lagrange interpolation onto V and we expect that

S,k :=

  Ek (T ) + δ(T ) ∇(ϕ − I ϕ)ωk (T )

T ∈Tk \T∗

gets small because of decreasing meshsize, whereas ∗ := S,k



 Ek (T ) + δ(T ) ∇(ϕ − I ϕ)ωk (T )

T ∈T∗

gets small thanks to condition (13) on the marking strategy. Here ωk (T ) = ωTk (T ) is the ring around T in Tk . We first deal with S,k . The lower bounds in (10), (11), and Theorem 1 entail that

 2 Ek (T ) + δ(T )  ∇(Uk − U∞ )2Ω + ∇(U∞ − u)2Ω  1

(19)

T ∈Tk \T∗

is uniformly bounded thanks to (15). Furthermore, standard error bounds for Lagrange interpolation on T yield

T ∈Tk \T∗

∇(ϕ − I ϕ)2ωk (T ) 

T ∈T \T∗

∇(ϕ − I ϕ)2ω (T )

 h χ 2L∞ (Ω) D 2 ϕ2L∞ (Ω) |Ω|.

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Hence the Cauchy-Schwarz inequality on the sum S,k and (14) imply S,k → 0 as  → ∞ uniformly in k.

(20)

∗ , condition (13) leads to For S,k

 2 Ek (T ) + δ(T ) = Ek (T )2 → 0 T ∈T∗

as k → ∞

T ∈T∗

 2 and, since T ∈T∗ ∇(ϕ − I ϕ)L2 (ωk (T )) is uniformly bounded, the CauchySchwarz inequality provides ∗ S,k → 0 as k → ∞ for any fixed .

(21)

To conclude, let  > 0 be arbitrary. We exploit (20) and (21) by first choosing  ∗ ≤ /2. Inserting this into (18) yields so that S,k ≤ /2 and next k ≥  so that S,k the desired convergence (17) and finishes the proof. & %

4 Convergent Marking Strategies In this concluding section, we discuss how to ensure condition (13) on the marking strategy, which is necessary and sufficient for the convergence of the AFEM. To this end, the following observation about a vanishing limit is helpful. If (Tk )k∈N0 is a sequence of elements satisfying Tk ∈ Tk \ Tk+1 , k ∈ N0 , then Ek (Tk ) + δ(Tk )  ∇(u − Uk )ωk (Tk ) ≤ ∇(U∞ − Uk )ωk (Tk ) + ∇(u − U∞ )ωk (Tk ) → 0

(22)

as k → ∞. In fact, the first term on the right-hand side vanishes thanks to (15) and the second term vanishes in view of ωk (Tk )  |Tk |  hk χk dL∞ (Ω) and property (14). Condition (13) is thus satisfied if we have   max δ(T  ) + Ek (T  ) ≤

T  ∈T

k

max

T  ∈T

k \Tk+1

  δ(T  ) + Ek (T  ) ,

which, thanks to Mk ⊂ Tk \ Tk+1 , can be achieved by requiring that at least one element with maximal combined indicators is marked: 9 C     Mk ∩ T ∈ Tk | δ(T ) + Ek (T ) ≥ max δ(T ) + Ek (T ) = ∅. (23) T  ∈ Tk

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This property is verified by most of the common marking strategies applied to the combined indicators δ(T ) + Ek (T ), T ∈ Tk . Important examples are maximum-, Dörfler-, and equal distribution strategy; cf. [8]. As an alternative to combined marking, one may mark the two indicators separately. Similarly as before, requiring (23) for the single indicator Ek (T ) instead of the combined indicator Ek (T ) + δ(T ) then implies ∀T ∈ T ∗

lim Ek (T ) = 0.

k→∞

To ensure also δ(T ) = 0 for all T ∈ T ∗ , one may employ again the respective counterpart of (23) or a different approach, which, as fast tree approximation [1], capitalizes on the locality and history of the oscillation indicator. The following simple consequence of Theorem 2, which is also of interest by its own, is useful in this context. ˜ ) be a modification Remark 3 (Modified Oscillation) For any simplex T in T, let δ(T ˜ ) = 0 0⇒ δ(T ) = 0. Then the AFEM, or approximation of δ(T ) such that δ(T ˜ ), converges if and only if limk→∞ Ek (T ) = 0 where each δ(T ) is replaced by δ(T ˜ ) = 0 for all T ∈ T ∗ . and δ(T Acknowledgements AV gratefully acknowledges the support of the GNCS, which is a part of the Italian INdAM.

References 1. P. Binev, R. DeVore, Fast computation in adaptive tree approximation. Numer. Math. 97(2), 193–217 (2004) 2. P. Binev, W. Dahmen, R. DeVore, P. Petrushev, Approximation classes for adaptive methods. Serdica Math. J. 28(4), 391–416 (2002). Dedicated to the memory of Vassil Popov on the occasion of his 60th birthday 3. C. Carstensen, M. Feischl, D. Praetorius, Axioms of adaptivity. Comput. Math. Appl. 67(6), 1195–1253 (2014) 4. A. Cohen, R. DeVore, R.H. Nochetto, Convergence rates of AFEM with H −1 data. Found. Comput. Math. 12(5), 671–718 (2012) 5. L. Diening, C. Kreuzer, R. Stevenson, Instance optimality of the adaptive maximum strategy. Found. Comput. Math. 16(1), 33–68 (2016) 6. F.D. Gaspoz, P. Morin, Approximation classes for adaptive higher order finite element approximation. Math. Comput. 83(289), 2127–2160 (2014) 7. C. Kreuzer, A. Veeser, Oscillation in a posteriori error estimation. (in preparation) 8. P. Morin, K.G. Siebert, A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18(5), 707–737 (2008) 9. R.H. Nochetto, A. Veeser, Primer of adaptive finite element methods, in Multiscale and Adaptivity: Modeling, Numerics and Applications. Lecture Notes in Mathematics, vol. 2040 (Springer, Heidelberg, 2012), pp. 125–225 10. K.G. Siebert, A convergence proof for adaptive finite elements without lower bound. IMA J. Numer. Anal. 31(3), 947–970 (2011)

Finite Element Methods for Parabolic Problems with Time-Dependent Domains: Application to a Milling Simulation Carsten Niebuhr and Alfred Schmidt

Abstract We consider the finite element discretization of PDEs on time-dependent domains. Approximation of boundary conditions is one of the crucial aspects, as well as an appropriate approach to adaptive mesh refinement. We present some numerical test results and the application to the thermomechanical simulation of a milling process, where the domain changes in time due to material removal.

1 Introduction Time-dependent domains arise in many real world situations and applications, examples are free boundary problems like fluid flow with free capillary surfaces, mechanics with finite deformations, or milling processes with material removal. We investigate the finite element discretization of parabolic and elliptic problems on time-dependent domains. An approximation of the domain can be done either with a moving mesh approach, or a subdomain approach with or without cutcells. Here, we concentrate on non-fitting meshes in a subdomain-setting. Boundary conditions are delicate in the case of non-fitting meshes. The approximation of Neumann boundary conditions is addressed in Sect. 2.1. We present numerical tests for the heat equation on a shrinking domain. They show that a suitable combination of a-posteriori and heuristic conditions is a good approach for adaptive mesh refinement. Applications with industrial background consist typically of coupled systems of PDEs. We consider in particular the simulation and optimization of thermal distortions in milling processes and present some results in Sect. 3.

C. Niebuhr · A. Schmidt () University of Bremen, Center for Industrial Mathematics (ZeTeM), Bremen, Germany MAPEX Center for Materials and Processes, Bremen, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_43

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Here we assume that the evolution of the workpiece domain Ω(t) is given, with an initial blank part Ω0 = Ω(0) and Ω(t2 ) ⊂ Ω(t1 ) for t2 ≥ t1 . In reality, the material removal by the milling tool interacts with the workpiece deformation and thus the cutting conditions (resulting in forces and heat production) and the domain are only given implicitly. This leads to shape errors in the final workpiece, which can be compensated by optimization of process parameters.  With stress tensor σ (u, θ ) = 2μ(θ )((u) + λ(θ )tr((u)) − 3α(θ − θ0 ) I and strain tensor (u) = 12 (∇u + ∇uT ), the thermomechanical problem with quasistationary mechanics is given in strong form as: Find temperature θ and deformation u such that θ˙ − div(κ∇θ ) = 0 in Ω(t),

(1)

− div σ (u, θ ) = 0 in Ω(t)

(2)

with initial condition θ (0) = θ0 and boundary conditions κ∇θ · ν = gN on ∂Ω(t),

σ (u, θ ) · ν = fN on ΓN (t),

u = 0 on ΓD .

(3)

The heat flux gN over the boundary is given by a cooling condition or by the flux produced during the milling process, as are the forces fN . The workpiece is clamped at ΓD and ∂Ω(t) = ΓD ∪ ΓN (t).

2 Discretization Various approaches are used for the discretization of time-dependent domains for finite element simulations. Depending on the problem considered, a moving mesh approach (based on parametrization over a reference domain, e.g.) can be appropriate and efficient. This is often the case for free-surface flows [2] or some other free boundary problems, where the motion of the boundary or interface is governed by the PDEs and the change of the domain is not too large. If deformations get large, remeshings are needed. A different approach is the use of a fixed ‘background’ domain Ω0 which contains the time-dependent domain Ω(t) for all times considered, with a corresponding triangulation S0 and using an approximation of S0 ∩ Ω(t) for the discretization at time t [1]. Here, the approximation of domain and solution can be improved by local mesh refinement or by the introduction of cut-cells [6]. For our application, the simulation of a milling process, a moving mesh approach is not really appropriate. There is no natural parametrization, and due to removal of a large amount of material, the deformation of a mesh would be very large and thus a remeshing would be needed quite often. A much more natural approach is to use a discretization of the initial workpiece and remove mesh elements (or parts of them) when the corresponding material is removed by the process. In order to keep the discretization simple, and as local

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483

adaptive mesh refinements should be used anyway, we do not want to use cut-cells but just keep resp. remove full mesh elements. Thus, the approximation of the system of PDEs on the time-dependent domain is done with finite elements 4 on non-fitting meshes: Given the domain Ω0 and a triangulation S0 (t) with {S ∈ S0 (t)} = Ω¯0 , we define the triangulation S(t) and discrete domain Ωh (t) by Ω¯ h (t) =

S(t) := {S ∈ S0 (t) : S ∩ Ω(t) = ∅},

3

{S ∈ S(t)}.

(4)

By construction, Ω(t) ⊂ Ωh (t) ⊂ Ω0 and the approximation of the domain is of the same order as the local mesh size near the boundary ∂Ω(t) \ ∂Ω0 .

2.1 Neumann Boundary Values on Unfitted Meshes The direct transfer of Neumann boundary data from ∂Ω to an approximation ∂Ωh may lead to totally wrong results, for example in case when the size of the discrete boundary does not converge to the continuous one. Thus, an adjustment of the boundary condition has to be done. For simplicity, we first consider the approximation of a non-time-dependent domain Ω. In the elliptic case, Babuška and Chleboun [1] consider the approximation of a general domain Ω ⊂ R2 by unfitted meshes in the case of Neumann boundary conditions for an elliptic problem with bilinear form aΩ given by an integral over Ω. They generalize the Neumann condition ∂u = gN on ∂Ω ∂ν resp. its representation in the weak formulation with test function ϕ, giving the term  1 2 ¯ ¯ g ∂Ω N ϕ do, by introducing a vector valued function G ∈ H0 (B) where G · ν = gN on ∂Ω, with Ω ⊂⊂ B, and replace 



¯ · ν ϕ do G

gN ϕ do = ∂Ω

∂Ω

with possible generalization for on-smooth boundary. Now, if Ωh is an approximation to Ω, they show that the weak solution u˜ h to the corresponding problem with bilinear form aΩh on Ωh ,  u˜ h ∈ H 1 (Ωh ) : aΩh (u˜ h , ϕ) =



¯ · νh ϕ do G

f ϕ dx + Ωh

∂Ωh

∀ϕ ∈ H 1 (Ωh )

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¯ are smooth enough, then they converges to the solution u on Ω. If ∂Ω, ∂Ωh , and G prove convergence of order u − u˜ h 2aΩ ≤ Ch, where h is a bound for the maximal distance of the two domain boundaries. More precisely, for the triangulation S, u − u˜ h 2aΩ ≤ C max(hS : S ∩ ∂Ωh = ∅). S∈S

For our situation, a simple correction of the boundary data is in many cases ¯ above: replace equivalent to the introduction of the function G   ¯ · νh ϕ do = (ν · νh ) g¯ ϕ do, G ∂Ωh

∂Ωh

where ν and νh are the unit normals to the continuous and discrete boundaries, respectively, and g¯ a smooth continuation of gN . If the continuous boundary is smooth and the mesh size is small enough, then the scalar product accounts just for the (typically longer) discrete boundary introduced by the approximation by mesh element boundaries, see Fig. 1. Combining the estimate above with standard a-posteriori finite element error estimates on Ωh ⊃ Ω, we easily get the error estimate u − uh Ω ≤ u − u˜ h Ω + u˜ h − uh Ωh 1  2 ≤ C max( hS : S ∩ ∂Ωh = ∅) + ηS2 .

(5)

S⊂Ω¯ h

Thus, a combination of standard adaptive refinement procedures with an additional refinement near the boundary can keep the overall discretization error small. We demonstrate this approach in the context of the parabolic heat equation on a time dependent domain.

Fig. 1 Boundary approximation and normal vectors

FEM for Parabolic Problems with Time-Dependent Domains

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2.2 Numerical Experiments We consider a moving boundary problem with travelling wave solution in 2D, where Ω0 = (0, 1)2 and the time-dependent domain is given by Ω(t) = {x ∈ Ω0 : nα · x > t}, where nα = (cos(α), sin(α)) for α = π/6. The solution u for a constant Neumann condition flux at the moving flat boundary is then u(x, t) = exp(nα ·x −t). We approximate the solution in the time interval (0.01, 1.01). In Fig. 2, we present the error measured by maxtn (u−uh L2 (Ω(tn )) ) for different discretization strategies, shown both against the number of unknowns (on the left) and the minimal mesh size (on the right).

Fig. 2 Convergence of L∞ (L2 ) error for different strategies. Top: global refinement, 2nd row: adaptive refinement without special treatment of the boundary, 3rd row: combined strategy after (5)

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The top row shows global refinements of the mesh; with N_DOF ≈ 1/ h2 in 2D. The graphs show a nearly linear convergence of the error with the mesh size h√(which is better than the estimate (5)) and an order of convergence err ≈ O(1/ N_DOF ). The second row indicates that the convergence rate does not improve when using the standard adaptive procedure using just the residual error indicators for the heat equation. The bottom row shows results from a combination of the adaptive refinement based on residual error indicators, with an additional refinement of the mesh elements near the cutting boundary ∂Ω(t) \ ∂Ω0 . All simulations here were done with the same tolerance for the standard adaptive procedure using an equidistribution marking strategy, but additional to that, the mesh near the cutting boundary was refined until a prescribed mesh size h∗ was reached. Different nodes in the graph correspond to different mesh sizes h∗ . As the number of mesh elements increases with smaller h∗ , the adaptive strategy produces somewhat finer meshes, too, due to the equidistribution strategy where the local error indicator ηS2 is asked to be around T OL2 /#S(tn ). Due to this combination, we observe an improved order of convergence, err ≈ O(1/N_DOF ). Thus, the combination of adaptive mesh refinement and special refinements near the moving boundary improves the overall numerical efficiency of the method.

3 Application: Milling Process The approach described above is used in an adaptive FEM simulation of milling processes. The overall simulation is built by combining a dexel model for material removal (an extension of a classical NC-Simulation) with a process model and a finite element discretization of the thermal and mechanical equations, see [3–5, 7] for details. The dexel model computes the local interaction of the cutting tool with the workpiece, giving the new geometry Ω(t) as well as cutting conditions like local chip thickness and tool velocity. The process model computes local forces and heat fluxes from the cutting conditions. These define the boundary conditions at the cutting boundary in the thermomechanical problem (3).

3.1 Reference Process The machining of a thin walled part made of 1.1191 steel has been analyzed (both numerically and experimentally) as a reference process in [3, 5, 7]. The blank part Ω0 is a rectangular block of 40 mm width, 40 mm thickness, and 195 mm length. The percentage of machined material is about 60%. The workpiece is clamped on both sides with one degree of freedom for torsion and translation and is fixed on

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a dynamometer, which allows to measure the fixture forces. The milling process is divided into two parts, a roughing and a finishing part. Roughing means the removal of large portions of material from the workpiece, while finishing is a detailed material removal of only few μm and is performed at the end. The machining strategy is z-level constant, the roughing process is divided in different steps for every z-level and the finishing of the thin wall is performed in one cut for each z-level.

3.2 Simulation Results The mathematical-numerical model to simulate the thermomechanical behavior of the workpiece during machining operations has been presented in [7]. It includes the description of heat equation and quasi-stationary linear elasticity equation on time-dependent domain with boundaries changing in each time step. This coupled system is implemented using the adaptive finite element toolbox ALBERTA [8]. The adaptive strategy used for the simulation reflects the approach presented in Sect. 2.1, with special refinement near the cutting boundary ∂Ω(t) \ ∂Ω0 . Figure 3 shows a locally refined mesh during the simulation of the cutting process, with nine refinement levels at the time-dependent boundary. Figure 4 shows some simulation results for temperature and thermomechanical deformation at the end of the milling process. These results have shown to be in accordance with experimental data for both temperature and deformation [4]. The thermomechanical simulation system has been used to simulate more detailed milling processes with thin walled workpieces for lightweight structures [7]. As mentioned in the introduction, the deformation of the workpiece leads in practice to a change of material removal and cutting conditions, and thus to shape errors of the produced part. If these are above tolerance, compensation strategies like an optimization of process parameters are needed. Some approaches and results are presented in [5, 9, 10].

Fig. 3 Adapted mesh during milling process with up to nine refinement levels on the timedependent boundary

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Fig. 4 Simulation result for reference process: Temperature [◦ C] and deformation (color represents its modulus in [mm]) with scaled factor 100

Acknowledgements The authors gratefully acknowledge the financial support by the German Research Foundation (DFG) via the project “Thermomechanical Deformation of Complex Workpieces in Drilling and Milling Processes” (MA1657/21-3) within the DFG Priority Program 1480 “Modeling, Simulation and Compensation of Thermal Effects for Complex Machining Processes”. Furthermore, we thank our project partners from ZeTeM Bremen and IFW Hannover for cooperation.

References 1. I. Babuška, J. Chleboun, Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions. Math. Comput. 71, 1339–1370 (2001) 2. E. Bänsch, Finite element discretization of the Navier-Stokes equations with a free capillary surface. Numer. Math. 88, 203–235 (2001) 3. B. Denkena, A. Schmidt, J. Henjes, D. Niederwestberg, C. Niebuhr, Modeling a Thermomechanical NC-Simulation, in Proceedings of 14th CIRP Conference on Modeling of Machining Operations (CIRP CMMO), Procedia CIRP, vol. 8 (2013), pp. 69–74

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4. B. Denkena, A. Schmidt, P. Maaß, D. Niederwestberg, C. Niebuhr, J. Vehmeyer, Prediction of temperature induced shape deviations in dry milling, in Proceedings of 15th CIRP Conference on Modeling of Machining Operations (CIRP CMMO), Procedia CIRP, vol. 31 (2015), pp. 340–345 5. B. Denkena, P. Maaß, A. Schmidt, D. Niederwestberg, J. Vehmeyer, C. Niebuhr, P. Gralla, Thermomechanical deformation of complex workpieces in milling and drilling processes, in Thermal Effects in Complex Machining Processes - Final Report of the DFG Priority Program, ed. by D. Biermann, F. Hollmann. Springer LNPE Series, vol. 1480 (Springer, Berlin, 2017), pp. 219–250 6. T.-P. Fries, A. Zilian, On time integration in the XFEM. Int. J. Numer. Methods Eng. 79, 69–93 (2009) 7. C. Niebuhr, FE-CutS—finite elemente modell für makroskopische Zerspanprozesse: modellierung, analyse und simulation, PhD thesis, University of Bremen, 2017 8. A. Schmidt, K.G. Siebert, Design of Adaptive Finite Element Software. Lecture Nodes in Computational Science and Engineering (Springer, Berlin, 2004) 9. A. Schmidt, E. Bänsch, M. Jahn, A. Luttmann, C. Niebuhr, J. Vehmeyer, Optimization of engineering processes including heating in time-dependent domains, in System Modeling and Optimization, ed. by L. Bociu, J.A. Desideri, A. Habbal. Springer IFIP AICT Series, vol. 494 (Springer, Berlin, 2017), pp. 452–461 10. A. Schmidt, C. Niebuhr, D. Niederwestberg, J. Vehmeyer, Modelling, simulation, and optimization of thermal deformations from milling processes, in Progress in Industrial Mathematics at ECMI 2016, ed. by P. Quintela, P. Barral, D. Gomez, F.J. Pena, J. Rodriguez, P. Salgado, M.E. Vazquez-Mendez. Springer Mathematics in Industry Series, vol. 26 (Springer, Berlin, 2017), pp. 337-343

Part XIII

Polyhedral Methods and Applications

Numerical Investigation of the Conditioning for Plane Wave Discontinuous Galerkin Methods Scott Congreve, Joscha Gedicke, and Ilaria Perugia

Abstract We present a numerical study to investigate the conditioning of the plane wave discontinuous Galerkin discretization of the Helmholtz problem. We provide empirical evidence that the spectral condition number of the plane wave basis on a single element depends algebraically on the mesh size and the wave number, and exponentially on the number of plane wave directions; we also test its dependence on the element shape. We show that the conditioning of the global system can be improved by orthogonalization of the local basis functions with the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES iterations for solving the discrete problem iteratively.

1 Introduction For the numerical approximation of the Helmholtz problem, it has been shown that, by using non-polynomial basis functions, it is possible to reduce the pollution effect in finite element approximations. One special class of such methods are Trefftz finite element methods, which use basis functions that are local solutions of the homogeneous problem under consideration. For the Helmholtz problem, a common choice of Trefftz basis functions are plane waves; when used in connection to a discontinuous Galerkin variational framework, they lead to the plane wave discontinuous Galerkin (PWDG) method [2–4, 7–9]. Containing information on the oscillatory behaviour of the solutions already in the approximation spaces, PWDG deliver better accuracy than standard polynomial finite element methods, for a comparable number of degrees of freedom. In addition, they involve only evaluation of basis functions on mesh interelement boundaries; hence, they can be easily used in connection with general polytopal meshes. However, it is well known that these

S. Congreve · J. Gedicke · I. Perugia () University of Vienna, Faculty of Mathematics, Vienna, Austria e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_44

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basis functions are ill conditioned for small mesh sizes, small wavenumbers and large numbers of plane wave directions [10, 11]. The aim of this paper is to numerically investigate the dependence of the elemental and global condition numbers of the PWDG system matrix on the size and shape of the local (convex) polygonal element, the wavenumber, and the number of plane wave directions in the local approximation spaces.

2 The PWDG Method for the Helmholtz Problem Let Ω ⊂ R2 be a bounded Lipschitz domain and k > 0 denote the wavenumber. We consider the homogeneous Helmholtz problem with impedance boundary condition: −Δu − k 2 u = 0 in Ω, ∇u · n + iku = g

(1)

on ∂Ω,

where i denotes the imaginary unit, n is the unit outward normal and g ∈ L2 (∂Ω) is given. The variational formulation of problem (1) reads as follows: find u ∈ H 1 (Ω) such that    (∇u · ∇ v¯ − k 2 uv)dx ¯ + ik uv¯ ds = g v¯ ds for all v ∈ H 1 (Ω). (2) Ω

∂Ω

∂Ω

Problem (2) is well posed by the Fredholm alternative argument [13]. We consider a shape-regular, uniform partition Th of the domain Ω into convex polygons K ∈ Th of diameter h. We define the mesh skeleton Fh = ∪K∈Th ∂K, and denote the interior mesh skeleton by FhI = Fh \∂Ω. For an element K ∈ Th we define the plane wave space PWp (K) of degree p as PWp (K) = {v ∈ L2 (K) : v(x) =

p

αj exp(ikd j · (x − x K )), αj ∈ C},

j =1

where x K is the mass center of K, and d j , |d j | = 1, j = 1, . . . , p, are p unique directions. Since, in general, small angles between those directions lead to bad conditioning of the basis, we consider equally spaced directions. The PWDG space is defined as PWp (Th ) = {vhp ∈ L2 (Ω) : vhp |K ∈ PWp (K)

for all K ∈ Th }.

The functions in PWp (Th ) are local solutions of the homogeneous Helmholtz problem; therefore, they exhibit the Trefftz property −Δvhp − k 2 vhp = 0

for all vhp ∈ PWp (K).

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We assume uniform local resolution, i.e., we employ the same uniformly distributed directions d j , j = 1, . . . , p on each element K ∈ Th . We use the standard notation for averages and normal jumps of traces across inter-element boundaries, namely {{·}} and [[·]], respectively, and denote by ∇h the elementwise application of ∇. Hence, we can formulate the PWDG method as follows [7–9]: find uhp ∈ PWp (Th ) such that Ah (uhp , vhp ) = h (vhp )

for all vhp ∈ PWp (Th ),

(3)

where   Ah (uhp , vhp ) := i −

FhI

 {{u}}[[∇h v]] ¯ ds +

FhI

{{∇h u}} · [[v]] ¯ ds

@  1 u∇h v¯ · n ds + ∇h u · nv¯ ds 2 ∂Ω ∂Ω   1 k + [[∇h u]][[∇h v]] ¯ ds + [[u]] · [[v]] ¯ ds 2k FhI 2 FhI   1 k + (∇h u · n)(∇h v¯ · n) ds + uv¯ ds, 2k ∂Ω 2 ∂Ω   1 i g∇h v¯ · n ds − g v¯ ds. h (v) := 2k ∂Ω 2 ∂Ω 1 − 2



The PWDG method (3) is unconditionally well-posed and stable [2, 3]. The h, p and hp convergence has been studied in [2, 7–9]. Let A ∈ CNh ×Nh denote the matrix associated with the sesquilinear form Ah (·, ·), and b ∈ CNh the vector associated with the functional h (·), for Nh := dim(PWp (Th )). Then, the algebraic linear system associated with the PWDG method (3) on the mesh Th is Au = b.

3 Conditioning of the Plane Wave Basis In this section, we investigate numerically the conditioning of the local plane wave basis. In order to do so, we consider the spectral condition number of the local mass matrix MK ∈ Cp×p on a single element K ∈ Th . From [6] we get MK,jj = |K|, and  MK,j l = eikd j ·(x−x K ) eikd l ·(x−x K ) dx K

=−

F ∈∂K∩Fh

ik(d j − d l ) · n 2 k (d j − d l ) · (d j − d l )



eik(d j −d l )·(x−x K ) ds, F

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condition

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Fig. 1 Spectral condition numbers of MK for regular n-polygons (left) and anisotropic rectangles (right) with h = 1 and k = 10

for j = l, 1 ≤ j, l ≤ p, which can be evaluated in closed form. Note that the entries of MK tend to |K| as k(d j − d l ) · (x − x K ) tends to zero; hence, small values of the element size h and wavenumber k, or a small angle between two plane wave directions, lead to ill conditioning.

3.1 Dependence on the Shape The initial numerical experiments investigate the conditioning of the basis for different shapes of the element, for a fixed wavenumber k = 10. Firstly, we consider regular n-polygons with element size h = 1; cf. Fig. 1 (left). We observe that the condition numbers grow in the number of plane waves directions p, but are smaller for larger n. In particular, the condition number is decreasing in the number of sides n and is asymptotically stable; hence, small edges do not cause any problems. It has been noted in [11] that the conditioning of the basis depends on the aspect ratio of the elements. Therefore, we consider a single anisotropic rectangle, with size h = 1, and vary its aspect ratio. We see that the condition number increases exponentially as the aspect ratio increases; cf. Fig. 1 (right).

3.2 Dependence on hk and p In this section we empirically determine the dependence of the condition number on hk and p. We restrict to the case of a single square element. The numerical experiments displayed in the first two graphs in Fig. 2 suggest that the condition number is algebraic with respect to hk and exponential with respect to p. To get a more precise answer, we fitted the data obtained from numerous numerical

10

p=7 p=11 p=15 p=19 p=23

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condition

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hk

hk=2 hk=3 hk=4 hk=5 hk=6

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h=1/2 h=1/4 h=1/8 h=1/16 h=1/32 p=5,...,23

8 6 4 2

0

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20

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p

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30

k

Fig. 2 Dependence of the condition number of Mk on hk and p, and verification of the approximation (4)

experiments to cond2 (MK ) ≈

2.34p ln p . (hk)p−1

(4) p ln p

In Fig. 2 (right) we show the values of cond2 (MK )/( 2.34 ) for values h = (hk)p−1

2−1 , . . . , 2−5 , k = 5, . . . , 30 and p = 5, . . . , 23. To obtain reliable data, we only plot data points for which cond2 (MK ) < 1015, due to double precision limitations, and for which hk < 10, due to the resolution condition. Hence, we could only cover a moderate range of values for h, k and in particular p. All the presented values of p ln p ) are between 1 and 10 (recall that the corresponding values cond2 (MK )/( 2.34 (hk)p−1 of cond2 (MK ) are between 1 and 1015), which confirms that the approximation (4) is reasonable, at least for moderate h, k and p.

4 Orthogonalization of the Plane Wave Basis In the previous section, we have observed that the condition number of the local basis is large for small hk or large p. In this section we aim at improving the conditioning of the local basis in order to lower the condition number of the global system matrix A. Therefore, we will investigate the effect of orthogonalization of the (local) basis functions on the conditioning of the (global) system matrix A. A different approach has been presented in [10], where improvement of the conditioning of the global system is achieved by suitably designed non-uniform distributions of p. We compare the condition numbers of the system matrix A with original basis 8 := QT AQ with orthogonalized functions with that of the system matrix A basis functions for the three different meshes displayed in Fig. 3. Here, Q ∈ CNh ×Nh denotes the transformation matrix obtained by modified Gram-Schmidt orthogonalization [15] of the (local) basis functions with respect to the Hermitian part of the local system matrix H (AK ) :=

AK +A¯ Tk 2

on each element K separately;

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Fig. 3 Three different meshes

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condition

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tri quad hex

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25

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Fig. 4 Spectral condition numbers for k = 10 in double precision arithmetic

cf. [1, 12, 14] for application of modified Gram-Schmidt to partition of unity methods, (polynomial) DG methods, and virtual element methods, respectively. In all experiments, we choose k = 10 and only investigate the effect on the critical dependence of the condition number on p. As a model problem, we consider problem (1) with Ω = (0, 1)2 , and exact solution 2 given by the Bessel function

of the third kind (Hankel function) u(x) = H01 (k (x1 + 1/4)2 + x22 ). In Fig. 4 (left), we observe, for all meshes, the expected increase of the condition number in p for the original system matrix A (dashed lines), which results in the loss of accuracy in the L2 error for p > 21, cf. Fig. 4 (right), when using a direct linear solver. We observe major improvements of the condition numbers for the matrix 8 (solid lines) until p = 21 when the (modified) Gram-Schmidt orthogonalization A breaks down, which directly correlates to the point when the direct solver fails to produce a more accurate solution. Note that, for the original matrix A, there is no such correlation. To further investigate these results, we also carried out the same experiments in single precision arithmetic. Figure 5 shows the results for single precision, where we observe that the loss of accuracy already occurs at p = 13. Note, again, that this loss is correlated to the failure of the orthogonalization, 8 at p = 13. indicated by the sudden increase of the condition numbers for A Finally, we are interested in the effect of the orthogonalization on the convergence of iterative solvers such as GMRES. From the convergence theory of GMRES [5], provided that H (A), the Hermitian part of the system matrix A, is positive

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p

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Fig. 5 Spectral condition numbers for k = 10 in single precision arithmetic Table 1 Eigenvalue approximations and GMRES iteration count for the original basis and the orthogonalized basis using the second mesh of Fig. 3 and k = 10 λmin (H (A)) λmin (H (A−1 )) GMRES (A) 8 λmin (H (A)) 8−1 )) λmin (H (A 8 GMRES (A)

p=5 7.75 · 10−1 3.83 · 10−2 60 1.10 · 10−1 5.02 · 10−1 47

p=7 2.56 · 10−1 2.70 · 10−2 96 8.45 · 10−2 5.00 · 10−1 52

p=9 1.16 · 10−2 2.09 · 10−2 135 8.29 · 10−2 5.00 · 10−1 58

p = 11 4.50 · 10−4 1.71 · 10−2 159 8.27 · 10−2 5.00 · 10−1 62

p = 13 6.73 · 10−6 1.44 · 10−2 193 7.44 · 10−2 5.00 · 10−1 68

p = 15 1.18 · 10−7 1.25 · 10−2 217 6.08 · 10−2 5.00 · 10−1 73

definite, the residual contraction factor, for the residual rj at iteration j , can be bounded as j/2 rj   ≤ 1 − λmin (H (A))λmin (H (A−1 )) . r0  Therefore, in Table 1, we report (λmin (H (A)), λmin (H (A−1))), for the system 8 λmin (H (A 8−1))), for the system matrix A with the original basis, and (λmin (H (A)), 8 matrix A obtained with the orthogonal basis, along with the number of GMRES iterations needed in order to reduce the residual of a factor 10−10 . We observe that, 8−1 )) are fairly constant, for increasing p, the values λmin (H (A−1 )) and λmin (H (A 8 while the values λmin (H (A)) and λmin (H (A)) decrease significantly. However, the 8 decrease much more slowly than those of λmin (H (A)), which values of λmin (H (A)) results in a far slower increase of GMRES iterations for the orthogonalized basis than for the original one. As we merely change the basis and do not precondition the system matrix, we cannot expect a constant number of GMRES iterations.

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5 Conclusions We have provided empirical evidence that, in 2D, the condition number of the plane wave basis is stable for large edge counts in regular polygons, and grows as the aspect ratio of the anisotropy of the elements increases. We also observed its algebraic dependence on the product hk, and exponential dependence on p. It has been demonstrated that the condition number of the global system matrix can be significantly lowered by a local modified Gram-Schmidt orthogonalization with respect to the Hermitian part of the local system matrix; this results in faster convergence of the GMRES solver. The improvement of the conditioning in 3D will be considered in future work. Acknowledgements The authors have been funded by the Austrian Science Fund (FWF) through the project P 29197-N32. The third author has also been funded by the FWF through the project F 65.

References 1. F. Bassi, L. Botti, A. Colombo, D.A. Di Pietro, P. Tesini, On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231(1), 45–65 (2012) 2. A. Buffa, P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation. M2AN Math. Model. Numer. Anal. 42(6), 925–940 (2008) 3. O. Cessenat, B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35(1), 255–299 (1998) 4. O. Cessenat, B. Després, Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. J. Comput. Acoust. 11(2), 227–238 (2003) 5. S.C. Eisenstat, H.C. Elman, M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20(2), 345–357 (1983) 6. C. Gittelson, Plane wave discontinuous Galerkin methods, Master’s thesis, SAM, ETH Zurich, Switzerland, 2008 7. C.J. Gittelson, R. Hiptmair, I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version. M2AN Math. Model. Numer. Anal. 43(2), 297–331 (2009) 8. R. Hiptmair, A. Moiola, I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49(1), 264–284 (2011) 9. R. Hiptmair, A. Moiola, I. Perugia, Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version. Found. Comput. Math. 16(3), 637–675 (2016) 10. T. Huttunen, P. Monk, J.P. Kaipio, Computational aspects of the ultra-weak variational formulation. J. Comput. Phys. 182(1), 27–46 (2002) 11. T. Luostari, T. Huttunen, P. Monk, Improvements for the ultra weak variational formulation. Int. J. Numer. Methods Eng. 94(6), 598–624 (2013) 12. L. Mascotto, Ill-conditioning in the virtual element method: stabilizations and bases. Numer. Methods Partial Differ. Equ. 34(4), 1258–1281 (2018) 13. J. Melenk, On generalized finite element methods, Ph.D. thesis, University of Maryland, 1995 14. M.A. Schweitzer, Stable enrichment and local preconditioning in the particle-partition of unity method. Numer. Math. 118(1), 137–170 (2011) 15. G.W. Stewart, Matrix Algorithms. Vol. I: Basic Decompositions (Society for Industrial and Applied Mathematics, Philadelphia, 1998)

The Virtual Element Method for the Transport of Passive Scalars in Discrete Fracture Networks S. Berrone, M. F. Benedetto, Andrea Borio, S. Pieraccini, and S. Scialò

Abstract Simulation of physical phenomena in networks of fractures is a challenging task, mainly as a consequence of the geometrical complexity of the resulting computational domains, typically characterized by a large number of interfaces, i.e. the intersections among the fractures. The use of numerical strategies that require a mesh conforming to the interfaces is limited by the difficulty of generating such conforming meshes, as a consequence of the large number of geometrical constraints. Here we show how this issue can be effectively tackled by resorting to the Virtual Element Method on polygonal grids. Advection-diffusion-reaction phenomena are considered, also in advection-dominated flow regimes.

1 Introduction of the Problem In the present work we consider the issue of the effective simulation of advectiondiffusion-reaction problems in networks of intersecting fractures, modeled by means of the Discrete Fracture Network (DFN) model. DFNs are a widely accepted tool to represent the system of fractures inside an otherwise impervious medium, where fractures are represented as planar polygons, possibly intersecting each other in the three dimensional space, and we call trace each intersection between exactly two fractures. Let Fi , i ∈ I = {1, . . . , N} be one of the fractures in the DFN 4 Ω = F i∈I i , and let Γm , m ∈ M = {1, . . . , M} be one of the traces, we denote by Mi , i ∈ I, the set of traces belonging to fracture Fi and by Im , for m ∈ M the pair of fracture indexes meeting at Γm . DFNs for practical applications typically involve a large number of fractures with arbitrary orientations in the three

S. Berrone · A. Borio () · S. Pieraccini · S. Scialò Politecnico di Torino, Turin, Italy e-mail: [email protected]; [email protected]; [email protected]; [email protected] M. F. Benedetto Universidad de Buenos Aires, Ciudad de Buenos Aires, Argentina © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_45

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dimensional space, and thus forming an intricate network of intersections (Fig. 1). Further, DFNs are generated stochastically, starting from probability distributions of fracture related quantities, such that a large number of simulations is required for a site of interest to obtain reliable statistics of the quantities of interest. All this calls for the development of reliable numerical methods, particularly robust to geometrical complexities. In particular, the generation of a mesh conforming to the intersections between fractures is an extremely challenging task for the huge number of geometrical constraints dictated by the conformity requirement. As a consequence, standard numerical approaches requiring conforming meshes have limited applicability in this context. Here we propose a review of the evolution of the results published in [1–4] about the use of Virtual Elements [6] to easily build polygonal conforming meshes of the whole network of fractures starting from triangular non conforming meshes on each fracture. The generation of the conforming mesh thus mostly reduces to a local problem, thus reducing drastically its complexity and cost. The bundle of methods developed in the cited papers is here finalized in a geothermal-like simulation. Let us split the boundary of Ω, ∂Ω, in a Dirichlet part ΓD = ∅ and a Neumann part ΓN = ∂Ω \ ΓD , and let us set ΓiD = ΓD ∩ ∂Fi , i ∈ I the portion of Dirichlet boundary of fracture Fi and similarly ΓiN = ΓN ∩ ∂Fi . An homogeneous Dirichlet boundary condition is assumed, for simplicity, on < ΓD and a Neumann function uN is prescribed on ΓN . The functional space V := i∈I H10,ΓDi (Fi ) is introduced and, given any function v ∈ V we indicate by vi its restriction to Fi . Then the advectiondiffusion-reaction problem in Ω is written as: find u ∈ V such that:         μi ∇ui , ∇vi F + βi · ∇ui , vi F + γi ui , vi = fi , vi F i i i F E ∂ui  , vi + uN , vi ΓNi +  , ∂ nˆ Mi Mi

Fig. 1 An example of discrete fracture network

(1)

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with the following matching conditions at the traces ∀m ∈ M, if Im = (i, j ):  BB uΓm = ui − uj B 

= 0,

(2)

∂uj ∂ui  +  = 0. ∂ nˆ Γm ∂ nˆ Γm

(3)

Γm

It is well known that, under standard assumptions on the data and on the coefficients the previous problem is well posed. In order to obtain a saddle point formulation for problem (1)–(3) we define, for each m ∈ M, with Im = (i, j ), an operator sΓm : Im → {0, 1} such that sΓm (i) = 1 and sΓm (j ) = 0, and then the following local bilinear forms

 G H  biM vi , ψ = (−1)sΓm (i) ψm , vi |Γm Γm

∀i ∈ I .

(4)

m∈Mi

Furthermore, setting J

I   M  bM v, ψ = bi vi , ψ = ψm , vΓm i∈I

m∈M

Γm

(5)

,

the continuity condition (2) can be rewritten as: .   1 bM v, ψ = 0 ∀ψ ∈ W = H− 2 (Γm ) . m∈M

Finally, introducing the bilinear form B : V × V → R, such that B (u, v) =

      μi ∇ui , ∇vi F + βi · ∇ui , vi F + γi ui , vi F , i

i∈I

i

i

(6)

and the operator F : V → R representing the right-hand side of problem (1) F (v) =

  fi , vi F + uN , vi ΓNi i∈I

i

∀v ∈ V .

problem (1)–(3) is equivalent to the following saddle-point problem: find the pair (u, λ) ∈ V × W such that ⎧ ⎨B (u, v) + bM (v, λ) = F (v) ∀v ∈ V , (7)   ⎩b M u, ψ = 0 ∀ψ ∈ W .

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2 VEM Discretization of the Problem The Virtual Element Method (VEM) is here used to discretize the elliptic operator B defined in Eq. (6). Let k ≥ 1 be a chosen integer and suppose we have discretized each fracture by a polygonal mesh Tδi , for all i ∈ I. First, for each polygon E ∈ Tδi we indicate by Pk (E) the space of polynomials of degree ≤ k and we define the H1 -orthogonal projection Πk∇ : H1 (E) → Pk (E) such that     ∇ =0 ∇ v − Πk v , ∇p

∀p ∈ Pk (E) ,

E

and ⎧  ⎪ ⎨ Π ∇ v, 1 = (v, 1)∂E  k ∂E ⎪ ⎩ Πk∇ v, 1 = (v, 1)E E

if k = 1 , if k ≥ 1 .

On each element E ∈ Tδi we define the Virtual Element space as VδE = {v ∈ H1 (E) : Δv ∈ Pk (E), v ∈ Pk (e) ∀e ⊂ ∂E, v|∂E ∈ C 0 (∂E),     ∀p ∈ Pk (E)/Pk−2 (E)} . v, p E = Πk∇ v, p E

The discrete space on each fracture Fi , i ∈ I, is   Vδi = v ∈ C 0 (Fi ) : v ∈ VδE ∀E ∈ Tδi , while the discrete subspace of the global space V with the VEM is defined as Vδ = < i∈I Vδi . It can be proven (see [6]) that a possible set of degrees of freedom for this space is given by: – the values at the vertices of the polygons; – if k ≥ 2, for each edge e of the mesh, the value of vδ at k − 1 internal points on e; – if k ≥ 2, the moments with respect to polygons  of degree ≤ k − 2 on each  1 polygon, i.e., ∀vδ ∈ Vδ , the quantities |E| vδ , p E ∀p ∈ Pk−2 (E). Using the above degrees of freedom, it is possible to compute, ∀vδ ∈ Vδ , the 0 v and Π 0 ∇v , being Π 0 2 projections Πk∇ vδ , Πk−1 δ δ k−1 k−1 the piecewise L -orthogonal projection on the space of polynomials of degree ≤ k − 1 on each polygon. In the case of vector functions, this projection is intended to be component-wise (see [5]).

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The discrete bilinear form of our problem is a SUPG-stabilized form of the classical VEM discrete bilinear form, as shown in [2]: Bδ (uδ , vδ ) =

E E E aE δ (uδ , vδ ) + bδ (uδ , vδ ) + cδ (uδ , vδ ) + dδ (uδ , vδ ) ,

(8)

E∈Tδ

where, for each element E of the mesh     0 0 0 0 aE β ·Π , v ∇u , Π ∇v + τ ∇u , β ·Π ∇v = μΠ (u ) δ δ δ δ E δ δ δ k−1 k−1 k−1 k−1 E E      7 7  7 72 + 7μ7∞,E + τE 7β 7∞,E S E I − Πk∇ uδ , I − Πk∇ vδ ,   0 0 bE , v ∇u , Π v , = β ·Π (u ) δ δ δ δ k−1 k−1 δ E   0 0 0 , cE δ (uδ , vδ ) = γ Πk−1 uδ , Πk−1 vδ + τE β ·Πk−1 ∇vδ E     0 0 dE , δ (uδ , vδ ) = τE −∇ · μΠk−1 ∇uδ , β ·Πk−1 ∇vδ E

τE being a stabilizing factor depending locally on the mesh-Péclet number PeE =

mE k

7 7 7β 7 hE 7 7 ∞,E , 7 7 2 μ ∞,E

through the relation ⎧ ⎫ 7 7 7β 7 ⎨ ⎬ hE 7 ∞,E 7 τE = 7 7 min 1, PeE , , ⎩ 2 7β 7∞,E hE 7γ 7∞,E ⎭ S E is the standard VEM stabilization operator, and mE k is a parameter depending on the order k as shown in [2]. The discrete version of the right-hand side is (see again [2]): ∀vδ ∈ Vδ , Fδ (vδ ) =

 i∈I E∈Tδi

0 0 fi , Πk−1 vδ + τE Πk−1 ∇vδ

 Fi

+ uN , vδ ΓNi .

3 The Virtual Element Method for DFN Simulations With the definitions given in Sect. 2, we discretize problem (7). Functions in the VEM space Vδ are explicitly known on the edges of mesh elements, and thus any operator involving VEM basis functions can be computed on the traces of the

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network provided that traces do not cross any mesh polygon. For any space Wδ ⊂ W of Lagrange multipliers on the traces, the discrete counterpart of the saddle point problem (7) reads as follows: find (uδ , λδ ) ∈ Vδ × Wδ such that ⎧ ⎨B (u , v ) + bM (v , λ ) = F (v ) δ δ δ δ δ δ δ   ⎩bM uδ , ψδ = 0

∀vδ ∈ Vδ ,

(9)

∀ψδ ∈ Wδ .

All the terms in the system are completely computable given the degrees of freedom, provided that the polygons of the mesh on each fracture are conforming to traces. In order to obtain such a conforming mesh in a reliable and efficient way, given a fracture Fi , i ∈ I, we start by building a regular triangulation independently of the traces and then the triangles crossed by traces are cut into polygons not crossing the traces (see also following [1, 3]). If a trace tip happens to be inside a triangle, the trace segment is prolonged up to the nearest edge keeping the trace tip as a node of the mesh, thus producing a polygon with two edges that form a flat angle: this is a situation that can be treated by Virtual Elements without any modifications to the method. The mesh obtained in such a way is called locally conforming mesh, since on each trace two different discretizations are induced by the meshes of the two fractures meeting there. An example of such construction is shown in Fig. 2a. With this kind of mesh it is possible to apply the Mortar Method, thus obtaining a weakly continuous solution and a piecewise polynomial approximation of the fluxes at traces (see [1]). Another possible approach, explored in [3], is to build a locally conforming mesh on each fracture, as described above, and then add nodes to both discretizations in such a way that the resulting induced discretizations coincide, thus obtaining a globally conforming mesh. In Fig. 2b we shown an application of such approach. This latter kind of mesh is suitable for obtaining a continuous solution, equating the degrees of freedom on the induced discretizations on each of pair of fractures that intersect at a trace. This corresponds to choosing the Lagrange multipliers as linear operators that select the values of those degrees of freedom. This procedure can sometimes produce badly shaped elements, and instabilities may arise with high order VEM. This issue was considered in [4], where a strategy based on local orthogonal polynomial bases is devised that is effective in many situations. (a)

(b) Fj

Fj

Γm Fi

Γm

Γm Fi

Γm

Fig. 2 Locally and globally conforming meshes on two intersecting fractures. (a) Locally conforming mesh. (b) Globally conforming mesh

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4 An Application of the Method In this section we apply the described method to a very simplified geothermal-like simulation on a DFN of 150 fractures: we assume that the surrounding rock matrix has a constant temperature, neglecting the cooling effect of the flow on the rock matrix, and we model the injection and extraction wells as very conductive fractures characterised by a high transmissivity. No other physical phenomena are included in the model. We are interested in solving the following non-stationary system of equations: ⎧  ⎨ ∂u , v + B (u, v) + b M (v, λ) = F (v) ∂t ⎩bM u, ψ  = 0

∀v ∈ V , ∀ψ ∈ W .

The fluid enters the system at a temperature of 15◦ C on the top edge of the injection fracture (inlet), see Fig. 3. Homogeneous Neumann boundary conditions

(a) t = 10 s

(b) t = 40 s

(c) t = 60 s

(d) t = 100 s

Fig. 3 Time evolution of the temperature distribution. (a) t = 10 s. (b) t = 40 s. (c) t = 60 s. (d) t = 100 s

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are imposed on the rest of the boundary. The velocity field β is obtained by means of the Darcy’s law, computing the hydraulic head distribution h, solution of: ⎧ ⎨



Ki ∇h, ∇v   ⎩bM h, ψ = 0 i∈I

 Fi

+ bM (v, λ) = F (v)

∀v ∈ V , ∀ψ ∈ W,

where Ki is the transmissivity of fracture Fi and there are no forcing terms (no distributed sources of pressure). For this problem, Dirichlet boundary conditions imposing h = 10 at the inlet boundary and h = 0 at the outlet boundary are used, whereas homogeneous Neumann boundary conditions are set on all other fracture edges. The transmissivity on the two fractures representing the wells is high (∼ 103 ), while the fractures inside the reservoir have varying transmissivities around 0.5. The values of the other parameters used are: f = γ TRock , γ = 0.1, TRock = 50◦ C and μ ∼ 10−5 . In Fig. 3 some snapshots are shown for a simulation run for 100 s, using a time-step of 0.1 s. It is possible to see how the physics of the phenomenon is well represented: indeed, the dead-ends in the network are the slowest regions where the cold water arrives, since the gradient of pressure is almost zero, there. Notice that here the SUPG stabilizing terms are of crucial importance, since we have meshPéclet numbers of the order of 104 . Acknowledgements This work has been partially supported by INdAM-GNCS and by Politecnico di Torino through project Starting Grant RTD. Computational resources were partially provided by HPC@POLITO (http://hpc.polito.it).

References 1. M. Benedetto, S. Berrone, A. Borio, S. Pieraccini, S. Scialò, A hybrid mortar virtual element method for discrete fracture network simulations. J. Comput. Phys. 306, 148–166 (2016) 2. M. Benedetto, S. Berrone, A. Borio, S. Pieraccini, S. Scialò, Order preserving SUPG stabilization for the virtual element formulation of advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 311, 18–40 (2016) 3. M. Benedetto, S. Berrone, S. Scialò, A globally conforming method for solving flow in discrete fracture networks using the virtual element method. Finite Elem. Anal. Des. 109, 23–36 (2016) 4. S. Berrone, A. Borio, Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method. Finite Elem. Anal. Des. 129, 14–31 (2017) 5. L. Beirão Da Veiga, F. Brezzi, L.D. Marini, A. Russo, The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci 24(8), 1541–1573 (2014) 6. L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2015)

On the Implementation of a Multiscale Hybrid High-Order Method Matteo Cicuttin, Alexandre Ern, and Simon Lemaire

Abstract A multiscale Hybrid High-Order method has been introduced recently to approximate elliptic problems with oscillatory coefficients. In this work, with a view toward implementation, we describe the general workflow of the method and we present one possible way for accurately approximating the oscillatory basis functions by means of a monoscale Hybrid High-Order method deployed on a finescale mesh in each cell of the coarse-scale mesh.

1 Introduction Let Ω be an open, bounded, connected polytopal subset of Rd , d ∈ {2, 3}, with some characteristic length scale Ω . We consider the model problem ⎧ ⎨ −div(Aε ∇uε ) = f

in Ω,

⎩

on ∂Ω,

uε = 0

(1)

where f ∈ L2 (Ω) is non-oscillatory, and Aε is an oscillatory, uniformly elliptic, bounded, symmetric matrix-valued field on Ω. The parameter ε  Ω encodes the fine-scale oscillations of the coefficients. An accurate, monoscale approximation of this problem would require an overwhelming number of degrees of freedom. In a multi-query context, where the solution is needed for a large number of right-hand

M. Cicuttin () · A. Ern Université Paris-Est, CERMICS (ENPC), Marne-la-Vallée Cedex, France Inria Paris, Paris, France e-mail: [email protected] S. Lemaire École Polytechnique Fédérale de Lausanne, FSB-MATH-ANMC, Lausanne, Switzerland Inria Lille - Nord Europe, Villeneuve d’Ascq, France © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_46

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sides, multiscale methods may be preferred. To this aim, different multiscale Finite Element Methods (msFEM) were proposed in the literature; we refer the reader to [8] for an overview. Other paradigms are also available to tackle multiscale problems, such as the Heterogeneous Multiscale Method (HMM) [1]. In the case of periodic coefficients, msFEM leads, on classical element shapes (e.g., simplices, quadrangles or hexahedra), in the lowest-order case, √ and without oversampling, to √ energy-error estimates of the form ( ε + H + ε/H ) where H represents the coarse-scale meshsize (the regime of interest is ε ≤ H  Ω ). Higher-order extensions using simplical Lagrange elements of degree k ≥ 1 were devised in [2], leading √in the same√setting of periodic coefficients to energy-error estimates of the form ( ε + H k + ε/H ). Recently the authors devised and analyzed a multiscale Hybrid High-Order (msHHO) method [3]. This method, which is a further development of the monoscale HHO method introduced in [7] for diffusion problems and in [6] for linear elasticity, uses as discrete unknowns polynomials of degree k ≥ 0 on the mesh faces and of degree l ≥ 0 in the mesh cells. The crucial difference with respect to the monoscale HHO method is that the msHHO method uses oscillatory basis functions to define the local reconstruction operator, which is the core ingredient in building HHO methods. Two variants of the msHHO method were developed in [3], the mixed-order one (l = k − 1, k ≥ 1) and the equal-order one (l = k, k ≥ 0). The motivations for considering HHO methods are that these methods support polytopal meshes, share a common design in any space dimension, are robust in various parametric regimes, and allow one to express basic conservation principles at the cell level, while offering computational efficiency since cell unknowns can be eliminated locally, leading to (compact-stencil) global problems with fewer unknowns. The monoscale HHO method has been bridged in [5] to the Hybridizable Discontinuous Galerkin (HDG) method and to the nonconforming Virtual Element Method (ncVEM). In the case of periodic coefficients, √ the msHHO method leads to √ energy-error estimates of the form ( ε + H k+1 + ε/H ). The msHHO method can be viewed as a higher-order, polytopal version of the msFEM à la Crouzeix–Raviart introduced in [10]. Other recent multiscale methods attaching discrete unknowns to the mesh faces include [9, 11, 12]. This contribution is organized as follows. In Sect. 2, we recall the oscillatory basis functions from [3] and discuss how they can be computed in each mesh cell. In Sect. 3, we outline the general workflow of the msHHO method. Finally, in Sect. 4, we present some numerical experiments illustrating how the accuracy in computing the oscillatory basis functions influences the accuracy of the msHHO method. For a detailed presentation of the implementation of the monoscale HHO method, we refer the reader to [4]. The present developments are part of the Disk++ library, available as open-source under MPL license (https://github.com/wareHHOuse/diskpp).

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511

2 Oscillatory Basis Functions Let GΩ := (TΩ , FΩ ) be a coarse-scale mesh discretizing Ω, where TΩ and FΩ are, respectively, the collection of the mesh cells and faces. The oscillatory basis functions are defined on each mesh cell T ∈ TΩ . Let F∂T := {Fn }0≤n:= K∈Th K P1 g v˜ h dx. Then we have: < gh −   K K g, vh > = K∈Th K (P1 g(v˜h − vh ) + (P1 g − g)vh ) dx and, since v˜ h − vh have zero mean value over K, we obtain:

 < gh − g, vh > = ((P1K g − P0K g)(v˜h − vh ) + (P1K g − g)(vh − v˜h )) dx K∈Th K

≤C



 P1K g − P0K g0,K + P1K g − g0,K v˜h − vh 0,K

K∈Th

⎛

≤ Ch ⎝

⎞1/2 |g|21,K ⎠

v˜h − vh 0,K .

K∈Th

(16)

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C. Chinosi

This estimate allows us to achieve the proper order of convergence in L2 , as the numerical results will show.

4 Numerical Results Some numerical results have been presented in [6] in order to assess the accuracy and the performance of the element of degree two, constructed by taking the L2 projection of the load onto P0 (K). In this section we compare this element, that we will denote with V EM0 , with the element, named V EM1 , constructed by using the L2 - projection of the load onto P1 (K). Moreover, we will make the comparison between the virtual element V EM1 and the finite element MITC9. As a test problem we take an isotropic and homogeneous plate Ω = [0, 1]2, clamped on the whole boundary, for which the analytical solution is explicitly known (see [6] for all the details). We consider an approximated H 1 -norm error for the variables θ and w (see [6]) and the usual relative errors in the discrete L2 norm. Comparison Between the Two Different Approximations of the Load We compare the element V EM0 with the element V EM1 . We consider a sequence Th1 of uniform decompositions into N × N, (N = 8, 16, 32, 64) squares and a sequence Th2 of distorted hexagonal grids. The coarsest mesh of every sequence is depicted on the left of Fig. 1. We keep the thickness sufficiently small (t = 0.001) and we analyze the behaviour of the relative H 1 -errors, e1 (θ ) and e1 (w), and the relative L2 -errors, e0 (θ ) and e0 (w) for the two mesh sequences, for the elements V EM0 and V EM1 . On the left of the Fig. 2 from top to bottom we report the convergence curves against the mesh diameter in log-log scale. We observe that the convergence rates of the H 1 -errors for both elements are in agreement with the theoretical predictions, while the convergence rates of the L2 -errors show that the P1 approximation of the load term lead to the optimal O(h3 ) order in L2 -norm, contrary to the P0 approximation that gives a O(h2 ) order. Comparison Between the V EM1 Element and the MITC9 Finite Element We compare the behaviour of the element V EM1 , with the finite element MITC9. We

Fig. 1 The coarsest grid of the meshes. Left: Th1 and Th2 . Right: Th3 and Th4

VEM-Mindlin

525

H1 - rotation errors, t=10-3

100

100

10-1

e1 ( )

e1 ( )

10-1

H1 - rotation errors t=10 -3

10-2 VEM0 -uniform

10-2

VEM0 -distorted hexagon VEM1 -uniform

10-3

MITC slightly distorted MITC strongly distorted VEM1 slightly distorted

10-3

VEM1 -distorted hexagon

VEM1 strongly distorted

h2

10-4 10-2

10-1

10-4 10-2

100

mean diameter 10

0

10-1

H1 - deflection errors t=10 -3

e1 (w)

10-2

10-2 VEM0 -uniform

10-3 MITC slightly distorted MITC strongly distorted VEM1 slightly distorted

VEM0 -distorted hexagon

10-3

10-4

VEM1 -uniform

VEM1 strongly distorted

VEM1 -distorted hexagon h

10-4 -2 10

h2

2

10-1

10-5 -2 10

100

mean diameter 100

100

mean diameter

H1 - deflection errors, t=10-3

10-1

e1 (w)

h2

10-1

10-1

100

mean diameter

L2 - rotation errors, t=10-3

10-1

10-1

L2 - rotation errors t=10-3

10-2

10-3

e0 ( )

e0 ( )

10-2

VEM0 -uniform

10-4

VEM0 -distorted hexagon VEM1 -uniform

10-5

10-3

10

MITC slightly distorted MITC strongly distorted VEM1 slightly distorted

-4

VEM1 strongly distorted

VEM1 -distorted hexagon

h3

h3

10

-6

10-2

h

2

10-1

10-5 10-2

100

mean diameter 100

L2 - deflection errors, t=10-3

10-1

100

L2 - deflection errors t=10-3

10-2

-2

10-3

VEM0 -uniform VEM0 -distorted hexagon

10-4

10-3 10-4

MITC slightly distorted MITC strongly distorted VEM1 slightly distorted

VEM1 -uniform VEM1 -distorted hexagon

10-5 10-6 10-2

e0 (w)

e0(w)

10-1 10

10-1

mean diameter

10-5

VEM1 strongly distorted

h3 h

10

h3

2

-1

mean diameter

10

0

10-6 -2 10

10-1

100

mean diameter

Fig. 2 Convergences curves. Left: VEM 0 compared with VEM 1 . Right: VEM 1 compared with MITC9

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100

H1 - rotation errors

10-1

H1 - deflection errors

MITC, t=10-3 MITC, t=10-4 MITC, t=10-5

10

-1

10-2

VEM1 , t=10-3

MITC, t=10-3

e1(w)

e1 ( )

VEM1 , t=10-4 VEM1 , t=10-5

MITC, t=10-4 MITC, t=10-5 VEM1 , t=10-3

10-3

10-2

VEM1 , t=10-4 VEM1 , t=10-5

10-3 -2 10

10-1

mean diameter

100

10-4 10-2

10-1

100

mean diameter

Fig. 3 Locking analysis: comparison between VEM 1 and MITC9 element on Th4 for different values of the thickness

consider two families of N × N distorted quadrilateral meshes: slightly distorted, named Th3 and strongly distorted, named Th4 . The coarsest mesh of every sequence is depicted on the right of Fig. 1. As in the previous case we keep the thickness (t = 0.001), we consider the relative H 1 -errors, e1 (θ) and e1 (w) for the V EM1 element, the usual relative errors in the discrete energy norm for the MITC9 element, and the relative L2 -errors, e0 (θ ) and e0 (w) for both elements. On the right of Fig. 2 from top to bottom we report the convergence curves against the mesh diameter in log-log scale. We observe that, as expected, the V EM1 element performs well and shows the same behaviour observed in the case of the meshes Th1 and Th2 . The MITC9 element, on the contrary, is sensitive to the mesh distorsion, in particular in the case of strongly distorsion. Remark 1 It is known that an important issue related to the finite element discretization of the Reissner-Mindlin plate model is the locking phenomenon. In [6] it has been proved that the Virtual Element Method here considered is robust with respect to the this phenomenon. Here, without being exhaustives, we will compare the behaviour of the V EM1 element with the MITC9 finite element when strongly distorted meshes are considered and the thickness becomes small. In Fig. 3 we plot the errors e1 (θ ) and e1 (w) for the values of the plate thickness: t = 10−i , i = 3, 4, 5 on the sequence of meshes Th4 . We note that, for both elements, the convergence curves do not significantly depend on t and the loss of convergence of the MITC9 element, already observed for t = 10−3 , does not increase when the thickness decreases. Acknowledgements This research has a financial support of the Università del Piemonte Orientale.

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References 1. L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo, Basic principles of virtual element methods. Math. Models Methods. Appl. Sci. 23(1), 199–214 (2013) 2. L. Beirão da Veiga, F. Brezzi, L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013) 3. L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016) 4. F. Brezzi, K.J. Bathe, M. Fortin, Mixed-interpolated elements for Reissner-Mindlin plates. Int. J. Numer. Methods Eng. 28, 1787–1801 (1989) 5. S. Cafiero, Virtual elements for the Reissner-Mindlin plates. Master’s degree thesis (Advisor: Beirão da Veiga L.), Università di Milano Bicocca, 2015 6. C. Chinosi, Virtual elements for the Reissner-Mindlin plate problem. Numer. Methods Partial Differ. Equ. 34(4), 1117–1144 (2018) https://doi.org/10.1002/num.22248 7. P.A. Raviart, J.M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics, vol. 606 (Springer, Berlin, 1977), pp. 292–315

Part XIV

Recent Advances in Space-Time Galerkin Methods

DGM for the Solution of Nonlinear Dynamic Elasticity Miloslav Feistauer, Martin Hadrava, Jaromír Horáˇcek, and Adam Kosík

Abstract The subject of the paper is the numerical solution of dynamic elasticity problems. We consider linear model and nonlinear Neo-Hookean model. First the continuous dynamic elasticity problem is formulated and then we pay attention to the derivation of the discrete problem. The space discretization is carried out by the discontinuous Galerkin method (DGM). It is combined with the backward difference formula (BDF) for the time discretization. Further, several numerical experiments are presented showing the behaviour of the developed numerical method in dependence on the coefficient in the penalty form. At the end the developed method is applied to the simulation of vibrations of 2D model of human vocal fold formed by four layers with different materials.

1 Continuous Elasticity Problems We assume that an elastic body is represented by a bounded domain Ω ⊂ R2 with boundary ∂Ω = ΓD ∪ ΓN . Let T > 0. We seek a displacement function u : Ω × [0, T ] → R2 such that ρ

∂ 2u ∂u − divP (F ) = f + cM ρ 2 ∂t ∂t u = uD

in Ω × [0, T ],

(1)

in ΓD × [0, T ],

(2)

M. Feistauer () · M. Hadrava · A. Kosík Charles University, Faculty of Mathematics and Physics, Praha 8, Czech Republic e-mail: [email protected]; [email protected] J. Horáˇcek Institute of Thermomechanics, The Academy of Sciences of the Czech Republic, Praha 8, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_48

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P (F ) · n = g N u(·, 0) = u0 ,

in ΓN × [0, T ],

(3)

in Ω.

(4)

∂u (·, 0) = z0 ∂t

Here f is outer volume force, ρ > 0 is material density, P denotes stress tensor and the quantity F depends on u as shown further. The expression cM ρ ∂u ∂t with cM > 0 represents structural damping. In the stationary case (static problem) we seek u : Ω → R2 such that −divP (F ) = f u = uD

1.1

on ΓD ,

in Ω,

(5)

P (F ) · n = g N

on ΓN .

(6)

Linear Elasticity

In case of linear elasticity the stress tensor depends linearly on the strain tensor e(u) = (∇u + ∇uT )/2 according to the relation P (F ) = σ (u) = λ tr(e(u))I + 2μe(u).

(7)

Here λ and μ are the Lamé parameters that can be expressed with the aid of the Young modulus E and the Poisson ratio ν: λ=

Eν , (1 + ν)(1 − 2ν)

μ=

E 2(1 + ν)

(8)

1.2 Nonlinear Elasticity In the case of nonlinear models we introduce the deformation mapping ϕ(x) = x + u(x), deformation gradient (i.e., the Jacobian matrix of the deformation mapping ϕ) F := ∇ϕ(x), the Jacobian of the deformation J = det F > 0 and the Green strain tensor E ∈ R2×2 defined by E=

 1 T F F −I , 2

E = (Eij )2i,j =1

(9)

with components 1 Eij = 2



∂uj ∂ui + ∂xj ∂xi



1 ∂uk ∂uk . 2 ∂xi ∂xj 2

+

k=1

(10)

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In the case of a nonlinear material we consider the Neo-Hookean model with the stress tensor P (F ) = μ(F − F −T ) + λlog(detF )F −T .

(11)

For a detailed description we can refer the reader to the monograph [3].

2 Discretization In the discretization of the structural problem we consider the displacement u and the deformation velocity z and split the basic system into two systems of first-order in time ρ

∂z + cM ρz − divP (F ) = f , ∂t

u(·, 0) = u0 ,

∂u −z=0 ∂t

in Ω × [0, T ],

(12)

u = uD

in ΓD × [0, T ],

(13)

P (F ) · n = g N

in ΓN × [0, T ],

(14)

in Ω.

(15)

z(·, 0) = z0

Let Ω be a polygonal domain.

2.1 Space DG Discretization Let Ω be a polygonal domain. We construct a partition Th of Ω into a finite number of closed triangles K with mutually disjoint interiors satisfying standard properties formulated in [2]. The approximate solution at every time instant t ∈ [0, T ] will be sought in the finite-dimensional space (cf. [4]) 2  S hs = v ∈ L2 (Ω); v|K ∈ Ps (K), K ∈ Th ,

(16)

where s > 0 is an integer and Ps (K) denotes the space of polynomials of degree ≤ s on K. By Fh we denote the system of all faces of all elements K ∈ Th and distinguish there sets of boundary, “Dirichlet”, “Neumann” and inner faces: FhB = = > > > = = Γ ∈ Fh ; Γ ⊂ ∂Ω , FhD = Γ ∈ Fh ; Γ ⊂ ΓD , FhN = Γ ∈ Fh ; Γ ⊂ ΓN and FhI = Fh \FhB . For each Γ ∈ Fh we define a unit normal vector nΓ . We assume that for Γ ∈ FhB the normal nΓ has the same orientation as the outer normal to ∂Ω. (R) By hΓ we denote the length of Γ . For ϕ ∈ S hs the symbols ϕ (L) Γ and ϕ Γ denote (L) (R) the traces of ϕ ∈ S hs on Γ from the sides of elements KΓ and KΓ adjacent to

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G H (L) Γ . We assume that nΓ is the outer normal to ∂KΓ . Further, ϕ Γ denotes average   (R) of the traces on Γ and ϕ Γ = ϕ (L) Γ − ϕ Γ is the jump of ϕ on Γ . The DG discretization in space is formulated with the use of following forms. Linear elasticity form: ah (u, ϕ)



 G H    = σ (u) · n · ϕ dS σ (u) : e(ϕ) dx − K∈Th K

Γ ∈FhI

(17)

Γ

 

 G H   σ (ϕ) · n · [u] dS σ (u) · n · ϕ dS − θ

−

Γ

Γ ∈FhD

 

−θ

Γ ∈FhD

Γ ∈FhI

Γ

 σ (ϕ) · n · u dS.

Γ

Here θ is chosen as 1, 0, −1 for SIPG, IIPG, NIPG, respectively, version of the elasticity form. Nonlinear IIPG elasticity form (θ = 0): ah (u, ϕ) =



P (F ) : ∇ϕ dx −

K∈Th K

−

  Γ ∈FhD

     P (F )n · ϕ dS

(18)

Γ

Γ ∈FhI

 P (F ) n · ϕ dS,

Γ

penalty form :

 CW

 CW   Jh (u, ϕ) = u · ϕ dS. [u] · ϕ dS + hΓ hΓ I Γ D Γ Γ ∈Fh

(19)

Γ ∈Fh

Here CW > 0 is a sufficiently large constant. In what follows it is shown that CW should have the order as the Young modulus. Right-hand side form:



h (ϕ)(t) =

f (t) · ϕ dx +

K∈Th K

 Γ ∈FhN

Γ

g N (t) · ϕ dS

 

 CW  σ (ϕ) · n · uD (t) dS + −θ uD (t) · ϕ dS, hΓ D Γ D Γ Γ ∈Fh

Finally, we set Ah = ah + Jh ,

Γ ∈Fh

   u, ϕ Ω = Ω u · ϕ dx.

(20)

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In the nonlinear case, it is not clear how to define the SIPG and NIPG versions of the elasticity forms.

2.2 Full Space-Time Discretization Because of the time discretization we introduce a partition 0 = t0 < · · · < tM = T of the time interval [0, T ] and set τm = tm − tm−1 for m = 1, . . . , M. By um h we denote the approximate solution at time instant tm . The time discretization is carried out with the use of a backward difference formula (BDF)

Dappr um ∂u h (tm ) ≈ = α0 um α um− . h + h ∂t Dt q

(21)

=1

Similar formula is used for the approximation of ∂z/∂t. The coefficients α depend on time steps τm , τm−1 , . . .. See, e.g. Table 1 in [1]. m Now we come to the complete BDF-DG discrete problem: Find um h , zh ∈ S hs such that for all ϕ h ∈ S hs , m = 1, . . . , M, 

Dappr zm h a) ρ , ϕh Dt



   m  + cM ρ z m h , ϕ h Ω + Ah uh , ϕ h

(22)

Ω

= h (ϕ h ) (tm ),

(23)

   Dappr um h , ϕh b) − zm h , ϕ h Ω = 0, Dt Ω     0 c) uh , ϕ h = u0 , ϕ h Ω , Ω     d) z0h , ϕ h = z0 , ϕ h Ω . 

Ω

(24) (25) (26)

In the stationary case we seek uh ∈ S hs such that Ah (uh , ϕ h ) = h (ϕ h )

∀ϕ h ∈ S hs .

(27)

2.3 Realization of the Discrete Dynamic Elasticity Problem We express the sought approximate solution=at >each time step as a linear combination of basis functions in the space S hs . Let ξ i , i = 1, . . . , 2N, form the basis of

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  m S hs . Then the sought solution couple um h , zh , m = 0, . . . , M, can be expressed as 1,m um )= h = uh (β

2N

βim ξ i ,

2,m zm )= h = zh (β

i=1

2N

m β2N+i ξi,

(28)

i=1

T   4N where β m = β 1,m , β 2,m = βim i=1 are the finite element coefficients. Plugging these expressions into (22) and (24) and setting v h = ξ i , i = 1, . . . , 2N, we arrive at the system of algebraic equations for unknowns β 1,m and β 2,m in the form 

k(β 1,m ) + (α0 + CM )ρMβ 2,m α0 Mβ 1,m − Mβ 2,m

 − bm = 0,

(29)

where :  ;2N = >2N M = mij i,j =1 = ξ j , ξ i

Ω i,j =1

,

:  ;2N k(β 1,m ) = Ah uh (β 1,m ), ξ j , j =1

 4N bm = bjm

j =1

⎧   q ⎪ m−l ⎨ h (ξ j )(tm ) − ρ α z , ξ , j = 1, . . . , 2N, l j h Ω l=1 = q m−l ⎪ , j = 1, . . . , 2N. ⎩ l=1 αl uh , ξ j Ω

Nonlinear discrete problems are solved by the Newton method. Linear systems are solved by the direct solver UMFPACK or iterative method GMRES.

3 Analysis and Application of the Developed Method 3.1 Choice of the Penalty Coefficient It appears that the quality of an approximate solution obtained by the DGM-BDF method depends on the choice of the parameter CW from the penalty form (19). For the analysis of this question we use a simple static problem (27) in the unit square with the Dirichlet boundary condition and such data that the exact solution has the form T  u = sin(2(x1 + 1)) cos(x2 + 1)2 , cos(π(x1 + 1)) sin(x2 + 1)2 .

(30)

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Fig. 1 IIPG: L2 -norm error, P1 approximation (left), L2 -norm error, P2 approximation (right)

Fig. 2 IIPG: L2 -norm error, P3 approximation

The material is characterized by the Young modulus E = 1.0 · 105 and the Poisson ratio ν = 0.3. We solve the stationary problem on different meshes with their sizes h, using polynomial orders s = 1, 2, 3. For every computation we estimate the error in L2 -norm in dependence on the parameter CW . In Figs. 1 and 2 we see that for linear elasticity the optimal value of the parameter CW ≈ E can be approximated by the Young modulus E. This value is also used in the nonlinear, nonstationary case.

3.2 Computation of Vocal Fold Vibrations This section is devoted to the solution of a dynamic nonlinear elasticity problem motivated by the simulation of the deformation of a domain representing a plane cut through a vocal fold. This domain is formed by several materials as shown in Fig. 3. On the bottom part of the boundary x1 ∈ [0, 0.0175]. The vertical component x2 ∈ [−0.009, −0.000726]. We prescribe the surface force acting on the body for short time and let the body vibrate further without any other influence. The following data are used: ρ = 1040 kg.m−3 , cM = 0, f = 0 in Ω × [0, T ], u0 = 0, v 0 = 0 in Ω. On the bottom, right and left straight part of the boundary we prescribe homogeneous Dirichlet boundary condition uD = 0 in ΓD × [0, T ], and on the curved part of the boundary we prescribe the Neumann boundary condition with

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Fig. 3 Nonhomogeneous model of vocal folds: visualization of the deformation at time instants 0.0, 0.002, 0.004, 0.006 s

g N = −pN n

in ΓN × [0, T ], where pN is the pressure force defined as ⎧ ⎪ ⎪ ⎪ ⎨

800 for x1 ∈ [0, 0.00875), t ∈ [0, 0.02), −1000 for x1 ∈ [0.00875, 0.015), t ∈ [0, 0.02), pN = ⎪ 200 for x1 ∈ [0.015, 0.0175], t ∈ [0, 0.02), ⎪ ⎪ ⎩ 0 else.

(31)

The computation was carried out with time steps τ = 10−3 and 10−4 . The tolerance 10−8 in the Newton method was achieved after 2 or 3 Newton iterations. In our computations the convergence of the Newton method was independent of the time step. The visualization of the solution (with τ = 10−4) at time instants 0.0, 0.002, 0.004, 0.006 using the neo-Hookean material is in Fig. 3. (The results are ordered in such a way that the upper half shows the patterns corresponding to the time instants t = 0.0, 0.002 s and the lower half corresponds to t = 0.004, 0.006 s.) Further, in Fig. 4 the displacement u of the point [0.01, −0.001] in the horizontal direction x1 and the vertical direction x2 is shown. Moreover, this figure also shows the Fourier analysis of the displacement in the x1 -direction computed by the linear and Neo-Hookean models. It is interesting that the amplitude of vibrations computed by the linear model is larger than in the nonlinear case. This shows that the Neo-Hookean material is more stiff. It will require further analysis using also other nonlinear elasticity models in order to find the reason for this phenomenon.

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Fig. 4 Nonhomogeneous model of vocal folds: comparison of the displacement of the point [0.01, −0.001] in the horizontal x1 -direction (above) and the vertical x2 -direction (middle) computed by the linear and Neo-Hookean models. Comparison of the Fourier analysis of the displacement in the x1 -direction of the point [0.01, −0.001] computed by the linear and Neo-Hookean models (below)

Acknowledgements This research was supported under the grants of the Czech Science Foundation No. 17-01747S (M. Feistauer, M. Hadrava, A. Kosík) and 16-01246S (J. Horáˇcek).

References ˇ 1. J. Cesenek, M. Feistauer, A. Kosík, DGFEM for the analysis of airfoil vibrations induced by compressible flow. Z. Angew. Math. Mech. 93(60–67), 387–402 (2013) 2. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications, vol. 4 (North-Holland, Amsterdam, 1978)

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3. P.G. Ciarlet, Mathematical Elasticity, Volume I, Three-Dimensional Elasticity. Studies in Mathematics and its Applications, vol. 20 (Elsevier Science Publishers B.V., Amsterdam, 1988) 4. V. Dolejší, M. Feistauer, Discontinuous Galerkin Method, Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics, vol. 48 (Springer, Cham, 2015)

Higher Order Space-Time Elements for a Non-linear Biot Model Manuel Borregales and Florin Adrian Radu

Abstract In this work, we consider a non-linear extension of the linear, quasistatic Biot’s model. Precisely, we assume that the volumetric strain and the fluid compressibility are non-linear functions. We propose a fully discrete numerical scheme for this model based on higher order space-time elements. We use mixed finite elements for the flow equation, (continuous) Galerkin finite elements for the mechanics and discontinuous Galerkin for the time discretization. We further use the L-scheme for linearising the system appearing on each time step. The stability of this approach is illustrated by a numerical experiment.

1 Introduction Flow in deformable porous media appears to be relevant for several important applications including groundwater hydrology, CO2 sequestration, geothermal energy, and subsidence phenomena. A commonly used mathematical model for flow in deformable porous media is the linear, quasi-static Biot model [7]. In this work, a generic, non-linear extension of the linear Biot model is studied. The volumetric strain in the mechanics deformation model and the fluid compressibility are assumed to be now non-linear. For a discussion concerning the considered model we refer to [8], where a discretization based on lowest order mixed finite elements, Galerkin finite elements (FE) and backward Euler was proposed and analysed. The non-linear Biot model consists on two fully coupled, non-linear partial differential equations. As a linearization we will use the robust, linearly convergent L-scheme [14, 17, 18]. The well-known fixed stress and fixed strain splitting schemes [1, 3, 9, 11, 12, 16] can be interpreted as particular cases of the L-scheme for coupled problems [10]. For the discretization in space and time we propose a higher order space-time method [5, 6]. We use the mixed finite element method (MFEM)

M. Borregales () · F. A. Radu Department of Mathematics, University of Bergen, Bergen, Norway e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_49

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for the flow equation, continuous Galerkin (cG) FE for the mechanics equation and discontinuous Galerkin (dG) FE for the discretization in time. We refer to [5] for a similar approach for the linear Biot model. In the future we plan to extend the methodology for more complex non-linear models and to the fully-dynamic BiotAllard system [15]. The paper is structured as follows. In the next section we briefly introduce the considered non-linear Biot model. In Sect. 3 we present the space-time discretization and announce a convergence result. Numerical simulations are shown in Sect. 4. Finally, concluding remarks are given in Sect. 5.

2 A Non-linear Biot’s Model We consider flow of a slightly compressible fluid in a non-linear elastic, homogeneous, isotropic, porous medium Ω ⊂ Rd , with d = 2, 3 being the spatial dimension. The medium is assumed to be saturated. We extend the linear Biot consolidation model by considering a non-linear volumetric strain and non-linear fluid compressibility. On the space-time domain Ω × (0, T ], where T denotes the final time, the governing equations read as follows: − ∇ · [2με(u) + h(∇ · u) − α(pI )] = ρb g,   ∂t b(p) + α∇ · u + ∇ · q = f,   q = −K ∇p − ρf g ,

(1) (2) (3)

where u is the displacement, p is the fluid pressure and q is the Darcy flux. We denoted by h(·) the non-linear volumetric stress in the mechanics equation (1). t is the linearised strain tensor, μ is the shear modulus, α is Further, ε(u) := ∇u+(∇u) 2 the Biot’s coefficient, ρb is the bulk density and g the gravity vector. The non-linear compressibility is denote by b(·), f is a volume source term, K is the permeability tensor divided by fluid viscosity and ρf is the fluid density. For simplicity, we assume homogeneous Dirichlet boundary conditions u = 0, p = 0 on ∂Ω × (0, T ]. At the initial time we assume u = u0 , p = p0 in Ω × {0}.

3 A Fully Discrete Higher Order Numerical Scheme Throughout our paper we use common notations of functional analysis. Let L2 (Ω) be the space of Lebesque measurable and square integrable functions on Ω and H m (Ω), m ≥ 1 be the space of L2 -functions having weak derivates up to order m in L2 (Ω). We denote by ·, · and · the inner product and norm in L2 (Ω). We are using bold letters for variables, functions or spaces which are vectors or tensors. For

High Order Time Discretization for Non-linear Biot Model

rank 2 tensors A, B ∈ Rd,d A : B := the spaces

 d

i,j =1

Ω

543

Aij Bij dx. Further, we consider

H10 (Ω) := {u ∈ H1 (Ω)| u = 0 on ∂Ω}, H(div; Ω) := {q ∈ L2 (Ω) | ∇ · q ∈ L2 (Ω)}. Let X0 ⊂ X ⊂ X1 be three reflexive Banach spaces with continuous embedding and let I = (0, T ) be the time interval. Following [5, 6] we consider the following set of Bochner spaces ⎧ ⎨

⎫ B ⎬ B T B L2 (I ; X) = w : (0, T ) → X B w(t)2X dt < ∞ , ⎩ ⎭ B 0 H 1 (I ; X0 , X1 ) = {w ∈ L2 (I ; X0 ) | ∂t w ∈ L2 (I, X1 )}, that are equipped with their natural norms and where the time derivative ∂t is understood in the sense of distributions. In particular, every function in H 1 (I ; X0 , X1 ) is continuous on I with values in X. For X0 = X = X1 we simply write H 1 (I ; X).

3.1 Variational Formulation of the Non-linear Biot Model We can now proceed and state the variational formulation of the considered nonlinear Biot model (1)–(3):       Find u ∈ H 1 I ; H1 (Ω) ∩ L2 I ; H10 (Ω) , q ∈ L2 I ; H div; Ω and p ∈   H 1 I ; L2 (Ω) such that: 

 ε(u) : ε(v)dτ +

2μ I

h(∇ · u), ∇ · vdτ I



 p, ∇ · vdτ =

−α I



K−1 q, zdτ − I









I

p, ∇ · zdτ = I

∂t (b(p) + α∇ · u), wdτ + I



I

∇ · q, wdτ = I

ρb g, vdτ,

(4)

ρf g, zdτ,

(5)

f, wdτ,

(6)

I

       for all v ∈ L2 I ; H10 (Ω) , z ∈ L2 I ; H div; Ω and w ∈ L2 I ; L2 (Ω) . We refer to [5] for the linear case of the above system.

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3.2 A L-Scheme Type Linearization The non-linear system above (4)–(6) can be solved monolithically [8] or by a splitting approach [5]. Following [8, 14], a monolithic version of the L-scheme applied to the system (4)–(6) reads as:   Given u0 = u0 , q0 = q0 and p0 = p0 , for s ≥ 1, find us ∈ H 1 I ; H1 (Ω) ∩        L2 I ; H10 (Ω) , qs ∈ L2 I ; H div; Ω and ps ∈ H 1 I ; L2 (Ω) such that there holds   s 2μ ε(u ) : ε(v)dτ + L2 (∇ · δus ) − αps , ∇ · vdτ I

I

 I



K−1 qs , zdτ − I



  ∂t L1 δps + α∇ · us , wdτ + I



for all v ∈ L2

ρb g, v, 

I



I

ps , ∇ · zdτ =



I

ρf g, zdτ,

∇ · qs , wdτ = I





h(∇ · us−1 ), ∇ · vdτ =

+

f − ∂t b(ps−1 ), wdτ, (7) I

      I ; H10 (Ω) , z ∈ L2 I ; H div; Ω and w ∈ L2 I ; L2 (Ω) ,

where δ(·)s := (·)s − (·)s−1 . The monolithic L-scheme introduced above can be modified by replacing ∇·us ≈ ∇ · us−1 in (7) to obtain a fixed stress type of splitting scheme. We refer to [8] for the details.

3.3 Discretization in Time: Discontinuous Galerking dG(r) The time interval (0, T ] is decomposed in N subintervals In = (tn−1 , tn ], where n=1, . . . , N , 0 = t0 < t1 < . . . < tn−1 < tn = T and τn = tn −tn−1 . Moreover τ = max1≤n≤N τn denotes the time discretizations parameter. In order to define a higher order dG scheme in time we need to introduce the space of piecewise polynomials of order r in time with coefficients in a Banach space X and the associated Bochner space X r (X): ⎧ ⎪ ⎨

⎫ B B ⎪ r ⎬

j B j Pr (In ; X) := ψn : In → X BBψn (t) = ξn t j , ξn ∈ X, j = 0, . . . , r , ⎪ ⎪ B ⎩ ⎭ j =0 : ; B B X r (X) := ψτ ∈ L2 (I ; X) B ψτ |In = ψn ∈ Pr (In ; X); ∀n ∈ {1, . . . , N} .

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We can now state a semi-discrete variational form of the system (1)–(3). We mention that the test functions semi-discrete scheme reads  vanishing on I \In . The  as:  ψn are   Find usτ ∈ X r H1 (Ω) , qsτ ∈ X r H div; Ω and pτs ∈ X r L2 (Ω) , such that    s s s 2μ ε(uτ ) : ε(vτ )dτ + L2 (∇ · δuτ ) + αpτ , ∇ · vτ dτ = ρb g, vτ dτ In

In

K−1 qsτ , zτ dτ

 −

In



In

h(∇ · us−1 τ ), ∇ · vτ dτ,

− 

In



 pτs , ∇

· zτ dτ =

In

  ∂t L2 δpτs + α∇ · usτ , wτ dτ +

In

ρf g, zτ dτ, In



∇ · qsτ , wτ dτ In

  +  L2 pτs + α∇ · usτ n−1 , wτ (tn+ ) = − 6 5 −  b(pτs−1 )

n−1

, wτ+ (tn−1 ) −

 ∂t b(pτs−1 ), wτ dτ In



f, wτ dτ, In

       for all vτ ∈ X r H10 (Ω) , zτ ∈ X r H div; Ω , wτ ∈ X r L2 (Ω) . We also

used the notations [wτ ]n−1 = wτ+ (tn−1 ) − wτ− (tn−1 ), wτ+ (tn−1 ) = wτ |In (tn−1 ) and wτ− (tn−1 ) = wτ |In−1 (tn−1 ). In the next we represent usτ |In , qsτ |In and pτs |In , in terms of the basis functions with        respect to the time variable of X r H1 (Ω) , X r H div; Ω and X r L2 (Ω) , j

respectively. For this, let tn , for j = 0, . . . , r be the (r + 1) Gauss quadrature points j on In . We define ψn to be the Lagrange polynomial of degree r, which satisfies j i ψn (tn ) = δˆj i , with δˆ being the Kronecker symbol. Now we can write usτ |In (t) =

r

j =0

s,j

j

un ψn (t), qsτ |In (t) =

r

j =0

s,j

j

qn ψn (t), pτs |In (t) =

r

s,j

j

pn ψn (t).

j =0

Then, by taking vτ = vψni , zτ = zψni and wτ = wψni ,i = 0, . . . , r in the semidiscrete problem above, we get the equivalent formulation on each time interval In :

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s,j

Find un ∈ H01 (Ω), qn 0, . . . , r such that

  s,j ∈ H div; Ω and pn ∈ L2 (Ω) for every j =

s,i s,i 2με(us,i n ), ε(v) + L2 ∇ · δun − αpn , ∇ · v i = ρb g, v − h(∇ · us−1 n (tn )), ∇ · v, s,i K−1qs,i n , z − pn , ∇ · z = ρf g, z, r 

 s,j s,j i αij L1δpn + α∇ · un , w + βii ∇ · qs,i n , w = βii f (tn ), w

j =0

−

r 

 I J j j + − ) α∇ · u− + b(p ), w , αij b (pns−1 (tn ))pns−1 (tn ), w − ψni (tn−1 n−1 n−1

j =0

  holds true ∀i = 0, .., r and for all v ∈ H10 (Ω), z ∈ H div; Ω , w ∈ L2 (Ω). The  j j+ coefficients above are defined by αij := In ∂t (ψn )ψni dt +ψn (tn−1 )ψni+ (tn−1 ) and  βii := In φni ψni dt, see [5, 6] for more details.

3.4 Discretization in Space by cG(p+1)-MFEM(p) We proceed by formulating now a fully discrete scheme for solving (1)–(3). Let Kh be a regular decomposition of Ω into d-simplices. We denote by h the mesh diameter. The lowest order of the discrete spaces cG(1)-MFEM(0) are given by d Vh := {vh ∈ H 1 (Ω) ; vh |K ∈ Pd1 , ∀K ∈ Kh }, Wh := {wh ∈ L2 (Ω) ; wh |K ∈ P0 , ∀K ∈ Kh } and Zh := {zh ∈ H (div; Ω) ; zh|K (x) = a + bx, a ∈ Rd , b ∈ R, ∀K ∈ Kh }. Further higher order cG(p+1)-MFEM(p) for p > 1 are described in [4, 6]. This space discretization is not uniformly inf-sup stable with respect to the physical parameters. In this regard, a small enough h is needed to avoid oscillations [19]. Then the fully discrete scheme for solving (1)–(3) on each time interval In reads as follows: s,i s,i For every i ∈ {0, . . . , r}, find us,i n,h , ∈ Zh , qn,h ∈ Vh and pn,h ∈ Wh , such that there holds for all vh ∈ Vh , zh ∈ Zh and wh ∈ Qh : s,i s,i 2με(us,i n,h ), ε(zh ) + L2 ∇ · δun,h − αpn,h , ∇ · vh  = ρb g, vh 

−h(∇ · us−1,i ), ∇ · vh , n s,i K−1 qs,i n,h , zh  − pn,h , ∇ · zh  = ρf g, zh , r  

s,j i , w  + βii ∇ · qs,i αij L1 δpn,h + α∇ · us,i h n,h n,h , wh  = βii f (tn ), wh  j =0

High Order Time Discretization for Non-linear Biot Model

−

547

r  J  I

s−1 j s−1 j + αij b (pn,h (tn ))pn,h (tn ), wh  − ψni (tn−1 ) α∇ · u− n−1 , wh j =0

J I + − −ψni (tn−1 ) b(pn−1 ), wh . We end this section by postulating the following convergence result. Theorem 1 Assuming that the functions h(·) and b(·) are Lipschitz continuous and that the time step is small enough, then the fully discrete scheme above converges for any L1 ≥ Lb and L2 ≥ Lh . The proof combines the ideas in [5] with the ones in [8].

4 Numerical Results We solve the non-linear Biot problem (1)–(3) in the unit-square Ω = (0, 1)2 and time interval I = [0, 1]. We consider K = νf = M = α = λ = μ = 1.0. The mesh size is h = 0.1 and the time step size is τ = 0.1 For all cases, we use as stopping criterion for the iterations δpi  + δqi  + δui  ≤ 10−8 . For dG(0) and dG(1), we investigated a range of values for L1 and L2 to assess the sensitivity of the proposed L-scheme with respect to these parameters. All numerical experiments were implemented using the open-source finite element library deal.II [2] and the DTM++ framework [5]. Figure 1 illustrates the number of iterations for the L-scheme for different values of L1 and L2 at the last time step. For both dG(0) and dG(1), the L-scheme is more sensitive on the choice of the stabilizing parameter L2 . The fastest convergence for dG(0) scheme is obtained when L1 ∼ Lb and L2 ∼ Lh . Nevertheless, for dG(1) the fastest convergence is obtained for a different value of L1 ∼ 0.5Lb .

Fig. 1 Performance of L-scheme for different values of L1 and L2 for test problem 1, b(p) = ep ; 3 5 h(∇ · u) = (∇ · u) + ∇ · u, dG(0) to the left and dG(1) to the right

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Fig. 2 Performance of L-scheme for different mesh size h, time step τ and order of the elements at the last time step

Figure 2 shows the performance of the L-scheme for different discretizations in space and time. We observe, in accordance with the theoretical results [8, 14], that a larger time step leads to a faster convergence. Furthermore, the convergence seems not to be affected by the mesh size h, but slightly depends on the order of the spatial discretization. We remark that the algebraic systems obtained by using the higher-order spacetime discretizaton are more challenging to solve, see also [5]. A preconditioner based on the splitting L-scheme is a promising choice to solve the algebraic system efficiently, see [8, 13, 20].

5 Conclusions We considered a generic non-linear Biot model to be used for simulation of flow in deformable porous media. We proposed a higher order space-time numerical scheme. Numerical results were shown to illustrate the performance of the scheme. A convergence result has been stated, a rigorous proof will be the subject of a followup paper. Acknowledgements This work was partially supported by the NFR-DAAD project EDIFY 255715 and the NFR project SUCCESS.

References 1. T. Almani, K. Kumar, A.H. Dogru, G. Singh, M.F. Wheeler, Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics. Comput. Methods. Appl. Mech. Eng. 311, 180–207 (2016)

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2. W. Bangerth, G. Kanschat, T. Heister, L. Heltai, G. Kanschat, The deal.II library version 8.4. J. Numer. Math. 24, 135–141 (2016) 3. M. Bause, Iterative coupling of mixed and discontinuous Galerkin methods for poroelasticity. arXiv:1802.03230 (2018) 4. M. Bause, U. Köcher, Variational time discretization for mixed finite element approximations of nonstationary diffusion problems. J. Comput. Appl. Math. 289, 208–224 (2015) 5. M. Bause, F. Radu, U. Köcher, Space–time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods. Appl. Mech. Eng. 320, 745–768 (2017) 6. M. Bause, F.A. Radu, U. Köcher, Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space. Numer. Math. 137(4), 773–818 (2017) 7. M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941) 8. M. Borregales, J.M. Nordbotten, K. Kumar, F.A. Radu, Robust iterative schemes for non-linear poromechanics. Comput. Geosci. 22, 1021–1038 (2018) 9. M. Borregales, K. Kumar, F.A. Radu, C. Rodrigo, F.J. Gaspar, A parallel-in-time fixed-stress splitting method for Biot’s consolidation model. arXiv:1802.00949 (2018) 10. J.W. Both, M. Borregales, J.M. Nordbotten, K. Kumar, F.A. Radu, Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017) 11. F.J. Gaspar, C. Rodrigo, On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics. Comput. Methods Appl. Mech. Eng. 326, 526–540 (2017) 12. J. Kim, H. Tchelepi, R. Juanes, Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits. Comput. Methods Appl. Mech. Eng. 200(13–16), 1591–1606 (2011) 13. U. Köcher, Space-time-parallel poroelasticity simulation. arXiv:1801.04984 (2018) 14. F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(2), 341–353 (2016) 15. A. Mikeli´c, M.F. Wheeler, Theory of the dynamic Biot-Allard equations and their link to the quasi-static Biot system. J. Math. Phys. 53(12), 123702 (2012) 16. A. Mikeli´c, M.F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 18(3–4), 325–341 (2013) 17. I. Pop, F. Radu, P. Knabner, Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168(1–2), 365–373 (2004) 18. F.A. Radu, J.M. Nordbotten, I.S. Pop, K. Kumar, A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015) 19. C. Rodrigo, X. Hu, P. Ohm, J.H. Adler, F.J. Gaspar, L. Zikatanov, New stabilized discretizations for poroelasticity and the Stokes’ equations. arXiv:1706.05169 (2017) 20. J.A. White, N. Castelletto, H.A. Tchelepi, Block-partitioned solvers for coupled poromechanics: a unified framework. Comput. Methods. Appl. Mech. Eng. 303, 55–74 (2016)

Iterative Coupling of Mixed and Discontinuous Galerkin Methods for Poroelasticity Markus Bause

Abstract We analyze an iterative coupling of mixed and discontinuous Galerkin methods for numerical modelling of coupled flow and mechanical deformation in porous media. The iteration is based on an optimized fixed-stress split along with a discontinuous variational time discretization. For the spatial discretization of the subproblem of flow mixed finite element techniques are applied. The spatial discretization of the subproblem of mechanical deformation uses discontinuous Galerkin methods. They have shown their ability to eliminate locking that sometimes arises in numerical algorithms for poroelasticity and causes nonphysical pressure oscillations.

1 Introduction and Mathematical Model We consider the quasi-static Biot system of flow in deformable porous media,   − ∇ · 2με(u) + λ∇ · uI − b pI = ρb g ,  1 p + ∇ · (bu) + ∇ · q = f , q = −K∇p , ∂t M

(1) (2)

in Ω × I for a bounded Lipschitz domain Ω ⊂ Rd , with d = 2, 3, and I = (0, T ]. For simplicity, it is supplemented by homogeneous initial and Dirichlet boundary conditions for p and u. We denote by u the unknown displacement field, ε(u) = (∇u + (∇u)? )/2 the linearized strain tensor, p the unknown fluid pressure, μ and λ the Lamé constants, b Biot’s coefficient, ρb the bulk density, M Biot’s modulus, q Darcy’s velocity and K the permeability field. Further, g denotes gravity or, in general, some body force and f is a volumetric source. We assume that g(0) = 0.

M. Bause () Helmut Schmidt University, Faculty of Mechanical Engineering, Hamburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_50

551

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M. Bause

The quantities μ, λ, b, ρb and M are positive constants. The matrix K is supposed to be symmetric and uniformly positive definite. Well-posedness of (1), (2) has been shown in the literature. Recently, iterative coupling schemes for solving the Biot system (1), (2) have attracted researchers’ interest and have shown their efficiency; cf., e.g., [1–3] and the references therein. Iterative coupling offers the appreciable advantage over the fully coupled method that existing and highly developed discretizations and algebraic solver technologies can be reused. Here, we use an “optimized fixedstress split” type iterative method; cf. [1–3]. For the time discretization of the arising subproblems of flow and mechanical deformation a discontinuous in time variational approach (cf. [1]) is applied. For the approximation of the pressure and flux unknown p and q within the iterative coupling approach we use a mixed finite element method. The displacement variable u is discretized by a discontinuous Galerkin method. Thereby the results herein represent the natural extension of the work of the author in [1] and further work in the literature, where the displacement field u was approximated by a continuous Galerkin method. The motivation for using a discontinuous Galerkin scheme for the discretization of the displacement u comes from combating the locking phenomenon, that sometimes arises in numerical algorithms for poroelasticity and manifests as spurious nonphysical pressure oscillations. In poroelasticity the locking-dominant parameter is the specific storage coefficient c0 = 1/M in Eq. (2). Locking primarily arises if c0 = 0 and, usually, does not appear for c0 = 0. For a more extensive discussion of locking in poroelasticity we refer to the literature; cf., e.g., [5, 6]. In [4] a coupling of mixed and discontinuous Galerkin finite methods is studied within a monolithic approach and an error analysis for the semi-discretization is space is given. By numerical experiments it is shown for the problem of Barry and Mercer, that a discontinuous Galerkin discretization of the displacement u is capable of eliminating spurious pressure oscillations related to locking arising in continuous discretizations of u. Therefore, it seems worth to study the combined mixed and discontinuous Galerkin approach also within a fixed-stress split iterative method which is done here. We use standard notation. In particular, we put W = L2 (Ω) and V = H (div; Ω) and denote by ·, · the inner product of W .

2 Iterative Coupling and Space-Time Discretization We consider solving the system (1), (2) by the following iteration scheme. Subproblem of Flow Let f k := f − b ∇ · ∂t uk + L ∂t pk ∈ L2 (I ; W ) be given. Find pk+1 ∈ H 1 (I ; W ), q k+1 ∈ L2 (I ; V ) such that pk+1 (0) = 0 and 

   1 +L ∂t pk+1 , w dt + ∇ · q k+1 , w dt = f k , w dt , M I I I

(3)

Mixed and Discontinuous Galerkin Methods for Poroelasticity



K −1 q k+1 , v dt − I

553

 pk+1 , ∇ · v dt = 0

(4)

I

for all w ∈ L2 (I ; W ) and v ∈ L2 (I ; V ). Subproblem of Mechanical Deformation Let pk+1 ∈ H 1 (I ; W ) be given. Find uk+1 ∈ H 1 (I ; H 1 (Ω)) ∩ L2 (I ; H 10 (Ω)) such that u(0) = 0 and 

 2με(uk+1 ), ε(z) dt + I

λ∇ · uk+1 , ∇ · z dt I



 = ρb

g, z dt + b I

(5)

pk+1 , ∇ · z dt I

for all z ∈ L2 (I ; H 10 (Ω)). In this scheme the artificial quantity L is a numerical parameter that was firstly introduced in [3] to accelerate the iteration process. The convergence of the iteration (3)–(5) is ensured for all L ≥ b2/(2λ); cf. [1, Thm. 2.1]. For the discretization we decompose the time interval (0, T ] into N subintervals In = (tn−1 , tn ], where n ∈ {1, . . . , N} and 0 = t0 < t1 < · · · < tN−1 < tN = T and τ = maxn=1,...N (tn − tn−1 ). We denote by Th = {K} a finite element decomposition of mesh size h of the polyhedral domain Ω into closed subsets K, int quadrilaterals in two dimensions and hexahedrals in three dimensions. Further, Eh, is the set of all interior edges (faces for d = 3). To each interior edge (or face) e ∈ Ehint we associate a fixed unit normal vector ν e . For the spatial discretization of (3) and (4) we use a mixed finite element approach. We choose the class of Raviart-Thomas elements for the two-dimensional case and the class of Raviart–Thomas–Nédélec elements in three space dimensions, where for s ≥ 0 the space Whs ⊂ W with Whs = {wh ∈ L2 (Ω) | wh|K ◦ TK ∈ Qs } and V sh ⊂ V denote the corresponding inf-sup stable pair of finite element spaces; cf. [1]. Here, Qs is the space of polynomials that are of degree less than or equal to s in each variable and TK is a suitable invertible mapping of the reference cube K to K of Th . For the spatial discretization of (5) we use a discontinuous Galerkin method with the space H lh = {zh ∈ L2 (Ω) | zh|K ◦ TK ∈ Qdl , zh|∂Ω = 0} . The fully discrete space-time finite element spaces are then defined by r,s Wτ,h = {wτ,h ∈ L2 (I ; W ) | wτ,h|In ∈ Pr (In ; Whs ) , wτ,h (0) ∈ Whs } ,

(6)

2 s s V r,s τ,h = {v τ,h ∈ L (I ; V ) | v τ,h|In ∈ Pr (In ; V h ) , v τ,h (0) ∈ V h } ,

(7)

2 2 l l Z r,l τ,h = {zτ,h ∈ L (I ; L (Ω)) | zτ,h|In ∈ Pr (In ; H h ) , zτ,h (0) ∈ H h } ,

(8)

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where Pr (In ; X) is the space of all polynomials in time up to degree r ≥ 0 on In with values in X. We choose l = s + 1 to equilibrate the convergence rates of the spatial discretization for the three unknowns p, q and u; cf. [4, p. 426, Thm. 1]. For short, we put Wh = Whs , V h = V sh and H h = H s+1 h . k ∈ W r,s , q k ∈ On each time interval In we expand the discrete functions pτ,h τ,h τ,h r,s+1 k in time in terms of Lagrangian basis functions ϕn,j ∈ V r,s τ,h and uτ,h ∈ Z τ,h Pr (In ; R) with respect to r + 1 nodal points tn,j ∈ In , for j = 0, . . . , r, k pτ,h |In (t) =

r

j,k

Pn,h ϕn,j (t) ,

zkτ,h |In (t) =

j =0

r

j,k

Z n,h ϕn,j (t) ,

(9)

j =0 j,k

j,k

for t ∈ In and z ∈ {q, u} with coefficient functions Pn,h ∈ Wh , Qn,h ∈ V h for j,k

z = q and U n,h ∈ H h for z = u and j = 0, . . . , r. The nodal points tn,j are chosen as the quadrature points of the (r+1)-point Gauss quadrature formula on In which is exact for polynomials of degree less or equal to 2r + 1. Solving the variational problems (3), (4) and (5) in the fully discrete function spaces (6)–(8) and using a discontinuous test basis in time with support on In then leads us to the following fully discrete iteration scheme, referred to as the MFEM(s)dG(s+1)–dG(r) splitting scheme. Fully Discrete Subproblem of Flow Let n ∈ {1, . . . , N}. Find coefficient i,k+1 functions Pn,h ∈ Wh and Qi,k+1 n,h ∈ V h for i = 0, . . . , r such that

1 j,k+1 j,k+1 j,k αij Pn,h , wh  + L αij Pn,h − Pn,h , wh  + τn βii ∇ · Qi,k+1 n,h , wh  M r

r

j =0

j =0

= τn βii f (tn,i ), wh  − b

r

j =0

+ γi

αij

j,k

∇ · U n,h , wh K

(10)

K∈Th

1 ∞ − − pτ,h (tn−1 ), wh  + γi b ∇ · u∞ τ,h (tn−1 ), wh K , M K∈Th

i,k+1 K −1 Qi,k+1 n,h , v h  − Pn,h , ∇ · v h  = 0

(11)

− for all wh ∈ Wh , v h ∈ V h and i = 0, . . . , r, where pτ,h (tn−1 ) = 0 for n = 1 and − − − k ∞ pτ,h (tn−1 ) = limk→∞ pτ,h (tn−1 ) for n > 1, and similarly for u∞ τ,h (tn−1 ).

Fully Discrete Subproblem of Mechanical Deformation Let n ∈ {1, . . . , N}. Find coefficient functions U i,k+1 n,h ∈ H h for i = 0, . . . , r such that

i,k+1 i,k+1 σ (U i,k+1 n,h ), ε(zh )K + Jδ (U n,h , zh ) − Jd (σ (U n,h ), zh )

K∈Th

=

K∈Th

i,k+1 i,k+1 bPn,h , ∇ · zh K − b Jp (Pn,h , zh )

(12)

Mixed and Discontinuous Galerkin Methods for Poroelasticity

555

for all zh ∈ H h and i = 0, . . . , r with effective stress σ (u) = 2με(u) + λ∇ · u I . In Eq. (12) we use the notation

δe [y h ], [zh ]e , |e|β int

Jδ (y h , zh ) =

Jp (wh , zh ) =

e∈Eh

{wh } ν e , [zh ]e ,

e∈Ehint

  {σ (y h )ν e }, [zh ]e + {σ (zh )ν e }, [y h ]e . Jd (y h , zh ) = e∈Ehint

We denote by ·, ·K and ·, ·e the L2 inner products on K and e, respectively. As usual, we let {w} = ((w|K )|e + (w|K  )|e )/2 for two adjacent elements K and K  with common edge (or face) e and, similarly, [w] = (w|K )|e − (w|K  )|e . The penalty term Jδ contains the numerical parameter δe that takes a constant value at each edge (or face) e ∈ Ehint with Lebesgue measure |e|. The power β is a positive number that depends on the dimension d. In [4], the choice β = (d − 1)−1 is proposed for the fully coupled semi-discretization of (1), (2). The coefficients αij , βii and γi are    (t) · ϕ (t) dt + γ · γ , β = defined by αij = In ϕn,j ϕ (t) · ϕn,i (t) dt and n,i i j ii n,i In + γi = ϕn,i (tn−1 ) for i, j = 0, . . . , r. For a detailed derivation of the Eqs. (10)–(12) we refer to [1] for the space-time issue and to [4] for the discontinuous Galerkin approximation in space of the displacement field u. Eq. (12) is referred to as the symmetric interior penalty (SIP) discontinuous Galerkin method. Finally, we note that in [4] an additional penalty term involving ∂t u is proposed for the semidiscretization in space of a coupled approach.

3 Convergence of the Iteration Scheme Here we prove the convergence of the iteration scheme (10), (11), and (12). For r,s r,s+1 × V r,s denote the fully discrete this, let {pτ,h , q τ,h } ∈ Wτ,h τ,h , uτ,h ∈ Z τ,h MFEM(s)dG(s+1)–dG(r) approximation of (1), (2), formally given by passing to the limit k → ∞ in the scheme (10)–(12). Analogously to (9), let {pτ,h , q τ,h , uτ,h } j j j on In be represented by coefficients Pn,h ∈ Wh , Qn,h ∈ V h and U n,h ∈ H h for j = 0, . . . , r. For n ∈ {1, . . . , N} and t ∈ In we put j,k Ep

=

j,k Pn,h

−

j Pn,h ,

epk (t)

=

r

j =0

j,k

j,k Ep ϕn,j (t) ,

Spi,k

=

r

j,k

αij Ep .

j =0

j,k

i,k The quantities E q , ekq , E u , eku and S i,k q , S u are defined analogously. r,s r,s+1 Theorem 1 Let {pτ,h , q τ,h } ∈ Wτ,h × V r,s denote the fully τ,h , uτ,h ∈ Z τ,h discrete space-time MFEM(s)dG(s+1)–dG(r) approximation of (1), (2). Let k {pτ,h , q kτ,h , ukτ,h } be defined by (9) with coefficients being given by (10), (11)

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M. Bause

and (12). Then, if the parameter L > 0 and penalty function δ in Jδ of (12) are chosen sufficiently large, the sequence {Spi,k }k , for i = 0, . . . , r, k (t ± ), q k (t ± ), uk (t ± )} converges in Wh . This implies the convergence of {pτ,h n τ,h n τ,h n ± ± ± to {pτ,h (tn ), q τ,h (tn ), uτ,h (tn )} in Wh × V h × H h for k → ∞ and n = 1, . . . , N, k , q k and uk in L2 (I ; W ) and L2 (I ; L2 (Ω)), respectively. as well as of pτ,h n n τ,h τ,h Proof We split the proof into several steps. 1. Step (Error equations). By substracting Eqs. (10)–(12) from the fully discrete monolithic space-time approximation MFEM(s)dG(s+1)–dG(r), given by pτ,h , q τ,h and uτ,h , of the Biot system (1), (2) we obtain for i = 0, . . . , r that r r

1 j,k+1 j,k+1 j,k αij Ep , wh  + L αij Ep − Ep , wh  M j =0

j =0

+ τn βii ∇ · E i,k+1 , wh  = −b q

r

αij

j =0

j,k

∇ · E u , wh K ,

K −1 E i,k+1 , v h  − Epi,k+1 , ∇ · v h  = 0 , q

σ (E i,k+1 ), ε(zh )K + Jδ (E i,k+1 , zh ) − Jd (E i,k+1 , zh ) u u u K∈Th

=

(13)

K∈Th

bEpi,k+1 , ∇ · zh k − b Jp (Epi,k+1 , zh )

(14)

(15)

K∈Th

for all wh ∈ Wh , v h ∈ V h , zh ∈ H h . 2. Step (Choice of test function in Eq. (13)). We test Eq. (13) with wh = r j,k+1 . By some calculations following [1, p. 756] we get that j =0 αij Ep  1 L  i,k+1 2 L L + S i,k+1 − Spi,k 2 − S i,k 2 Sp  + Mβii 2βii 2βii p 2βii p b i,k+1 + τn ∇ · E i,k+1 , Spi,k+1  = − ∇ · S i,k K . q u , Sp βii

(16)

K∈Th

3. Step (Summation of Eq. (14) and choice of test function). Changing the index i in Eq. (14) to j , multiplying this equation with αij , summing up from j = 0 to r and testing with v h = τn E i,k+1 ∈ V h we have that q , E i,k+1  − τn Spi,k+1 , ∇ · E i,k+1  = 0. τn K −1 S i,k+1 q q q Adding Eq. (17) to Eq. (16) implies that  1 L  i,k+1 2 L L Sp  + + S i,k+1 − Spi,k 2 − S i,k 2 Mβii 2βii 2βii p 2βii p b i,k+1 + τn K −1 S i,k+1 , E i,k+1 =− ∇ · S i,k K . q q u , Sp βii K∈Th

(17)

(18)

Mixed and Discontinuous Galerkin Methods for Poroelasticity

557

4. Step (Summation of Eq. (15) and choice of test function). Changing the index i in Eq. (15) to j , multiplying the resulting equation with αij , summing up from j = 0 to r and choosing zh = βii−1 S i,k u ∈ H h we find that 1 1 1 σ (S i,k+1 ), ε(S i,k Jδ (S i,k+1 , S i,k Jd (S i,k+1 , S i,k u u )K + u u )− u u ) βii βii βii K∈Th

=

b b i,k+1 Sp , ∇ · S i,k Jp (Spi,k+1 , S i,k u K − u ). βii βii

(19)

K∈Th

Adding Eq. (19) to Eq. (18) leads to  1 L  i,k+1 2 L + S i,k+1 − Spi,k 2 + τn K −1 S i,k+1 , E i,k+1  Sp  + q q Mβii 2βii 2βii p 1 1 Jδ (S i,k+1 , S i,k σ (S i,k+1 ), ε(S i,k (20) + u u )+ u u )K βii βii K∈Th

=

L b 1 S i,k 2 − Jp (Spi,k+1 , S i,k Jd (S i,k+1 , S i,k u )+ u u ). 2βii p βii βii

5. Step (Formation of incremental equation for (15), summation and choice of test function.) Firstly, we write Eq. (15) for two consecutive iterations, k and k +1, and substract the resulting equations from each other. Secondly, we change the index i in the thus obtained equations to j , multiply them with αij and sum up from j = 0 to r to obtain that

i,k+1 σ (S i,k+1 − S i,k − S i,k u u ), ε(zh )K + Jδ (S u u , zh ) = b

K∈Th

Spi,k+1

K∈Th

− Spi,k , ∇ · zh K − bJp (Spi,k+1 − Spi,k , zh ) + Jd (S i,k+1 − S i,k u u , zh )

(21)

for all zh ∈ H h . Choosing zh = S i,k+1 − S i,k u u ∈ H h in (21), dividing by βii > 0 and summing up the resulting identity from i = 0 to r, we find that r

1  i,k+1 i,k+1 i,k+1 σ (S i,k+1 − S i,k − S i,k − S i,k u u ), ε(S u u )K + Jδ (S u u , Su βii i=0

K∈Th



− S i,k u ) =

r

1  bSpi,k+1 − Spi,k , ∇ · (S i,k+1 − S i,k u u )K βii i=0

K∈Th

(22)

 i,k+1 i,k+1 − b Jp (Spi,k+1 − Spi,k , S i,k+1 − S i,k − S i,k − S i,k u u ) + Jd (S u u , Su u ) .

558

M. Bause

Further, from Eq. (21) with zh = S i,k+1 −S i,k u u we get by means of the inequalities of Cauchy–Schwarz and Cauchy–Young that

i,k+1 i,k+1 i,k+1 σ (S i,k+1 − S i,k − S i,k − S i,k u u ), ε(S u u )K + Jδ (S u u , Su

K∈Th

− S i,k u )≤

λ b 2 i,k+1 2 Sp ∇ · (S i,k+1 − Spi,k 2 + − S i,k u u )K 2λ 2

(23)

K∈Th

i,k+1 i,k+1 − b Jp (Spi,k+1 − Spi,k , S i,k+1 − S i,k − S i,k − S i,k u u ) + Jd (S u u , Su u ).

The terms Jp and Jd can be bounded by means of (cf. [4, p. 429, p. 431]) c 1 wh 2 + Jδ (zh , zh ) , δmin R B B (24) c 1 B B {σ (y h )ν e }, [zh ]e B ≤ σ (y h ), ε(y h )K + Jδ (zh , zh ) B δmin R int int |Jp (wh , zh )| ≤

e∈Eh

e∈Eh

for R ∈ N, R > 1 and some constant c > 0, such that they can be absorbed by the left-hand side, if the penalty parameter δmin is chosen sufficiently large. 6. Step (Summation of Eq. (20) and combination with Eq. (22)). Using in (20) that 4x, y = x + y2 − x − y2 , summing up the resulting equation over i and using (22) together with (24)) we get that r 

L  i,k+1 2 L 1 Sp  + + S i,k+1 − Spi,k 2 Mβii 2βii 2βii p i=0

r  1 i,k+1 i,k+1 + τn K −1 S i,k+1 , E  + Jδ (S i,k+1 + S i,k + S i,k q q u u , Su u ) 4βii i=0

r r

1 L i,k+1 σ (S i,k+1 + S i,k + S i,k S i,k 2 + u u ), ε(S u u )K ≤ 4βii 2βii p i=0

K∈Th

i=0

r r

1 b2 i,k+1 i,k + bSpi,k+1 − Spi,k , ∇ · (S i,k+1 − S ) + c S K u u 4βii βii p i=0

− Spi,k 2 −

K∈Th r

i=0

 1  i,k+1 i,k ) − J (S , S ) . bJp (Spi,k+1 , S i,k d u u u βii

i=0

(25)

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For the second term T2 on the right-hand side, (23) and (24) yield that |T2 | ≤ c

r

b2 S i,k+1 − Spi,k 2 . λβii p i=0

On the left-hand side, the third term can be rewritten by [1, p. 760, Eq. (4.29)]. The fourth and fifth term are rewritten by first using the identity 2x, y = x, x + y, y − x − y, x − y and then applying (23) with (24). 7. Step (Contraction). We combine the results of the 5th and 6th step to r  

L  i,k+1 2 L 1 + Spi,k+1 − Spi,k 2 Sp  + Mβii 2βii 2βii i=0

τn τn + − 2 2 K −1/2 ek+1 K −1/2 ek+1 q (tn ) + q (tn−1 ) 2 2 r r

1 1 i,k+1 + σ (S i,k+1 ), ε(S ) + Jδ (S i,k+1 , S i,k+1 ) K u u u u 2βii 2βii +

i=0

+

r

i=0

−

i=0

r r

1 1 i,k i,k σ (S i,k Jδ (S i,k u ), ε(S u )K + u , Su ) 2βii 2βii i=0

≤

K∈Th

K∈Th

L S i,k 2 + c 2βii p

i=0

r

i=0

b2 S i,k+1 − Spi,k 2 λβii p

(26)

r r

b 1 Jp (Spi,k+1 , S i,k ) + Jd (S i,k+1 , S i,k u u u ). βii βii i=0

i=0

Using (24) the last two terms on the right-hand side of (26) can be absorbed by terms on the left-hand side, if L > 0 and the penalty function δ in Jδ of (12) are chosen sufficiently large, i.e. L > cb2 /λ. Inequality (26) then shows the convergence of the iterates Spi,k in Wh , for i = 0, . . . , r. From (26) along with + ) to 0 for the convergence of Spi,k we get the convergence of ekq (tn− ) and ekq (tn−1 k − k → ∞. Eq. (14) together with the convergence of eq (tn ) yields the convergence of epk (tn− ) to 0 for k → ∞. Finally, Eq. (15) implies the convergence of eku (tn− ) to 0. The rest follows as in [1, Cor. 4.6]. 

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References 1. M. Bause, F. Radu, U. Köcher, Space-time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods Appl. Mech. Eng. 320, 745–768 (2017) 2. J.W. Both, M. Borregales, J.M. Nordbotton, K. Kundan, F.A. Radu, Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017) 3. A. Mikeli´c, M. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17, 479–496 (2013) 4. P. Philips, M. Wheeler, A coupling of mixed and discontinuous Galerkin finite element methods for poroelasticity. Comput. Geosci. 12, 417–435 (2008) 5. P. Philips, M. Wheeler, Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput. Geosci. 13, 5–12 (2009) 6. S.-Y. Yi, A study of two modes of locking in poroelasticity. SIAM J. Numer. Anal. 55, 1915– 1936 (2017)

Stability of Higher-Order ALE-STDGM for Nonlinear Problems in Time-Dependent Domains Monika Balázsová and Miloslav Vlasák

Abstract In this paper we investigate the stability of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear convection-diffusion problem in time-dependent domains. At first we define the continuous problem and reformulate it using the Arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so called ALE-derivative and an additional convective term. Then the problem is discretized with the aid of the ALE space-time discontinuous Galerkin method (ALE-STDGM). The discretization uses piecewise polynomial functions of degree p ≥ 1 in space and q > 1 in time. Finally in the last part of the paper we present our results concerning the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties, namely its continuity in the ·L2 -norm and in special ·DG -norm.

1 Introduction Problems in time-dependent domains are very important in many areas of science and technology, for example, fluid-structure interaction problems. In this paper we deal with the stability analysis of the ALE-STDGM with arbitrary polynomial degree in space as well as in time, applied to a nonstationary, nonlinear convection-diffusion problem equipped with initial and Dirichlet boundary condition. The ALE-STDGM analyzed here corresponds to the technique used in [3] and [4] for the numerical simulation of airfoil vibrations induced by compressible flow, which means that the ALE mapping is not prescribed globally in the whole time interval, but separately for each time slab.

M. Balázsová () · M. Vlasák Faculty of Mathematics and Physics, Charles University, Praha 8, Czech Republic e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_51

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We present here new technique of theoretical analysis in contrast to [1] and [2], where we proved the unconditional stability of the ALE-STDGM with arbitrary polynomial degree in space, but only linear approximation in time. The new technique is based on generalization of the discrete characteristic function in timedependent domains.

2 Formulation of the Continuous Problem We consider an initial-boundary value nonstationary, nonlinear convection-diffusion problem in a time-dependent bounded domain Ωt , t ∈ (0, T ): Find a function u = u(x, t) with x ∈ Ωt , t ∈ (0, T ) such that ∂u ∂fs (u) + − div(β(u)∇u) = g ∂t ∂xs d

in Ωt , t ∈ (0, T ),

(1)

s=1

u = uD

on ∂Ωt , t ∈ (0, T ),

u(x, 0) = u (x), 0

x ∈ Ω0 .

(2) (3)

We assume that fs , β, g, uD , u0 are given functions, |fs | ≤ Lf , s = 1, . . . , d, and function β is Lipschitz-continuous and bounded: β : R → [β0 , β1 ] where 0 < β0 < β1 < ∞. Problem (1)–(3) can be reformulated using the Arbitrary Lagrangian-Eulerian (ALE) method. First we consider a standard ALE formulation prescribed globally in the whole time interval, used in a number of works (cf., e.g., . . . ). It is based on a regular one-to-one ALE mapping of the reference domain Ω0 onto the current configuration Ωt : At : Ω 0 → Ω t ,

X ∈ Ω 0 → x = x(X, t) = At (X) ∈ Ω t ,

t ∈ [0, T ].

(4)

Usually it is supposed that the ALE mapping is sufficiently regular, e.g., A ∈ W 1,∞ (0, T ; W 1,∞ (Ωt )). Now we introduce the domain velocity z˜ (X, t) =

∂ At (X), z(x, t) = z˜ (A−1 t (x), t), t ∈ [0, T ], X ∈ Ω0 , x ∈ Ωt , ∂t (5)

and define the ALE derivative of a function f = f (x, t) for x ∈ Ωt and t ∈ [0, T ] using the chain rule as ∂f Df = + z · ∇f, Dt ∂t

(6)

Stability of Higher-Order ALE-STDGM

563

which allows us to reformulate problem (1)–(3) in the ALE form: Find u = u(x, t) with x ∈ Ωt , t ∈ (0, T ) such that Du ∂fs (u) + − z · ∇u − div(β(u)∇u) = g Dt ∂xs d

in Ωt , t ∈ (0, T ),

(7)

s=1

u = uD

on ∂Ωt , t ∈ (0, T ),

u(x, 0) = u0 (x),

x ∈ Ω0 .

(8) (9)

Moreover we assume the following properties of the domain velocity: There exists a constant cz > 0 such that |z(x, t)|, |divz(x, t)| ≤ cz

for x ∈ Ωt , t ∈ (0, T ).

(10)

3 ALE–Space Time Discretization We consider a time partition 0 = t0 < t1 < · · · < tM = T and set τm = tm − tm−1 , Im = (tm−1 , tm ) for m = 1, . . . , M. The space-time discontinuous Galerkin method (STDGM) has an advantage that on every time interval I m = [tm−1 , tm ] it is possible to consider a different space partition. Here we also use this property of the STDGM in the ALE framework: we consider an ALE mapping separately on each time interval [tm−1 , tm ) for m = 1, . . . , M. The resulting ALE mapping in [0, T ] may be discontinuous at time instants tm , which means that A(tm −) = A(tm +) in general. Such situation appears in the numerical solution of fluidstructure interaction problems, when both the ALE mapping and the approximate flow solution are constructed successively on time intervals Im by the STDGM (see [6]).

3.1 ALE Mappings and Triangulations For every m = 1, . . . , M we consider a standard conforming triangulation Tˆh,tm−1 in Ωtm−1 , where h ∈ (0, h), h > 0 and introduce a one-to-one ALE mapping onto

Am−1 h,t : Ω tm−1 −→ Ω t

for t ∈ [tm−1 , tm ), h ∈ (0, h).

(11)

We assume that Am−1 h,t is in space a piecewise affine mapping, continuous in space variable X ∈ Ωtm−1 as well as in time t ∈ [tm−1 , tm ) and Am−1 h,tm−1 = Id (identical mapping). For every t ∈ [tm−1 , tm ) we define the conforming triangulation   ˆ Kˆ ∈ Tˆh,tm−1 in Ωt . Th,t = K = Am−1 ( K); h,t

(12)

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At t = tm we define the one-sided limit Am−1 h,tm − , and introduce the corresponding triangulation. As we see, for every t ∈ [0, T ] we have a family {Th,t }h∈(0,h) of triangulations of the domain Ωt .

3.2 Discrete Function Spaces ˆ the space of all polynomials on Kˆ of degree Let p ≥ 1 be an integer and P p (K) ≤ p. Then for every m = 1, . . . , M we consider the space p,m−1

Sh

  ˆ ∀ Kˆ ∈ Tˆh,tm−1 . = ϕ ∈ L2 (Ωtm−1 ); ϕ|Kˆ ∈ P p (K)

(13)

p,m−1

Further, for q ≥ 1 by P q (Im ; Sh ) we denote the space of mappings of the p,m−1 which are polynomials of degree ≤ q in time. time interval Im into the space Sh We set   p,q p,m−1 q Sh,τ = ϕ; ϕ(t) ◦ Am−1 ), m = 1, . . . , M . (14) h,t |Im ∈ P (Im ; Sh p,q

This means that if ϕ ∈ Sh,τ , then q   ϕ Am−1 (X), t = ϑi (X) t i , h,t

p,m−1

ϑi ∈ Sh

, X ∈ Ωtm−1 , t ∈ I m .

(15)

i=0

3.3 Some Notation and Important Concepts Over a triangulation Th,t , for each positive integer k, we define the broken Sobolev space H k (Ωt , Th,t ) = {v; v|K ∈ H k (K) ∀K ∈ Th,t }. By Fh,t we denote the system of all faces of all elements K ∈ Th,t . It consists of I and the set of all boundary faces F B . Each Γ ∈ F the set of all inner faces Fh,t h,t h,t (L)

(R)

will be associated with a unit normal vector nΓ . By KΓ and KΓ ∈ Th,t we I . Moreover, for Γ ∈ F B the denote the elements adjacent to the face Γ ∈ Fh,t h,t (L)

element adjacent to this face will be denoted by KΓ . We shall use the convention, that nΓ is the outer normal to ∂KΓ(L). (L) (R) If v ∈ H 1 (Ωt , Th,t ) and Γ ∈ Fh,t , then vΓ and vΓ will denote the traces of v on Γ from the side of elements KΓ(L) and KΓ(R) , respectively.We set hK =  diam K for K ∈ Th,t , h(Γ ) = diam Γ for Γ ∈ Fh,t and vΓ =

(L) vΓ

(R) − vΓ ,

for Γ ∈

I . Fh,t

1 2

(L)

(R)

vΓ + vΓ

, [v]Γ =

Stability of Higher-Order ALE-STDGM

565

3.4 Discretization Let t ∈ (0, T ) be an arbitrary but fixed time instant. For u, ϕ ∈ H 2 (Ωt , Th,t ), θ ∈ IR and cW > 0 we introduce the following forms ah (u, ϕ, t) := −

−

 G I Γ ∈Fh,t

Γ

B Γ ∈Fh,t

Γ

 K∈Th,t

β(u)∇u · ∇ϕ dx K

 H G H β(u)∇u · nΓ [ϕ] + θ β(u)∇ϕ · nΓ [u] dS

   β(u)∇u · nΓ ϕ + θβ(u)∇ϕ · nΓ u − θβ(u)∇ϕ · nΓ uD dS,

Jh (u, ϕ, t) := cW

h(Γ )−1



[u] [ϕ] dS + cW Γ

I Γ ∈Fh,t

h(Γ )−1

B Γ ∈Fh,t

 u ϕ dS, Γ

Ah (u, ϕ, t) = ah (u, ϕ, t) + β0 Jh (u, ϕ, t), d

 ∂ϕ bh (u, ϕ, t) := − fs (u) dx ∂xs K +

K∈Th,t

 I Γ ∈Fh,t

Γ



(R) H (u(L) Γ , uΓ , nΓ ) [ϕ] dS +

B Γ ∈Fh,t

d



dh (u, ϕ, t) := −

K∈Th,t

lh (ϕ, t) :=

s=1

 K∈Th,t

K s=1

zs

Γ

(L) H (u(L) Γ , uΓ , nΓ ) ϕ dS,

 ∂u ϕ dx = − (z · ∇u)ϕ dx, ∂xs K

gϕ dx + β0 cW

K

K∈Th,t

B Γ ∈Fh,t

h(Γ )−1

 uD ϕ dS. Γ

Let us note that in integrals over faces we omit the subscript Γ . We consider θ = 1, θ = 0 and θ = −1 and get the symmetric (SIPG), incomplete (IIPG) and nonsymmetric (NIPG) variants of the approximation of the diffusion terms, respectively. In bh (u, ϕ, t), H is a numerical flux which is Lipschitz-continuous, consistent and conservative. 4 For a function ϕ defined in M m=1 Im we denote ± ϕm = ϕ(tm ±) = lim ϕ(t) t →tm ±

and {ϕ}m = ϕ(tm +) − ϕ(tm −).

Now we define an ALE-STDG approximate solution of our problem.

(16)

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Definition 1 A function U is an approximate solution of problem (7)–(9), if U ∈ p,q Sh,τ and  Im

   Dt U, ϕ Ωt + Ah (U, ϕ, t) + bh (U, ϕ, t) + dh (U, ϕ, t) dt

+ +({U }m−1 , ϕm−1 )Ωtm−1 =

U0− ∈ Sh , p,0

 lh (ϕ, t) dt Im

p,q

∀ϕ ∈ Sh,τ ,

(17)

m = 1, . . . , M,

(U0− − u0 , vh ) = 0 ∀vh ∈ Sh . p,0

(18)

4 Analysis of the Stability In the space H 1 (Ωt , Th,t ) we define the norm  · DG,t by the relation ϕ2DG,t =  2 K∈Th,t |ϕ|H 1 (K) + Jh (ϕ, ϕ, t). Moreover, over ∂Ωt we define the norm of the   Dirichlet boundary condition by uD 2DGB,t = cW Γ ∈F B h(Γ )−1 Γ |uD |2 dS. h,t If we use ϕ := U as a test function in (17), we get the basic identity  Im

  (Dt U, U )Ωt + Ah (U, U, t) + bh (U, U, t) + dh (U, U, t) dt

+ +({U }m−1 , Um−1 )Ωtm−1

(19)

 =

lh (U, t) dt. Im

4.1 Important Estimates Here we estimate forms from (19) individually. The proofs can be carried out similarly as in [1]. For a sufficiently large constant cW we obtain the coercivity of the diffusion and penalty terms. Lemma 2 Let cW ≥ cW ≥ cW ≥

β12 β02 β12 β02

cM (cI + 1)

for θ = −1 (NIPG),

cM (cI + 1)

for θ = 0 (IIPG),

16β12 β02

cM (cI + 1)

for θ = 1 (SIPG),

Stability of Higher-Order ALE-STDGM

567

where constants cM and cI are from the multiplicative trace inequality and the inverse inequality, respectively. Then  Ah (U, U, t) dt ≥ Im

β0 2

 Im

U 2DG,t dt −

β0 2

 Im

uD 2DGB,t dt.

(20)

Further, we estimate the convection terms and the right-hand side term: Lemma 3 For each k1 , k2 , k3 > 0 there exist constants cb , cd > 0 such that we have    β0 2 |bh (U, U, t)|dt ≤ U DG,t dt + cb U 2Ωt dt. (21) 2k1 Im Im Im    β0 cd |dh (U, U, t)| dt ≤ U 2DG,t dt + U 2Ωt dt. (22) 2k2 Im 2β0 Im Im     1 g2Ωt + U 2Ωt dt (23) |lh (U, t)| dt ≤ 2 Im Im   β0 k 3 β0 2 + uD DGB,t dt + U 2DG,t dt. 2 2k3 Im Im Finally we need to estimate the term with the ALE derivative. The proof is based on the Reynolds transport theorem and on (10). Lemma 4 It holds that    1 + − 2 2 2 Um Ωtm − Um−1 Ωt (Dt U, U )Ωt dt ≥ − cz U Ωt dt , (24) m−1 2 Im Im   + {U }m−1 , Um−1 (25) 

Ωtm−1

 1 + 2 − , Um−1 Ωt = + {U }m−1 2Ωt − Um−1 2Ωt m−1 m−1 m−1 2 Theorem 5 There exists a constant CT > 0 such that  β0 − 2Ωt + {U }m−1 2Ωt + U 2DG,t dt (26) Um− 2Ωtm − Um−1 m−1 m−1 2 Im     2 2 2 ≤ CT gΩt dt + uD DGB,t dt + U Ωt dt . Im

Im

Im

Proof From (19), by virtue of (24), (20), (21), (22), (25) and (23), after some manipulation and choosing k1 = k2 = k3 = 6, we get (26) with CT = max{1, 7β0, cz + 1 + cd /β0 + 2cb }. 

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4.2 Discrete Characteristic Function In our further considerations, the concept of a discrete characteristic function will play an important role. Here it is generalized to time-dependent domains. For m = 1, . . . , M we use the following notation: U = U (x, t), x ∈ Ωt , t ∈ Im , will denote the approximate solution in Ωt , and U˜ = U˜ (X, t) = U (At (X), t), X ∈ Ωtm−1 , t ∈ Im , denotes the approximate solution transformed to the reference domain Ωtm−1 . Definition 6 The discrete characteristic function to U˜ at a point s ∈ Im is defined p,m−1 as U˜s = U˜s (X, t) ∈ P q (Im ; S ) such that h

 Im

(U˜s , ϕ)Ωtm−1 dt =



s tm−1

(U˜ , ϕ)Ωtm−1 dt

p,m−1

∀ϕ ∈ P q−1 (Im ; Sh

+ + ) = U˜ (X, tm−1 ), X ∈ Ωtm−1 . U˜s (X, tm−1

),

(27) (28)

Further, we introduce the discrete characteristic function Us = Us (x, t), x ∈ Ωt , t ∈ p,q Im to U ∈ Sh,τ at a point s ∈ Im : Us (x, t) = U˜s (A−1 t (x), t), x ∈ Ωt , t ∈ Im .

(29)

p,q

Hence, in view of (14), Us ∈ Sh,τ and for X ∈ Ωtm−1 we have Us (X, tm−1 +) = U (X, tm−1 +).

(30)

In what follows, we prove that the discrete characteristic function mapping U → Us is continuous with respect of the norms  · L2 (Ωt ) and  · DG,t . (1) (2) , cCH > 0, such that Theorem 7 There exist constants cCH

 Im

 Im

(1) Us 2Ωt dt ≤ cCH

Us 2DG,t

dt ≤

(2) cCH

 

Im

Im

U 2Ωt dt

(31)

U 2DG,t dt

(32)

for all s ∈ Im , m = 1, . . . , M and h ∈ (0, h). Proof The proof is very long and technical. It is based on three steps. At first, the discrete characteristic function Us is transformed to the reference domain, i.e. U˜s = Us ◦ At . In the second step we apply continuity properties from [5] of the discrete characteristic function in the reference (fixed) domain. Finally in the last step we transfer it back to the current configuration. 

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569

Using the definition and properties (31)–(32) of the discrete characteristic function, we can prove the following theorem. The proof is very long and technical. Theorem 8 There exist constants C, C ∗ > 0 such that       − 2 2 2 2 gΩt + uD DGB,t dt U Ωt dt ≤ C τm Um−1 Ωt + m−1

Im

(33)

Im

provided 0 < τm < C ∗ . Finally we arrive to our main result concerning the unconditional stability of the method. Theorem 9 Let 0 < τm ≤ C ∗ for m = 1, . . . , M. Then there exists a constant CS > 0 such that Um− 2Ωtm + ⎛

m

{Uj −1 }2Ωt

j =1

≤ CS ⎝U0− 2Ωt + 0

j−1

m 

+

m  β0 U 2DG,t dt 2 Ij j =1

⎞

Rt dt ⎠ ,

m = 1, . . . , M, h ∈ (0, h),

j =1 Ij

where Rt = (CT + C τj ) (g2Ωt + uD 2DGB,t ) for t ∈ Ij . Proof The proof is based on (26), (33) and the use of the discrete Gronwall inequality.  Acknowledgements This research was supported by the project GA UK No. 127615 of the Charles University (M. Balázsová) and by the grant 17-01747S of the Czech Science Foundation (M. Vlasák, who is a junior member of the University Centre for Mathematical Modeling, Applied Analysis and Computational Mathematics - MathMAC).

References 1. M. Balázsová, M. Feistauer, On the stability of the space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains. Appl. Math. 60, 501– 526 (2015) 2. M. Balázsová, M. Feistauer, On the uniform stability of the space-time discontinuous Galerkin method for nonstationary problems in time-dependent domains, in ALGORITMY 2016, 20th Conference on Scientific Computing, Vysoké Tatry - Podbanské, Slovakia, March 13–18, 2016, ed. by A. Handloviˇcová, D. Ševˇcoviˇc (Slovak University of Technology, Bratislava, 2016), pp. 84–92 ˇ 3. J. Cesenek, M. Feistauer, J. Horáˇcek, V. Kuˇcera, J. Prokopová, Simulation of compressible viscous flow in time-dependent domains. Appl. Math. Comput. 219, 7139–7150 (2013)

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ˇ 4. J. Cesenek, M. Feistauer, A. Kosík, DGFEM for the analysis of airfoil vibrations induced by compressible flow. ZAMM Z. Angew. Math. Mech. 93(6–7), 387–402 (2013) 5. V. Dolejší, M. Feistauer, Discontinuous Galerkin method – Analysis and Applications to Compressible Flow (Springer, Berlin, 2015) 6. A. Kosík, M. Feistauer, M. Hadrava, J. Horáˇcek, Numerical simulation of the interaction between a nonlinear elastic structure and compressible flow by the discontinuous Galerkin method. Appl. Math. Comput. 267, 382–396 (2015)

Part XV

PDE Software Frameworks

Implementation of Mixed-Dimensional Models for Flow in Fractured Porous Media Eirik Keilegavlen, Alessio Fumagalli, Runar Berge, and Ivar Stefansson

Abstract Models that involve coupled dynamics in a mixed-dimensional geometry are of increasing interest in several applications. Here, we describe the development of a simulation model for flow in fractured porous media, where the fractures and their intersections form a hierarchy of interacting subdomains. We discuss the implementation of a simulation framework, with an emphasis on reuse of existing discretization tools for mono-dimensional problems. The key ingredients are the representation of the mixed-dimensional geometry as a graph, which allows for convenient discretization and data storage, and a non-intrusive coupling of dimensions via boundary conditions and source terms. This approach is applicable for a wide class of mixed-dimensional problems. We show simulation results for a flow problem in a three-dimensional fracture geometry, applying both finite volume and virtual finite element discretizations.

1 Introduction Simulation models for real-life applications commonly must represent objects with high aspect ratios embedded in the domains. This includes both objects of codimension 1, exemplified by fractures in a porous rock, and co-dimension 2 models such as reinforced concrete and root systems. Although the embedded objects occupy a small part of the simulation domain, they can have a decisive impact on the system behavior, thus their representation in the simulation model is critical. The small object sizes, and in particular the high aspect ratio, make an equidimensional representation computationally prohibitively expensive. The standard

E. Keilegavlen () · A. Fumagalli · R. Berge · I. Stefansson University of Bergen, Department of Mathematics, Bergen, Norway e-mail: [email protected]; [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_52

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simulation technique has therefore been to apply homogenization to arrive at an, ideally, equivalent upscaled model. In recent years, advances in computational power, modeling approaches and numerical methods have made resolving the objects more feasible. This calls for the development of new simulation tools for mixed-dimensional problems that allow for a high degree of reuse of software designed for mono-dimensional simulations. Here, we explore the implementation of a simulation model based on a newly developed modeling framework for flow and transport in fractured porous media [2, 3]. The model considers fractures as manifolds of co-dimension 1 that are embedded in the simulation domain, and further allows for intersections of fractures as objects of co-dimension 2 and 3. Central to our implementation is the representation of the mixed-dimensional problem as a graph, where the nodes represent monodimensional problems, and the edges represent couplings between subdomains. Discretization internal to each subdomain is then a matter of iterating over the nodes of the graph, and apply standard numerical methods by invoking, possibly legacy, mono-dimensional discretizations. Discretization of mixed-dimensional dynamics, which is commonly not handled by existing software, is associated with the edges of the graph. Depending on how the interactions are modeled, the implementation of the subdomain couplings comes down to treatment of boundary conditions and source terms, both of which are standard in most numerical tools. Focusing on locally conservative methods, which are prevailing in porous media applications, we consider both virtual element and finite volume approaches for flow, in addition to finite volume techniques for transport. The simulation model discussed is implemented in the software framework PorePy [9], and is available at www.github.com/pmgbergen/porepy.

2 Mixed-Dimensional Flow in Fractured Porous Media Let us consider a N-dimensional domain Ω ⊂ RN , typically N = 2 or 3, with outer boundary ∂Ω. Ω represents the porous medium, which is composed of a N-dimensional domain Ω N (the rock matrix) and lower-dimensional domains Ω N−1 , . . . , Ω 0 , representing fractures and possibly objects of lower dimensions such as fracture intersections and intersections of fracture intersections. We assume that Ω d−1 ⊂ Ω d for d = 0, . . . , N, and Ω = ∪d Ω d . Let Γ denote the set of internal boundaries between subdomains of different dimension. In each dimension, we consider the flow of a single-phase incompressible fluid, with governing equations stated on mixed form as ud + K∇pd = 0,

in Ω d , d > 0

∇ · ud − [[ud+1 · nd+1 ]] = f d , in Ω d , d ≤ N ud · nd + κ(pd−1 − pd ) = 0

on Γ.

(1)

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The unknowns are the Darcy velocity u and the pressure p, and the data are the permeability K, the normal vector on the internal boundary nd+1 , the effective normal permeability κ between the subdomains, and f is sources and sinks. We define uN+1 = 0. The coupling between pairs of subdomains one dimension apart can be seen in the conservation statement, where the divergence of the flux in the dimension under consideration is balanced by source terms in the same dimension, and the contribution from the higher dimension via the jump [[ud+1 · nd+1 ]], which represent the net flux into the domain. From the higher dimension, this term will appear as the leakage into lower dimensions, which may be interpreted as an internal boundary. The last equation in (1) is a mixed-dimensional Darcy law with effective permeability κ, which models the flux between two subdomains separated by the interface Γ . On outer boundaries of each Ω d we assign Dirichlet or Neumann conditions. For more details on the mathematical model, we refer to [3–6, 8, 10– 12]. We also note that a similar model can be used to express transport of a scalar in mixed-dimensional geometries, see [7].

3 Discretization of Mixed-Dimensional Problems To device design principles for an implementation of the model (1), we first observe that interaction between subdomains takes the form of boundary conditions from lower to higher dimension and conversely source terms from higher to lower dimensions. Moreover, this should apply also for more general classes of models, including most, if not all, that are built upon conservation principles [2]. Thus a versatile implementation should be based on independent discretizations on the subdomains, together with appropriate coupling conditions. Below we describe how this is naturally achieved by representing the computational grid as a graph, and discretization as an iteration over its nodes and edges. This abstraction allows for reuse of an existing code base, and if combined with a flexible interface between the grids, it can be extended to heterogeneous discretizations and multi-physics modeling. In Sect. 3.1 we present the grid structure needed for the mixed-dimensional description of (1), while in Sect. 3.2 we briefly introduce the numerical discretizations.

3.1 Grid Structure For simplicity, we require that the computational mesh is fully conforming to all objects Ω d in all dimensions. This condition can be relaxed, e.g. by applying mortar discretizations in the transition between subdomains [3], however, we will not pursue this herein. To illustrate the mixed-dimensional grid structure, we consider the hierarchy of grids depicted in Fig. 1. The main domain is Ω = [−1, 1]3. Define the fracture Ω12

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Fig. 1 Hierarchy of grids derived from the meeting of three 2D objects embedded in 3D space. The mixed-dimensional grid is naturally represented as a graph, with individual grids forming nodes

is embedded in the xy-plane, with the coordinate axes excluded, and similarly let Ω22 and Ω32 be embedded in the xz- and yz-plane, respectively. The intersection between pairs of fractures defines intersection lines Ωi1 , i = {1, 2, 3} along the coordinate 0 axes, with the origin excluded. Further, the  intersect to define Ω 4 intersection 4 lines d 3 in the origin. Finally, define Ω = Ω \ d=0,1,2 i Ωi . To apply a discretization scheme on this grid structure requires iterations over all subdomains and their connections. This is readily implemented by considering the subdomains as nodes in a graph, with the connections forming edges, as illustrated in Fig. 1. We define nodes(·) as a function such that given a dimension d it returns the nodes in the graph of the same dimension. Similarly, edges(·, ·) is a function that, given consecutive dimensions d and d − 1, returns the edges in the graph

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associated with nodes of dimension d and d − 1. This abstraction is particularly suitable for existing software frameworks that can handle grids of a single but flexible dimension. Discretization of a mixed-dimensional problem can then be defined by two iterations; (1) an iteration on the nodes of the graph to invoke the standard discretization on each subdomain, and (2) an iteration on the edges of the graph to impose coupling conditions on subdomain boundaries. Given a suitable numerical scheme for the discretization of (1), the system obtained from the first iteration gives the following block-diagonal matrix diag(AN , AN−1 , . . . , A1 , 0), where Ad is related to the set nodes(d) of nodes in the graph. The matrices Ad are block diagonal, with one block for each node defining the mono-dimensional discretization of each sub-grid. The interface conditions between d and d − 1 can be written as   H d,d−1 C d,d−1 . C d,d−1 Ld,d−1 Here, the matrix H d,d−1 represent the discretization of the coupling condition seen from the higher dimension, in terms of variables in the higher dimension. Similarly, Ld,d−1 discretizes the part of the coupling condition in the lower dimension that is associated with lower dimensional variables. Finally, C d,d−1 gives the crosscoupling terms. All three matrices have a block structure, with one block for each edge in edges(d, d − 1). For a three-dimensional problem, the general structure of the global matrix is ⎡

A3 + H 3,2 C 3,2 0 ⎢ 3,2 2 A + L3,2 + H 2,1 C 2,1 ⎢ C ⎢ ⎣ A1 + L2,1 + H 1,0 0 C 2,1 0 0 C 1,0

⎤ 0 ⎥ 0 ⎥ ⎥. 1,0 C ⎦ L1,0

The matrix in Fig. 2 illustrates the block structure associated with the mesh of Fig. 1.

3.2 Conservative Discretizations We consider two discretization schemes for the mixed-dimensional pressure equation (1). The simplest option is a finite volume scheme built as a two-point flux approximation (TPFA), which is standard in commercial porous media simulators. This scheme is easy to implement, and, with the data structures outlined above, a simple extension to mixed-dimensional problems is fairly straightforward; more complex approaches involving mortar variables are currently under investigation.

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3D

2D

1D

0D Fig. 2 Structure of linear system resulting from a discretization of the mixed-dimensional problem on the geometry shown in Fig. 1. The diagonal blocks represent the discretization of each grid, while the off diagonal blocks represent the interface condition between two grids. The top and bottom colors of each coupling block correspond to the higher and lower dimensional grids, respectively

However, TPFA is consistent only for K-orthogonal grids, and can be expected to suffer from poor accuracy for the complex grids needed to cover realistic fracture networks. Our second approach applies the dual form of the virtual element method [1], with the extension to mixed-dimensional problems, as discussed in [7]. The virtual element method puts almost no restrictions on the cell shape, and is thus ideally suited for handling rough grids. This also makes it possible to merge simplex cells into general polyhedral shapes, and thus reduce the number of degrees of freedom. We do not consider non-simplex cells here, see [7] for details. Given a flux field, a tracer advection problem can be discretized, e.g. by a firstorder upstreaming technique, with a procedure similar to the one described above.

4 Example Simulation In this part we present an example to assess the above models and numerical schemes. The source code of the example is available online in the PorePy repository. We consider a 3D fracture network with heterogeneous permeabilities in the fractures and on their intersections exploiting the mixed-dimensional structure of the model. The domain is Ω = [0, 2] × [0, 1] × [0, 1] and the geometry of the network, containing 7 fractures, is depicted at the top of Fig. 3. The permeability in the rock matrix is set to the identity matrix. The fracture marked with a red color in the figure is high permeable with permeability equal to 104 , while the other fractures behave as barriers having permeabilities equal to 10−4 . The fracture aperture is constant and equal to 10−2 for all fractures. All 1D intersections inherit the highest permeability of the intersecting fractures. Thus, the ones involving the highly permeable fracture are permeable, and the others behave as impermeable

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Fig. 3 Top: representation of the considered geometry for the example. Centre: pressure field for the TPFA (left) and VEM (right). Bottom: concentration at the final time with the discharge computed by the TPFA and VEM. The flow is from the right to the left. In all the figures a “Blue to Red Rainbow” colour map is used in the range [0, 1]

paths. The grid is composed by 23,325 tetrahedra for the 3d rock matrix, 2624 triangles for the fractures, and 48 segments for the fracture intersections. We impose a pressure gradient by setting the pressure on the left boundary to 0 and the right boundary to 1. The other boundaries are given a zero flux condition. We compute the pressure field and the discharge (flux) using the TPFA and VEM on the same grid. The discharge is a face variable, obtained by back-calculation from the pressures in the TPFA method and directly computed for the VEM. The computed solution is represented in the middle row of Fig. 3. The low permeable fractures at the right end of the domain do not affect the pressure field much, as the conductive fracture makes a high permeable connection between the right and the center part of the domain. However, at the left part the low permeable fractures force a pressure gradient between the small gap between two of the fractures (orange and green). The solutions obtained from both methods are in good agreement. Once the discharge is computed we consider a pure transport problem, where the advective field is given by the discharge. We inject a tracer with concentration 1 on the right part of the domain with outflow on the left part. The tracer thus follows the discharge, flowing in the high permeable fracture first and then propagating in the rock matrix, avoiding the low permeable fractures in the left part of the domain. An implicit Euler scheme is applied for the time discretization with time step equal to 0.01. The final time of the simulation is 3. As in the case of the pressures, the tracer solutions for the discharge computed by the TPFA or by the VEM are in agreement.

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5 Concluding Remarks We have discussed the design of simulation tools for mixed-dimensional equations, in the setting of flow in fractured porous media. We presented a data hierarchy where the mixed-dimensional structure is a graph, with each node representing a standard mono-dimensional domain. This allows for extensive reuse of existing code designed for mono-dimensional problems, and also facilitates simple implementation of new discretization schemes. Numerical examples of flow and transport in a three-dimensional fractured medium illustrate the capabilities of a simulation tool based on this approach. Acknowledgements We acknowledge financial support from the Research Council of Norway, project no. 244129/E20 and 250223.

References 1. L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM: Math. Model. Numer. Anal. 50(3), 727–747 (2016) 2. W.M. Boon, J.M. Nordbotten, J.E. Vatne, Mixed-dimensional elliptic partial differential equations. arXiv:1710.00556 (2017) 3. W.M. Boon, J.M. Nordbotten, I. Yotov, Robust discretization of flow in fractured porous media. SIAM J. Numer. Anal. 56(4), 2203–2233 (2018) 4. K. Brenner, J. Hennicker, R. Masson, P. Samier, Gradient discretization of hybrid-dimensional Darcy flow in fractured porous media with discontinuous pressures at matrix-fracture interfaces. IMA J. Numer. Anal. 37(3), 1551–1585 (2017). 5. F.A. Chave, D. Di Pietro, L. Formaggia, A hybrid high-order method for Darcy flows in fractured porous media. Tech. report, HAL archives:hal-01482925, 2017 6. C. D’Angelo, A. Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: Math. Model. Numer. Anal. 46(2), 465–489 (2012) 7. A. Fumagalli, E. Keilegavlen, Dual virtual element methods for discrete fracture matrix models. Tech. report, arXiv:1711.01818, 2017 8. A. Fumagalli, A. Scotti, An efficient XFEM approximation of Darcy flows in arbitrarily fractured porous media. Oil Gas Sci. Technol. Revue d’IFP Energies Nouvelles 69(4), 555– 564 (2014) 9. E. Keilegavlen, A. Fumagalli, R. Berge, I. Stefansson, I. Berre, Porepy: an open source simulation tool for flow and transport in deformable fractured rocks. Tech. report, arXiv:1712.00460, 2017 10. V. Martin, J. Jaffré, J.E. Roberts, Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005) 11. N. Schwenck, B. Flemisch, R. Helmig, B. Wohlmuth, Dimensionally reduced flow models in fractured porous media: crossings and boundaries. Comput. Geosci. 19(6), 1219–1230 (2015) 12. L. Formaggia, A. Scotti, F. Sottocasa, Analysis of a mimetic finite difference approximation of flows in fractured porous media. ESAIM: Math. Model. Numer. Anal. 52, 595–630 (2018)

Fast Matrix-Free Evaluation of Hybridizable Discontinuous Galerkin Operators Martin Kronbichler, Katharina Kormann, and Wolfgang A. Wall

Abstract This paper proposes a new algorithm for fast matrix-free evaluation of linear operators based on hybridizable discontinuous Galerkin discretizations with sum factorization, exemplified for the convection-diffusion equation on quadrilateral and hexahedral elements. The matrix-free scheme is based on a formulation of the method in terms of the primal variable and the trace. The proposed method is shown to be up to an order of magnitude faster than the traditionally considered matrix-based formulation in terms of the trace only, despite using more degrees of freedom. The impact of the choice of basis on the evaluation cost is discussed, showing that Lagrange polynomials with nodes co-located with the quadrature points are particularly efficient.

1 Introduction Discontinuous Galerkin methods are highly attractive discretization schemes for a wide variety of problems, in particular for problems with strong transport character as they allow for upwinding and are amenable to efficient explicit time stepping. They are also increasingly used in problems dominated by elliptic terms where linear systems need to be solved. Whereas classical discontinuous schemes have often been criticized as being less efficient than continuous ones due to more degrees of freedom and a wider stencil for delivering the same accuracy, the hybridizable discontinuous Galerkin (HDG) method developed by Cockburn et al. [3, 10] has addressed this deficiency by expressing the connectivity in the final system matrix only through a variable for the trace of the solution on the mesh skeleton, into

M. Kronbichler () · W. A. Wall Institute for Computational Mechanics, Technical University of Munich, Garching, Germany e-mail: [email protected]; [email protected] K. Kormann Max Planck Institute for Plasma Physics, Garching, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_53

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which all volume terms are absorbed by a static-condensation-like technique. This reduced number of unknowns has been shown to render the method highly efficient when compared to continuous and other discontinuous Galerkin discretizations, in particular for higher polynomial degrees [4, 12]. When using modern iterative solvers with matrix-free implementations, however, matrix-based HDG is still lagging behind formulations in the primal variable amenable to matrix-free operator evaluation [7, 8]. This work closes this gap by developing a new matrix-free scheme for a particular HDG variant in the primal variable u and the trace u that was recently proposed by Cockburn [2], relying on the algorithms from the generic finite element library deal.II [1, 5, 6]. We show that operator evaluation is more than an order of magnitude faster with the proposed implementation than matrix-based HDG for moderate polynomial degrees of four to five in three spatial dimensions. The beneficial properties of the new method motivate future work on efficient preconditioners for iterative solvers with this scheme. This article is structured as follows. Section 2 introduces the HDG discretization and details the two possible matrix formulations. In Sect. 3 the sum factorization ingredients and efficient bases are presented. Section 4 shows our computational results and gives an outlook to future work.

2 Hybridizable Discontinuous Galerkin Discretization We consider the linear convection-diffusion equation on a bounded computational domain Ω ⊂ Rd in d space dimensions, ∇ · (cu) − ∇ · (κ∇u) = f u = gD

on ∂ΩD ,

in Ω,

(−k∇u + cu) · n = gN

on ∂ΩN ,

(1)

where c is the direction of transport, κ > 0 is the diffusivity, and f is a given forcing. The solution u is found subject to a Dirichlet value gD on ∂ΩD ⊂ ∂Ω and a Neumann value gN on ∂ΩN = ∂Ω \ ∂ΩD . We assume a tesselation Th that partitions the computational domain Ω into ne elements Ωe of characteristic size h. We specialize algorithms for quadrilateral or hexahedral elements on which sum factorization is straight-forwardly implemented. An element Ωe is the image of the reference domain [−1, 1]d under a polynomial mapping of degree k. We denote by Vh,k the space of admissible solutions that are polynomials of tensor degree k. The HDG method [10], which rewrites the convection-diffusion Eq. (1) into a system in the primal variable u and the flux q = −κ∇u, uses this space for both u and each component of q. A third variable u is introduced on the mesh skeleton, the faces Fh of the triangulation, with the associated solution space Mh,k as the (d − 1)-dimensional tensor product polynomials on the faces Fh that are zero on the Dirichlet boundaries ∂ΩD . The

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d resulting weak form is to find a triple (q, u, u) ∈ Vh,k × Vh,k × Mh,k holds G H (w, κ −1 q)Th −(∇ · w, u)Th + w · n, u ∂Th \∂ΩD = − w · n, gD ∂Ω D G H − (∇v, cu + q)Th + v, (c u + q) · n + τ (u − u) ∂ T =(v, f )Th h G H G H μ, (cu+q) · n + τ (u − u) ∂Th \∂ΩD = μ, gN ∂Ω N

such that it d ∀w∈Vh,k ,

∀v ∈ Vh,k , ∀μ∈Mh,k . (2)

We denote by (·, ·)Th the bilinear form associated with cell integrals in the usual finite element fashion and by ·, ·∂Th the bilinear form for face integrals, where each interior face is visited twice, involving different fields u and q but the same u on the two sides. The stabilization parameter τ can come in various flavors adapted to the problem nature [10]. We exemplify the implementation with a centered scheme as τ = |c · n| + κ with  = 5. The weak forms in (2) give rise to the following matrices, see also [10], Aij = (ϕ i , κ −1 ϕ j )Th ,

I J Bij = −(∇ϕi , ϕ j )Th + ϕi , ϕ j · n

I J Cij = ψi , ϕ j · n

H G Dij = −(∇ϕi , cϕj )∂Th + ϕi , τ ϕj ∂ T ,

G

∂ Th

,

G

H

∂ Th

,

h

H

(3)

Eij = ϕi , (c · n − τ )ψj ∂ T , Gij = ψi , τ ϕj ∂ T , h h G H Hij = ψi , (c · n − τ )ψj ∂ T , h

d , ϕ are basis where ϕ i denotes vector-valued shape functions spanning Vh,k i functions spanning Vh,k , and ψi are the basis functions of Mh,k . The right hand side of (2) gives rise to the three vectors

H H H G G G Ri = ϕ i · n, gD ∂Ω , Fi = (ϕi , f )Th + ϕi , τgD ∂Ω , Li = ψi , gN ∂ T . D

D

h

(4) With these ingredients, the discrete HDG method seeks for the coefficient vectors (Q, U , U ) through the linear system ⎛

⎞ ⎞⎛ ⎞ ⎛ A −B T C T −R Q ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎝B D E ⎠ ⎝ U ⎠ = ⎝ F ⎠ . L U C G H

(5)

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2.1 Formulation in Terms of the Trace  u An attractive feature of the HDG method is the possibility to statically condense out some of the degrees of freedom before solving the linear system [3]. This can significantly speed up the solution of the final linear system in implicit methods, besides the beneficial convergence properties of HDG fluxes [3, 10]. In Eq. (5), the matrices A, B, D are block-diagonal over elements because the coupling is expressed solely in terms of the trace variable u. Thus, they can be inverted element by element through a Schur complement, eliminating q and u from the system. This reduces the matrix-vector product complexity per element from O(k 2d ) to O(k 2d−2 ). The resulting linear system in the trace reads KU = T ,

(6)

where the matrix K and right hand side vector T are given by  −1    −1     A −B T   A −B T CT −R K=H− CG , T =L− C G . B D E B D F The trace formulation has been used in a large number of works on HDG, including the performance comparisons [4, 12]. The static condensation with inverse matrices, however, mandates to resort to an explicit sparse matrix storage.1

2.2 Formulation in Terms of u and  u An alternative formulation of HDG proposed recently [2] is to express the final linear system in the two variables u and u, rather than the trace only, ⎡ ⎤           ⎣ D E − B A−1 −B T C T ⎦ U = F + B A−1 R. (7) C C L GH U The advantage of this compact form is that only an inverse of a vector mass matrix A weighted by the diffusion coefficient appears, for which fast matrix-free inversion methods with lower complexity than the 2d(k + 1)2d arithmetic operations in the polynomial degree k of the full matrix inversion are available as detailed in the next section.

1 Note that we do not consider the narrow case where the mesh is Cartesian, c is axis-aligned and element-wise constant, and κ is constant, when the matrix for (q, u) is separable and can be expressed as a Kronecker product of 1D matrices with the inverse given by the fast diagonalization method [9].

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3 Sum Factorization Algorithms for HDG We exemplify the structure of a matrix-free evaluation kernel on the multiplication by the cell integral contribution to the matrix B T in 3D, i.e., the integral −(ϕ i , ∇ϕj )Th . The shape functions are assumed to be the product of three onedimensional shape functions ϕi (ξ ) = ϕi1D (ξ1 )ϕi1D (ξ2 )ϕi1D (ξ3 ) for i = 1, . . . , (k + 1 2 3 q q q 3 1) . Integrals are evaluated with Gaussian quadrature in the points (ξ1 , ξ2 , ξ3 ), q = 1, . . . , nq . Here, we choose nq = (k + 1)3 points which ensures exact integration of constant-coefficient weak forms on affine geometries. While this is usually enough also for representing integrals for curved geometries accurately, variable coefficients or nonlinearities often require more points to avoid aliasing effects. Our methods also apply to those cases, albeit at slightly reduced efficiency. For matrix-free evaluation, we associate the input vector U with the representa(k+1)3 (e) (e) tion uh (x) = j =1 ϕj (x)Uj on a cell Ωe . This quantity is then tested by all functions ϕ i . The integral is computed by a loop over all cells Ωe . On each cell, the approximation reads 

n3

Ωe

ϕ i (x) · ∇uh (x)dx ≈

q

q=1

⎛

3 (k+1)

⎜ ϕ i (ξ q ) · J −T q ⎝

⎞

(e) ⎟ ∇ξ ϕj (ξ q )Uj ⎠ det(J q )wq ,

j =1

where J = J (ξ ) is the Jacobian of the transformation from the reference to the real cell and ∇ξ denotes the gradient with respect to the reference coordinates. Moreover, wq and ξ q denote the quadrature weights and points, respectively. The three steps to evaluate the integrals for all test functions on an element are (a) the evaluation of the solution gradient in all quadrature points, (b) the multiplication by the inverse −T (ξ ) and the factor det(J )w , and (c) the multiplication by Jacobian J −T q q q q = J the values ϕ i and summation of quadrature points for all i. Steps (a) and (c) can be recast as matrix-vector products of size 3nq × (k + 1)3 and 3(k + 1) × nq , respectively. However, the potential cost of (k + 1)6 can be reduced to (k + 1)4 (or to d(k + 1)d+1 in space dimension d) by sum factorization, an algorithm that transforms the summations into one-dimensional kernels along all spatial directions by using the tensor product structure in the shape functions. Sum factorization has its origin in spectral elements [11] and has found widespread use in both continuous and discontinuous Galerkin implementations. The matrix defining the interpolation of ∇ξ u(e) h in the quadrature points reads ⎛

⎞ D1co ⊗ I2 ⊗ I3   ⎜ ⎟ (e) ⎝I1 ⊗ D2co ⊗ I3 ⎠ S1 ⊗ S2 ⊗ S3 U . I1 ⊗ I2 ⊗ D3co

(8)

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  In this formula, the product by the first matrix S1 ⊗ S2 ⊗ S3 —with (S∗ )ij = ϕj1D (ξi ) the matrix of 1D shape functions evaluated in the 1D quadrature points—is a basis change [6] into the Lagrange basis defined in the points of Gaussian quadrature. This basis with collocated node and quadrature points reduces evaluation costs because it only involves 2d tensor product kernels to get all components of the gradient, as opposed to d 2 tensor product kernels in a naive implementation. The matrices D∗co are the derivatives of the Lagrange polynomials in the Gauss points, i.e., in the collocation basis.

3.1 Matrix-Vector Product for General Bases Using algorithms in the lines of (8) for both cell and face integrals as appropriate f (e.g., using a matrix S1 ⊗ S2 ⊗ S3 for an interpolation of values onto the face f f with normal in ξ3 direction with a 1 × (k + 1) matrix S3 ), all integrals appearing d in Eq. (7) can be evaluated with complexity O(k ) operations for face integrals and O(k d+1 ) operations for the cell integrals. The computations are done by the following algorithmic steps in a loop over all cells (see also [8, Sec. 4.1] for a partly matrix-free variant): 1. Compute cell integral P = −(B c )T U (e) , with B c the cell integral part of B = B c + B f in Eq. (3). 2. Loop over allfacesof cell e and add the face contribution, P = P +   U (e) −(B f )T C T (e) . U 3. Compute Q(e) = A−1 P by using the tensor product inverse matrix algorithm from [6, Sec. 4.4.1] (change to diagonal basis). 4. Compute cell integral contribution in V (e) = D c U (e) − B c Q(e) . 5. Loop over all faces of cell e (e)

(a) Add face integral V (e) = V (e) + D f U (e) + E U − (B f )Q(e) . (e) (e) = GU (e) + H U − CQ(e) and add into the (b) Compute face integral V respective position in the global result vector V . When the loop over faces is completed, the cell contribution of V (e) contains the result for the primal variable V .

3.2 Matrix-Vector Product for Collocation Basis The algorithm for a generic basis contains a large number of multiplications by matrices of the form S1 ⊗ S2 ⊗ S3 , their transpose, and the respective face versions.

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In essence, these steps interpolate from the coefficients in the chosen basis to the values in the points of Gaussian quadrature. For the HDG method in (u, u) format, a basis with collocated node and quadrature points where S1 , S2 , S3 are unit matrices is very beneficial because a large number of interpolations turn into the identity operation. In particular, the collocation basis is orthogonal with respect to the L2 inner product and thus results in a diagonal mass matrix, considerably simplifying the multiplication by A−1 , in contrast to a generic tensor product basis with algorithm according to [8, Sec. 4.4.1]. Table 1 lists the number of tensor product calls for the collocation basis compared to a general basis. Both the number of cell sum factorization kernels and face sum factorization kernels are significantly reduced, which results in a significant saving also in our highly tuned implementation using advanced SIMD (vectorized) instructions [6], see Fig. 1. In particular for the face integrals, the only sum factorization calls are for the interpolation of contributions onto the faces. Note that besides the operations listed in the table, the algorithm also does O((k + 1)3 ) operations on quadrature points of cells and O((k + 1)2 ) on quadrature points of faces. Table 1 Number of tensor product kernel calls for evaluation of HDG in (u, u) for 3D convectiondiffusion operator of cost (k + 1)4 (cells) and (k + 1)3 (faces) Cell terms BT A−1 15 18 3 —

Generic basis Collocation basis

2D mesh primal DoFs/s

Face terms BT, CT 6 × 14 6×4

B, D 18 3

109

109

Second face loop 6 × 24 6×4

3D mesh

108 108 1

2

3

4

5

6

7

8

9

10

107

1

2

3

Polynomial degree trace matrix

(u, u) matrix

4

5

6

7

8

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4 Numerical Experiments and Outlook We run experiments of an MPI-only parallelized code on all 28 cores of a dual-socket Intel Xeon E5-2690 v4 (Broadwell) system running at 2.6 GHz to present the memory bound sparse matrix kernels in a fair way as compared to the more compute-heavy algorithms developed in this contribution. The implementation makes use of the optimized sum factorization kernels available in the deal.II library [1, 5, 6] and uses the Trilinos Epetra library for the sparse matrix-vector products with the trace matrix. Our test problem is Eq. (1) with solution u(x, y, z) = sin(10x) sin(12y) cos(10z), diffusion κ = 1, and convection c(x, y, z) = (−y, x, 0). The right hand side function f is set such that the given analytical solution is obtained. We solve the problem on half a spherical ball of radius 1 in the positive hemisphere z ≥ 0 with Dirichlet conditions set on {z = 0} and Neumann conditions on the spherical surface. We also run a 2D experiment with the z component ignored on half a circular disk. Figure 1 displays the computational throughput on experiments recorded at around 10 million unknowns with varying k. The throughput is measured as the number of primal degrees of freedom, (k + 1)d nelements , per second (primal DoFs/s) to make the HDG matrix system (6) in the trace comparable to the larger (u, u) system. The 2D results show a similar performance in the matrix-free implementation as compared to the sparse matrix for the trace. In 3D, however, the matrix-free implementation is more than an order of magnitude faster already for k = 5. The gap in performance is explained by computer architectural properties: the matrix-free implementation replaces the memory-bound sparse matrix-vector product by a less memory-intensive algorithm and achieves a higher arithmetic intensity [6]. The fact that the gap widens as k is increased might seem surprising since both the trace matrix and the matrix-free implementation have a complexity of O((k+1)4) per cell or O(k+1) per primal degree of freedom. The explanation is that the matrix-free implementation is dominated by (face) terms of cost O(k 3 ) for the considered range of polynomial degrees, see the cost estimates in Table 1, a fact also observed in matrix-free implementations for continuous and discontinuous elements [5, 6]. Figure 1 also shows that the matrix-free formulation in a generic basis is 15– 40% slower than the specialized version for collocated node and quadrature points. Furthermore, our experiments illustrate that a matrix formulation for (u, u) instead of the matrix-free kernels is significantly slower than the trace matrix. In other words, the (u, u) formulation demands a matrix-free implementation. Finally, the throughput at up to 5 × 108 primal DoFs/s with the 3D matrix-free implementation on a general mesh is similar to the performance recorded for the symmetric interior penalty method [7]. The significant potential of matrix-free HDG implementations revealed by the experiments in this work motivate the development of iterative solver schemes that can utilize the vastly faster matrix-vector products for the (u, u) formulation. Novel developments are necessary because multigrid techniques established for other formulations [7] are not directly applicable.

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Acknowledgements This work was supported by the German Research Foundation (DFG) under the project “High-order discontinuous Galerkin for the exa-scale” (ExaDG) within the priority program “Software for Exascale Computing” (SPPEXA), grant agreement no. KO5206/1-1, KR4661/2-1 and WA1521/18-1.

References 1. D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin, D. Wells, The deal.II library, version 8.5. J. Numer. Math. 25, 137– 145 (2017) 2. B. Cockburn, Static condensation, hybridization, and the devising of the HDG methods, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, ed. by G.R. Barrenechea, F. Brezzi, A. Cangiani, E.H. Georgoulis (Springer, Cham, 2016), pp. 129–177 3. B. Cockburn, J. Gopalakrishnan, R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic equations. SIAM J. Numer. Anal. 47, 1139–1365 (2009) 4. R.M. Kirby, S.J. Sherwin, B. Cockburn, To CG or to HDG: a comparative study. J. Sci. Comput. 51, 183–212 (2012) 5. M. Kronbichler, K. Kormann, A generic interface for parallel cell-based finite element operator application. Comput. Fluids 63, 135–147 (2012) 6. M. Kronbichler, K. Kormann, Fast matrix-free evaluation of discontinuous Galerkin finite element operators (2017). arXiv preprint arXiv:1711.03590 7. M. Kronbichler, W.A. Wall, A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers (2016). arXiv preprint arXiv:1611.03029 8. M. Kronbichler, S. Schoeder, C. Müller, W.A. Wall, Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation. Int. J. Numer. Methods Eng. 106, 712–739 (2016) 9. R.E. Lynch, J.R. Rice, D.H. Thomas, Direct solution of partial difference equations by tensor product methods. Numer. Math. 6, 185–199 (1964) 10. N.C. Nguyen, J. Peraire, B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations. J. Comput. Phys. 228, 3232–3254 (2009) 11. S.A. Orszag, Spectral methods for problems in complex geometries. J. Comput. Phys. 37, 70– 92 (1980) 12. S. Yakovlev, D. Moxey, R.M. Kirby, S.J. Sherwin, To CG or to HDG: a comparative study in 3D. J. Sci. Comput. 67, 192–220 (2016)

Part XVI

Numerical Methods for Simulating Processes in Porous Media

Numerical Benchmarking for 3D Multiphase Flow: New Results for a Rising Bubble Stefan Turek, Otto Mierka, and Kathrin Bäumler

Abstract Based on the benchmark results in Hysing et al (Int J Numer Methods Fluids 60(11):1259–1288, 2009) for a 2D rising bubble, we present the extension towards 3D providing test cases with corresponding reference results, following the suggestions in Adelsberger et al (Proceedings of the 11th world congress on computational mechanics (WCCM XI), Barcelona, 2014). Additionally, we include also an axisymmetric configuration which allows 2.5D simulations and which provides further possibilities for validation and evaluation of numerical multiphase flow components and software tools in 3D.

1 Introduction The aim of this short note is to present reference results for a 3D rising bubble benchmark which is based on the former 2D benchmark configuration in [1]. In a first step, we describe an ‘easier’ setting in an axisymmetric configuration which allows the rigorous comparison and validation of the used 3D methodology and software based on reference results obtained by a highly accurate 2.5D approach in [2]. Then, in the second step, we demonstrate the numerical convergence behaviour for the 3D configuration for (at least) three successively refined spatial meshes and time steps which demonstrate that the proposed reference values are (almost) grid independent. Finally, we compare the new results with the previously published results in [3] which are slightly improved in terms of accuracy so that a new validated set of reference data is available now which can be found and downloaded from www.featflow.de.

S. Turek () · O. Mierka Institute for Applied Mathematics (LS3), TU Dortmund University, Dortmund, Germany e-mail: [email protected]; [email protected] K. Bäumler Department of Radiology, Stanford University, Stanford, CA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_54

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1.1 Definition of the Benchmark Configuration The presented benchmark considers isothermal, incompressible flow of two immiscible fluids. The conservation of momentum and mass is described by the NavierStokes equations    ∂u + (u · ∇)u = −∇p + ∇ · μ(x)(∇u + (∇u)T ) + ρ(x)g ρ(x) ∂t 

∇ ·u=0 in a fixed space-time domain Ω × [0, T ], where Ω ⊂ R3 . Here, ρ(·) and μ(·) denote the density and viscosity of the fluids, u the velocity, p the pressure, and g the external gravitational force field. It is assumed that fluid 1 occupies the domain Ω1 and that it completely surrounds fluid 2 in Ω2 (see Fig. 1), in particular Γ := ∂Ω1 ∩ ∂Ω2 , Ω = Ω1 ∪ Γ ∪ Ω2 , and ∂Ω2 ∩ ∂Ω = ∅. Surface tension effects are taken into consideration through the following force balance at the interface Γ B B [u]|Γ = 0, [−pI + μ(∇u + (∇u)T )]B · nˆ = σ κ nˆ Γ

where nˆ is the unit normal at the interface, σ is the surface tension coefficient, κ is the curvature of the interface (Table 1). The configuration of this benchmark problem is designed as an extension of its 2D predecessor benchmark introduced by Hysing et al. [1]. Besides the dimensional difference (2D/3D) the only difference between the two benchmarks is the boundary condition imposed on the vertical sides of the domain being no-slip in the 3D case instead of the free-slip condition in the 2D case. This difference originates from Fig. 1 Geometry with the corresponding L1 mesh and initial condition of the 3D rising bubble benchmark

Numerical Benchmarking of a 3D Rising Bubble Table 1 Dimensionless physical parameters and geometrical statistics of the 3D bubble benchmark

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ρ2 100 #nel 2280 18.240 145.920

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#doftotal 78,762 596,958 4,647,990

the pioneering work of Adelsberger et al. [3], where such an adaptation has been chosen. The geometric description of the benchmark at initial condition is displayed at Fig. 1. All walls of the domain are characterized by no-slip boundary conditions. The considered benchmark quantities are selected as for the original 2D benchmark, namely: – Bubble Size—Size of the bubble in two different directions, namely in the rise direction (z) expressed as Rz /R0 (normalized w.r.t. initial bubble size R0 ) and in a perpendicular-to-rise direction being aligned with one of the cartesian axis (x or y) expressed as Rx,y /R0 . – Bubble Sphericity—The “degree of sphericity” in R3 is defined as 4πR02 A0 = . A A Here, A0 denotes the area of a sphere with the initial bubble diameter R0 which has a volume equal to that of the bubble with area A. – Rise Velocity—The mean velocity with which the bubble is rising or moving and which is defined as  Ω u dx Uc =  2 Ω2 1 dx where Ω2 denotes the region that the bubble occupies. The velocity component in the direction opposite to the gravity vector is then denoted as rise velocity Vc , for which the stationary limit is called terminal velocity.

1.2 Short Description of the Used Numerical Techniques The 3D results are based on a specific extension (‘FeatFlower’) of the FEM based open-source software package FeatFlow (http://www.featflow.de) which is

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a 3D multiphase CFD package (see also [4]) associated with the following key attributes: – – – –

parallelized on the basis of domain decomposition techniques equipped with geometrical (Newton)-multigrid solvers using higher order Q2 /P1 iso-parametric elements using semi-implicit surface tension treatment on the basis of the LaplaceBeltrami transformation – using interface-aligned, moving meshes in the framework of PDE based mesh deformation techniques together with the Arbitrary-Lagrangian-Eulerian method The main components of the 2.5D software by K. Bäumler which is described in [2] can be shortly listed as follows:

– – – – –

iso-parametric P2 /P1 finite elements semi-implicit treatment of surface tension via Laplace-Beltrami transformation interface-aligned meshes which are moving in an ALE framework use of a reference frame fixing the center of mass of the bubble subspace projection method for implementing interface conditions

2 Numerical Simulation Results First, the validation of the 3D code w.r.t. the 2.5D approach for the axisymmetric configuration will be presented, demonstrating the temporal and spatial convergence behaviour, before switching to the fully 3D benchmark test case.

2.1 Validation of the 3D Results via 2.5D Configuration The corresponding results in Fig. 2 provide the results for different mesh levels and time steps and demonstrate that the 3D results are more or less independent of the chosen spatial refinement levels and that they agree, for a sufficiently small time step, very well with the results by the 2.5D approach. Based on these results and the corresponding numerical analysis, we claim that the applied 3D code is validated and is able to reproduce the ‘reference results’ which have been calculated via the special 2.5D code.

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2.2 3D Benchmark Results Having validated the 3D multiphase flow solver via the shown axisymmetric configuration, we perform the analogous simulations for the described full 3D benchmark case. Again, we provide in Table 2 the results for different mesh levels in space and time for selected benchmark quantities and mark the resulting reference results: Here, we use the L3 results with the smallest time step size as reference values (corresponding line is marked with bold in Table 2) since the simulations show that the higher mesh level L4 leads to almost identical values as compared with level L3. Moreover, we provide in Fig. 3 the corresponding temporal plots for the described benchmark quantities. Finally, we compare in Fig. 4 the ‘new’ reference results with the previously published results in [3]. Table 2 Convergence of the monitored quantities at T = 1.0 and at final time T = 3.0 w.r.t. temporal and spatial refinements L T = 1.0 2 2 2 3 3 3 3 4 4 T = 3.0 2 2 2 3 3 3 3 4 4

Δt 1e−3

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0.27509 0.27553 0.27553 0.27513 0.27560 0.27568 0.27583 0.27514 0.27561

0.35653 0.35694 0.35707 0.35652 0.35695 0.35707 0.35709 0.35650 0.35695

0.78862 0.78778 0.78763 0.78858 0.78763 0.78739 0.78708 0.78855 0.78757

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0.34779 0.34855 0.34875 0.34775 0.34851 0.34871 0.34876 0.34775 0.34851

0.73614 0.73424 0.73375 0.73622 0.73430 0.73382 0.73368 0.73623 0.73431

1.15893 1.15676 1.15660 1.15919 1.15806 1.15778 1.15776 1.15932 1.15820

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3 Conclusions In this short paper, we have described the settings for a quantitative 3D Rising Bubble benchmark which is based on the previous studies in [1] and [3]. After validating the 3D code via an axisymmetric configuration (allowing to compare with corresponding highly accurate 2D simulations), we provide (new) reference benchmark quantities which all can be downloaded from www.featflow.de. A more detailed numerical analysis of the benchmark simulations as well as a more detailed description of the used methodology and codes will be part of a forthcoming paper.

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Acknowledgements The financial support of DFG (SPP 1740) is gratefully acknowledged (TU 102/53-1). The computations have been carried out on the LiDOng cluster at TU Dortmund University. We would like to thank the LiDOng cluster team for their help and support.

References 1. S. Hysing, S. Turek, D. Kuzmin, N. Parolini, E. Burman, S. Ganesan, L. Tobiska, Quantitative benchmark computations of two-dimensional bubble dynamics. Int. J. Numer. Methods Fluids 60(11), 1259–1288 (2009) 2. K. Bäumler, Simulation of single drops with variable interfacial tension, Ph.D. thesis, FriedrichAlexander-Universität Erlangen-Nürnberg, 2014 3. J. Adelsberger, P. Esser, M. Griebel, S. Groß, M. Klitz, A. Rttgers, 3D incompressible two-phase flow benchmark computations for rising droplets, in Proceedings of the 11th World Congress on Computational Mechanics (WCCM XI), Barcelona, 2014 4. S. Turek, O. Mierka, S. Hysing, D. Kuzmin, Numerical study of a high order 3D FEM-Level Set approach for immiscible flow simulation, in Computational Methods in Applied Sciences, ed. by S. Repin, T. Tiihonen, T. Tuovinen (Springer, Dordrecht, 2013), pp. 65–91

A Linear Domain Decomposition Method for Two-Phase Flow in Porous Media David Seus, Florin A. Radu, and Christian Rohde

Abstract This article is a follow up of our submitted paper (D. Seus et al, Comput Methods Appl Mech Eng 333:331–355, 2018) in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of two-phase in porous media and the convergence of the proposed domain decomposition method is rigorously shown.

1 Introduction Soil remediation, enhanced oil recovery, CO2 storage and geothermal energy are among the most important applications of porous media research and are notable examples of multiphase flow processes through porous media. In these situations mathematical modelling and simulations are often the only tools available to predict subsurface processes as well as to assess feasibility and risk of envisioned technology, since measurements below surface are very difficult, expensive or not possible at all. Specifically when considering layered soil with very different porosity and permeability in each layer, the mathematical and computational problems appearing are most challenging as soil parameters may even be discontinuous and the appearing coupled nonlinear partial differential equations change type and degenerate. In these settings Newton-based solvers can struggle with robustness and convergence.

D. Seus () · C. Rohde University of Stuttgart, Institute of Applied Analysis and Numerical Simulation, Stuttgart, Germany e-mail: [email protected] F. A. Radu University of Bergen, Department of Mathematics, Bergen, Norway e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_55

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To overcome the difficulties in robustness, the L-type linearisation, which replaces the Newton solver by a fixed point type iteration, has been proposed and tested in various model settings. We refer to [9, 13] for an overview of the application of the L-scheme to the Richards equation as well as comparisons to other methods and only mention [14] and [10], where the L-scheme was used in combination with mixed finite elements. More recently, the same ideas have been extended to two-phase flow in porous media, cf. [11] for finite volumes and [12] for mixed finite elements. While most of the mentioned papers assume a Lipschitz continuous dependency of the water saturation on the pressure, error estimates for the Hölder continuous case were first introduced in [12], which is highly relevant due to van Genuchten-Mualem parametrisations falling into this category. The Lscheme has also been applied to other models and coupling problems. Karpinski et al. analyses the method in [7] for the case of two-phase flow including dynamic capillary pressure effects. Borregales et al. and Both et al. [4, 5] propose an optimised Fixed Stress Splitting method, based on an L-type linearisation technique to solve robustly a coupling of flow and geomechanics, modelled by linearised Biot’s equation. The added robustness that L-type linearisations offer, comes at the price of a merely linear convergence rate. Aside from using the L-scheme as a preconditioner as mentioned above, another way of improving convergence speed is by combining the L-scheme with a model based domain decomposition ansatz. In doing so, the decomposition can be performed in a physically relevant manner, taking into account e.g. actual layers of the soil. In [2, 3], the authors considered a substructuring of the Richards equation along the soil layers and applied monotone multigrid methods to solve the substructured problems. (Optimized) Schwarz-Waveform methods for Richards equation were considered in [1], where also a posteriori error estimates and stopping criteria were discussed. For the full two-phase flow system domain decomposition based on mortar finite elements and Newton-based solvers have been considered in [15–17]. In contrast to the existing approaches, we propose a combination of a nonoverlapping domain decomposition ansatz and an L-scheme linearisation for twophase flow in porous media, originally dating back to the work of Schwarz and later Lions, see e.g. [8]. The scheme, called LDD, is formulated independently of a concrete space discretisation and uses a Robin type interface condition to decouple both subdomain problems. Decoupling and linearisation is achieved in one step instead of two separate ones and the use of Newton’s method is avoided. Maintaining the form of the equations in physical variables makes the method particularly accessible for application in the engineering context. In Sect. 2, we introduce the problem formulation, notation and formulate the iterative scheme. The ideas presented here follow closely [13]. Section 3 is devoted to the formulation and proof of the main result of this article, the convergence of the scheme. Finally, we draw conclusions and give a brief outlook on future work.

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2 Problem Description Let Ωl ⊂ Rd (l = 1, 2) be two Lipschitz domains connected through the interface Γ . We consider the flow of two immiscible, incompressible fluids in an isotropic, non-deformable porous medium which is governed by the equations, [6], k     i,l Φl ∂t Sl (pnw,l , pw,l )− ∇· kw,l Sl (pnw,l , pw,l ) ∇ pw,l − zw = fw,l (1) μw k     i,l −Φl ∂t Sl (pnw,l , pw,l )− ∇· knw,l 1 − Sl (pnw,l , pw,l ) ∇ pnw,l − znw = fnw,l μnw (2) on Ωl × (0, T ) and the coupling conditions pα,1 = pα,2 ,

Fα,1 · n1 = Fα,2 · n1

α = w, nw

on Γ × (0, T ).

(3)

All notation will be introduced in Notation 1. We adopt a pressure-pressure formulation with wetting and non-wetting pressures pl,w , pl,nw as primary variables, together with the continuity of pressures and fluxes (cf. Notation 1) as coupling conditions over the interface. Throughout the article, we adhere to the following notational conventions and abbreviations. Notation 1 Sl is the water saturation and is assumed to be a function of the phase pressures via the capillary pressure saturation relationship pc,l (Sl ) = pnw,l − pw,l , [6], i.e., it is assumed that pc,l is invertible, cf. Assumptions 1. In particular, we exclude hysteresis effects. pw,l and pnw,l are the pressures of the wetting and nonwetting phases on Ωl , respectively. Note that because of the definition of the capillary pressure it is known, that the dependence of Sl on the  pressures is of the −1  pnw,l − pw,l , which will become form Sl (pnw,l , pw,l ) = Sl (pnw,l − pw,l ) = pc,l essential in the proof, as we will see. n , pn n := n · τ and S k = pw,l nw,l denote the pressures at time step t l k , pk ). Here, Φ are the constant porosities on each Ω , ρ denote Φl Sl (pnw,l l l α w,l the densities of the phases, μα are the viscosities and assuming an intrinsic permeability of the form Kl = ki,l Ed (Ed the identity matrix, dropped in the notation, ki,l ∈ L∞ (Ω, R) ∩ C 0,1 (Ω, R)), we abbreviate   ki,l k k kw,l Sl (pnw,l , pw,l ), μw  ki,l n,i n,i  := kw,l Sl (pnw,l , pw,l ), μw

  ki,l k k knw,l 1 − Sl (pnw,l , pw,l ), μnw  ki,l n,i n,i  := knw,l 1 − Sl (pnw,l , pw,l ), μnw

k kw,l :=

k knw,l :=

n,i kw,l

n,i knw,l

(4) n,i where kα,l are the relative permeability functions. The pressures pα,l are the iterates in our scheme, henceforth called the LDD-scheme. Finally, we define (already

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used in (3)) β

  ki,l β β   β kw,l Sl (pnw,l , pw,l ) ∇ pw,l − zw , μw   ki,l β β   β := − knw,l 1 − Sl (pnw,l , pw,l ) ∇ pnw,l − znw μnw

Fw,l := − β

Fnw,l

(5)

for the fluxes, where we abbreviated zα = ρα gx3 for the gravitational term. β can be empty, meaning the continuous case, as well as β = k, meaning the pressure  n,i n,i−1  n,i iterate at time step k. In the case β = n, i, we define Fα,l := −kα,l ∇ pα,l − zα instead, which is used in the i-th iteration of the LDD-scheme. For later use we also n,i n,i define Sln,i := Φl Sl (pnw,l , pw,l ). Furthermore, denoting the whole domain by Ω := Ω1 ∪ Γ ∪ Ω2 , the following spaces will be used. L2 (Ω) is the space of Lebesgue measurable, square integrable functions over Ω. H 1 (Ω) contains functions in L2 (Ω) having also weak derivatives 1 in L2 (Ω). H01 (Ω) = C0∞ (Ω)H , where the completion is with respect to the standard H 1 norm and C0∞ (Ω) is the space of smooth functions with compact support in Ω. The definition for H 1 (Ωl ) (l = 1, 2) is similar. With Γ being a (d −1) ¯ H 12 (Γ ) contains the traces of H 1 functions on Γ Given dimensional manifold in Ω, u ∈ H 1 (Ω), its trace on Γ is denoted by u|Γ . We abbreviate   B Vl := u ∈ H 1 (Ωl ) B u|∂Ωl ∩∂Ω ≡ 0 ,   B V := (u1 , u2 ) ∈ V1 × V2 B u1|Γ ≡ u2|Γ ,

(6) (7)

B = > 1/2 H00 (Γ ) = ν ∈ H 1/2(Γ ) B ν = w|Γ for a w ∈ H01 (Ω) .

(8)

Note, that V = H01 (Ω). H00 (Γ ) denotes the dual space of H00 (Γ ). ·, ·X will denote the L2 (X) scalar product, with X being one of the sets Ω, Ωl (l = 1, 2) or Γ . Whenever self understood, the notation of the domain of integration X will be G H 1/2 dropped. Furthermore, ·, · Γ stands also for the duality pairing between H00 (Γ ) 1/2

1/2

1/2

and H00 (Γ ). After a backward Euler discretisation in time with time step τ := N ∈ N0 , the coupled two-phase flow problem in weak form reads

T N

for some

n n Problem 1 (Semi-Discrete Coupled Two-Phase Flow System) Find (pα,1 , pα,2 ) 1/2

n ·n ∈H ∈ V, α ∈ {w, nw}, such that Fα,l l 00 (Γ ) and

G

H G n H G n H H G n Sln − Sln−1 , ϕw,l − τ Fw,l , ∇ϕw,l + τ Fw,3−l · nl , ϕw,l |Γ Γ = τ fw,l , ϕw,l (9)

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G G n H G n H G n H H − Sln − Sln−1 , ϕnw,l − τ Fnw,l , ∇ϕnw,l +τ Fnw,3−l · nl , ϕnw,l |Γ Γ = τ fnw,l , ϕnw,l (10) are satisfied for all (ϕw,l , ϕnw,l ) ∈ V. Note, that the pressure coupling is implicitly contained in the weak form, cf. [13]. The following general assumptions will be used throughout the rest of the article. Assumptions 1 For l = 1, 2 we assume that1 a) the relative permeabilities of the wetting phases kw,l : [0, 1] → [0, 1] are strictly monotonically increasing and Lipschitz continuous functions with Lipschitz constants Lkw,l . The relative permeabilities of the non-wetting phases knw,l : [0, 1] → [0, 1] on both domains are strictly monotonically decreasing and Lipschitz continuous functions with Lipschitz constants Lknw,l . k k b) there exists m ∈ R such that i,lμαα,l ≥ m > 0, for α = w, nw. c) The capillary pressure saturation relationships pcl (Sl ) := pnw,l − pw,l are monotonically decreasing functions and therefore the saturations, Sl pcl =   Sl pnw,l − pw,l are also monotonically decreasing as functions of pcl and moreover assumed to be Lipschitz continuous with Lipschitz constants LSl . Note, that by an abuse of notation, we will actually denote by Lkα,l the Lipschitz k  kα,l in the proof of Theorem 3 below. Completely constant of the function i,l μ∞ α analogous to [13], we introduce an iteration scheme to solve Problem 1 that linearises and decouples simultaneously. To this end, let λα ∈ (0, ∞) and assume  n−1 n−1  n,0 n−1 , pα,2 ∈ V is given for α ∈ {w, nw}. Set pα,l := pα,l as well as that pα,1 0 := F n−1 · n − λ pn−1 | and assume that for some i ∈ N the approximations gα,l α α,l Γ 0 l α,l = n,k >i−1 = k >i−1 (solutions of Problem 2) pα,l as well as g are already known for α,l k=0 k=0 l = 1, 2 and α ∈ {nw, w}. Choosing additionally constants Lα,l > 0 for l = 1, 2, the constraints of which will be made precise in Theorem 3, the LDD iteration scheme consists of solving the following  n,i n,i  , pα,2 ∈ V such that Problem 2 (LDD-Scheme) Find pα,1 G n,i H G n,i H G H n,i i Lα,l pα,l , ϕα,l − τ Fα,l , ∇ϕα,l + τ λα pα,l |Γ + gα,l , ϕα,l |Γ Γ G i−1 H G H G n H = Lα,l pα,1 , ϕα,l + (−1)δαw Sln,i−1 − Sln−1 , ϕα,l + τ fα,l , ϕα,l G i G H H i−1 i−1 with gα,l , ϕα,l |Γ Γ := − 2λα pα,3−l |Γ − gα,3−l , ϕα,l |Γ Γ ,

(11) (12)

is fulfilled for all (ϕα,1 , ϕα,2 ) ∈ V, l = 1, 2, α = w, g. Here, the Kronecker symbol δab is used. 1 Similar assumptions are used in the literature, cf. [10]. More recently, the case of Hölder continuity has been treated, see [12].

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n,i n i By taking the formal limit in Problem 2, assuming that pα,l → pα,l and gα,l → gα,l , for some function gα,l , the limit system of the LDD-scheme is

G n G H G n H G H H n (−1)δαw +1 Sln − Sln−1 , ϕα,l − τ Fα,l , ∇ϕα,l + τ λα pα,l |Γ + gα,l , ϕα,l |Γ Γ = τ fα,l , ϕα,l

(11’) G

where

gα,l , ϕα,l |Γ

H

G

Γ

:= − 2λα pα,3−l |Γ − gα,3−l , ϕα,l |Γ

H Γ

.

(12’) This can be shown to be equivalent to Problem 1 analogously to [13, Lemma 2]. The next section will be devoted to showing, that the LDD-scheme actually converges to this limit system.

3 Convergence of the Scheme We are now ready to formulate and prove our main result, the convergence of the LDD-scheme. n , pn ) ∈ V, α ∈ {nw, w}, Theorem 3 Assume there exists a unique solution (pα,1 α,2  n to Problem 1 that additionally satisfy supl,α ∇ pα,l − zα L∞ ≤ M < ∞.  Let λα > 0 and Lα,l ∈ R satisfy L1S − α 2L1α,l > 0 for l = 1, 2. For l

n,0 := νl,α ∈ Vl , (l = 1, 2, α = w, nw), arbitrary starting pressures pα,l = n,i = n,i n,i > n,i > let pw,1 , pw,2 i∈N , pnw,1 , pnw,2 i∈N ∈ VlN be a sequence of solutions to 0 = i > 0 Problem 2, gα,l being defined by (12). Assume, that the time step τ satisfies i∈N 0

C(LSl , Lα,l , M, m) :=

1

L2kα,l M 2 1 >0 − −τ LSl 2Lα,l 2m α α

(13)

n,i n in V and g i → g  for l = 1, 2. Then, pα,l → pα,l l α,l in Vl as i → ∞ for l = 1, 2 α,l and both phases. α,i := Proof For α ∈ {w, nw} and l = 1, 2, we introduce the iteration errors ep,l n − pn,i as well as e α,i := g i n n pα,l α,l − gα,l . Adding Lα,l pα,l , ϕα,l  − Lα,l pα,l , ϕα,l  α,l g,l to Eq. (11’) and subtracting Eq. (11), we arrive at G α,i G α,i H G α,i H H Lα,l ep,l , ϕα,l + τ λα ep,l |Γ , ϕα,l |Γ Γ + τ eg,l , ϕα,l |Γ Γ 5I J6   n,i n,i−1  n n,i−1  n n + τ − Fα,l −kα,l ∇ pα,l − zα + kα,l ∇ pα,l − zα + Fα,l , ∇ϕα,l G α,i−1 H G H G H = Lα,l ep,l , ϕα,l + (−1)δαw Sln − Sln−1 , ϕα,l − (−1)δαw Sln,i−1 − Sln−1 , ϕα,l . *+ , ) G H n,i−1 n (−1)δαw Sl −Sl

,ϕα,l

(14)

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α,i Inserting ϕα,l := ep,l in Eq. (14) and noting the identity

I J L 57 7 7 α,i−1 72 7 α,i 7 6 α,l 7 α,i 72 α,i α,i−1 α,i 7 + 7e − eα,i−1 72 , Lα,l ep,l ep,l − 7ep,l − ep,l , ep,l = p,l p,l 2 (15) yields 7 7 7 6 7 7 G H Lα,l 57 7eα,i 72 − 7eα,i−1 72 + 7eα,i − eα,i−1 72 + τ λα eα,i |Γ , eα,i |Γ p,l p,l p,l p,l p,l p,l Γ 2 G n H G α,i α,i α,i H = Sl − Sln,i−1 , (−1)δαw ep,l , ep,l |Γ Γ − τ eg,l   J I  n,i−1 α,i n n ∇ pα,l − τ kα,l − kα,l − zα , ∇ep,l J I n,i−1 α,i α,i ∇ep,l , ∇ep,l . (16) − τ kα,l G w,i−1 nw,i−1 H Summing up Eq. (16) over α = w, g and adding Sln − Sln,i−1 , ep,l − ep,i yields

Lα,l 57 α,i 72 7 α,i−1 72 7 α,i 7 6 G H 7 + 7e − eα,i−1 72 + S n − S n,i−1 , ew,i−1 − enw,i−1 7e 7 − 7e l p,l p,l p,l p,l l p,l p,i 2 *+ , ) α I1

G G  nw,i−1 w,i−1 w,i nw,i H α,i α,i α,i H = Sln − Sln,i−1 , ep,l − ep,l − ep,i − ep,l −τ |Γ + eg,l , ep,l |Γ Γ λα ep,l *+ , ) α −τ )

I α



=:I2

J

n − k n,i−1 ∇ p n − z , ∇eα,i − τ kα,l α α,l α,l p,l

*+

=:I3

,

)

J

I n,i−1 α,i α,i kα,l ∇ep,l , ∇ep,l . α

*+

=:I4

,

(17) We estimate terms  the assigned   I1 –I4 from  (17) one by one and start with I1 . Recall Sl pw,l , pnw,l = pc−1 pnw,l − pw,l and that pc < 0 so that we actually     have the dependence Sl pw,l , pnw,l = Sl pnw,l − pw,l where Sl is monotonically decreasing. Thereby we have B2 B    n,i−1 B n,i−1 B n n − pw,l − pw,l − Sl pnw,l B BSl pnw,l BB B  B   n,i−1 B n,i−1 BB nw,i−1 w,i−1 B n n − Sl pnw,l ≤ LSl BSl pnw,l − pw,l − pw,l − ep,l BBep,i B       n,i−1 n,i−1  w,i−1 nw,i−1 n n = LSl Sl pnw,l − Sl pnw,l ep,l − pw,l − pw,l − ep,i

(18)

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as a result of the Lipschitz continuity of Sl . Therefore, by integrating (18), we estimate I1 by 7 H G 1 7 7S n − S n,i−1 72 ≤ S n − S n,i−1 , ew,i−1 − enw,i−1 . l l l l p,l p,i LSl Young’s inequality |xy| ≤ |x|2 +

1 2 4 |y|

(19)

with  > 0 applied to the term I2 , gives

B L 7 BG 7  nw,i−1 B w,l 7 w,i−1 w,i−1 w,i nw,i HB w,i 72 |I2 | = B Sln − Sln,i−1 , ep,l − ep,l − ep,i − ep,l ep,l − ep,l B≤ 2   7 n n,i−1 72 7  7 Lnw,l 7 nw,i−1 1 1 nw,i 72 7S −S 7 , ep,i + − ep,l + + l l 2 2Lw,l 2Lnw,l

(20) L

where we chose α = 2α,l for α = w, nw. For I3 , consider the estimation of the summands BI   JB 7  77 α,i 7 B n n,i−1 α,i B n,i−1   n n n 7 − zα , ∇ep,l − kα,l ∇ pα,l − zα 77∇ep,l B kα,l − kα,l ∇ pα,l B ≤ 7 kα,l 77 α,i 7 7 7 7 7 L M7 7 ≤ Lk Mα,l 7S n − S n,i−1 72 + kα,l 7∇eα,i 72 . ≤ Lkα,l M 7Sln − Sln,i−1 77∇ep,l l α,l l p,l 4α,l

(21)  n Here, we used the Lipschitz-continuity of kα,l and the assumption supl,α ∇ pα,l −  zα ∞ ≤ M. α,l will be chosen later. I3 can therefore be estimated as |I3 | ≤ τ

α

Lkα,l M 7 α,i 72 7 72 7∇e 7 . Lkα,l Mα,l 7Sln − Sln,i−1 7 + τ p,l 4 α,l α

(22)

J I n,i−1 α,i α,i > ∇ep,l , ∇ep,l Finally, by Assumption 1b), we estimate I4 by τ kα,l 7 α,i 72 τ m7∇ep,l 7 . Using this and the estimates (19), (20) and (22), Eq. (17) becomes

G

Lα,l 57 α,i 72 7 α,i−1 72 6 7 7 α,i α,i α,i H 7e 7 − 7e 7 + 1 7S n − S n,i−1 72 + τ |Γ + eg,l , ep,l |Γ Γ λα ep,l l p,l p,l l 2 L S l α α   7

 1

Lkα,l M 72 7 α,i 72 7 . ≤ + τ Lkα,l Mα,l 7Sln − Sln,i−1 7 + τ − m 7∇ep,l 2L 4 α,l α,l α α

(17’) G H G α,i α,i H , ep,l Γ , recall that ·, · Γ denotes In order to deal with the interface terms τ eg,l

both scalar product in H00 (Γ ) and dual pairing for functionals in H00 (Γ ) . 1/2

1/2

LDD-Scheme for Two Phase Flow

611

α,i α,i−1 α,i−1 Subtracting (12) from (12’), i.e. obtaining eg,l = −2λα ep,3−l − eg,3−l , we get

7 7 7 α,i 72 7 7 G H  7e 7 = 1 7eα,i+1 72 − 7eα,i 72 − 4λα eα,i , eα,i p,l Γ g,l Γ p,l g,l Γ . g,3−l Γ 4λ2α

(23)

Inserting Eq. (23) in Eq. (17’), we arrive at 

  7

 1

Lkα,l M 7 α,i 72 1 2 n,i−1 7 n 7∇e 7 7 7 Sl − Sl − + τ Lkα,l Mα,l +τ m− p,l L Sl 2Lα,l 4α,l α α

1 7 α,i 72

Lα,l 57 α,i−1 72 7 α,i 72 6 7 7  7e 7 − 7e 7 + τ 7e 7 − 7eα,i+1 72 . ≤ p,l p,l g,l g,3−l Γ Γ 2 4λα α α

(24) Lk

M

Lk

M

α,l such that m − 4α,lα,l = m2 > 0 for both l and α. Now choose α,l = 2m Taking into account that by assumption Lα,l have been chosen large enough that 1 1 − α 2Lα,l > 0 and that (13) holds, summing (24) first over l = 1, 2 and then L Sl over iterations i = 1, . . . , r then leads to

2 r

i=1 l=1

r

7 72 7 m7 7∇eα,i 72 C(LSl , Lα,l , M, m)7Sln − Sln,i−1 7 + τ p,l 2 i=1 α,l

1 7 α,1 72 7 α,r+1 72 

Lα,l 57 α,0 72 7 α,r 72 6 7e 7 −7e 7 + τ 7e 7 −7e 7 , ≤ p,l p,l g,l Γ g,l Γ 2 4λα α,l

α,l

(25) where the telescopic nature of the sums on the right hand side have been exploited. This implies the estimates 2 r

i=1 l=1

1 7 α,1 72 72 Lα,l 7 α,0 72 7 7e 7 , 7e 7 +τ C(LSl , Lα,l , M, m)7Sln −Sln,i−1 7 ≤ p,l 2 4λα g,l Γ l,α

l,α

(26) τ

r

Lα,l 7 α,0 72

1 7 α,1 72

7 m7 7e 7 . 7∇eα,i 72 ≤ 7e 7 + τ p,l p,l 2 2 4λα g,l Γ i=1 α,l

l,α

(27)

l,α

7 Since the right hand sides are independent of r, we thereby conclude that 7Sln − 7 7 7 α,i 7 −→ 0 as i → ∞. Due to the partial homogeneous Dirichlet Sln,i−1 7, 7∇ep,l boundary, the Poincaré inequality is applicable for functions in Vl so that (27) further 7 α,i 7 7 −→ 0 as i → ∞. implies 7ep,l

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α,i In order to show that eg,l → 0 in Vl , we subtract (11) from (11’) for both phases respectively and consider only test functions in ϕα,l ∈ C0∞ (Ωl ), i.e.

I J G α,i H G α,i−1 H G H n,i n −τ Fα,l − Fα,l , ∇ϕα,l = −Lα,l ep,l , ϕα,l + Lα,l ep,l , ϕα,l + (−1)δαw Sln − Sln,i−1 , ϕα,l .

(28)  n n,i  Thus, ∇· Fα,l − Fα,l exists in L2 (Ωl ) and  α,i    n n,i  α,i−1  − (−1)δαw Sln − Sln,i−1 − Fα,l = Lα,l ep,l − ep,l −τ ∇· Fα,l

(29)

almost everywhere, from which we deduce for ϕα,l now taken to be in Vl BI  JB L 7 7 7 77 7 7 α,l 7 α,i B B n,i  α,i−1 7 n 77ϕα,l 7 + 1 7S n − S n,i−1 77ϕα,l 7. ep,l − ep,l − Fα,l , ϕα,l B ≤ B ∇· Fα,l l l τ τ (30) B n,i  B Introducing the abbreviation BΨα,l ϕα,l B for the left hand side of (30), the limit sup ϕα,l ∈Vl

ϕα,l =0

B n,i  B BΨ ϕα,l B l

ϕα,l Vl

≤

7 7 7 Lα,l 7 7eα,i − eα,i−1 7 + 1 7S n − S n,i−1 7 −→ 0 (i → ∞) l p,l p,l l τ τ (31)

7 7 follows as a consequence of (26) and (27). In other words 7Ψln,i 7V  → 0 as i → ∞. l On the other hand, starting again from (14) (without the added zero term), this time however inserting ϕα,l ∈ Vl and integrating by parts, keeping in mind (29), one notices J I H G α,i H G α,i n,i  n eg,l , ϕα,l |Γ Γ = −λα ep,l , ϕα,l |Γ Γ + Fα,l − Fα,l · nl , ϕα,l |Γ . (32) Γ

7 α,i 7 7 → 0 as i → 0 and we will use the continuity of the We already know, that 7ep,l Vl H G α,i , ϕα,l |Γ Γ . For the last summand in (32) we trace operator to deal with the term ep,l have by the integration by parts formula I I J J n,i  n,i n,i n n Fα,l − Fα,l · nl , ϕα,l |Γ = Ψα,l (ϕα,l ) + Fα,l − Fα,l , ∇ϕα,l , Γ

(33)

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613

and the second term can be estimated by BI JB   B B n  n n,i−1  n,i B kα,l ∇ pα,l + zα − kα,l ∇ pα,l + zα , ∇ϕα,l B BI JB  B B n n,i−1   n n,i−1 α,i ∇ pα,l + zα − kα,l − kα,l ∇ep,l , ∇ϕα,l B ≤ B kα,l 7 77 7 7 α,i 77 7 77ϕα,l 7 , ≤ Lkα,l M 7Sln − Sln,i−1 77ϕα,l 7V + Mkα,l 7∇ep,l Vl l (34) where we used the same reasoning as in (21) and max |kα,l | ≤ Mkα,l . With this, we get BI J B B B n,i  n − Fα,l · nl , ϕα,l B B Fα,l

sup ϕα,l ∈Vl

Γ

ϕα,l Vl =1

7 n,i 7 7 7 7 7 7  + Lk M 7S n − S n,i−1 7 + Mk 7∇eα,i 7 −→ 0 ≤ 7Ψα,l l l p,l V α,l α,l

(35)

l

as i → ∞ from (33). Finally, we deduce from (32) and the continuity of the trace ˜ on Lipschitz domains operator (with constant C) sup ϕα,l ∈Vl

ϕα,l =0

BG α,i H B B e , ϕα,l |Γ B g,l

ϕα,l Vl

Γ

7 7 7 α,i 7 7 + 7Ψ n,i 7  ≤ λα C˜ 7ep,l α,l V V l

l

7 7 7 α,i 7 7 −→ 0, + Lkα,l M 7Sln − Sln,i−1 7 + Mkα,l 7∇ep,l

α,i as i → ∞. This shows eg,l → 0 in Vl for l = 1, 2 and α = w, nw and concludes the proof.

4 Conclusion We proposed and analysed a fully implicit non-overlapping domain decomposition method for efficiently solving two-phase flow in heterogeneous porous media. The developed scheme gains its efficiency by decoupling and linearising the decomposed problem in one step and additionally avoids using Newton’s method. Moreover, for sufficiently small time steps, the method converges irrespectively of the initial guess. The generalisation to several soil layers, the analysis for a specific discretisation in space as well as thorough numerical testing are left for future work. Acknowledgements This work was partially supported by the NFR supported project CHI #25510 and by the VISTA project #6367.

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References 1. E. Ahmed, S. Ali Hassan, C. Japhet, M. Kern, M. Vohralík, A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types (2017). https://hal.inria.fr/hal-01540956 2. H. Berninger, O. Sander, Substructuring of a Signorini-type problem and Robin’s method for the Richards equation in heterogeneous soil. Comput. Vis. Sci. 13(5), 187–205 (2010) 3. H. Berninger, R. Kornhuber, O. Sander, A multidomain discretization of the Richards equation in layered soil. Comput. Geosci. 19(1), 213–232 (2015) 4. M. Borregales, F.A. Radu, K. Kumar, J.M. Nordbotten, Robust iterative schemes for non-linear poromechanics. Comput. Geosci. 22(4), 1021–1038 (2018) 5. J. Both, M. Borregales, J. Nordbotten, K. Kumar, F. Radu, Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017) 6. R. Helmig, Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems (Springer, Berlin, 1997) 7. S. Karpinski, I.S. Pop, F.A. Radu, Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two-phase flow in porous media with dynamic capillarity effects. Int. J. Numer. Methods Eng. 112(6), 553–577 (2017) 8. P.-L. Lions, On the Schwarz alternating method, in Proceedings of the 1st International Symposium on Domain Decomposition Methods for Partial Differential Equations, ed. by R. Glowinski, G.H. Golub, G.A. Meurant, J. Periaux (SIAM, Philadelphia, 1988), pp. 1–42 9. F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(2), 341–353 (2016) 10. I.S. Pop, F.A. Radu, P. Knabner, Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168(1–2), 365–373 (2004) 11. F.A. Radu, J.M. Nordbotten, I.S. Pop, K. Kumar, A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015) 12. F.A. Radu, K. Kumar, J.M. Nordbotten, I.S. Pop, A robust, mass conservative scheme for twophase flow in porous media including Hölder continuous nonlinearities. IMA J. Numer. Anal. 38, 884–920 (2017) 13. D. Seus, K. Mitra, I.S. Pop, F.A. Radu, C. Rohde, A linear domain decomposition method for partially saturated flow in porous media. Comput. Methods Appl. Mech. Eng. 333, 331–355 (2018) 14. M. Slodiˇcka, A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media. SIAM J. Sci. Comput. 23(5), 1593–1614 (2002) 15. I. Yotov, A mixed finite element discretization on non–matching multiblock grids for a degenerate parabolic equation arizing in porous media flow. East-West J. Numer. Math. 5, 211–230 (1997) 16. I. Yotov, M.F. Wheeler, Domain decomposition mixed methods for multiphase groundwater flow in multiblock aquifers, in Computer Methods in Water Resources XII. Transactions on Ecology and the Environment, vol. 17 (Wit Press, Ashurst, 1998), pp. 59–66 17. I. Yotov, Scientific computing and applications, ed. by P. Minev, Y. Lin (Nova Science Publishers, Commack, 2001), pp. 157–167

A Numerical Method for an Inverse Problem Arising in Two-Phase Fluid Flow Transport Through a Homogeneous Porous Medium Aníbal Coronel, Richard Lagos, Pep Mulet, and Mauricio Sepúlveda

Abstract In this paper we study the inverse problem arising in the model describing the transport of two-phase flow in porous media. We consider some physical assumptions so that the mathematical model (direct problem) is an initial boundary value problem for a parabolic degenerate equation. In the inverse problem we want to determine the coefficients (flux and diffusion functions) of the equation from a set of experimental data for the recovery response. We formulate the inverse problem as a minimization of a suitable cost function and we derive its numerical gradient by means of the sensitivity equation method. We start with the discrete formulation and, assuming that the direct problem is discretized by a finite volume scheme, we obtain the discrete sensitivity equation. Then, with the numerical solutions of the direct problem and the discrete sensitivity equation we calculate the numerical gradient. The conjugate gradient method allows us to find numerical values of the flux and diffusion parameters. Additionally, in order to demonstrate the effectiveness of our method, we present a numerical example for the parameter identification problem.

A. Coronel () Departamento de Ciencias Básicas, Universidad del Bío-Bío, Chillán, Chile e-mail: [email protected] R. Lagos Departamento de Matemática y Física, Facultad de Ciencias, Universidad de Magallanes, Punta Arenas, Chile e-mail: [email protected] P. Mulet Departament de Matemàtiques, Universitat de València, Burjassot, Spain e-mail: [email protected] M. Sepúlveda CI2MA & Departamento de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_56

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1 Introduction In this paper, we deal with an inverse problem that originates from parameter identification in two-phase fluid flow through porous media. In a broader sense, the inverse problem is motivated by practical situations, in which, on the one hand, state variables can be measured with relative ease, on the other hand, however, the experimental determination of parameters in the coefficients is very costly or even infeasible. We assume that the two-phase fluid flow through the porous media can be modeled with a one-dimensional nonlinear convection-diffusion equation with a weak degeneration. Additionally, we consider that we have a measurement of the recovery response which is defined at fixed time by the total mass of the wetting phase on the reservoir. Thus, in the inverse problem, we want to perform an estimation of the nonlinear model coefficients: flux and diffusion. We formulate the inverse problem as an optimization problem. By making use of this formulation, the natural numerical approach of the inverse problem is the application of numerical optimization methods such as gradient, Newton or quasiNewton. In all these techniques, the cornerstone step is the calculation of the numerical gradient, which is not a trivial task. Now, following the ideas of optimal control theory, we have at least two ways for the gradient calculation: introducing an adjoint equation or introducing a sensitivity system. The adjoint-based gradient calculation for convection-diffusion equations with strong degeneration was introduced in [6], in the context of modelling sedimentation theory. The formal calculation presented in [6] can be straightforwardly adapted to the case of the convection-diffusion equation modelling fluid flow through porous media. Under additional regularity conditions, we have that the adjoint state is given by a backward linear weakly-degenerate convection-diffusion equation. Furthermore, the exact numerical gradient can be calculated via the adjoint state following the discretize-then-differentiate approach, see [2, 6] for details. In this work, we use the sensitivity methodology to deduce the gradient. The continuous sensitivity system is an initial boundary value problem for an uncoupled linear, weakly-degenerate convection-diffusion system. Therefore, the gradient calculation based on the differentiate-then-discretize methodology faces theoretical and numerical difficulties similar to those found when employing the continuousadjoint-state-based calculation. Due to this fact, we adopt a discrete sensitivity system for the calculation of exact discrete gradients. In addition, we show that, in the case of fluid flow through porous media, we can give a rigorous mathematical analysis of the sensitivity methodology. The same analytical framework permits us to prove convergence of the numerical method. The outline of this work is the following: in Sects. 2 and 3 we state precisely the direct and inverse problems, respectively. In Sect. 4 we present the sensitivity equation calculation of gradients. In Sect. 5, we present the numerical method. In Sect. 6, we summarize some convergence results. Finally, in Sect. 7, we give a numerical example.

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2 Direct Problem Two-phase flow in porous media can be described by the solution of the following initial boundary value problem [1]: Φ(x)ut + b(u)x = (K(x)α(u)x )x , (x, t) ∈ QT = [0, 1] × [0, T ], u(0, t) = 1, u(1, t) = 0, u(x, 0) = u0 (x),

(1)

t ∈ [0, T ], x ∈ I :=]0, 1[, (2)

where u is the saturation of the wetting phase, Φ is the porosity of the porous medium, K is the absolute permeability, b and α are nonlinear functions which depend on the relative permeability, the viscosity and the capillary pressure. Modelling aspects and the mathematical analysis require the following assumptions: Φ, K ∈ L∞ (I ), Φ(x) ∈ [Φ− , Φ+ ], K(x) ∈ [K− , K+ ], b ∈ C 1 (I ), α ∈ C 2 (I ), a := α  , a(0) = a(1) = 0, a(s) > 0 for s ∈]0, 1[, b  (u) > 0, u0 (x) ∈ [0, 1],

(3)

u0 , Kα(u0 )x ∈ L∞ (I ) ∩ BV (I ),

for some Φ± , K± ∈]0, 1[ and where BV (I ) denotes the space of functions of finite total variation in I . In particular, by the condition a(0) = a(1) = 0 and a(s) > 0 for s ∈]0, 1[, it follows that (1) is weakly degenerate. Definition 1 Let us consider the space of functions W = {w ∈ H 1 (I ) : w(1) = 0}, W0 = {u ∈ L2 (I ) : α(u) ∈ W, u(0) = 1}, V = {v ∈ C 1 (0, T , C01 (I )) : v(., T ) = 0}. If the following properties u :]0, T [→ W0 , u ∈ [0, 1] a.e. in QT , ut ∈ L2 (0, T ; W  ), α(u) ∈ L2 (0, T ; W ),  5 6  Φuvt + (b(u) − Kα(u)x )vx + Φu0 v(x, 0)dx = 0, v ∈ V , QT

I

are satisfied, then u is called a weak solution of (1) and (2). Theorem 2 ([8]) If the assumptions given in (3) are satisfied, there exists a unique weak solution, in the sense of Definition (1), for (1) and (2).

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3 Inverse Problem For the inverse problem, we suppose that we have some measurements of the recovery response and we want to determine the coefficients f and a in the mathematical model (1) and (2). Indeed, we start by considering that the m quantities E1 , . . . , Em are the total saturation measurements of the wetting phase (recovery response) taken at times t1 , . . . , tm , respectively. In practice, the data are adjusted by a curve, i.e. one defines the observed recovery response as a function E obs : [0, T ] → R such that E obs (ti ) = Ei . Then, the inverse problem of determining the flux and diffusion functions is formulated in an abstract setting as the optimization problem J (b, a) := J (ub,a ), min b, a ∈ Uad

subject to ub,a satisfying (1) and (2),

(4)

  where Uad = (b, a) ∈ C 1 (I ) × C 1 (I ) : b, a satisfying (3) , 1 J (u) = 2



T 0

B2 B B B BEu (t) − E obs (t)B dt,

 Eu (t) =

u(x, t)dx. I

Notice that Eu is the continuous recovery response. It is well known that the functions b and a can be described by a finite number of parameters (see Sect. 7). Then, the inverse problem (4) is reduced to the called parameter identification problem. Indeed, we assume that the flux and diffusion functions are parametrized having a parameter vector e = (e1 , . . . , ed ) ∈ Rd , which means that b(·) = b(·; e) and a(·) = a(·; e). In consequence, the parameter identification is formulated by the following optimization problem in several variables: min Jˆ(e), e ∈ U ad

subject to ue satisfying (1) and (2),

(5)

where U ad = {e ∈ Rd : b(·; e), a(·; e) ∈ Uad } and Jˆ(e) = J (b(·; e), a(·; e)). The existence of solutions for (4) is guaranteed by assuming that (b, a) belongs to a compact subset of Uad and applying the following continuous dependence result (ub1 ,a1 − vb2 ,a2 )(., t)L1 (I ) ≤ C1 (u − v)(., 0)L1 (I ) + C2 b1 − b2 Lip(R) √ √ +C3  a1 − a2 Lip(R) , a.e. t ∈ [0, T ], which implies that J : Uad → R is a continuous functional, see [6] for details. The uniqueness conditions for this inverse problem are difficult to find, even in the case of a linear diffusion term. Thus, in the general case, the inverse problem is ill-posed.

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However, in practical applications of parameter identification, we expected that we have additional information such that we have a unique local solution.

4 The Continuous Sensitivity Gradient Calculation In this subsection, the identification method based on the sensitivity equations is derived. The idea of the sensitivity equations is to track how small changes in the parameters wi := ∂ei u for i = 1, . . . , M affect the solution and thus the cost function. Formally, we differentiate the model equations (1) and (2) with respect to the individual parameters ei to obtain Φ(x)(wi )t + (b (u)wi )x = (K(x)(a(u)wi )x )x + (K(x)αei (u)x − bei (u))x ,

(6)

wi (1, t) = wi (0, t) = wi (x, 0) = 0,

(7)

t ∈ [0, T ],

x ∈ I.

for i = 1, . . . , M. The system (6) and (7) is called the sensitivity system. We notice that the equations in (6) form an uncoupled linear system where the coefficients are calculated by evaluation of the direct problem solution. The Eq. (6) is a linear degenerate parabolic equation (a(u) = 0 for u ∈ {0, 1}) with a source term. These kinds of equations are well posed in the sense of the theory of weak solutions presented in [9], even in the case of low regularity of the coefficients. The gradient of the cost function for the parameter identification problem can be (formally) computed as ∇ Jˆ(e) =



T

   obs (Eu (t) − E (t)) w(x, t)dx dt,

0

(8)

I

where w := (w1 , . . . , wM ) is a solution of the sensitivity system (6) and (7). This gradient can be either used as a first-order necessary condition for a minimum, ∇ Jˆ(e) = 0 or, more practically, to employ a gradient scheme after a discretization. However, there is no obvious way how to directly discretize the perturbation equation since the analytic solution for the direct problem it is not available. Thus, the subsequent strategy is to first discretize the governing equations and then optimize the discrete cost function.

5 Numerical Method For the direct problem a homogeneous discretization of space and time is introduced. For simplicity of the presentation we consider only an explicit finite volume scheme and remark that the ideas can be straightforwardly generalized to implicit and implicit-explicit methods like the ones considered in [7]. Indeed, let us consider

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J, N ∈ N and recall the standard notation of finite difference schemes. We denote by Δx := 1/J and Δt := T /N, the size of space and time steps and denote by λ := Δt/Δx and μ := Δt/(Δx)2 the ratios of these quantities. We define the grid points as (xj , t n ) := (j Δx, nΔx) and denote by unj the numerical solution on the finite volume cell Qnj := (xj −1/2 , xj +1/2 ) × (t n , t n+1 ), where xj +1/2 = (xj + xj +1 )/2 for j ∈ {0, J − 1} with x−1/2 = x0 and xJ +1/2 = xJ . Thus, by the standard arguments of finite volume methods, the following scheme can be stated as follows 6 λ 5 n b(uj ) − b(unj−1 ) Φj 6 μ 5 Kj +1/2 (α(unj+1 ) − α(unj )) − Kj −1/2 (α(unj ) − α(unj−1 )) , + Φj

un+1 =unj − j

(9)

for j = 0, . . . , J , where u0j for j ∈ {0, . . . , J } is the average of u0 over (xj −1/2 , xj +1/2 ), α(un−1 ) := α(un0 ), α(unN+1 ) := α(unN ), Kj +1/2 ≈ K(xj +1/2), Φj ≈ Φ(xj ) and b(un−1 ) := b(1). Observe that the convective part is approximated by the upwind Godunov numerical flux, which is possible since we assume that b satisfies (3). We denote by Δ = (Δx, Δt) and by uΔ the piecewise constant numerical solution of (1) and (2) obtained by the finite volume scheme (9), i.e. uΔ (x, t) = unj , for all (x, t) ∈ QT , if (x, t) ∈ Qnj . Let us denote by Enobs the average of E obs over (t n , t n+1 ) and by En =  Δx Jj=0 unj , n = {0, . . . , N} the recovery response calculated from the solution of the direct problem obtained with the scheme (9) at t = t n . Then, a natural definition of the discrete cost function is given by JΔ (uΔ ) =

N Δt |En − Enobs |2 . 2

(10)

n=0

We define JΔ and JˆΔ by the following relations JΔ (b, a) = JΔ (uΔ (·; b, a)) and JˆΔ (e) = JΔ (b(·, e), a(·, e)). Introducing the notation wnj := ∂e unj for the sensitivity variables and following the lines of the discrete gradient computation given in [4, 5], we deduce that ∇ JˆΔ (e) := (∂e1 JˆΔ (e), . . . , ∂eM JˆΔ (e)) is given by ∇ JˆΔ (e) = Δt

N 

(En − Enobs )

n=0

J

 wnj ,

(11)

j =0

where wnj , is the solution of the following first order perturbation scheme analogue to the continuous system (6) and (7) wn+1 = wnj − j +

6 λ 5 δe b(unj ) − δe b(unj−1 ) Φj

(12)

6 μ 5 Kj +1/2 (δe α(unj+1 ) − δe α(unj )) − Kj −1/2 (δe α(unj ) − δe α(unj−1 )) , Φj

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for j = 0, . . . , J . The initial condition is w0j = 0, for j = {0, . . . , J }, and the derivative operators δe are computed by the following relations δe b(unj ) = b (unj )wnj + ∂e b(unj ) and δe α(unj ) = a(unj )wnj + ∂e α(unj ). Similarly to uΔ , we denote by wΔ the piecewise constant discrete function such that wΔ (x, t) = wnj χQnj (x, t), where χQnj denotes the characteristic function of Qnj .

6 Stability and Convergence Results The rigorous convergence of the numerical methods for the direct problem and sensitivity equation can be developed by application of similar ideas to those introduced by the authors in [3]. We can summarize the required properties in the following stability proposition and convergence theorem. K

K

j−1/2 Proposition 3 Let C1 = supj (Φj )−1 , C2 = supj (max{ j+1/2 Φj , Φj }), and C denotes some (generic) positive constant independent of Δ. Assuming that (3) and the following CFL condition

CF L := λC1 b L∞ (I ) + 2μC2 aL∞ (I ) ≤ 1,

(13)

holds, we have the following estimates unj ∈ [0, 1],

J

j =0

n+1 |un+1 j +1 − uj | ≤

J

j =0

|unj+1 − unj |,

J

Δx|un+1 − unj | ≤ CΔt, j

j =0

J B B

B B BKj +1/2 (α(unj+1 ) − α(unj )) − Kj −1/2 (α(unj ) − α(unj−1 ))B ≤ CΔx, j =0

B B B B sup BKj +1/2 (α(unj+1 ) − α(unj ))B ≤ CΔx.

j =0

Theorem 4 Assuming that (3) and the CFL (13) condition are satisfied, we have that uΔ → u in L1 (QT ) and wΔ → w in V when Δ → 0. We remark that, by Theorem 4, we can expect the convergence of the discrete gradient to the continuous ones. However, to obtain this kind of result we would need to introduce an assumption implying that the cost function is locally convex, i.e. such that the subgradient is well-defined. We remark that this kind of hypothesis is hard to deduce since it is equivalent to the proof of local uniqueness of the inverse problem.

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Fig. 1 The observed, initial guess and identified recovery responses obtained with parameters eobs = (1, 3, 3), eig = (2, 2, 2), and e∞ = (0.9001, 3.1043, 3.1101), respectively

7 A Numerical Example In this section we consider the following parametric form of b and a [1]: b(u) =

μo

uγ

μo u γ , + 2μw (1 − u)γ

a(u) =

μo

uγ

uγ (1 − u)γ p (u), + 2μw (1 − u)γ c

with pc (u) = − (1 − u)u−1 and we assume that Φ(x) = K(x) = 0.2, T = 0.5, CF L = 0.98 (see (13)), u0 (0) = 1 and u0 (x) = 0 for x ∈]0, 1]. Also, we consider that e = (μw , μ0 , γ ). Now, in order to obtain E obs , we consider a direct problem simulation with J = 200, the observation parameters eobs = (1, 3, 3) and introduce a random noise perturbation of 5%. Then, considering J = 100 and starting the gradient from the initial guess parameters eig = (2, 2, 2) we identify the parameters e∞ = (0.9001, 3.1043, 3.1101). The numerical comparison of different recovery responses is given on Fig. 1. We remark that, for the identification algorithm we have used the BFGS quasi-Newton method with linear search satisfying the strong Wolfe condition. Acknowledgements We thank the anonymous reviewer for their insightful comments and suggestions. AC thanks to DIUBB 172409 GI/C, DIUBB 183309 4/R, and FAPEI at U. del Bío-Bío, Chile. RL thanks to PY-F1-01MF16 at U. de Magallanes, Chile. PM thanks to Spanish MINECO projects MTM2014-54388-P and MTM2017-83942-P and Conicyt PAI-MEC folio 80150006. MS thanks to Fondecyt 1140676 and BASAL project CMM, U. de Chile and CI2 MA, U. de Concepción, and by Conicyt project Anillo ACT1118 (ANANUM).

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References 1. M. Afif, B. Amaziane, On convergence of finite volume schemes for one-dimensional two-phase flow in porous media. J. Comput. Appl. Math. 145(1), 31–48 (2002) 2. R. Bürger, A. Coronel, M. Sepúlveda, On an upwind difference scheme for strongly degenerate parabolic equations modelling the settling of suspensions in centrifuges and non-cylindrical vessels. Appl. Numer. Math. 56(10), 1397–1417 (2006) 3. R. Bürger, A. Coronel, M. Sepúlveda, A semi-implicit monotone difference scheme for an initial-boundary value problem of a strongly degenerate parabolic equation modeling sedimentation-consolidation processes. Math. Comput. 75(253), 91–112 (2006) 4. R. Bürger, A. Coronel, M. Sepúlveda, A numerical descent method for an inverse problem of a scalar conservation law modelling sedimentation, in Proceedings of ENUMATH 2007, Graz, Austria, September 2007, ed. by K. Kunisch, G. Of, O. Steinbach, pp. 225–232 (Springer, Heidelberg, 2008) 5. R. Bürger, A. Coronel, M. Sepúlveda, Numerical solution of an inverse problem for a scalar conservation law modelling sedimentation, in Hyperbolic Problems: Theory, Numerics and Applications. Proceedings of Symposia in Applied Mathematics, vol. 67, Part 2 (American Mathematical Society, Providence, 2009), pp. 445–454 6. A. Coronel, F. James, M. Sepúlveda, Numerical identification of parameters for a model of sedimentation processes. Inverse Prob. 19(4), 951–972 (2003) 7. R. Donat, F. Guerrero, P. Mulet, Implicit-explicit methods for models for vertical equilibrium multiphase flow. Comput. Math. Appl. 68(3), 363–383 (2014) 8. Y. Jingxue, On the uniqueness and stability of BV solution for nonlinear diffusion equations. Commun. PDE 15, 1671–1683 (1990) 9. O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type (Translated from the Russian by S. Smith). Translations of Mathematical Monographs, vol. 23 (AMS, Providence, 1968)

A Fully-Implicit, Iterative Scheme for the Simulation of Two-Phase Flow in Porous Media Anna Kvashchuk and Florin Adrian Radu

Abstract In this work, we present a new implicit scheme for two-phase flow in porous media. The proposed scheme is based on the iterative IMPES (IMplicit Pressure Explicit Saturation) method and, therefore, preserves its efficiency in treatment of nonlinearities, while relaxing the time step condition common for explicit methods. At the same time, it does not involve costly computation of Jacobian matrix required for generic Newtons type methods. Implicit treatment of capillary pressure term ensures the stability and convergence properties of the new scheme. This choice of stabilization is supported by mathematical analysis of the method which also includes the rigorous proof of convergence. Our numerical results indicate that the scheme has superior performance compared with standard IMPES and fully implicit methods on benchmark problems.

1 Introduction Mathematical models for two-phase flow in porous media are of a high practical relevance. CO2 storage, nuclear waste management, and ground water remediation are only a few examples of their applications. From a mathematical point of view the two-phase flow models can be represented as a system of coupled, nonlinear, partial differential equations (PDEs). These make possible development and implementation of robust numerical schemes for two-phase flow in porous media a challenging task.

A. Kvashchuk () The National IOR Centre of Norway/UiS, Stavanger, Norway e-mail: [email protected] F. A. Radu University of Bergen, Bergen, Norway e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_57

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In this work we consider a two-phase flow system, with the phases being incompressible and immiscible, and the skeleton matrix non-deformable. An averaged pressure formulation is adopted [14]. One of the most common solution procedure for solving the considered model is the IMplicit Pressure Explicit Saturation method (IMPES) [14], which eliminates nonlinearities by taking advantage of the structure of the system. This common approach provides a good tool for finding the numerical solution of the system. However, an explicit solving of the saturation equation leads to stability problems and imposes restrictions on the size of the time step. The alternatives to IMPES are the fully implicit schemes [3–5, 10, 11, 17] which do not have any restrictions on the time step. The system, arising from applying a fully implicit scheme, is nonlinear and one additionally needs an efficient algorithm for solving it. A common approach is Newton’s method [6, 9] which has quadratic convergence. This comes at a price of a costly computation of derivatives at each iteration. Additionally, the mentioned convergence properties demand that the initial guess is sufficiently close to the true solution, which, in its turn, may imply additional requirements on the step size, see [16]. In this work we propose a new IMPES-based solver that maintains cheap computational costs of IMPES while relaxing its stability constraints. Previously improved and stabilized versions of IMPES scheme were presented in [2, 7, 8, 13]. Our scheme is an iterative IMPES solver with extra stabilization in the capillary pressure term. The stabilization allowed us to get more accurate results with the larger time step compared to IMPES. Also, the new scheme does not involve costly derivative computations, which brings advantages compare to fully implicit formulation with Newton solver. What is more, the linear systems to be solved at each iteration step for the new scheme are better conditioned compared with the one resulting for Newton’s method.

2 Two Phase Flow Model in the Averaged Pressure Formulation The two phase flow model in averaged pressure formulation can be easily derived from general two-phase flow model, which consists of mass balance equation and Darcy’s law for each phase, and constitutive equations [14]. We assume the incompressibility of both porous media and fluids involved. Also we assume fluids are immiscible and the gravity can be neglected. We begin by defining the averaged pressure as: p=

pn + pw , 2

(1)

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where subscript n and w denotes nonwetting and wetting fluids respectively. Combining it with the capillary pressure definition pc = pn − pw we can easily get an expression for the pressures pn = p + 12 pc and pw = p − 12 pc . The computational domain Ω ⊂ R d (d ≥ 1), with d being the dimension of the space, is assumed to have a Lipschitz continuous boundary Γ . Let us denote T as the final computational time, then the two-phase flow model in the averaged pressure formulation can be written as:   1 − ∇ · k λΣ ∇p + λdif ∇pc = Fpr , on Ω × (0, T ) 2 (2)   ∂Sw 1 − ∇ · kλw ∇(p − pc ) = Fsat , on Ω × (0, T ) φ ∂t 2 Sw (x, 0) = Sw0 (x), Sw |Γ = SwΓ ,

p(x, 0) = p0 (x),

on Ω (3)

p|Γ = p Γ , 

Fα Fw , Fsat = , φ is the ρα ρw porosity of the porous medium, k is the permeability, pα , Sα , uα , λα , Fα , and ρα are the pressure, saturation, the volumetric flow rate, phase mobility, external mass flow rate, and the density of each fluid phase α, respectively. where λΣ = λn + λw , λdif = λn − λw , Fpr =

α=n,w

3 Discretization In this study we use linear Galerkin finite elements. We cover our domain Ω with a set Th = T1 , . . . , Tm of non-overlapping simplices Ti , such that Ω = ∪T . We T ∈ Th

define a finite-dimensional space Vh ⊂ H 1 of piecewise linear continuous functions Vh = {υ : υ is continuous on Ω, linear on each Ti , υ = 0 on Γ }. The time mesh 0 ≤ t0 < t1 < . . . < tN = T , tn = nδt, where δt is the time step. Using the above introduced notation, we can write the nonlinear fully discrete variational formulation of our system (2) and (3) at the time tn . n+1 Find p n+1 ∈ Vh , where superscript n + 1 indicates the solution at time h , Sh tn+1 , such that the following equations are satisfied ∀υh ∈ Vh : L K   G H 1 n+1 n+1 n+1 n+1 k λΣ (Sh )∇p h + λdif (Sh )∇pc (Sh ) , ∇υh = Fpr , υh , 2 K

(4)

L E F Shn − Shn+1 1 n+1 n+1 n+1 φ , υh + kλw (Sh )∇(p h − pc (Sh )), ∇υh = Fsat , υh  . Δt 2 (5)

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The system (4) and (5) is still nonlinear, and one option to solve it is Newton’s method, which is quadratical, but only locally, convergent. Let us now introduce the new implicit scheme. In the averaged pressure formulation of the two-phase flow model (2) and (3) the capillary pressure appears under the gradient and depends on the unknown Shn+1,i+1 . We linearize it as follows: ∇pc (Shn+1,i+1 ) ∼ pc (Shn+1,i )∇Shn+1,i+1 ,

(6)

with i denoting the iteration index. This linearization is similar to [8] where it has been shown to be effective for a different formulation of IMPES. Assume that the discrete solution at the fixed time tn is known, then in order to find a solution at the next time step we start the iteration process with the solution at the previous time step, i.e. Shn+1,0 = Shn , and the new iteration scheme to solve (2) and (3) reads as follows. Given Shn and Shn+1,i , find pn+1,i+1 , Shn+1,i+1 such that h K  L  1 n+1,i n+1,i+1 n+1,i n+1,i k λΣ (Sh )∇ph + λdif (Sh )∇pc (Sh ) , ∇υh = Fpr , υh , 2 (7) E

F Shn+1,i+1 − Shn , υh (8) Δt   F E 1  n+1,i n+1,i+1 p − (S )∇S , ∇υ = Fsat , υh . + kλw (Shn+1,i ) ∇p n+1,i+1 h h h 2 c h

φ

The stopping criteria are ||Shn+1,i − Shn+1,i+1 || ≤ T OL, ||pn+1,i − pn+1,i+1 || ≤ h h T OL, with T OL being a tolerance.

3.1 Convergence Analysis In this section we present the convergence proof of the scheme (7) and (8). The techniques used are similar with the ones in [12, 15, 17]. We will make the following assumptions: dpc (Sw ) and ∇pc are Lipschitz continuous dSw n+1 and ∇p h is bounded in the domain Ω, (A2) pc is a decreasing function, and, as a consequence, pc ≤ 0.

(A1) The functions λw , λn , pc , pc =

In the proof we will work with the following errors: epi+1 = p n+1 − p n+1,i+1 , h h esi+1 = Shn+1 − Shn+1,i+1 .

A Fully-Implicit, Iterative Scheme for the Two-Phase Flow

629

Theorem 1 Under the assumptions (A1) and (A2) the scheme (7) and (8) converges linearly when the time step satisfies (16). Proof Subtracting (7) from (4), and (8) from (5) we get E  F  n+1,i n+1,i+1 n+1 n+1 k λΣ (Sh )∇ph − λΣ (Sh , ∇υh )∇ph E +

F 1 1 kλdif (Shn+1 )∇pc (Shn+1 ) − kλdif (Shn+1,i )∇pc (Shn+1,i ), ∇υh = 0, 2 2

(9)

F J E   φ I i+1 n+1,i n+1,i+1 n+1 n+1 , ∇υh e , υh + k λw (Sh )∇ph − λw (Sh )∇ph Δt s J 1I + kλw (Shn+1,i )pc (Shn+1,i )∇Shn+1,i+1 − kλw (Shn+1 )pc (Shn+1 )∇Shn+1 , ∇υh . 2 (10) Equation (9) can be rewritten as: J I J n+1,i i+1 + λ (λΣ (Shn+1 ) − λΣ (Shn+1,i ))∇p n+1 , ∇υ (S )∇e , ∇υ h Σ h p h h L K  1 1 λdif (Shn+1 ) − λdif (Shn+1,i ) ∇pc (Shn+1 ), ∇υh + 2 2 E F 1 + λdif (Shn+1,i )∇(pc (Shn+1 ) − pc (Shn+1,i )), ∇υh = 0. (11) 2 I

Let us test (11) with υh = epi+1 . Using the Lipschitz continuity of the functions λw , λn , λΣ , λdif , ∇pc and that the domain Ω is bounded, we get: 1 λ0Σ ||∇epi+1 ||2 ≤ LλΣ Mp ||esi || ||∇epi+1|| + Lλn Mpc ||esi || ||∇epi+1 || 2 1 1 + Lλw Mpc ||esi || ||∇epi+1|| + λ0dif Lpc ||esi || ||∇epi+1 ||. 2 2 Here and later, for a function η the constants Lη , Mη and η0 are its Lipschitz constant, maximum and minimum values respectively. After additional algebraic manipulations, we get the following estimation for the pressure error: ⎛ ||∇epi+1 || ≤ ⎝

LλΣ Mp λ0Σ

+

Lλn + Lλw 2λ0Σ

Mpc +

λ0dif Lpc 2λ0Σ

⎞ ⎠ ||esi ||.

(12)

Let us return to the Eq. (10) and perform a similar procedure. We test it with υh = esn+1 and use the assumption (A2) together with (A1). As the function pc is negative,

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its smallest value is also negative and we denote it as min(pc ) = −Mpc , where Mpc ≥ 0. It follows φ i+1 2 Δt ||e || + Mλw Mpc ||∇esi+1 ||2 k s 2 ≤ ΔtMλw ||∇epi+1 || ||∇esi+1|| + ΔtLλw Mp ||esi || ||∇esi+1|| +

Δt (Mpc Lλw + Mλw Lpc )MS ||esi || ||∇esi+1 ||. 2

Now using (12), we can estimate the saturation error as: φ i+1 2 Δt ||e || + Mλw Mpc ||∇esi+1 ||2 ≤ ΔtC||esi || ||∇esi+1 ||, k s 2

(13)

where  C = Mλw

LλΣ Mp λ0Σ

+

Lλn + Lλw 2λ0Σ

Mpc +

λ0dif Lpc



2λ0Σ

1 + Lλw Mp + (Mpc Lλw + Mλw Lpc )MS ≥ 0. 2

(14)

εb2 a2 + for all ε > 0 and Poincaré 2ε 2 inequality ||u||L2(Ω) ≤ CΩ ||∇u||L2 (Ω) . The last holds for any u ∈ H01 (Ω). With ε C in Young’s inequality being ε = we obtain the following from (13): Mλw Mpc Further we use Young’s inequality ab ≤

||esi+1 ||2 ≤

ΔtkC 2 ||ei ||2 . 2Mλw Mpc φ s

(15)

This proves that our scheme (7) and (8) linearly converges if the coefficient in front of ||esi ||2 in (15) is less than one, which implies into the following mild restriction on the time step: Δt ≤

2φMλw Mpc kC 2

.

(16)

We remark the appearance of Δt in the convergence rate in (15), which indicates a very fast convergence.

A Fully-Implicit, Iterative Scheme for the Two-Phase Flow

631

3.2 Numerical Results In this section we compare numerically the new iterative scheme’s performance with IMPES and Newton’s method. In all the test cases we choose the right-hand side functions such that (2) and (3) admits an analytical solution in the domain Ω = (0 ≤ x1 ≤ 1, 0 ≤ x2 ≤ 1) given by: p(x, t) = tx1 (1 − x1 )x2 (1 − x2 ), S(x, t) =

1 + tx1 (1 − x1 )x2 (1 − x2 ). 2

All computations are done on the time interval t ∈ [0.0, 1.0]. We chose van Genuchten parametrization [18] for the relative permeability and capillary pressure functions, as it is sufficient to capture the real processes [1]: kr,n (Sw ) = 1 − Sw [1 − Sw1/m ]2m , kr,w = Sw [1 − (1 − Sw1/m )m ]2 , (17) pc (Sw ) = pe (Sw−1/m − 1)1/n ,

(18)

where m = 1 − 1/n and n, pe are the van Genuchten parameters equal to n = 2, pe = 2 MPa. Table 1 shows the improvement in a time step of the new iteration scheme comparing to IMPES for the mesh size h = 0.1 and h = 0.05. The different mesh size does not influence the convergence of the two implicit schemes, but requires a smaller time step for IMPES. Examples where Newton’s method is not converging but an stabilized iterative scheme does, can be found in [12] for Richard’s equation and in [5] for the two-phase flow model with dynamic capillarity. Regarding the CPU times for the schemes, we see that the new iterative scheme is much faster than IMPES. Despite the fact that the implicit scheme needs to complete a few iterations at each time step, the improvement in the size of the time step made it more efficient compare to IMPES scheme. Even more interesting is the fact that we have a shorter computational time even comparing with Newton’s method that needs fewer iteration due to the second order of convergence. One possible reason is that the Jacobian matrix, needed in Newton’s method, is large and it takes quite a long time to compute it. Another important factor is the condition

Table 1 Convergence of IMPES and implicit schemes w.r.t. different time steps Numerical method h = 0.1 IMPES Implicit scheme Newton’s scheme h = 0.05 IMPES Implicit scheme Newton’s scheme

Δt = 0.2

Δt = 0.1

Δt = 1e-2

Δt = 1e-3

Δt = 1e-4

Δt = 1e-5

No No Yes

No Yes Yes

No Yes Yes

Yes Yes Yes

Yes Yes Yes

Yes Yes Yes

No No Yes

No Yes Yes

No Yes Yes

No Yes Yes

Yes Yes Yes

Yes Yes Yes

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A. Kvashchuk and F. A. Radu

numbers of the matrices of the linear systems that appear at each iteration of the methods. We computed the condition number estimates for the both schemes using the MATLAB function condest(), see Fig. 1. Clearly, the lower condition number of the new iterative scheme could contribute to the shorter CPU time. To sum up, we presented the rigorous convergence proof of the new scheme (Sect. 3.1), and also compared its practical performance with most commonly used schemes on synthetic test cases. The new scheme is faster than IMPES and has better conditioned matrices than fully implicit scheme with Newton solver (Fig. 2).

Condition Number w.r.t. |.|

1

Pressure

16637

4

10

29520

63092 47205

7359 1857 977

1753

2796 3810

435 105

2

10

New Scheme Newton

1

1/h

10

Fig. 1 Condition number of pressure linear system for several mesh sizes, average over all iterations 47546

IMPES 23680

Newton New Scheme

4

11041

10

CPU time (s)

4217 2368 1567

3

10

956

797 615 346

318 148

2

10

81 56

12

1

10

1/h

Fig. 2 CPU time of the implicit scheme, IMPES and Newton’s scheme for several mesh sizes, dtimp. = dtN. = 0.1, dtI MP ES = 10−4

A Fully-Implicit, Iterative Scheme for the Two-Phase Flow

633

Acknowledgements The first author acknowledges the Research Council of Norway and the industry partners; ConocoPhillips Skandinavia AS, Aker BP ASA, Eni Norge AS, Maersk Oil Norway AS, Statoil Petroleum AS, Neptune Energy Norge AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS and DEA Norge AS of The National IOR Centre of Norway for support.

References 1. B. Amaziane, M. Jurak, A. Žgalji´c Keko, Modeling and numerical simulations of immiscible compressible two-phase flow in porous media by the concept of global pressure. Transp. Porous Media 84, 133–152 (2010) 2. Z. Chen, G. Huan, B. Li, An improved IMPES method for two-phase flow in porous media. Transp. Porous Media 54, 361–376 (2004) 3. C. Frepoli, J.H. Mahaffy, K. Ohkawa, Notes on the implementation of a fully-implicit numerical scheme for a two-phase three-field flow model. Nucl. Eng. Des. 225, 191–217 (2003) 4. B. Ganis, K. Kumar, G. Pencheva, M.F. Wheeler, I. Yotov, A multiscale mortar method and two-stage preconditioner for multiphase flow using a global Jacobian approach, in SPE Large Scale Computing and Big Data Challenges in Reservoir Simulation Conference and Exhibition, 2014 5. S. Karpinski, I.S. Pop, F.A. Radu, Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two-phase flow in porous media with dynamic capillarity effects. Int. J. Numer. Methods Eng. 112(6), 553–577 (2017) 6. P. Knabner, L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations (Springer, New York, 2003) 7. J. Kou, S. Sun, A new treatment of capillarity to improve the stability of IMPES two-phase flow formulation. Comput. Fluids 39, 1923–1931 (2010) 8. J. Kou, S. Sun, On iterative IMPES formulation for two phase flow with capillarity in heterogeneous porous media. Int. J. Numer. Anal. Model. 1(1), 20–40 (2010) 9. B.H. Kueper, E. Frind, Two-phase flow in heterogeneous porous media: 1. Model development. Water Resour. Res. 27(6), 1049–1057 (1991) 10. S. Lacroix, Y. Vassilevski, J. Wheeler, M.F. Wheeler, Iterative solution methods for modeling multiphase flow in porous media fully implicitly. SIAM J. Sci. Comput. 25, 905–926 (2006) 11. T. Lee, M. Leok, N.H. McClamroch, Geometric numerical integration for complex dynamics of tethered spacecraft, in Proceedings of the 2011 American Control Conference, 2011 12. F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(2), 341–353 (2016) 13. B. Lu, M.F. Wheeler, Iterative coupling reservoir simulation on high performance computers. Pet. Sci. 6, 43–50 (2009) 14. J.M. Nordbotten, M.A. Celia, Geological Storage of CO2: Modeling Approaches for LargeScale Simulation (Wiley, Hoboken, 2012) 15. I.S. Pop, F.A. Radu, P. Knabner, Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168, 365–373 (2004) 16. F.A. Radu, I.S. Pop, Newton method for reactive solute transport with equilibrium sorption in porous media. J. Comput. Appl. Math. 234(7), 2118–2127 (2010) 17. F.A. Radu, J.M. Nordbotten, I.S. Pop, K. Kumar, A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015) 18. M.T. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(5), 892–898 (1980)

Mass Lumping for MHFEM in Two Phase Flow Problems in Porous Media Jakub Solovský and Radek Fuˇcík

Abstract This work deals with testing of the Mixed-Hybrid Finite Element Method (MHFEM) for solving two phase flow problems in porous media. We briefly describe the numerical method, it’s implementation, and benchmark problems. First, the method is verified using test problems in homogeneous porous media in 2D and 3D. Results show that the method is convergent and the experimental order of convergence is slightly less than one. However, for the problem in heterogeneous porous media, the method produces oscillations at the interface between different porous media and we demonstrate that these oscillations are not caused by the mesh resolution. To overcome these oscillations, we use the mass lumping technique which eliminates the oscillations at the interface. Tests on the problems in homogeneous porous media show that although the mass lumping technique slightly decreases the accuracy of the method, the experimental order of convergence remains the same.

1 Introduction Mathematical modeling of two phase flow in porous media can be used in many applications. For instance prediction of contaminant transport can be used for protection of water resources or for sanitation of dangerous substances leakage. Except for special cases, there is no known way how to solve these problems exactly but with numerical methods, we can find at least a good approximation of the solution. This paper focuses on the verification of the numerical method based on the Mixed-Hybrid Finite Element Method (MHFEM). The method is implemented in parallel using MPI [9]. Firstly, we test the method on two phase flow problems in homogeneous porous media in 2D and 3D. We further proceed with a problem

J. Solovský () · R. Fuˇcík CTU in Prague, FNSPE, Praha 1, Czech Republic e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_58

635

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J. Solovský and R. Fuˇcík

in heterogeneous porous media and show that the flow across material interfaces is simulated incorrectly. Therefore, we propose a modification of the numerical scheme using the mass lumping technique and show that it helps to solve problems in heterogeneous porous media correctly. Finally, we compare both approaches on problems with known exact solution to show how the mass lumping modification of the numerical scheme affects the accuracy and the convergence.

2 Numerical Method Here, we briefly describe the numerical method. A detailed description of the method together with a different approach to parallelism (on GPU using CUDA) is described in [5]. The method can be used for solving a system of n partial differential equations in the following coefficient form: ⎡ n

j =1

Ni,j

⎛

∂Zj ⎢ + ∇ · ⎣mi ⎝− ∂t

n

⎞⎤ ⎥ D i,j ∇Zj + wi ⎠⎦ = fi ,

(1)

j =1

where Zj = Zj (x, t), j = 1, . . . , n, are unknown functions (∀t > 0, ∀x ∈ Ω), Ω ⊂ Rd is the computational domain, and d is the spatial dimension, d ∈ {1, 2, 3}. Ni,j , fi , and mi are scalar coefficients, wi are vector coefficients and D i,j are symmetric, second order tensors. The coefficients can be functions of time t and spatial coordinates x, but also of the unknown functions Zj . The method was implemented in C++ and for the parallel implementation, MPI was used. Serial implementation of the method is described in detail in [5], parallel implementation in 2D, using MPI, is described in [9]. The parallelism in 3D which is used in this paper is a direct extension of the 2D case. All meshes used in this paper were generated using Gmsh [6].

2.1 Coefficients in General Formulation All benchmark problems presented here are represented by the following choice of coefficients in the general formulation of the method given by Eq. (1): 

w −Φρw dS 0 dpc N= dρn dSw −Φρn dpc ΦSn dp n   λt K −λt K , D= 0 λt K





,

 ρw λλwt m= , ρn λλnt   −λt ρw Kg w= , λt ρn Kg



 −fw f = , fn

Mass Lumping for MHFEM in Two Phase Flow Problems in Porous Media

637

where: Φ Sα ρα fα g K krα μα k λα = μrαα pα α ∈ {w, n}

(–) (–) (kg m−3 ) (kg m−3 s−1 ) (m s−2 ) (m2 ) (–) (kg m−1 s−1 ) (kg−1 m s) (Pa)

is the porosity, is the α-phase saturation, is the α-phase density, are the sinks/sources, is the gravity vector, is the permeability tensor, is relative permeability, is dynamic viscosity of the phase α, is the α-phase mobility (λt = λw + λn ), is the α-phase pressure, denotes the wetting or non-wetting phase.

These coefficients represent mass conservation law and Darcy’s law for both phases, refer to [3] for details.

3 Homogeneous Porous Media In this section, we verify the numerical method on benchmark problems in 2D and 3D in homogeneous porous media. For these problems, the exact solution can be found and, therefore, we can compute the errors of the numerical solution and the experimental order of convergence.

3.1 Benchmark Problems The benchmark problem used in this section is the extension of the McWhorter and Sunada [8] problem into an arbitrary dimension. Here, we only briefly describe the configuration of the problem, a more detailed description together with the method to find the exact solution can be found in [4, 8]. We assume a radially symmetric domain with the prescribed initial saturation Si and the inflow at the origin in the form: Q0 (t) = At

d−2 2

,

(2)

and we consider the cases d = 2 and d = 3. The problem is defined in the whole R2 or R3 but due to the assumed radial symmetry, we restrict ourselves only to the first quadrant in 2D or the first octant in 3D, respectively. We also have to restrict ourselves to a domain of finite size and compare the results at a certain time for which the head of the solution does not reach the boundary representing infinity.

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In this paper, the computational domains are a square with 1 m long side and a cube with 1 m long edge in 2D and 3D, respectively. In both cases, we compare the solutions at time t = 20,000 s. The problem formulation for which the exact solution can be obtained, requires prescribing a point-wise flux at the origin. Numerical method used in this paper cannot handle to prescribe a flux in one point, therefore, we approximate the inflow condition via a boundary condition by prescribing the flux through all element boundaries (edges, faces) that are adjacent to the origin. The corresponding value of the Neumann boundary condition is computed so that the total volume injected through the boundary is the same as the volume given by Eq. (2). 3 We set coefficients A = 10−5 m2 s−1 for the 2D case and A = 10−7 m3 s− 2 for the 3D case. Initial saturation in the domain is Si = 0.95 for both cases.

3.2 Numerical Analysis In this paper, the Brooks–Corey and van Genuchten models for capillary pressure together with the Burdine and Mualem models for relative permeability, respectively, are used [7]. Properties of the meshes used here are given in Table 1 the following notation is used: h

is the mesh element size. To compute h, we circumscribe a circle (ball) to each triangle (tetrahedron) of the mesh and take h as the radius of the largest such circle (ball).

Eh,Sn p is the Lp norm of the difference between the exact and numerical solution of the saturation Sn on mesh with element size h. eocSn ,p

is the experimental order of convergence in Lp norm, see [5] for details.

Table 1 Mesh properties for the benchmarks described in Sect. 3.1

Mesh ID

Elements

h

Degrees of freedom

2D1

@

6.71 × 10−2

242

766

@ 2D2 @ 2D3 @ 2D4 @ 2D5 @ 3D1 @ 3D2 @ 3D3 @ 3D4 @ 3D5

3.49

× 10−2

944

2912

1.64

× 10−2

3714

11,302

8.73 × 10−3

14,788

44,684

4.23 × 10−3

59,336

178,648

× 10−1

1312

5874

1.27 × 10−1

3697

15,546

6.29 × 10−2

29,673

121,678

2.13

× 10−2

240,372

973,750

1.84 × 10−2

1,939,413

7,807,218

3.48

Mass Lumping for MHFEM in Two Phase Flow Problems in Porous Media

639

Table 2 Errors of the numerical solution and experimental orders of convergence in 2D for the benchmark problem described in Sect. 3.1 Brooks & Corey Id. Eh,Sn 1 eocSn ,1 @ 2D1 1.45 × 10−2 0.92 @ 2D2 7.94 × 10−3 0.78 @ 2D3 4.40 × 10−3 0.95 @ 2D4 2.41 × 10−3 0.85 @ 2D5 1.30 × 10−3

Eh,Sn 2 eocSn ,2 3.17 × 10−2 0.78 1.91 × 10−2 0.60 1.21 × 10−2 0.69 7.84 × 10−3 0.66 4.85 × 10−3

van Genuchten Eh,Sn 1 eocSn ,1 1.42 × 10−2 0.98 7.51 × 10−3 0.86 3.93 × 10−3 1.05 2.03 × 10−3 0.90 1.06 × 10−3

Eh,Sn 2 eocSn ,2 2.12 × 10−2 0.94 1.15 × 10−2 0.84 6.11 × 10−3 1.03 3.19 × 10−3 0.89 1.68 × 10−3

Table 3 Errors of the numerical solution and experimental orders of convergence in 3D for the benchmark problem described in Sect. 3.1 Brooks & Corey Id. Eh,Sn 1 eocSn ,1 @ 3D1 1.12 × 10−2 0.69 @ 3D2 7.82 × 10−3 0.84 @ 3D3 4.35 × 10−3 1.03 @ 3D4 2.37 × 10−3 0.82 @ 3D5 1.41 × 10−3

Eh,Sn 2 eocSn ,2 3.38 × 10−2 0.60 2.47 × 10−2 0.72 1.49 × 10−2 0.92 8.63 × 10−3 0.79 5.23 × 10−3

van Genuchten Eh,Sn 1 eocSn ,1 1.21 × 10−2 0.77 8.13 × 10−3 0.93 4.25 × 10−3 1.14 2.17 × 10−3 1.04 1.12 × 10−3

Eh,Sn 2 eocSn ,2 2.43 × 10−2 0.73 1.66 × 10−2 0.90 8.84 × 10−3 1.12 4.56 × 10−3 1.02 2.39 × 10−3

With the exact solution known, we can compute errors of the numerical solution and the experimental order of convergence. Results for 2D and 3D are shown in Tables 2 and 3, respectively. Different results for the Brooks–Corey and van Genuchten models are caused by different capillary pressure—saturation relationships for the near-water-saturated state.

4 Heterogeneous Porous Media In this section, we focus on problems in heterogeneous porous media. In this section we show that the numerical method cannot correctly capture the effects at the interface between two different porous media. Oscillations appear in the solution and are more apparent in the case of flow from finer to coarser sand. To demonstrate the oscillations in this work, we use the benchmark problem originally proposed in [7]. We consider three layers of sands, the middle one finer than the remaining two, initially fully saturated with water. NAPL is injected through the upper boundary with a given flux and the gravity acts in the negative vertical direction.

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We use the numerical solution obtained using the vertex centered finite volume method in 1D on a very fine mesh as a reference solution to which we compare our numerical results. The 1D solution taken from [2] is in a good match with the results provided in [7]. We want to compare our 2D results with this 1D solution but instead of plotting the values over a single crossection through the center of the domain, we plot superposed values from all the elements of the mesh using their y position of the center. Numerical results for the original variant of the method are shown in Fig. 1a,c, and e. We can see the oscillations that are present for several mesh refinements and, therefore, are not caused by the mesh resolution.

4.1 Mass Lumping To overcome the oscillations at the material interface we use the mass lumping technique [1]. One of the steps of the MHFEM is to discretize numerical fluxes between elements. This is done by computing matrices B i,j,K with elements defined by the following integral [5]:  Bi,j,K,E,F = K

ωTK,F D −1 i,j ωK,E ,

(3)

where K denotes the element, ωK,F and ωK,E are the basis functions of the lowest order Raviart-Thomas-Nédélec space. Element K is a simplex (triangle or tetrahedron depending on the dimension of the problem) and the functions integrated in Eq. (3) are polynomials of the second order and, therefore, the integral in Eq. (3) can be computed exactly. The value of this integral can be also approximated using a quadrature rule [1]. We use the following quadrature rule to approximate the integral of some arbitrary function over simplex K: 

1 f ≈ |K| f (x i ), k K k

(4)

i=1

where k is the number of vertices of the simplex (k = 3 for triangle and k = 4 for tetrahedron) and x i are the positions of the vertices. Numerical solutions using the mass lumping technique are shown in Fig. 1b,d, and f. Clearly, the use of the mass lumping technique, when compared to the exactly computed integrals, reduces the oscillations at the material interface.

Mass Lumping for MHFEM in Two Phase Flow Problems in Porous Media 1

1D reference solution Exact integration

0.8

0.8

0.6

0.6

Sn [−]

Sn [−]

1

0.4 0.2

0.4

0

0

0.1

1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

y [m]

y [m]

(a)

(b) 1

1D reference solution Exact integration

0.8

0.8

0.6

0.6

Sn [−]

Sn [−]

1D reference solution Mass lumping

0.2

0

0.4 0.2

0.4

0.5

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Fig. 1 Comparison between the exact integration and the mass lumping technique on various meshes for the superposed solution in a heterogeneous porous medium. (a) 1506 @, exact integration. (b) 1506 @, mass lumping. (c) 5886 @, exact integration. (d) 5886 @, mass lumping. (e) 23,308 @, exact integration. (f) 23,308 @, mass lumping

5 Mass Lumping in Homogeneous Porous Media In the previous section, we showed that use of the mass lumping reduces oscillations at material interfaces. In this section, we investigate whether the mass lumping technique affects the accuracy of the method in the case of homogeneous porous

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Table 4 Errors of the numerical solution and experimental orders of convergence in 2D for the mass lumping variant of the method Brooks & Corey Id. Eh,Sn 1 eocSn ,1 @ 2D1 1.48 × 10−2 0.91 @ 2D2 8.17 × 10−3 0.77 @ 2D3 4.56 × 10−3 0.96 @ 2D4 2.49 × 10−3 0.86 @ 2D5 1.33 × 10−3

Eh,Sn 2 eocSn ,2 3.22 × 10−2 0.76 1.96 × 10−2 0.59 1.25 × 10−2 0.69 8.10 × 10−3 0.68 4.96 × 10−3

van Genuchten Eh,Sn 1 eocSn ,1 1.44 × 10−2 0.98 7.59 × 10−3 0.86 3.95 × 10−3 1.04 2.04 × 10−3 0.90 1.06 × 10−3

Eh,Sn 2 eocSn ,2 2.16 × 10−2 0.95 1.17 × 10−2 0.85 6.15 × 10−3 1.04 3.20 × 10−3 0.89 1.68 × 10−3

Table 5 Errors of the numerical solution and experimental orders of convergence in 3D for the mass lumping variant of the method Brooks & Corey Id. Eh,Sn 1 eocSn ,1 @ 3D1 1.13 × 10−2 0.67 @ 3D2 7.96 × 10−3 0.82 @ 3D3 4.50 × 10−3 1.01 @ 3D4 2.47 × 10−3 0.83 @ 3D5 1.44 × 10−3

Eh,Sn 2 eocSn ,2 3.46 × 10−2 0.61 2.52 × 10−2 0.72 1.53 × 10−2 0.92 8.64 × 10−3 0.79 5.26 × 10−3

van Genuchten Eh,Sn 1 eocSn ,1 1.22 × 10−2 0.77 8.22 × 10−3 0.93 4.30 × 10−3 1.13 2.20 × 10−3 1.04 1.15 × 10−3

Eh,Sn 2 eocSn ,2 2.49 × 10−2 0.74 1.70 × 10−2 0.91 8.97 × 10−3 1.12 4.63 × 10−3 1.02 2.41 × 10−3

media where we can compare the results with exact solutions. We use the benchmark problem described in Sect. 3.1, solve it with the mass lumping variant of the method, and compare the results with those given in Sect. 3.2. Errors of the solution and the experimental orders of convergence for the 2D and 3D cases are shown in Tables 4 and 5, respectively. Results show that in both 2D and 3D cases, the errors of the mass lumping variant of the method are slightly worse than those without mass lumping but the method is still convergent with the same experimental order of convergence.

6 Conclusion In this work, we tested the MHFEM numerical method for solving two phase flow problems in porous media. We showed that for homogeneous porous media, the method is convergent for both 2D and 3D cases with the experimental order of convergence slightly less than one. In the case of heterogeneous porous media, the method produces oscillations at the interface between different porous media when exact evaluation of the integrals in matrix B is used. To overcome these difficulties, we used the mass lumping technique which reduces the oscillations and only very

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slightly affects the accuracy of the method as was shown in the comparison of the solutions using the benchmark problems in 2D and 3D with known exact solutions. Acknowledgements The work was supported by the Czech Science Foundation project no. 1706759S and by grant No. SGS17/194/OHK4/3T/14 of the GA, CTU in Prague.

References 1. G. Chavent, J. Roberts, A unified physical presentation of mixed, mixed-hybrid finite elements and usual finite differences for the determination of velocities in waterflow problems, Research Report RR-1107, INRIA (1989) 2. R. Fuˇcík, Advanced numerical methods for modelling two-phase flow in heterogeneous porous media, Ph.D. thesis, FNSPE of Czech Technical University Prague, 2010 3. R. Fuˇcík, J. Mikyška, Mixed-hybrid finite element method for modelling two-phase flow in porous media. J. Math. Ind. 3, 9–19 (2011) 4. R. Fuˇcík, T.H. Illangasekare, M. Beneš, Multidimensional self-similar analytical solutions of two-phase flow in porous media. Adv. Water Resour. 90, 51–56 (2016) 5. R. Fuˇcík, J. Mikyška, J. Solovský, J. Klinkovský, T. Oberhuber, Multidimensional mixed hybrid finite element method for compositional two phase flow in heterogeneous porous media and its massively parallel implementation on GPU. Comput. Phys. Commun. (2017, in review) 6. C. Geuzaine, J.F. Remacle, GMSH: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79, 1309–1331 (2009) 7. R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, A Contribution to the Modelling of Hydrosystems (Springer, Berlin, 1997) 8. D.B. McWhorter, D.K. Sunada, Exact integral solutions for two-phase flow. Water Resour. Res. 26, 399–413 (1990) 9. J. Solovský, R. Fuˇcík, A parallel mixed-hybrid finite element method for two phase flow problems in porous media using MPI. Comput. Methods Mater. Sci. 17(2), 84–93 (2017)

Uncertainty Quantification in Injection and Soil Characteristics for Biot’s Poroelasticity Model Menel Rahrah and Fred Vermolen

Abstract As demand for water increases across the globe, the availability of freshwater in many regions is likely to decrease due to a changing climate, an increase in human population and changes in land use and energy generation. Many of the world’s freshwater sources are being drained faster than they are being replenished. To solve this problem, new techniques are developed to improve and optimise renewable groundwater sources, which are an increasingly important water supply source globally. One of this emerging techniques is rainwater storage in the subsurface. In this paper, different methods for rainwater infiltration are presented. Furthermore, Monte Carlo simulations are performed to quantify the impact of variation in the soil characteristics and the infiltration parameters on the infiltration rate. Numerical results show that injection pulses may increase the amount of water that can be injected into an aquifer.

1 Introduction Access to freshwater is considered a universal human right [12]. However, climate change and the growing population and industrial activity put water resources worldwide under severe pressure [8]. This issue has already resulted in desiccation in large areas of the world and in a global freshwater crisis. For this reason, authorities, knowledge institutions and private companies collaborate closely to develop new approaches to the growing freshwater demand. On the other hand, heavy rainfall regularly leads to flooding, damaging buildings and infrastructure, and to erosion of valuable top soil. A simple and cheap solution for both global problems is rainwater storage. Infiltration of large amounts of rainwater into the shallow subsurface can have great value for battling flooding and for the underground storage of freshwater.

M. Rahrah () · F. Vermolen Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_59

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Groundwater is currently the primary freshwater supply source for approximately two billion people and the dependence on it will increase into the future [4]. A prerequisite for effective storage of rainwater in periods of extreme precipitation is that water can be stored quickly. During rainwater infiltration, different injection methods could be used. Our aim is to investigate the impact of these injection methods on the percolation fluid velocity of the injected water into the aquifer. Two injection methods are considered in this paper: constant injection and pulsed injection. Furthermore, to be able to determine the fluid flow through the pores of the aquifer, taking into account the local displacement of the skeleton of the porous medium, Biot’s model for poroelasticity [2] is used in this study. In addition, the impact of uncertainty in the soil characteristics and in the injection parameters is quantified using Monte Carlo techniques and statistical analysis. The poroelasticity equations are often solved using finite element methods [5, 6]. In this study, a finite element method based on Taylor-Hood elements, with linear and quadratic basis functions, has been developed for solving the system of incompressible poroelasticity equations. We remark that the Taylor-Hood elements are suitable as a stable approach for this problem. Spurious oscillations are diminished but not completely removed for small time steps. To fully remove the non-physical oscillations, one may use the stabilisation techniques as considered in [9].

2 Governing Equations Assuming that the deformations are very small, the model provided by Biot’s theory of linear poroelasticity with single-phase flow [2] is used to determine the local displacement of the skeleton of a porous medium, as well as the fluid flow through the pores. We assume that the deformable fluid-saturated porous medium has a linearly elastic solid matrix and is saturated by an incompressible Newtonian fluid. The solid matrix is assumed to be fully connected. Let Ω ⊂ R3 denote the domain occupied by the porous medium with boundary Γ , and x = (x, y, z) ∈ Ω. Furthermore, t denotes time, belonging to a half-open time interval I = (0, T ], with T > 0. The initial boundary value problem for the consolidation process of a fluid flow in a deformable porous medium is stated as follows [1, 13]: equilibrium equations: ∇ · σ  − (∇p + ρge z ) = 0

on Ω × I ;

(1a)

on Ω × I,

(1b)

Biot’s constitutive equations: σ  = λ(∇ · u)I + μ(∇u + ∇uT );

(2)

continuity equation:

∂ (∇ · u) + ∇ · v = 0 ∂t

where σ  and v are defined by the following equations

κ Darcy’s law: v = − (∇p + ρge z ). η

(3)

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In the above relations, σ  , p, ρ, g, u, v, λ and μ; κ and η respectively, denote the effective stress tensor for the porous medium, the pore pressure, the fluid density, the gravitational acceleration, the displacement vector of the porous medium, Darcy’s velocity, the Lamé coefficients, the permeability of the porous medium, and the fluid viscosity. To complete the formulation of a well-posed problem, appropriate boundary and initial conditions are specified in Sect. 3. In this study, we focus on the interaction between the mechanical deformations and the fluid flow during an infiltration process. Therefore, we consider the spatial dependency of the porosity and the permeability of the porous medium. The permeability can be determined using the Kozeny-Carman equation [14] κ(x, t) =

ds2 θ (x, t)3 , 180 (1 − θ (x, t))2

(4)

where ds is the mean grain size of the soil and the porosity θ is computed from the displacement vector using the porosity-dilatation relation (see [11]) θ (x, t) = 1 −

1 − θ0 , exp(∇ · u)

(5)

with θ0 the initial porosity, which is treated in the current paper as a given constant in each sample computation. Problem (1) is solved as a fully coupled system. At each time step, after having obtained the numerical approximations for u and p, we update the porosity using Eq. (5). Subsequently, the Kozeny-Carman relation (4) is used to calculate the permeability. The new value for the permeability is then used for the next time step.

3 Problem Formulation The infiltration of a fluid through a filter into an aquifer is shown in Fig. 1a. We assume that the flow pattern is axisymmetric, hence for the azimuthal coordinate θˆ holds ∂ˆ (.) = 0. Therefore, we determine the solution for a fixed azimuth. The ∂θ computational domain Ω is an L-shaped two-dimensional surface with cylindrical coordinates r = (r, z), as depicted in Fig. 1b. In order to solve this problem, Biot’s consolidation model, as described in Sect. 2, is applied on the computational domain Ω with radius R and height H . The fluid is injected into the soil through a filter placed on boundary segment Γ3 . Furthermore, the injection tube (boundary segments Γ2 , Γ3 and Γ4 ) is fitted with a casing (boundary segments Γ2 and Γ4 ) and a perforated section (boundary segment Γ3 ) to prevent loose material from entering and potentially clogging the

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Γ1 Ω

Γ2 H z

Γ3 Γ4 Γ5

Γ7

r

Γ6 (b)

(a)

Fig. 1 Sketch of the setup for the aquifer problem: (left) physical problem and (right) numerical discretisation. Taking advantage of the symmetry of geometry and boundary conditions, only the grey region is discretised

injection tube. More precisely, the boundary conditions for this problem are given as follows: p = ρg(H − z)

on r ∈ Γ1 ∪ Γ6 ∪ Γ7 ;

(6a)

κ (∇p + ρge z ) · n = 0 η

on r ∈ Γ2 ∪ Γ4 ∪ Γ5 ;

(6b)

on r ∈ Γ3 ;

(6c)

on r ∈ Γ1 ∪ Γ6 ;

(6d)

on r ∈ Γ2 ∪ Γ3 ∪ Γ4 ∪ Γ5 ;

(6e)

u·n ≤0

on r ∈ Γ2 ∪ Γ3 ∪ Γ4 ;

(6f)

u·n=0

on r ∈ Γ5 ;

(6g)

u=0

on r ∈ Γ7 ,

(6h)

Qin (t) κ ∇p · n = η 2πRf Lf σ n = 0 (σ  n) · t = 0

where t is the unit tangent vector at the boundary, n the outward unit normal vector and Qin is a prescribed injection velocity due to the injection of the fluid through a filter with radius Rf and length Lf . Figure 1b shows the definition of the boundary segments. Note that the boundary conditions on boundary segment Γ5 appear as a result of symmetry. Initially, the following condition is fulfilled: u(r, 0) = 0 for r ∈ Ω.

(7)

To infiltrate a fluid into the soil, several injection methods can be used. In this paper, we present two different injection methods in order to investigate the impact of the injection on the water flow in the aquifer. Constant Injection Velocity We start with the simulation of problem (1) with a constant injection velocity Qin,c (t) = Qin,c . Note that the injection rate is continuous over time.

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Pulsed Injection In this method, the prescribed injection velocity will have a pulsing behaviour rather than being constant. In this case, the injection velocity is represented by a combination of the Heaviside step functions. A rectangular pulse wave with period TQ and pulse time τQ is defined as Qin,p (t) = γQ

NQ

(H(t − kTQ ) − H(t − kTQ − τQ )),

(8)

k=0

where γQ is the maximum injection velocity and NQ is the number of periods. Note that the injection rate contains discontinuities over time.

4 Numerical Results The Galerkin finite element method [10], with triangular Taylor-Hood elements, is employed for the solution of the discretised quasi-two-dimensional problem (1). The displacements are spatially approximated by quadratic basis functions, whereas a continuous piecewise linear approximation is used for the pressure field. For the time integration, the backward Euler method is applied. The numerical investigations are carried out using the matrix-based software package MATLAB (version R2011b). The computational domain is an L-shaped surface with radius R = 5.0 m, height H = 5.0 m, filter radius Rf = 20.0 cm and filter length Lf = 1.0 m. The filter is placed between z = 2 and z = 3. The domain is discretised using a regular triangular grid, with Δr = Δz = 0.1. Mesh refinement did not yield any significant changes of the numerical solution. In addition, values for some model parameters have been chosen based on literature (see Table 1). Furthermore, the Lamé coefficients λ and μ are related to Young’s modulus E νE E and Poisson’s ratio ν by Aguilar et al. [1]: λ = (1+ν)(1−2ν) and μ = 2(1+ν) . In this paper, we will investigate the impact of two different injection methods on the flow of water. Subsequently, the Monte Carlo method is applied to the values of the soil characteristics E, ν, θ0 and ds , using samples of uniform distributions with boundaries found in the literature [7]. Hence, for these simulations, 300 samples from the following uniform distributions are generated: E ∼ U(30 × 106, 160 × 106 ), ν ∼ U(0.15, 0.40), θ0 ∼ U(0.23, 0.46), ds ∼ U(0.05 × 10−3 , 50.0 × 10−3 ). (9) Table 1 An overview of the values of the model parameters

Property Fluid viscosity Fluid density Gravitational acceleration

Symbol η ρ g

Value 1.307 × 10−3 1000 9.81

Unit Pa s kg/m3 m/s2

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Finally, to quantify the difference between a pulsed injection and a constant injection of water into the aquifer, the contributions of the variations in the values of the pulse wave characteristics to the volume flow rate are analysed. These contributions are examined by applying Monte Carlo method to γQ , TQ and the relative pulse time τ τ˜ , defined as τ˜ = TQQ (see Formula (8)). The number of periods NQ is equal to five in all simulations. In each simulation, the constant injection velocity is chosen equal to the total pulsed injection velocity over time, in order to be able to draw reliable conclusions. Hence, the constant injection velocity is computed by Qin,c = τ˜ γQ . For the simulations, 300 samples from the following uniform distributions are generated: γQ ∼ U(10/3600, 40/3600),

TQ ∼ U(1, 10),

τ˜ ∼ U(0.1, 0.9).

(10)

The impact of the injection methods on the water flow is defined in this publication as the impact on the time average of the volume flow rate Qout at a distance R − Rf from the injection filter. In the generations of the simulation results, the time T step size Δt is determined by Δt = 10Q . Let ‘p’ be an abbreviation for pulsed injection and ‘c’ for constant injection. The Pearson correlation coefficients ri , with i ∈ {p, c}, are presented together with the associated pi -values in Table 2. From Table 2 we can conclude that the variation in the soil characteristics does not have a large impact on the numerical results, while variation in the injection parameters influence the time average of the volume flow rate at a distance R − Rf from the injection filter significantly. The low impact of the soil characteristics on Qout may be attributed to boundary condition (6c). If a pressure was prescribed on the filter instead of the volumetric flow, then the significance may change. We will deal with this case in a later study. Furthermore, the results in Table 2 show that injection pulses with small pulse periods TQ lead to a major increase in the volume flow rate, while it increases slightly by increasing the relative pulse time τ˜ and the maximum injection velocity γQ . To compare the injection methods, the difference between Qout as result of the pulsed injection and Qout as result of the constant injection is computed by Qdiff = Qout,p − Qout,c . Table 2 The Pearson correlation coefficients together with the associated p-values

corr(E, Qout ) corr(ν, Q out ) corr(θ0 , Qout ) corr(ds , Qout ) corr(γQ , Qout ) corr(TQ , Qout ) corr(τ˜ , Qout )

rp pp 0.09 0.11 0.05 0.36 0.01 0.94 0.10 0.10 0.31 < 0.05 −0.69 < 0.05 0.45 < 0.05

rc

pc 0.10 0.10 0.07 0.21 0.01 0.85 0.09 0.13 0.28 < 0.05 −0.66 < 0.05 0.49 < 0.05

A p-value less than 0.05 means that the two paired sets of data are most probably related, at the significance level 0.05

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1

140

0.9 120 0.8 100

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Fig. 2 The histogram and the cumulative distribution function of the difference in the time average of the volume flow rate Qdiff = Qout,p − Qout,c

In Fig. 2, the histogram and the cumulative distribution function of this difference are presented. Since P (Qdiff > 0) ≈ 0.98, we can conclude that the pulsed injection has a beneficial effect on the water flow in the aquifer.

5 Conclusions In this work, a numerical model for the quasi-static Biot’s consolidation problem for poroelasticity has been developed, considering nonlinear permeability. The model is discretised by a continuous Galerkin finite element method based on TaylorHood elements, combined with the implicit Euler scheme for time stepping. The study contains Monte Carlo simulations to quantify the impact of variation in the soil characteristics and the injection parameters on the time average of the volume flow rate at a particular distance from the injection filter. Furthermore, two different injection methods are tested and compared with each other, to determine the best infiltration method that can be used for the storage of rainwater in the shallow subsurface. To reduce the Monte Carlo error, simulations should be performed with thousands of samples. However, as in our case each sample simulation takes more than 1 h, we instead adopted 300 samples. Another approach is to use Multilevel Monte Carlo methods [3]. By applying these methods, relatively few simulations are needed at high mesh resolutions, whereas one performs large numbers of simulations at lower resolutions.

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Numerical simulations of pulsed injection pointed out that injection pulses with small relative pulse periods lead to a major increase in the volume flow rate, while an increasing pulse time and maximum injection velocity result in a slight increase of the flow rate. On the other hand, variation in the values of the soil characteristics indicated that these parameters do not have a large impact on the volume flow rate. Most importantly, we can conclude that, regardless of the type of soil into which we inject, by applying pulsed injection we can increase the amount of rainwater that can be stored quickly in the underground. Acknowledgements This project is supported by the Dutch Technology Foundation STW (project number 13263) and the members of foundation O2DIT (Foundation for Research and Development of Sustainable Infiltration Techniques).

References 1. G. Aguilar, F. Gaspar, F. Lisbona, C. Rodrigo, Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation. Int. J. Numer. Methods Eng. 75, 1282–1300 (2008) 2. M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941) 3. K.A. Cliffe, M.B. Giles, R. Scheichl, A.T. Teckentrup, Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3–15 (2011) 4. Z.W. Kundzewicz, P. Döll, Will groundwater ease freshwater stress under climate change? Hydrolog. Sci. J. 54(4), 665–675 (2009) 5. R.W. Lewis, B. Schrefler, A fully coupled consolidation model of the subsidence of Venice. Water Resour. Res. 14(2), 223–230 (1978) 6. I. Masters, W.K.S. Pao, R.W. Lewis, Coupling temperature to a double-porosity model of deformable porous media. Int. J. Numer. Methods Eng. 49(3), 421–438 (2000) 7. A. McCauley, C. Jones, J. Jacobsen, Basic Soil Properties. Soil and Water Management Module 1(1), 1–12 (2005) 8. A.S. Richey, B.F. Thomas, M.-H. Lo, J.T. Reager, J.S. Famiglietti, K. Voss, S. Swenson, M. Rodell, Quantifying renewable groundwater stress with GRACE. Water Resour. Res. 51(7), 5217–5238 (2015) 9. C. Rodrigo, F.J. Gaspar, X. Hu, L.T. Zikatanov, Stability and monotonicity for some discretizations of the Biot’s consolidation model. Comput. Methods Appl. Mech. Eng. 298, 183–204 (2016) 10. A. Segal, Finite Element Methods for the Incompressible Navier-Stokes Equations (Delft Institute of Applied Mathematics, Delft, 2012) 11. T.-L. Tsai, K.-C. Chang, L.-H. Huang, Body force effect on consolidation of porous elastic media due to pumping. J. Chin. Inst. Eng. 29(1), 75–82 (2006) 12. United Nations Committee on Economic, Social and Cultural Rights, General Comment No. 15. The Right to Water (UN Economic and Social Council, Geneva, 2003) 13. H.F. Wang, Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology (Princeton University Press, Princeton, 2000) 14. S.-J. Wang, K.-C. Hsu, Dynamics of deformation and water flow in heterogeneous porous media and its impact on soil properties. Hydrol. Process. 23, 3569–3582 (2009)

Reactivation of Fractures in Subsurface Reservoirs—A Numerical Approach Using a Static-Dynamic Friction Model Runar L. Berge, Inga Berre, and Eirik Keilegavlen

Abstract Fluid-induced slip of fractures is characterized by strong multiphysics couplings. Three physical processes are considered: Flow, rock deformation and fracture deformation. The fractures are represented as lower-dimensional objects embedded in a three-dimensional domain. Fluid is modeled as slightly compressible, and flow in both fractures and matrix is accounted for. The deformation of rock is inherently different from the deformation of fractures; thus, two different models are needed to describe the mechanical deformation of the rock. The medium surrounding the fractures is modeled as a linear elastic material, while the slip of fractures is modeled as a contact problem, governed by a static-dynamic friction model. We present an iterative scheme for solving the non-linear set of equations that arise from the models, and suggest how the step parameter in this scheme should depend on the shear modulus and mesh size.

1 Introduction Slip or reactivation of fractures and faults in the earth’s crust can be caused by natural changes in the stress field in the rock, but concerns also arise related to human induced seismicity. Over 700 cases of seismicity related to human activities have been reported, caused by a variety of different activities [4]. In this paper, we focus on fracture reactivation due to injection of fluids in underground reservoirs, which concerns applications such as CO2 storage, enhanced oil recovery and production of geothermal energy. The seismic events related to these types of activities are usually not noticeable except by very sensitive equipment, but events of magnitude up to M3.5 have been recorded [12].

R. L. Berge () · I. Berre · E. Keilegavlen University of Bergen, Department of Mathematics, Bergen, Norway e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_60

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Reactivation of fractures is a strongly coupled problem involving disparate physical processes, including fluid flow, deformation of rock surrounding the fractures and the plastic deformation of fractures. In this work we will couple models for each of these sub-problems to simulate fracture reactivation and corresponding aperture changes. One of the main challenges of modeling fractures in the subsurface is the absence of scale separation between fractures, which can range from sub-resolution micro-fractures to fractures and faults spanning the whole reservoir. In the model presented here, small scale fractures are assumed to be up-scaled into effective matrix parameters, such as permeability, while the large macroscopic fractures are explicitly resolved. To reduce the computational cost further, fractures are represented as lower-dimensional domains with an associated aperture. We apply this mixed dimensional hybrid approach to model fluid flow following the ideas of [3]. The fluid pressure will act as a trigger mechanism for fracture reactivation, which is governed by a Mohr-Coulomb criterion: A fracture slips when the shear traction equals the coefficient of friction times the effective normal traction. This bound on shear traction introduces a non-linear inequality constraint to the system of equations, and various techniques have been employed to handle this constraint [1, 7]. We use a static-dynamic friction model where the coefficient of friction drops instantaneous when a fracture slips. Results from this model should be interpreted with caution, as it falls into the category of “inherently discrete” models due to this discontinuous drop [9]. Nevertheless, it has been shown to give feasible results in modeling of fracture reactivation as a consequence of fluid injection at elevated pressures [8]. Finding the regions of slip on a fracture is one of the challenges of this model, due to the discontinuous change in the coefficient of friction between stick and slip regions. We will give a mathematical formulation of the friction problem, and then present a simple solution strategy following the ideas Ucar et al. [11], who use an idea of excess shear stress to estimate fracture slip based on the shear and normal stress. They approximated the slip length, at each time step, based on how much the Mohr-Coulomb criterion is violated, multiplied by a “fracture stiffness” parameter. In the current work, we improve this approach by suggesting how this stiffness parameter should depend on the shear modulus and mesh size.

2 Governing Equations We consider three processes in the subsurface; fluid flow, rock deformation, and fracture deformation. Each of the problems will be defined in different domains of different dimensions, and the domains and processes are coupled together. As mentioned above, fractures are represented as lower dimensional objects in the reservoir. The intersection of two fractures is, correspondingly, represented by a line-segment. The reservoir is in this way divided into domains Ω d of different dimension d; the rock matrix Ω 3 , fractures Ω 2 , intersection of fractures Ω 1 , and the intersection points of these lines Ω 0 , as shown in Fig. 1.

Reactivation of Fractures in Subsurface Reservoirs

Ω1

Ω0

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Ω2

Ω3

Fig. 1 A mixed dimensional domain. The red cube represents the 3D domain, the green squares represent the 2D domains, the blue lines represent 1D domains, and the black dot the 0D domain

2.1 Flow We will assume that the fluid flow follows Darcy’s law in all domains, and extend the approach of Boon et al. [2] to compressible flow. For a detailed explanation of notation we refer the reader to the same work. The mixed-dimensional formulation for the conservation of mass is given by φcp

∂p − ∇ · K∇p + T = q ∂t

in {Ω d }d=0,...,3 ,

(1)

where φ is the porosity, cp the fluid compressibility, K the effective permeability (taking into account the scaling with fracture aperture and viscosity), p the fluid pressure, T the transfer term between domains of different dimensions, and q the source and sink term.

2.2 Rock Deformation The rock is modeled as an elastic medium, and assumed to always be in quasi-static equilibrium. The conservation of momentum can be written as ∇ · σ = 0,

σ = G(∇u + (∇u)? ) + λI ∇ · u in Ω 3 ,

(2)

where as for the flow, we have neglected the gravitational term. The variable σ is the stress tensor, G and λ are the Lamé parameters, I the identity matrix, and u the displacement vector. Because a fracture will have exactly two interfaces with Ω 3 , we adopt the notation that Γ + defines one side, while Γ − the other. We will neglect any elastic effects of contacting asperities of the fracture surfaces and potential gouge filling the fractures, thus, the traction on the two fracture boundaries must balance T + + T − = 0,

(3)

where the traction is defined as T = σ · n, with n being the normal vector of the surface.

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In addition to boundary conditions on the reservoir domain, we need a total of six equations defined on the fracture interfaces since we have three degrees of freedom on each fracture side. The first three are obtained from the force balance Eq. (3) and the last three equations are defined by the coupling to the fracture deformation model, which governs the fracture response to rock stress and strain.

2.3 Fracture Deformation For two surfaces in contact, the magnitude of the shear traction is bounded by the coefficient of friction times the effective normal traction |T s | ≤ μ(Tn − p),

(4)

where T s is the shear traction, μ the coefficient of friction, and Tn the normal traction. We use a static-dynamic friction model, where the coefficient takes one value when the fracture is not slipping μ = μs (static friction), and another, lower value when it is slipping μ = μd (dynamic friction). We introduce the slip distance d which defines the relative fracture displacement d = u+ − u− , where u+ and u− are the displacements on the positive and negative side of a fracture. When a fracture slips, the aperture can increase due to asperities on the fracture surfaces, and we approximate this increase by letting the aperture depend linearly on slip distance. According to the Mohr-Coulomb criterion, slip on a fracture is triggered when Inequality (4) reaches equality, using the static coefficient of friction. We define Γs to be the part of the fracture that is slipping, and enforce Inequality (4) as an equality in this domain, using the dynamic coefficient of friction. Further, when a fracture is slipping it should slip in the direction of shear traction. To sum up, the friction condition on the fractures can be formulated as |T s | < μs (Tn − p)

x ∈ Ω 2 \ Γs

(5a)

d=0

x ∈ Ω 2 \ Γs

(5b)

|T s | − μd (Tn − p) = 0

x ∈ Γs

(5c)

d = χT s , χ > 0

x ∈ Γs

(5d)

Γs = {x ∈ Ω 2 : d = 0},

(5e)

where we have included the definition of the slip region to stress the fact that Γs is not known, but one of the unknowns in this system.

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3 Spatial Discretization Both the flow equation and the elasticity problem are discretized using finite volume schemes. The flow equation is discretized using the two-point flux approximation, while the elasticity is discretized using the multi-point stress approximation [5, 10]. The implementation has been done in PorePy, which is an open source Python code for simulating fractured and deformable porous media [6].

4 Solution Strategy We will use an iterative scheme to solve the set of Eqs. (1), (2) and (5). At each step in the scheme we will first estimate the slip region Γs and then solve the equations using a linearized approximation of Eq. (5c). We start by solving Eq. (1) using backward Euler. To solve (2) and (5), we first assume that the fractures do not slip at the current time step, Γs = ∅, thus, the current slip vector d k+1 must equal the slip vector at the previous time step d k . Together with the stress condition, T + = −T − , the number of unknowns equals the number of equations and we have a closed system. After we obtain the displacement uk+1 , we can calculate the stress, and from this traction, on the fractures from Eq. (2). We check if the Mohr-Coulomb criterion (5a) is satisfied; if it is, our assumption that we have no slip is good, and we continue to the next time step. If the condition is violated, we update the slip region Γs by including all faces that violate the condition. We define the “excess shear stress” to be the residual of Eq. (5c), Te = |T s | − μd (Tn − p)

x ∈ Γs ,

where we use the dynamic coefficient of friction. From Eq. (5d), we know that the slip should be in the direction of shear stress, but the variable χ is unknown. We make a simple estimate of this distance by assuming that all degrees of freedom are fixed except on a single face Fi on the fracture. The displacement gradient ∇ui at this face then grows linearly with slip distance, and from Eq. (2) we obtain that the corresponding change in shear traction grows as ΔT s ∼

G di, |Fi |1/2

where |Fi | is the area of the face. For each face that is slipping, we therefore set our new guess for the slip vector to = d ki + d k+1 i

γ |Fi |1/2 Te τ , G

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where the constant γ is a dimensionless numerical parameter which should be chosen of the order of magnitude 1 and τ the tangential vector pointing in the direction of maximum shear traction. We now have managed to reduce the equations to a linear system, but its computational cost is still high as the linear system is about three times as large as the system for the fluid flow. Choosing the optimal step length parameter is therefore crucial to obtain a viable scheme. After updating the slip distance for all faces violating the Mohr-Coulomb criterion, we go back to solving Eq. (2), and iterate in this way until Eqs. (5a) and (5c) is satisfied. Note that we might update the slip distance for a fracture face several times per time step before the excess shear stress is reduced to zero.

5 Numerical Results In this section, we will present simulations of two different reservoirs, one with a single fracture, and the second with several fractures. The purpose of the first example is twofold; we wish to validate that the slip length in fact scales with shear modulus and the mesh size, and we will try to find an optimal γ . Meshes of two different sizes are used, one fine and one coarse, and two different shear modulus parameters G. We will vary γ and count the number of iterations per time step and look at the error in approximating Eq. (5). We stop the iteration procedure when Te /G < 1e−5. The geometric domain is described by a single circular fracture with a radius of 1500 m. Fluid is injected at the center of the fracture at a constant rate of 1 L/s. The total number of iterations at each time step is shown in Fig. 2. The number of iterations is relatively low at the beginning of the injection, but increases with time. At the first time steps, only a few cells in the vicinity of the injection cells slip. As the pressure front moves radially out from the center, more cells slip at each time step. When several cells slip in the same region, the assumption we made when updating the slip distance becomes more erroneous, and we need additional iterations to reach the stop criterion. For the coarse grid, the slip is more or less limited to one or two rows of cells around the pressure front, so the iteration numbers stay low. For the fine grid many more cells slip, which causes the number of iterations to increase drastically. Increasing the step parameter γ does as expected reduce the number of iterations. However, as γ increases, the overshoot of the slip distance can also increase (meaning that the excess shear stress Te becomes negative). For the fine grid, we did not include a simulation of γ = 4 as the scheme did not converge for this value. The maximum overshoot error |Te |/G were on the order of magnitude 1e−4 for γ = 3, 4 and 1e−5 for γ = 2. Typically, the error for only one or two cells came close to the maximum value, while the error for the other cells were much smaller. In the second case, we apply the algorithm to a small fracture network of 8 fractures. The size of the fractures varies from a few hundred meters up to 1500 m. We inject water at the center of the largest fracture with a constant rate of 30 L/s

Reactivation of Fractures in Subsurface Reservoirs 45

γ γ γ γ γ γ

40

# of iterations

35 30 25

= = = = = =

2, 2, 3, 3, 4, 4,

G G G G G G

= = = = = =

659 9

10GPa 20GPa 10GPa 20GPa 10GPa 20GPa

8 7 6 5

20

4

15

3

10

2

5

1

0

0

1

2

3 4 5 6 Time Step

7

8

9

0

0

1

2

3

4 5 6 Time Step

7

8

9 10

Fig. 2 Number of iterations needed for each time step to find the slip distance d. The left and right figures show the number of iteration needed on a fine grid (852 fracture cells) and a coarse grid (226 fracture cells) respectively

2e-5

1e-5

Slip Distance (m)

2.9e-5

0

Fig. 3 Total slip at equlibrium after shut-in of well for case 2. The well is located at the center of the biggest fracture

for 30 min. Figure 3 depicts the total slip distance at the end of the simulation. We register slip after the shut-in of the well as the pressure migrates through the fracture network. At the end time equilibrium has been reached, and we observe how the slip distance varies significantly between different fractures. This is due to the fracture’s

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orientation relative to the background stress field, and how far they are located from the injection well.

6 Concluding Remarks This paper has presented a coupled model of compressible fluid flow, elastic rock deformation and plastic fracture deformation. A simple iterative scheme to handle the non-linear coupling between the rock and fracture deformation were discussed, and we suggested an improvement to this scheme by showing how the step length parameter should depend on the mesh size and shear modulus. The numerical results support the choice of step length. Acknowledgements This work was partially funded by the Research Council of Norway, TheMSES project, grant no. 250223, and travel support funded by Statoil, through the Akademia agreement.

References 1. B.T. Aagaard, M.G. Knepley, C.A. Williams, A domain decomposition approach to implementing fault slip in finite element models of quasi-static and dynamic crustal deformation. J. Geophys. Res. Solid Earth 118(6), 3059–3079 (2013) 2. W.M. Boon, J.M. Nordbotten, I. Yotov, Robust discretization of flow in fractured porous media (2016), ArXiv e-prints 3. P. Dietrich, R. Helmig, M. Sauter, H. Hötzl, J. Köngeter, G. Teutsch, Flow and Transport in Fractured Porous Media, vol. 1 (Springer, Berlin, 2005) 4. G.R. Foulger, M.P. Wilson, J.G. Gluyas, B.R. Julian, R.J. Davies, Global review of humaninduced earthquakes. Earth Sci. Rev. 178, 438–514 (2017) 5. E. Keilegavlen, J.M. Nordbotten, Finite volume methods for elasticity with weak symmetry. Int. J. Numer. Methods Eng. 112(8), 939–962 (2017) 6. E. Keilegavlen, A. Fumagalli, R. Berge, I. Stefansson, I. Berre, PorePy: an open-source simulation tool for flow and transport in deformable fractured rocks (2017), ArXiv e-prints 7. N. Kikuchi, J. Oden, Contact Problems in Elasticity (Society for Industrial and Applied Mathematics, Philadelphia, 1988) 8. M.W. McClure, R.N. Horne, Investigation of injection-induced seismicity using a coupled fluid flow and rate/state friction model. Geophysics 76(6), WC181–WC198 (2011) 9. J.R. Rice, Spatiotemporal complexity of slip on a fault. J. Geophys. Res. Solid Earth 98(B6), 9885–9907 (1993) 10. E. Ucar, E. Keilegavlen, I. Berre, J.M. Nordbotten, A finite-volume discretization for deformation of fractured media (2016), ArXiv e-prints 11. E. Ucar, I. Berre, E. Keilegavlen, Three-dimensional numerical modeling of shear stimulation of naturally fractured rock formations (2017), ArXiv e-prints 12. A. Zang, V. Oye, P. Jousset, N. Deichmann, R. Gritto, A. McGarr, E. Majer, D. Bruhn, Analysis of induced seismicity in geothermal reservoirs an overview. Geothermics 52(Supplement C), 6–21 (2014); Analysis of Induced Seismicity in Geothermal Operations

Part XVII

Model Reduction Methods for Simulation and (Optimal)Control

POD-Based Economic Model Predictive Control for Heat-Convection Phenomena Luca Mechelli and Stefan Volkwein

Abstract In the setting of energy efficient building operation, an optimal boundary control problem governed by a linear parabolic advection-diffusion equation is considered together with bilateral control and state constraints. To keep the temperature in a prescribed range with the less possible heating cost, an economic model predictive control (MPC) strategy is applied. To speed-up the MPC method, a reduced-order approximation based on proper orthogonal decomposition (POD) is utilized. A-posteriori error analysis ensures the quality of the POD models. A numerical test illustrates the efficiency of the proposed strategy.

1 Introduction In this paper an optimal boundary control problem is considered for a linear parabolic advection-diffusion equation which describes the evolution of the temperature in a room. The objective is of economic type because it only contains the control variable. Moreover, the problem involves bilateral control and state constraints. Since it is expected that the physical data for the optimal control problem changes due to the long time horizon, we apply an economic model predictive control (MPC) strategy (see [6, Chapter 8] and references therein). In each iteration of the MPC method the open-loop problem is solved by the primal dual active set strategy (PDASS); cf. [9]. To speed-up the numerical optimization we apply model-order reduction based on proper orthogonal decomposition (POD); see [10]. In [12] it is explained how to solve the open-loop problems in each MPC iteration by the PDASS and the POD Galerkin approximation. In particular, the state constraints are regularized by a virtual control concept [11] which allows for a Lavrentiev regularization [14]. POD a-posteriori error analysis is utilized to control the accuracy of the POD reduced-order approximation. This error estimator

L. Mechelli () · S. Volkwein University of Konstanz, Department of Mathematics and Statistics, Konstanz, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_61

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is deduced in [12] and follows from [5, 15]. If the error estimator is too large, we build a new POD basis utilizing information from the current MPC iteration. Let us mention that POD is used for MPC methods in [1, 4], but the authors do not consider economic MPC. The paper is organized as follows: In Sect. 2 we introduce the optimal control problem and the Lavrentiev regularization. We describe the economic MPC algorithm in Sect. 3 and we propose two POD variants. In Sect. 4 we show numerical tests. Conclusions are drawn in Sect. 5.

2 The Optimal Control Problem We consider an optimal control problem with a long time horizon [0, T ] with T 3 0. Let Ω ⊂ Rd , d ∈ {2, 3}, be a bounded domain with Lipschitz-continuous boundary Γ = ∂Ω. We suppose that Γ is split into the two disjoint subsets Γc and Γo , where at least Γc has nonzero (Lebesgue) measure. Let us set Q = (0, T ) × Ω, Σc = (0, T ) × Γc , and Σo = (0, T ) × Γo . Moreover, let H = L2 (Ω), V = H 1 (Ω), U = L2 (0, T ; Rm ), and H = L2 (0, T ; H ). Recall the Hilbert space = > W (0, T ) = ϕ ∈ L2 (0, T ; V ) | ϕt ∈ L2 (0, T ; V  ) , where V  denotes the dual space of V ; cf. [3]. Then, the product space X = W (0, T ) × U × H is a Hilbert space as well. By b1 , . . . , bm ∈ L∞ (Γc ) we denote given control shape functions. We consider the following optimal control problem: min J (x) =

σw σ u2U + w2H 2 2

(1a)

subject to x = (y, u, w) ∈ X, which satisfies the state equation yt (t, x) − Δy(t, x) + v(t, x) · ∇y(t, x) = 0,

a.e. in Q,

m  ∂y (t, s) + y(t, s) = ui (t)bi (s), a.e. on Σc , ∂n i=1

∂y (t, s) + γ y(t, s) = γ yout (t), ∂n y(0, x) = y◦ (x),

(1b)

a.e. on Σo , a.e. in Ω

and the inequality constraints uai (t) ≤ ui (t) ≤ ubi (t), ya (t, x) ≤ y(t, x) + εw(t, x) ≤ yb (t, x)

i = 1, . . . , m a.e. in [0, T ],

(1c)

a.e. in Q.

(1d)

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Assumption 1 In (1) we suppose that σ, σw > 0, v = (v1 , . . . , vd ) ∈ L∞ (0, T ; L∞ (Ω; Rd )), γ ≥ 0, yout ∈ L2 (0, T ), y◦ ∈ H , ua = (uai )1≤i≤m , ub = (ubi )1≤i≤m ∈ U, ε > 0, and ya , yb ∈ H. Moreover, ∂y/∂n stands for the normal derivative of y with respect to the outward normal. Remark 1 1) The variable w can be interpreted as a virtual control; cf. [11]. It allows us to write the pure state constraints ya (t, x) ≤ y(t, x) ≤ yb (t, x)

a.e. in Q

in the regularized form (1d), which admits regular Lagrange multipliers in H; cf. [14]. 2) Since the cost functional J contains only control variables, problem (1) is called an economic optimal control problem. ♦ For given admissible control u we are interested in weak solutions to state equation (1b). For this purpose, we define the time-dependent mapping F (t; ·, ·) : V × Rm → V  as     ∇φ · ∇ϕ + v(t) · ∇φ ϕ dx − γ ϕφ ds F (t; φ, u), ϕV  ,V = − Ω



Γo



−

ϕφ ds + γ yout (t) Γc

ϕ ds + Γo

m

i=1



ui

bi ϕ ds Γc

for φ, ϕ ∈ V , u = (ui ) ∈ Rm a.e. in [0, T ]. Now, a weak solution y ∈ W (0, T ) to (1b) satisfies the dynamical system yt (t) = F (t; y(t), u(t)) ∈ V  a.e. in (0, T ],

y(0) = y◦ in H.

(2)

It is known that (2) admits a unique solution y ∈ W (0, T ); cf. [3]. We introduce the set of admissible solutions, which is given by B = > Xεad = x = (y, u, w) ∈ X B x satisfies (2), (1c) and (1d) . Now, (1) can be expressed as min J (x) subject to

x ∈ Xεad .

(Pε )

Note that (Pε ) is a linear-quadratic, strictly convex programming problem. Therefore, there exists a unique optimal solution which is denoted by x¯ ε = (y¯ ε , u¯ ε , w¯ ε ).

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Remark 2 1) Problem (Pε ) can be formulated as pure control constrained problem; see [12] for more details. 2) Due to the long time horizon, not all data in (Pε ) can be assumed to be fixed or known over the whole interval [0, T ]. Hence, we are interested to compute closed-loop controls for (Pε ) rather than open-loop controls. ♦

3 Model Predictive Control (MPC) Motivated by Remark 2-2) we apply an MPC strategy. The basic idea of MPC is to predict, stabilize and optimize a given dynamical system—like (2)—by reconstructing the optimal control u(t) = Φ(t, y(t)) in a feedback form. In order to do that, we solve repetitively open-loop optimal control problems on smaller time horizons NΔT  T , N ∈ N, with a primal-dual active set strategy (PDASS); cf. [9, 12]. Then, the first part of the open-loop control is stored and applied to (2), before solving the next open-loop problem on a shifted time horizon. A general theory can be found in [6, 13], for instance. For chosen 0 ≤ tn < tnN ≤ T with tnN = tn + NΔT and yn ∈ H we consider (2) on the time horizon [tn , tnN ]: yt (t) = F (t; y(t), u(t)) ∈ V  a.e. in (tn , tnN ],

y(tn ) = yn in H.

(3)

Further, instead of (1c)–(1d) we consider the inequality constraints uai (t) ≤ ui (t) ≤ ubi (t), ya (t, x) ≤ y(t, x) + εw(t, x) ≤ yb (t, x)

i = 1, . . . , m a.e. in [tn , tnN ], a.e. in Qn = (tn , tnN ) × Ω.

(4)

Next we define the function spaces related to [tn , tnN ] Un = L2 (tn , tnN ; Rm ),

Hn = L2 (tn , tnN ; H ),

Xn = W (tn , tnN ) × Un × Hn .

Further, let the set of admissible solutions be given as B = > B Xε,n ad = x = (y, u, w) ∈ X x satisfies (3) and (4) . Now, the open-loop problem can be adapted by choosing the following cost: Jn (x) =

σ σw u2Un + w2Hn 2 2

for x = (y, u, w) ∈ Xn .

The MPC method is summarized in Algorithm 1. If Φ N is computed by the MPC algorithm, then the state y¯ N solves (2) for the closed-loop control u¯ N = Φ N (· ; ynε (·)) with a given initial condition y◦ . Another advantage of MPC

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Algorithm 1 MPC method Require: Initial state y◦ , time horizon NΔt and regularization parameter ε > 0; 1: Put y0 = y◦ and t0 = 0; 2: Compute a POD basis {ψi }i=1 ⊂ V of rank ; 3: for n = 0, 1, 2, . . . do 4: Set tnN = tn + NΔt; 5: Compute a POD suboptimal solution x¯nε, = (y¯nε, , u¯ ε, ¯ nε, ) to n ,w min Jn (x)

6:

subject to x ∈ Xε,n ad ;

(Pεn )

Get the MPC state from the solution  (y¯nε )t (t) = F (t; y¯nε (t), u¯ ε, n (t)) ∈ V a.e. in (tn , tn + Δt],

y¯nε (tn ) = yn in H ;

7: Define the MPC feedback law Φ N (t; y¯nε (t)) = u¯ ε, n (t) for t ∈ (tn , tn + Δt]; 8: Put yn+1 = y¯nε (tn + Δt) and tn+1 = tn + Δt. 9: end for

is that we can update the data during the for-loop. For example, suppose that we have a good forecast for the outside temperature until a certain time t˜ ∈ (0, T ). Then, we can incorporate a new forecast at that time and update the data for the outside temperature in the next open-loop solves. The same can be done for the time-dependent velocity field v by solving Navier-Stokes equations with the new temperature and pressure information obtained at time t˜. This updating strategy can not be done when we solve (Pε ) with PDASS used in [12]: we can not ensure the convergence of PDASS while changing the data during the iteration. For numerical realization of Algorithm 1, we have to discretize (Pεn ): for the temporal discretization we utilize the implicit Euler method, while the spatial variable is approximated utilizing a reduced-order approach; cf. [2, 12]. Here, we apply the method of proper orthogonal decomposition (POD), where the snapshots are computed from piecewise linear finite element (FE) solutions of the state and dual variables; cf. [7, 8, 10]. For more details we refer to [12]. For the MPC algorithm we concentrate on the following two POD strategies: – Method 1: In Algorithm 1 we solve (Pεn ) by using the FE Galerkin scheme for n = 0. Then, we take the state y¯0ε and the associated adjoint variable p¯ 0ε to build a POD basis of rank  which is orthonormal in V = H 1 (Ω). Now, (Pεn ) is solved for all n > 0 applying its (from now on) fixed POD Galerkin approximation. – Method 2: We follow the same procedure of Method 1, but we add the aposteriori error estimator from [12]. If the a-posteriori error is too big, we solve the current problem (Pεn ) by using the FE Galerkin scheme and recompute the POD basis using the obtained optimal FE state and associated adjoint. In our numerical experiments we do not change the POD rank  (basis extension) in both methods.

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4 Numerical Tests All the tests in this section have been made on a Notebook Lenovo ThinkPad T450s with Intel Core i7-5600U CPU @ 2.60GHz and 12GB RAM. Let T = 1, Ω = (0, 1) × (0, 1) ⊂ R2 . Moreover, the bi ’s are chosen to be 9 b1 (x) = 9 b3 (x) =

1 if x ∈ {0} × [0.0, 0.25], 0 otherwise, 1 if x ∈ {1} × [0.5, 0.75], 0 otherwise,

9 b2 (x) = 9 b4 (x) =

1 if x ∈ [0.25, 0.5] × {1}, 0 otherwise, 1 if x ∈ [0.5, 0.75] × {0}, 0 otherwise.

The FE discretization on a triangular mesh has 625 degrees of freedom. We choose Δt = 0.01 as time step. We set also γ = 0.03 and the initial condition y◦ (x) = | sin(2πx1 ) cos(2πx2)| for x = (x1 , x2 ) ∈ Ω; cf. Fig. 1. In the first part of this tests we suppose to have as velocity field v = (v1 , v2 ): 9 v1 (t, x) =

9

−1.6 if x ∈ VF , 0 otherwise,

v2 (t, x) =

0.5 if x ∈ VF , 0 otherwise

(5)

B = > with VF = x = (x1 , x2 ) B 12x2 + 4x1 ≥ 3, 12x2 + 4x1 ≤ 13 . The outside temperature is chosen to be yout (t) = −1 for all t ∈ [0, T ]. As state constraints we take ya (t) = 0.5 + min(2t, 2), yb = 3 and ε = 0.01 and as control constraints we take uai = 0 and ubi = 7 for i = 1, . . . , 4. Further, σ = σw = 1. Now we solve the open-loop optimal control problem with PDASS, like in [12]. To illustrate that the MPC approach improves the control strategy if the data is changed during the

a

b

u2

c

u2 u3

u3

2 1.5 1 0.5 0 1 0.8

u1

u4

u1

0

u4

0.6 0.2

0.4

0.4 0.6

0.8

0.2 1 0

Fig. 1 Spatial domain Ω with the four boundary controls and the velocity fields (grey); initial condition y◦ . (a) v1 (t, x). (b) v2 (t, x). (c) y◦ (x)

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optimization, suppose that at time t˜ = 0.5 we measure again the outside temperature and the velocity field obtaining the new data be y˜out (t) = 1 and v˜ = (v˜1 , v˜2 ) with 9 v˜1 (t, x) =

−0.6 if t ≥ 0.5, x ∈ V˜ F , v˜2 (t, x) = 0 otherwise

9

1.5 if t ≥ 0.5, x ∈ V˜ F , 0 otherwise

B = > with V˜ F = x = (x1 , x2 ) B x1 + x2 ≥ 0.5, x1 + x2 ≤ 1.5 (cf. (5)). Suppose that x¯ ε = (y¯ ε , u¯ ε , w¯ ε ) solves (Pε ) computed with the data yout and v. If we now solve the state equation utilizing the control u = u¯ ε and the new data y˜out , v˜ for t ∈ [t˜, T ], the value of the cost J is 9.86. On the other hand, the value of the cost J utilizing the MPC control u¯ N is 8.797, i.e., significantly smaller. This illustrates that MPC takes care of changing data in the problem. This is the main reason, why we have decided to combine the previous work with a MPC approach, which improves the results and permits also to treat long-time horizon problem that are too costly, in terms of computer’s memory used and computational time, to solve with PDASS alone. In Table 1, the results for the MPC algorithm in this scenario are shown: regarding the MPC-POD, Methods 1 and 2 produce a good approximation of the MPC-FE system, with a reasonable speed-up. Since we evaluate the a-posteriori error estimate in each iteration of the MPC algorithm, Method 2 is slower than Method 1, because of the computation of the error estimate, which requires FE state and adjoint solves; see [12, 15]. For the a-posteriori error estimator we choose the tolerance 0.3 uPOD U . Let us define the quantities rel-err(y) =

y FE − y POD H , y FE H

rel-err(u) =

uFE − uPOD U . uFE U

Method 2 does what we expected: improving the Method 1 approximation of the FE optimal state and control (Table 2). Table 1 Results of the MPC algorithm with FE and POD

Spatial discretization MPC-FE MPC-POD, Method 1 MPC-POD, Method 1 MPC-POD, Method 1 MPC-POD, Method 2 MPC-POD, Method 2 MPC-POD, Method 2

Rank  – 8 12 16 8 12 16

J (x) 8.797 8.888 8.801 8.799 8.928 8.800 8.798

εwH 0.0182 0.0180 0.0183 0.0182 0.0192 0.0182 0.0182

Speed-up – 3.59 3.63 3.32 2.49 2.69 2.62

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Table 2 Relative errors between the MPC FE and its POD approximation Spatial discretization MPC-POD, Method 1 MPC-POD, Method 1 MPC-POD, Method 1 MPC-POD, Method 2 MPC-POD, Method 2 MPC-POD, Method 2

Rank  8 12 16 8 12 16

rel-err(y) 0.02304 0.00252 0.00248 0.00290 0.00208 0.00193

rel-err(u) 0.07415 0.01815 0.01771 0.02264 0.01622 0.01540

Basis updates – – – 6 3 1

5 Conclusion We have proposed a MPC algorithm, where in each iteration the open-loop problems are solved by PDASS and a POD Galerkin scheme. Due to an a-posteriori error estimator, we are able to adjust the POD approximation in such a way that the obtained optimal controls are sufficiently accurate for the FE Galerkin model of the considered dynamical system. If the error estimator is too large in a current MPC iteration, it turns out that the results are improved when the POD basis are recomputed. Acknowledgments L. Mechelli gratefully acknowledges support by the German Research Foundation DFG grant Reduced-Order Methods for Nonlinear Model Predictive Control.

References 1. A. Alla, S. Volkwein, Asymptotic stability of POD based model predictive control for a semilinear parabolic PDE. Adv. Comput. Math. 41, 1073–1102 (2015) 2. P. Benner, A. Cohen, M. Ohlberger, K. Willcox, Model Reduction and Approximation: Theory and Algorithms (SIAM Publications, Philadelphia, 2017) 3. R. Dautray, J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Evolution Problems I, vol. 5 (Springer, Berlin, 2000) 4. J. Ghiglieri, S. Ulbrich, Optimal flow control based on POD and MPC and an application to the cancellation of Tollmien-Schlichting waves. Optim. Softw. 29, 1042–1074 (2014) 5. E. Grimm, M. Gubisch, S. Volkwein, Numerical analysis of optimality-system POD for constrained optimal control. Lect. Notes Comput. Sci. Eng. 105, 297–317 (2015) 6. L. Grüne, J. Pannek, Nonlinear Model Predictive Control:Theory and Algorithms, 2nd edn. (Springer, London, 2016) 7. M. Gubisch, S. Volkwein, Proper orthogonal decomposition for linear-quadratic optimal control, in Model Reduction and Approximation: Theory and Algorithms, chap. 1 (SIAM Publications, Philadelphia, 2017) 8. M. Gubisch, I. Neitzel, S. Volkwein, A-posteriori error estimation of discrete POD models for PDE-constrained optimal controls, in Model Reduction of Parametrized Systems, ed. by P. Benner, M. Ohlberger, A.T. Patera, G. Rozza, K. Urban. Springer-Series Modeling, Simulation and Applications, vol. 17, 213–234 (2017) 9. M. Hintermüller, K. Ito, K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2003)

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10. P. Holmes, J.L. Lumley, G. Berkooz, C.W. Rowley. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, 2nd edn. (Cambridge University Press, Cambridge, 2012) 11. K. Krumbiegel, A. Rösch, A virtual control concept for state constrained optimal control problems. Comput. Optim. Appl. 43, 213–233 (2009) 12. L. Mechelli, S. Volkwein, POD-based economic optimal control of heat-convection phenomena (2017, Submitted). Preprint download at: http://nbn-resolving.de/urn:nbn:de:bsz:352-2-au6ei3apyzpv0 13. J.B. Rawlings, D.Q. Mayne. Model Predictive Control: Theory and Design (Nob Hill Publishing, Madison, 2009) 14. F. Tröltzsch, Regular Lagrange multipliers for control problems with mixed pointwise controlstate constraints. SIAM J. Optim. 22, 616–635 (2005) 15. F. Tröltzsch, S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44, 83–115 (2009)

Real-Time Optimization of Thermal Ablation Cancer Treatments Zoi Tokoutsi, Martin Grepl, Karen Veroy, Marco Baragona, and Ralph Maessen

Abstract Motivated by thermal ablation treatments for prostate cancer, the current work investigates the optimal delivery of heat in tissue. The problem is formulated as an optimal control problem constrained by a parametrized partial differential equation (PDE) which models the heat diffusion in living tissue. Geometry and material parameters as well as a parameter entering through the boundary condition are considered. Since there is a need for real-time solution of the treatment planning problem, we introduce a reduced order approximation of the optimal control problem using the reduced basis method. Numerical results are presented that highlight the accuracy and computational efficiency of our reduced model.

1 Introduction Thermal ablation cancer treatments aim at delivering heat locally in the cancer volume with the purpose of burning the tumor but at the same time leaving surrounding healthy tissue and neighboring sensitive risk structures undamaged [1]. The optimal volume heat source which induces the desired temperature change can be used to identify patient specific tissue parameters [8] and place the ablation device [3]. The optimal heat source results from solving a distributed optimal control problem, constrained by an elliptic partial differential equation (PDE). The PDE Z. Tokoutsi () AICES, RWTH Aachen University, Aachen, Germany Philips Research, Eindhoven, The Netherlands e-mail: [email protected] M. Grepl · K. Veroy AICES, RWTH Aachen University, Aachen, Germany e-mail: [email protected]; [email protected] M. Baragona · R. Maessen Philips Research, Eindhoven, The Netherlands e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_62

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constraint is parametrized with respect to material and geometrical parameters. We employ the reduced basis method [6] in order to derive a surrogate model for the elliptic distributed optimal control problem, based on the method presented in [5]. Thereby we achieve real-time efficient computation of the optimal heat source and corresponding temperature for different parameter values.

2 Problem Statement For the purposes of this work we consider a simplified computational domain. In real-life applications our model can be replaced with patient specific data. We introduce the relevant parameters and we specify the steady state problem determining the heat source which produces the desired temperature change.

2.1 Parametrized Domain and Biophysical Model The prostate and the tumor volumes are approximated by two ellipsoids of appropriate size. During prostate ablations it is common to inserted a cooling catheter via the urethra. Thus we can assume that the urethra adopts a cylindrical shape. Exploiting the almost rotational symmetry around the catheter, we can define the distance l of the tumor from the cooling catheter by its radial distance with respect to the centerline of the cylinder. A schematic description of the parametrized domain is presented in Fig. 1. The Pennes bioheat equation [7] describes the heat transfer in living tissue and approximates the cooling effect of blood circulation as an isotropic heat sink, Fig. 1 The parametrized domain Ω(l ) = ΩT (l ) ∪ ΩH (l ), where ΩT is an ellipsoid with principal semi-axes of length 2.2 × 2.2 × 1.2 mm and denotes the tumor volume, whereas ΩH is an ellipsoid with principal semi-axes of length 22.8×16.25×19.6 mm and represents the healthy prostate tissue. The cooling catheter is a cylinder of 4 mm radius

z

tumor x

urethra l

y

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proportional to the blood flow rate and the difference between the body core and the local tissue temperature. We model the cooling catheter Γcool using a convection boundary condition [2], and we assume that the rest of the boundary Γprostate = ∂Ω \ Γcool remains at body temperature. The resulting bioheat equation is −∇kt i ∇T + ρbl Cbl wbl (T − Tcore ) = Q,

in Ω(l ),

kt i ∇ν T + h(T − Tcool) = 0,

on Γcool ,

T = Tcore ,

(1)

on Γprostate.

Here, kt i = 0.54 J s−1 m−1 K−1 is the thermal conductivity, ρbl = 1050 kg m−3 is the blood density, Cbl = 3617 J kg−1 K−1 is the specific heat capacity of blood, wbl = 0.002857 s−1 is the blood perfusion rate, h = 250 W m−2 K−1 is the convection coefficient, T is the tissue temperature in K, Tcore = 310.15 K (37 ◦ C) is the core body temperature, and Tcool in K is the temperature of the cooling catheter surface. Finally, the heat source Q is measured in J m−3 s−1 and the differential operator ∇ has units m−1 . For simplicity, the material parameters are considered to be constant over the whole domain and the different tissue types; this assumption can be easily relaxed. Non-dimensionalization Each term in (1) is measured in J m−3 s−1 . Canceling out the units of each term via an appropriate substitution balances the difference in magnitude of the temperature and heat source and improves the scaling and stability of the numerical optimization. Setting y = (T − Tcore )/Tcore , ycool = (Tcool − Tcore )/Tcore , k = kti /kti = 1, x˜ = x/Lref , c = ρbl Cbl wbl L2ref /kti , c2 = L2ref /Tcore /kti , u = c2 Q, h˜ = hLref /kti , where the reference length Lref is set to 13 mm, and omitting the tilde notation for simplicity, the non-dimensional stationary bioheat equation becomes −kΔy + cy = u,

in Ω(l ),

k∇ν y + h(y − ycool ) = 0,

on Γcool,

y = 0,

(2)

on Γprostate.

Parametrized PDE The distance l of the tumor from the urethra serves as one parameter and varies between 0.5 and 7.88 mm. Two additional parameters are the blood perfusion parameter c ranging between 3608 and 18230 W m−3 K−1 , and the cooling catheter temperature ycool in the range of 17–37 ◦C.1 The weak formulation corresponding to the non-dimensional bioheat equation (2) is a(y, v; μ) = b(u, v; μ) + f (v; μ),

(3)

the parameter values are presented in SI units and ◦ C for the reader’s convenience. However all computations use the non-dimensional quantities.

1 Here

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for all v ∈ Y := {v ∈ H 1 (Ω) : v|Γprostate ≡ 0}. Here the heat source u lies 2 in U = L (Ω), the bilinear  form a(·, ·; μ) : Y × Y → R, with a(y, u; μ) = k∇y∇v + cyv dΩ + coercive, both the bilinear Ω(l ) Γcool hyv dS is continuous and  form b(·, ·; μ) : U × Y → R, with b(u, v; μ) = Ω(l ) uv dΩ, and the linear  form f (·; μ) : Y → R, with f (v; μ) = Γcool hycoolv dS, are continuous, and μ = (l, ycool, c) ∈ P = [0.03846, 0.6062] × [−0.0645, 0.0] × [1.132, 5.66] (nondimensional intervals) represents the parameters of interest.

2.2 Parametrized Optimal Control Problem The volume heat source is determined as the solution to the following parametrized distributed optimal control problem. For any μ ∈ P, solve min J (y, u; μ) =

y∈Y,u∈U

1 λ |y − yd |2D(μ) + u2U (μ) 2 2

s.t. a(y, v; μ) = b(u, v; μ) + f (v; μ),

(4)

∀v ∈ Y,

where λ > 0 is the regularization parameter, D(μ) = L2 (ΩT (l)), and U (μ) = L2 (Ω(l)). The target temperature for the thermal ablation yd is homogeneous and constant in the tumor volume and must be at least 62 ◦C which represent a conservative estimate of the ablated volume in the steady state case. Here, however, yd is set to 0.171 corresponding to 90 ◦C. This choice of a higher temperature ensures that temperatures above the target are not penalized. From our assumptions for the PDE constraint and the function spaces Y and U , it follows that there exists a unique optimal solution to (4); see e.g. [9]. We next define the continuous bilinear form d (·, ·; μ) : L2 (ΩT )×L2 (ΩT ) → R which is the L2 inner product restricted on ΩT and is symmetric and positive semi-definite, and c (·, ·; μ) : U × U → R, which √ is the energy inner product of U . The associated semi-norm is then | · |D(μ) = d(·, ·; μ)√– here specifically | · |D(μ) ≡  · L2 (ΩT ) – and the energy norm is  · U (μ) = c(·, ·; μ). Finally, all linear and bilinear forms in the cost functional J and the PDE constraint (4) are affine parameter dependent, e.g. a(·, ·; μ) can be written in the form a(y, v; μ) = Q a q q q q=1 Θa (μ)a (y, v), where Qa is a (preferably) small integer, Θa : P → R depend continuously on μ, and a q (·, ·) are parameter independent continuous bilinear forms for q = 1, · · · , Qa . Due to the geometric parametrization of the domain, the affine decomposition of the affected bilinear forms is not trivial. The domain has to be subdivided in simple volume components which can be affinely mapped to a reference domain for any value of the distance parameter l. First Order Optimality Conditions We employ the Lagrangian approach for solving the PDE constrained optimization problem (4). The obtained first order optimality conditions are necessary (and sufficient) and result in the following state,

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adjoint and optimality equations. Given μ ∈ P, the optimal solution (y ∗ , p∗ , u∗ ) ∈ X = Y × Y × U satisfies a(y ∗ , φ; μ) = b(u∗ , φ; μ) + f (φ; μ),

∀φ ∈ Y,

a(ϕ, p∗ ; μ) = d(yd (μ) − y ∗ , ϕ; μ),

∀ϕ ∈ Y,

∗

∗

λ c (u , ψ; μ) = b(ψ, p ; μ),

(5)

∀ψ ∈ U.

The numerical solution to the optimal control problem (4) is computed using the finite element (FE) approximations of the function spaces Y and U . The domain Ω(μ) is discretized using a tetrahedral mesh, which is locally refined around the tumor. We employ piecewise linear and continuous basis functions and denote by 1 (Ω), dim(Y ) = N = 79,813 and by U 2 YN ⊂ Y ≡ H(0) Y N N ⊂ U ≡ L (Ω), dim(UN ) = NU = 82,324 the FE state and control spaces and their dimensions, ∗ , p∗ , u∗ ) and is determined respectively. The FE solution to (4) is denoted by (yN N N solving the coupled equations (5); a problem of dimension 2NY + NU = 241,950. The need to adjust the ablation treatment plan may occur during the treatment. For example measurements of patient specific parameters, like the perfusion parameter, are not possible preoperatively. Furthermore in soft tissues like the prostate, the distance of the tumor from the cooling device may vary from the estimate based on preoperative images. This can be due to the insertion of the cooling device, the positioning and state of the patient, or the insertion of the ablation device – in the case of percutaneous ablations. Finally the cooling temperature may have to be adjusted during the treatment according to the physician’s judgment. Therefore real-time updates of the optimal heat source are very relevant.

3 Reduced Basis Approximation The efficient and reliable real-time solution of the optimal control problem (4) for different parameters μ is achieved by employing the Reduced Basis (RB) method; see e.g. [6]. Using an adjusted version of the well established greedy sampling algorithm, we determine a sample set PN = {μ1 , · · · , μN } ⊂ P, the associated ∗ (μn ), p∗ (μn ), 1 ≤ n ≤ N}, and the RB integrated RB space YN = span{yN N ∗ n control space UN = span{uN (μ ), 1 ≤ n ≤ N}, 1 ≤ N ≤ Nmax  N . Note that we integrate both state and adjoint snapshots in the space YN , hence dim(YN ) = 2N, whereas dim(UN ) = N. The reduced basis solution to the optimal control problem (4) ensues from the solution of the reduced order state, adjoint and ∗ ∈ Y , adjoint p∗ ∈ Y , optimality equations: given μ ∈ P, the optimal state yN N N N ∗ and control uN ∈ UN satisfy ∗ , φ; μ) = b(u∗N , φ; μ) + f (φ; μ), a(yN

∀φ ∈ YN ,

∗ ∗ ; μ) = d(yd,N (μ) − yN , ϕ; μ), a(ϕ, pN

∀ϕ ∈ YN ,

λ c(u∗N , ψ; μ)

=

∗ b(ψ, pN ; μ),

∀ψ ∈ UN ,

(6)

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which is of significantly smaller dimension 5N than the FE problem, assuming N  NY , NU . The greedy algorithm utilizes rigorous and (online-)efficient a posteriori error bounds obtained by manipulating the error residual equations of the optimality system (6). For any μ ∈ P the residuals of the state, adjoint and optimality equations (6) are defined by ∗ , φ; μ), ry (φ; μ) = f (φ; μ) + b(u∗N , φ; μ) − a(yN

∀φ ∈ Y,

∗ , ϕ; μ) − a(ϕ, p∗ ; μ), d(yd,N (μ) − yN N ∗ ∗ b(ψ, pN ; μ) − λ c(uN , ψ; μ),

∀ϕ ∈ Y,

rp (ϕ; μ) = ru (ψ; μ) =

(7)

∀ψ ∈ U,

It has been shown in [5] that the following proposition holds. Proposition 1 For any μ ∈ P, the error in the energy norm  · U (μ) satisfies ∗

uN − u∗N U (μ)

≤

ΔuN (μ)

:= c1 (μ) +

2

c1 (μ)2 + c2 (μ),

where c1 (μ) and c2 (μ) depend on the lower bound αaLB (μ) of the coercivity constant of a(·, ·; μ), the upper bound γbUB (μ) of the continuity constant of UB for CD (μ) := supu∈Y |u|D(μ) /vY ≥ 0, and the b(·, ·; μ), the upper bound CD dual norms of the error residuals corresponding to the Eq. (6) as follows   γbUB (μ) 1 c1 (μ) = rp (·; μ)Y  , ru (·; μ)U (μ) + LB 2λ αa (μ)  2   UB (μ) 1 1 CD 2 ry (·; μ)Y  rp (·; μ)Y  + ry (·; μ)Y  c2 (μ) = . λ αaLB (μ) 4 αaLB (μ) Observing that the coercivity constant αa (μ) is independent of ycool and takes its minimum over P at the smallest value of l and c, we use a global lower bound for αaLB (μ) = 0.287. Alternatively, the successive constraint method can be used [4].

4 Numerical Results and Summary The finite element solutions to the optimal control problem (4) are presented in Figs. 2 and 3 for different parameter values μ and regularization parameters λ. We visualize the optimal volume heat and temperature on the surface of a domain cutout. In addition we render the tumor surface in white and the 62 ◦ C-isosurface in red. Note in Fig. 2 that for small distance parameters the tumor target temperature is not achieved. Reducing the regularization parameter further weakens the penalization

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Fig. 2 Solutions to problem (4) for λ = 10−4 , where μ = (0.4652, −0.0608, 4.9951) on the top, and μ = (0.0606, −0.053, 5.3914) on the bottom. On the left we plot the optimal heat, while on the right is the resulting temperature and the 62 ◦ C-isovolume

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Fig. 3 Solutions to problem (4) for λ = 10−5 , where μ = (0.4652, −0.0608, 4.9951) on the top, and μ = (0.0606, −0.053, 5.3914) on the bottom

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Table 1 Maximum relative error, error bounds and effectivities as a function of N N

λ = 10−4 u εN,max rel

ΔuN,max rel

u ηN

λ = 10−5 u εN,max rel

ΔuN,max rel

u ηN

5 10 20 30 38

2.75e − 02 3.05e − 03 1.70e − 04 3.10e − 05 1.85e − 06

4.93e − 01 4.64e − 02 1.67e − 03 2.17e − 04 7.69e − 05

21.4 17.5 13.9 12.6 26.4

4.23e − 02 5.96e − 03 2.89e − 04 3.31e − 05 3.40e − 06

1.42e + 00 1.88e − 01 5.82e − 03 4.18e − 04 1.47e − 04

33.3 32.4 27.4 19.0 32.1

of the heat input and allows for a better approximation of the target temperature, see Fig. 3. The reduced basis spaces YN ⊂ YN and UN ⊂ UN are constructed for two regularization parameters λ = 10−4 and λ = 10−5 using a greedy sampling procedure and the energy norm error bound of Proposition 1. In each step of the greedy algorithm the error bound is evaluated over the train set Ξtrain ⊂ P consisting of ntrain = 20 × 5 × 10 = 1000 parameter points over P. These are sampled equidistantly over ycool and c and over the union of a linear and a logarithmic equidistant grid in l. The sampling tolerance is set to εtol,min = 5×10−5, the resulting reduced spaces have dimension Nmax = 38 for λ = 10−4 and Nmax = 42 for λ = 10−5 . In Table 1, we present the error, error bounds, and corresponding mean effectivities of the RB approximations evaluated over a test set of 20 random points in P for various values of N. We note that the error bounds converge rapidly and the corresponding effectivities are small for both regularization parameters. For the parameter setting considered in this paper, the dual norm of the state is slightly larger than the dual norms of both the adjoint and the control. Thus it has the strongest influence on the convergence of the error bound. On average over the test set it takes ( 400 s (λ = 10−4 ) and ( 500 s (λ = 10−5 ) to compute the FE solution to the optimal control problem, while the RB solution including the error bound computation requires 0.2–2.4 ms, depending on the RB dimension. As a consequence, we achieve speed up rates of ( 105 . The offline stage takes ( 274 min (λ = 10−4 ) and ( 431 min (λ = 10−5 ). Summary A thermal cancer treatment planning problem is formulated as a parametrized elliptic distributed optimal control problem. A surrogate model is created using the certified reduced basis method. Numerical results for small regularization parameters are presented. The achieved accuracy and the significant computational speed up of the RB model presented here constitute promising results for the application of our proposed method to real-time treatment planning. Acknowledgements This work is supported by the European Commission through the Marie Sklodowska-Curie Actions (EID, Project Nr. 642445). We would like to thank the anonymous reviewer for helpful comments.

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References 1. K.F. Chu, D.E. Dupuy, Thermal ablation of tumours: biological mechanisms and advances in therapy. Nat. Rev. Cancer 14, 199–208 (2014) 2. S. Davidson, M. Sherar, Theoretical modeling, experimental studies and clinical simulations of urethral cooling catheters for use during prostate thermal therapy. Phys. Med. Biol. 48(6), 729–744 (2003) 3. Y. Feng, D. Fuentes, Model-based planning and real-time predictive control for laser-induced thermal therapy. Int. J. Hyperth. 27(8), 751–761 (2011) 4. D.B.P. Huynh, G. Rozza, S. Sen, A.T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. 345(8), 473–478 (2007) 5. M. Kärcher, Z. Tokoutsi, M. Grepl, K. Veroy, Certified reduced basis methods for parametrized distributed optimal control problems. J. Sci. Comput. 75, 276–307 (2017) 6. A.T. Patera, G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. Pappalardo Graduate Monographs in Mechanical Engineering (MIT, Cambridge, 2007) 7. H.H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm. Am. Physiol. Soc. 1(2), 93–122 (1948) 8. H. Tiesler, Identification of material parameters from temperature measurements in radio frequency ablation, Ph.D. thesis, University of Bremen, 2011 9. F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112 (American Mathematical Society, Providence, 2010)

Parametric Model Reduction via Interpolating Orthonormal Bases Ralf Zimmermann and Kristian Debrabant

Abstract In projection-based model reduction (MOR), orthogonal coordinate systems of comparably low dimension are used to produce ansatz subspaces for the efficient emulation of large-scale numerical simulation models. Constructing such coordinate systems is costly as it requires sample solutions at specific operating conditions of the full system that is to be emulated. Moreover, when the operating conditions change, the subspace construction has to be redone from scratch. Parametric model reduction (pMOR) is concerned with developing methods that allow for parametric adaptations without additional full system evaluations. In this work, we approach the pMOR problem via the quasi-linear interpolation of orthogonal coordinate systems. This corresponds to the geodesic interpolation of data on the Stiefel manifold. As an extension, it enables to interpolate the matrix factors of the (possibly truncated) singular value decomposition. Sample applications to a problem in mathematical finance are presented.

1 Introduction Model reduction (MOR) is a branch of applied mathematics that is concerned with the emulation of large-scale dynamical systems via a highly reduced number of degrees of freedom. The resulting reduced order model (ROM) is expected to be much faster to evaluate, but less accurate than the original model. Ideally, it comes with inherent error indicators/estimators that allow the user to control the trade-off between the numerical efficiency and the numerical accuracy. Subspace-Based Model Reduction Among the most prominent model reduction techniques are projection-based methods. Here, the key idea is to construct a

R. Zimmermann () · K. Debrabant University of Southern Denmark, Odense, Denmark e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_63

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low-dimensional subspace of solution candidates and to restrict all successive computations to this subspace, e. g. via a projection. We explain this procedure on an example. A full survey of methods can be found in [2]. Consider a spatio-temporal dynamical partial differential equation (PDE) system in semi-discrete form ∂ yq (t) = f (t, yq (t)), ∂t

yq (t) ∈ Rn ,

f : [t0 , te ] × Rn → Rn .

Here, we assume that the spatial dimensions are resolved in the discretization with a grid of n ∈ N points. The index q indicates additional system parameters q = (q1 , . . . , qd )T ∈ Rd on which the system may depend. Subspace construction is based on sampled snapshot solutions at m selected sample points: y 1 = yq (t1 ), . . . , y m = yq (tm ). It is assumed that the system dimension n exceeds the number of sampled snapshots m by several orders of magnitude, n 3 m. The subspace of solution candidates is to represent the span of the sampled snapshots but truncated to the essential irredundant information. This is achieved via a compact singular value decomposition (SVD), which corresponds to the most basic form of a proper orthogonal decomposition [9] of the input data, SV D

U ΣV T = Y := (y 1 , . . . , y m ),

(1)

with U ∈ Rn×m , V ∈ Rm×m , Σ = diag(σ1 , . . . , σm ), followed by a truncation of U, V to the first p ≤ m columns that are associated with the most significant non-zero singular values. A popular measure for this notion of significance is the  p

σj

relative information content, RI C(p) = j=1 . In practice, p is chosen according m k=1 σk to a user-defined threshold ε such that RI C(p) ≥ 1 − ε. Perfect recovery of the snapshot data (up to numerical errors) corresponds to ε = 0. The process of snapshot sampling and basis construction is referred to as the off-line stage of the ROM and is assumed to be computationally intense, since solutions to the original system are required. The truncated U ∈ Rn×p can be interpreted as a coordinate system for solution candidates for the ROM: y˜q (t) = U yˆq (t), where yˆq ∈ Rp is the vector of basis coefficients with respect to the coordinate system induced by U and y˜q is the ROM approximation of the exact solution yq . One possible way to obtain the reduced coefficient vector yˆq is via Galerkin projection of the original system onto the reduced coordinates: d yˆq (t) = U T f (t, U yˆq (t)). dt

(2)

Yet, several other approaches to determine the coefficient vector yˆq (t) exist, including Petrov-Galerkin projection, the discrete empirical interpolation method (DEIM) [6], interpolation [4, 11] and residual optimization [5, 8, 14]. The process

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of determining yˆq (t) is referred to as the on-line stage of the ROM and is designed such that it is independent of the original system dimension n or scales at most linear in n. Problem Statement The main focus of this work is not on approaches to determine the reduced state yˆq (t) but on the variation of the ROM itself under changes of a scalar system parameter q ∈ R. We say that q specifies the operating condition. Our objective is to move from q to q˜ without repeating the off-line process of snapshot sampling and coordinate system construction. To this end, we parameterize this process via interpolation, but on the level of the orthogonal bases that define the ROM candidate solution subspaces, where we assume that the snapshot matrix q → Y = Y (q) depends differentiably on the operating condition q. Consider two snapshot matrices Y (q0 ) and Y (q1 ) with possibly truncated SVDs SV D

U (q0 )Σ(q0 )V T (q0 ) ≈ Y (q0 ),

SV D

U (q1 )Σ(q1 )V T (q1 ) ≈ Y (q1 ),

(3)

where the approximation holds up to a specified relative information content. More precisely, we assume that the reduced subspace dimension p is chosen such that RI C(p) ≥ 1 − ε at both operating conditions q0 , q1 . In particular, we require that U (q0 ) and U (q1 ) share the same dimensions. From this perspective, the task to construct a parametric ROM essentially reduces to the task of computing a trajectory q → U (q) that starts in U (q0 ) and ends in U (q1 ) without having to enter a new off-line stage for every q. The main obstacle is that this trajectory is a curve in the set of orthogonal coordinate systems. This set forms a curved matrix manifold, the so-called Stiefel manifold [1, 7], St (n, p) = {U ∈ Rn×p |

U T U = Ip }.

Our original contribution is a method for quasi-linear geodesic interpolation on the Stiefel manifold. Interpolation procedures on matrix manifolds have been considered for parametric model reduction before, see [2, §4] and references therein. The standard technique is to (1) first map the matrix data onto the flat tangent space of the manifold, (2) perform the interpolation in the tangent space, (3) map back the result to the matrix manifold. This procedure is illustrated in Fig. 1. However, to the best of our knowledge, interpolation of data on the Stiefel manifold has not yet been treated in the literature. A partial explanation is that conducting the backand-forth mapping between manifold data and tangent vectors requires a practical method for computing both the Riemannian exponential [1, §5.4] on the Stiefel manifold and its inverse, the Riemannian logarithm. While an explicit formula for computing the exponential on the Stiefel manifold exists for almost two decades [7], an efficient algorithm for computing the Riemannian logarithm has only recently been developed [10, 13].

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Fig. 1 Interpolation of orthogonal bases. The curved line represents the Stiefel manifold; the straight lines represent the tangent vectors in the tangent space at U (q0 ) and U (q1 ), respectively

2 The Stiefel Manifold in Numerical Schemes In this section, we recap the essentials of working with data on the Stiefel manifold in numerical procedures. For more details, the reader is referred to [1, 7]. The tangent space TU St (n, p) at a point U ∈ St (n, p) is represented by   TU St (n, p) = Δ ∈ Rn×p | U T Δ = −ΔT U   = U A + (I − U U T )T | A ∈ Rp×p skew, T ∈ Rn×p ⊂ Rn×p . St The Riemannian exponential t → ExpU (tΔ), which gives the geodesic curve 0 starting at t = 0 in U0 with velocity Δ can be computed with the standard matrix exponential as a building block: Let QR = Δ be the (compact) QR-decomposition of the tangent vector, then St U˜ = ExpU (Δ) = U0 M + QN ∈ St (n, p), where 0 ⎛ ⎞     T A −R ⎠ Ip M , A = U0T Δ, := expm ⎝ R 0 0 N

(4a) (4b)

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see [7, §2.4.2]. A matrix-algebraic algorithm for computing Δ, given two points U, U˜ ∈ St (n, p), was introduced in [13]. This scheme produces a sequence of skew-symmetric matrices Ak ∈ Rp×p and matrices Rk ∈ Rp×p such that in the limit, St ˜ Δ = LogU (U ) = U0 A∞ + QR∞ ,

(5)

where Q stems from the QR-decomposition of (I − U U T )U˜ . For full details and MATLAB code, see [13].

3 Quasi-Linear Interpolation of Orthogonal Bases Assume that we are given two snapshot ensembles Y (q0 ), Y (q1 ) with SVDs as in (3). With an efficient algorithm for the Riemannian ExpSt and Log St , we may perform quasi-linear interpolation on St (n, p) to obtain a trajectory of orthogonal bases q → U (q) ∈ St (n, p). To this end, we use U (q0 ) as a base point and map U (q1 ) to TU (q0 ) St (n, p). In this way, we obtain a velocity vector Δ that corresponds to the geodesic on St (n, p) that starts in U (q0 ) and eventually crosses U (q1 ). Since geodesics on curved manifolds are the natural generalization of straight lines in Euclidean spaces, we interpret this as quasi-linear interpolation. The algorithmic procedure is summarized in Algorithm 3.1. It may be extended to interpolating the complete SVD, see Algorithm 3.2. Algorithm 3.1 Geodesic interpolation on St (n, p) Input: U (q0 ), U (q1 ) ∈ St (n, p), s ∈ [q0 , q1 ] 1: Δ = LogUSt(q0 ) (U (q1 )) 2: q(s) =

%velocity vector, see (5), [13, Alg. 1]

s−q0 q1 −q0

  St Output: U (q(s)) := ExpU (q0 ) q(s)Δ

Algorithm 3.2 Geodesic interpolation of SVD data Input: U (q0 ), Σ(q0 ), V (q0 ), U (q1 ), Σ(q1 ), V (q1 ), s ∈ [q0 , q1 ] 1: ΔU = LogUSt(q0 ) (U (q1 )), ΔV = LogVSt(q0 ) (V (q1 )), ΔΣ = Σ(q1 ) − Σ(q0 ) 2: q(s) =

s−q0 q1 −q0

T    St St Output: ExpU (q0 ) (q(s)ΔU ) · Σ(q0 ) + q(s)ΔΣ · ExpV (q0 ) (q(s)ΔV )

% see (4a), (4b)

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4 Numerical Experiments In this section, we present an academic application to computational option pricing. The value function y(T , S; K, r, σ ) that gives the fair price for a European call option is determined via the Black-Scholes-equation [3], yt (t, S) =

1 2 2 σ S ySS (t, S) + rSyS (t, S) − ry(t, S), 2

yt (0, S) = max{S − K, 0},

S ≥ 0,

0 < t ≤ T,

S ≥ 0.

It is a parabolic PDE that depends on time t, the stock value S, also called the underlying, and a number of additional system parameters, namely the strike price K, the interest rate r, the volatility σ and the exercise time T . In this experiment, we consider a fixed interest rate of r = 0.01 and an exercise time of T = 2 units. The dependency on the underlying S ∈ [50, 150] is resolved via a discretization of the interval by equidistant steps of ΔS = 0.01, while the strike price K ∈ [30, 170] is discretized in steps of ΔK = 1. The volatility σ will act as the operating condition. We are interested in approximating the option price y as well as its derivative ∂σ y. This quantity is also called ‘vega’ and belongs to the set of the ‘greeks’, i.e., to the partial derivatives of the option value function with respect to the system parameters. The Black-Scholes equation for a single underlying has a closed-form solution. Yet, here, we will approach it via a numerical scheme in order to mimic the corresponding procedure for real-life problems. Application of a finite volume scheme to the Black-Scholes PDE yields snapshot matrices   Y (σ ) = Ys,k (σ ) s=1,...,10001 , ∂σ Y (σ )s=1,...,10001, k=1,...,141

k=1,...,141

for σ ∈ [0, 0.1, 0.2, . . . , 1.5]. The off-line calculation time amounts to 19+30 min per snapshot matrix (solutions + derivatives). The POD/SVD of a snapshot matrix Y (σ ) ∈ Rn×m , n = 10001, m = 141 yields a compressed representation Y (σ ) = U (σ )Σ(σ )V T (σ ), with U (σ ) ∈ St (n, p), Σ(σ ) ∈ diag(p, p), V (σ ) ∈ St (m, p) and consumes ca. 0.07 s on a laptop computer. First Experiment We take the snapshot matrices Y (0.9), Y (1.1) ∈ Rn×m as sample data. The goal is to compute a low-rank-trajectory q → Yˆ (q) of snapshot matrix approximants. We exemplify this by predicting the full snapshot ensemble Y (1.0) via Yˆ (1.0). To this end, we compute the SVDs of Y (0.9), Y (1.1). The original column dimension m = 141 is reduced to p = 5, which corresponds to a relative information content of RI C(p) = 1.0 − 10−7 . We interpolate each corresponding SVD matrix factor via Algorithm 3.2 in order to obtain interpolants ˆ Uˆ (1.0), Σ(1.0), Vˆ (1.0). We assess the accuracy of the approach by the following means: ˆ (1) When we recompose the matrix Yˆ (1.0) = Uˆ (1.0)Σ(1.0) Vˆ (1.0), we obtain a relative error of Yˆ (1.0) − Y (1.0)/Y (1.0) = 0.00267 with respect

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to the reference solution. For comparison purposes, we perform direct, noncompressed linear interpolation of the snapshot matrices, i.e., we compute YdLI (1.0) = 0.5(Y (0.9) − Y (1.1)). The result features a higher relative accuracy of YdLI (1.0) − Y (1.0)/Y (1.0) = 0.00146. In fact, we cannot expect that the quasi-linear interpolation of the compressed data outperforms the direct linear interpolation of the full data. (2) When we only consider the interpolated coordinate system Uˆ = Uˆ (1.0) and project the full snapshot data Y (1.0) onto the associated subspace, we obtain Uˆ Uˆ T Y (1.0) − Y (1.0)/Y (1.0) = 8.532 × 10−9 . In contrast, when we first compute YdLI (1.0) and the associated SVD factor UdLI and truncate T Y (1.0) − this to the same dimension as Uˆ , the projection error is UdLI UdLI −8 Y (1.0)/Y (1.0) = 4.759 × 10 , which is roughly 5.6 times higher. Second Experiment Here, we use ∂σ Y (0.9), ∂σ Y (1.1) as sampled input data to predict ∂σ Y (1.0). We repeat the same steps as in the first experiment with RI C(p) = 1.0 − 10−7 , p = 7. For brevity, we state only the results: (1)∂σ Yˆ (1.0) − ∂σ Y (1.0)/∂σ Y (1.0) ∂σ YdLI (1.0) − ∂σ Y (1.0)/∂σ Y (1.0) (2)Uˆ Uˆ T ∂σ Y (1.0) − ∂σ Y (1.0)/∂σ Y (1.0) T UdLI UdLI ∂σ Y (1.0) − ∂σ Y (1.0)/∂σ Y (1.0)

=0.00380, =0.00396, =1.454 × 10−8 , =2.410 × 10−8 .

The results are displayed in Figs. 2 and 3. In both experiments, the quality of the quasi-linear low-rank SVD interpolation is of the same accuracy order as the direct linear interpolation of the given data matrices. The prediction capabilities of the geodesically interpolated coordinate system are slightly better than for its directly computed counterpart. It is remarkable that even for this rather academic example, performing a single SVD of a snapshot ensemble takes longer than conducting Algorithm 3.2 from scratch, including the iterative Stiefel log procedure [13, Alg. 1] and re-assembling the output matrix (∼ 0.07 s vs. ∼ 0.02 s). Note that step 1 of Algorithm 3.2 has to be performed only once and that some of the quantities that are required for the Stiefel exponential may also be pre-computed, see [12]. Excluding these operations, the computation time reduces to ∼ 0.007 s, ten times less than for performing the SVD.

5 Summary and Conclusion We propose to use the Riemannian exponential and logarithm mappings to conduct quasi-linear interpolation on the Stiefel manifold. This results in a method for parametric model reduction that is completely data-driven in the sense that it relies only on output data samples of a given simulation model but does not require any intrinsic system modifications.

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Fig. 2 Left: Batch SVD interpolation of Y (1.0). Right: exact reference

Fig. 3 Left: Batch SVD interpolation of ∂σ Y (1.0). Right: exact reference

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The approach can be extended to an interpolation scheme for the singular value decomposition. In this form, it results in a ‘batch’ method: it directly gives approximations of the trajectory of the full, parameter-dependent snapshot matrices but relies on low-rank quantities exclusively. Yet, it may also be used as a building block in more sophisticated schemes: We may only interpolate the low-rank coordinate system U (q) and use the corresponding subspace as the space of solution candidates in combination with Galerkin projection, DEIM, residual optimization or other model reduction techniques. The method leads to a gain in efficiency if the snapshot input data allows for a high level of compression (p  m). Otherwise, one could directly interpolate the snapshot data matrices and the numerical approximations obtained in this way should be of higher accuracy than when interpolating every matrix factor in a lowrank SVD separately.

References 1. P.-A. Absil, R. Mahony, R. Sepulchre, Optimization Algorithms on Matrix Manifolds (Princeton University Press, Princeton, 2008) 2. P. Benner, S. Gugercin, K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015) 3. F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973) 4. T. Bui-Thanh, M. Damodaran, K. Willcox, Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics. AIAA J. 42(8), 1505–1516 (2004) 5. K. Carlberg, C. Farhat, J. Cortial, D. Amsallem, The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242(0), 623–647 (2013) 6. S. Chaturantabut, D. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010) 7. A. Edelman, T.A. Arias, S.T. Smith, The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1999) 8. P.A. LeGresley, J.J. Alonso, Investigation of non-linear projection for POD based reduced order models for aerodynamics, in Paper 2001-0926, 39th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 2001 9. R. Pinnau, Model reduction via proper orthogonal decomposition, in Model Order Reduction: Theory, Research Aspects and Applications, ed. by W.H.A. Schilders, H.A. Van der Vorst, J. Rommes. Springer Series Mathematics in Industry, vol. 13 (Springer, Berlin, 2008), pp. 95– 109 10. Q. Rentmeesters, Algorithms for data fitting on some common homogeneous spaces, Ph.D. thesis, Université Catholique de Louvain, Louvain, Belgium, July 2013 11. R. Zimmermann, Gradient-enhanced surrogate modeling based on proper orthogonal decomposition. J. Comput. Appl. Math. 237(1), 403–418 (2013) 12. R. Zimmermann, Local Parametrization of Subspaces on Matrix Manifolds via Derivative Information, pp. 379–387 (Springer, Cham, 2016) 13. R. Zimmermann, A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric. SIAM J. Matrix Anal. Appl. 38(2), 322–342 (2017) 14. R. Zimmermann, A. Vendl, S. Görtz, Reduced-order modeling of steady flows subject to aerodynamic constraints. AIAA J. 52(2), 255–266 (2014)

A Spectral Element Reduced Basis Method in Parametric CFD Martin W. Hess and Gianluigi Rozza

Abstract We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14,259 degrees of freedom. The steady-state snapshot solutions define a reduced order space, which allows to accurately evaluate the steady-state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation (Karniadakis and Sherwin, Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, Oxford, 2005) in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

1 Introduction The use of spectral element methods in computational fluid dynamics [1] allows highly accurate computations by using high-order spectral element ansatz functions. Typically, an exponential error decay can be observed under p-refinement. See [2–6] for an introduction and overview of the applications. This work is concerned with the reduced basis method (RBM, [7]) of a channel flow, governed by the Navier-Stokes equations, and discretized with the spectral element method into 14,259 degrees of freedom. In particular, we are interested in computing the steady-state solutions for varying Reynolds number with a reduced order model, guaranteeing competitive computational performances.

M. W. Hess () · G. Rozza SISSA mathLab, International School for Advanced Studies, Trieste, Italy e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_64

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Section 2 introduces the governing equations and used fixed-point iteration algorithm. Section 3 introduces the spectral element discretization, while Sect. 4 describes the model reduction approach. Numerical results are provided in Sect. 5, while Sect. 6 summarizes and concludes the work by also providing new perspectives.

2 Problem Formulation Let Ω ∈ R2 be the computational domain. Incompressible, viscous fluid motion in a spatial domain Ω over a time interval (0, T ) is governed by the incompressible Navier-Stokes equations with velocity u, pressure p, kinematic viscosity ν and a body forcing f , (1)–(2): ∂u + u · ∇u = −∇p + νΔu + f, ∂t ∇ · u = 0.

(1) (2)

Boundary and initial conditions are prescribed as u=d

on ΓD × (0, T ),

(3)

∇u · n = g

on ΓN × (0, T ),

(4)

u = u0

in Ω × 0,

(5)

with d, g and u0 given and ∂Ω = ΓD ∪ΓN , ΓD ∩ΓN = ∅. The Reynolds number Re depends on the viscosity ν through the characteristic velocity U and characteristic length L via Re = UνL , [8]. In particular, we are interested in computing the steady states for varying viscosity ν, such that ∂u ∂t = 0. A solution u(ν1 ) for a parameter value ν1 , can be used as an initial guess for a fixed point iteration to obtain the steady state solution u(ν2 ) at a parameter value ν2 , provided that the solution u(ν) depends continuously on ν in the interval [ν1 , ν2 ].

2.1 Oseen-Iteration The Oseen-iteration is a secant modulus fixed-point iteration, which in general exhibits a linear rate of convergence [9]. Given a current iterate (or initial condition) uk , the linear system − νΔu + (uk · ∇)u + ∇p = f in Ω,

(6)

∇ · u = 0 in Ω,

(7)

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u=d

on ΓD ,

(8)

∇u · n = g

on ΓN ,

(9)

is solved for the next iterate uk+1 = u. A typical stopping criterion is that the relative change between iterates in the H 1 norm falls below a predefined tolerance. An initial solution u0 (ν0 ) is computed by time-advancement of (1)–(2) from zero initial conditions at a parameter value ν0 , and the whole parameter domain is then explored by using a continuation method with the Oseen-iteration.

3 Spectral Element Discretization The Navier-Stokes problem is discretized with the spectral element method. The spectral/hp element software framework used is Nektar++ in version 4.3.5, [10].1 The discretized system to solve in each step of the Oseen-iteration is given by (10) as ⎤ ⎡ ⎤ ⎤⎡ T fbnd B A −Dbnd vbnd ⎥ ⎢ ⎥ ⎢ ⎥⎢ 0 −Dint ⎦ ⎣ p ⎦ = ⎣ 0 ⎦ , ⎣ −Dbnd T C B˜ T −Dint vint fint ⎡

(10)

where vbnd and vint denote velocity degrees of freedom on the boundary and in the interior, respectively. Correspondingly, fbnd and fint denote forcing terms on the boundary and interior, respectively. The matrix A assembles the boundary-boundary coupling, B the boundary-interior coupling, B˜ the interior-boundary coupling and C assembles the interior-interior coupling of elemental velocity ansatz functions. In the case of a Stokes system, it holds that B = B˜ T , but this is not the case for the Oseen equation, since the linearization term (uk · ∇)u is present in (6). The matrices Dbnd and Dint assemble the pressure-velocity boundary and pressurevelocity interior contributions, respectively. The linear system (10) is assembled in local degrees of freedom, resulting in ˜ C, Dbnd and Dint , each block corresponding to a spectral block matrices A, B, B, element. In particular, this means that the system is singular in this form. To solve the system, the local degrees of freedom need to be gathered into the global degrees of freedom [1]. Since C contains the interior-interior contributions, it is invertible

1 See

www.nektar.info.

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and the system can be statically condensed into ⎡

⎡

⎤ T − DT A − BC −1 B˜ T BC −1 Dint 0 bnd ⎢ ⎥ T 0⎦ ⎣ Dint C −1 B˜ T − Dbnd −Dint C −1 Dint T B˜ T −Dint C

⎤ vbnd ⎢ ⎥ ⎣ p ⎦ vint

⎤ fbnd − BC −1 fint ⎥ ⎢ = ⎣ Dint C −1 fint ⎦ . (11) fint ⎡

By taking the top left 2 × 2 block and reordering the degrees of freedom such that the mean pressure mode of each element is inserted into the corresponding block of Aˆ results in      Aˆ Bˆ b fˆbnd = , (12) pˆ Cˆ Dˆ fˆp where Dˆ is invertible, such that a second level of static condensation can be employed. We have: 

Aˆ − Bˆ Dˆ −1 Cˆ 0 Cˆ Dˆ

    b fˆbnd − Bˆ Dˆ −1 fˆp . = pˆ fˆp

(13)

When the vector b is computed, which contains the velocity boundary degrees of freedom and the mean pressure modes, the remaining solution components are computed by reverting the steps of the static condensations [1]. The main computational effort lies in solving the final system (14) ˆ = fˆbnd − Bˆ Dˆ −1 fˆp . (Aˆ − Bˆ Dˆ −1 C)b

(14)

Additionally, the matrices C and Dˆ need to be inverted, which due to the elemental block structure requires inverting submatrices in the size of the degrees of freedom per element for each submatrix.

4 Reduced Order Modelling The reduced order model (ROM) aims to represent the full order solution accurately in the parameter domain of interest. Two ingredients are essential to RB modelling, a projection onto a low order space of snapshot solutions and an offline-online decomposition for computational efficiency, [11]. A set of snapshots is generated by solving (14) over a coarse discretization of the parameter domain and used to define

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a projection space U of size N. The proper orthogonal decomposition computes a singular value decomposition of the snapshot solutions to 99.9% of the most dominant modes [7], which defines the projection matrix U ∈ RNδ ×N to project system (14) and obtain the reduced order solution bN .

4.1 Offline-Online Decomposition The offline-online decomposition [7] allows fast input-output evaluations independent of the original model size Nδ . It is a crucial part of an efficient reduced order model but since the static condensation includes the inversion of the parameterdependent matrix C, an intermediate projection is introduced. The reduced order model considers the top left 2 × 2 block of (11), i.e., one level of static condensation [1]. During the offline phase, full-order solutions have been computed over the parameter domain of interest, which now serve as a projection space to define the reduced order setting. This projection space incorporates the transformation of local velocity boundary degrees of freedom to global velocity boundary degrees of freedom and the reordering of mean pressure degrees of freedom. The projection space then takes the form U = P MV with a permutation matrix P to reorder the degrees of freedom and a transformation M from local to global degrees of freedom. The collected offline data V contain the gathered velocity and mean pressure modes as well as interior pressure modes. The projected system is then of the form   T − DT BC −1 Dint A − BC −1 B˜ T T bnd AN = U U, (15) T Dint C −1 B˜ T − Dbnd −Dint C −1 Dint and upon its solution, the interior velocity dofs can be computed by resubstituting into (11) at the reduced order level. To achieve fast reduced order solves, the offlineonline decomposition expands (15) in the parameter of interest and computes the parameter independent projections offline to be stored as small-sized matrices of the order N × N. Since in an Oseen-iteration each matrix is dependent on the previous iterate, the submatrices corresponding to each basis function is assembled and then formed online using the reduced basis coordinate representation of the current iterate. This is analogous to reduced order assembly of the nonlinear term in the Navier-Stokes case, [11].

5 Model and Numerical Results We consider a channel flow in the domain considered in Fig. 1, similar to the model considered in [12]. The rectangular domain Ω(x, y) = [0, 36] × [0, 6] is decomposed into 32 spectral elements. The spectral element expansion uses modal

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Fig. 1 Rectangular domain for the channel flow, shown is the 4 × 8 spectral element grid

Fig. 2 Full order, steady-state solution for ν = 0.0075. Top is the velocity in x-direction, below is the velocity in y-direction

Legendre polynomials of order p = 12 in the velocity. The pressure ansatz space is chosen of order p − 2 to fulfill the inf-sup stability condition, [13, 14]. The inflow is defined for y ∈ [2.5, 3.5] as ux (0, y) = (y − 2.5)(3.5 − y). At x = 36 is the outflow boundary, everywhere else are zero velocity walls. Note that the velocity boundary degrees of freedom are along the boundaries of the spectral elements and not only the domain boundary, resulting in 3072 local degrees of freedom for this problem. This is a simplified model of a contraction-expansion channel [8], where flow occurs though a narrowing of variable width. Variations in the width have been moved to variations in the Reynolds number and only the section after the narrowing comprises the computational domain. The relation to the Reynolds number is established with U = 14 as the maximum inflow velocity and L = 1 as 1 the width of the narrowing as Re = 4ν . Consider a parametric variation in the viscosity ν, ranging from ν = 0.0075 to ν = 0.0025, which corresponds to Reynolds numbers between 33 and 100. The solution for ν = 0.0075 is shown in Fig. 2. It is slightly unsymmetrical, which marks the onset of the Coanda effect [15, 16], which is a known phenomenon characterized as a ‘wall-hugging’ effect occurring at these Reynolds numbers. The solution for ν = 0.0025 is shown in Fig. 3. Here, the Coanda effect is fully developed as the flow orients itself along the boundaries. Using model reduction with the form (11), which allows the offline-online decomposition or using form (14), which has the lowest full-order system size, resulted in similar computational results. Shown in

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Relative velocity approximation error

Fig. 3 Full order, steady-state solution for ν = 0.0025. Top is the velocity in x-direction, below is the velocity in y-direction

10–2

10–5

10–8

30

40

50

60

Re

70

80

90

100

Fig. 4 Relative error in the velocity over the parameter domain

Fig. 4 is the relative H01 (Ω) error in the velocity between the full order and reduced order model. While the full-order solves were computed with Nektar++, the reduced-order computations were done in a separate python code. To compare computational gains, compute times between a full order solve and a reduced order solve both implemented in python are taken. The compute times reduce by a factor of 50, i.e., for a single iteration step from about 40 s to under 1 s. Current work also aims to extend the software to make it available as a SEM-ROM software framework within the AROMA-CFD project (see Acknowledgment) as ITHACA-SEM.

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6 Conclusion and Outlook It has been shown that the reduced basis technique generates accurate reduced order models of small size for channel flow discretized with spectral elements up to a Reynolds number of 100. The use of basis functions obtained by the spectral element method suggests a potential important synergy between high-order and reduced basis methods, see also [6]. Due to the multilevel static condensation used here, particular care must be taken to achieve an offline-online decomposition. The domain decomposition into spectral elements shows a resemblance to reduced basis element methods (RBEM), [17, 18]. A comparison of both approaches could be the subject of further investigation. Acknowledgements This work was supported by European Union Funding for Research and Innovation through the European Research Council (project H2020 ERC CoG 2015 AROMA-CFD project 681447, P.I. Prof. G. Rozza).

References 1. G. Karniadakis, S. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. (Oxford University Press, Oxford, 2005) 2. C. Canuto, M.Y. Hussaini, A. Quarteroni, Th.A. Zhang, Spectral Methods Fundamentals in Single Domains (Springer – Scientific Computation, New York, 2006) 3. C. Canuto, M.Y. Hussaini, A. Quarteroni, Th.A. Zhang, Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics (Springer – Scientific Computation, New York, 2007) 4. A.T. Patera, A spectral element method for fluid dynamics; laminar flow in a channel expansion. J. Comput. Phys. 54(3), 468–488 (1984) 5. H. Herrero, Y. Maday, F. Pla, RB (Reduced Basis) for RB (Rayleigh–Bénard). Comput. Methods Appl. Mech. Eng. 261–262, 132–141 (2013) 6. L. Fick, Y. Maday, A. Patera, T. Taddei, A reduced basis technique for long-time unsteady turbulent flows. J. Comput. Phys. (submitted). arxiv: https://arxiv.org/pdf/1710.03569.pdf 7. J.S. Hesthaven, G. Rozza, B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics (Springer, Berlin, 2016) 8. G. Pitton, A. Quaini, G. Rozza, Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: applications to Coanda effect in cardiology. J. Comput. Phys. 344, 534–557 (2017) 9. M. Burger, Numerical Methods for Incompressible Flow. Lecture Notes (UCLA, Los Angeles, 2010) 10. C.D. Cantwell, D. Moxey, A. Comerford, A. Bolis, G. Rocco, G. Mengaldo, D. de Grazia, S. Yakovlev, J.-E. Lombard, D. Ekelschot, B. Jordi, H. Xu, Y. Mohamied, C. Eskilsson, B. Nelson, P. Vos, C. Biotto, R.M. Kirby, S.J. Sherwin, Nektar++: an open-source spectral/hp element framework. Comput. Phys. Commun. 192, 205–219 (2015) 11. T. Lassila, A. Manzoni, A. Quarteroni, G. Rozza, Model Order Reduction in Fluid Dynamics: Challenges and Perspectives, vol. 9, ed. by A. Quarteroni, G. Rozza. Reduced Order Methods for Modelling and Computational Reduction (Springer International Publishing, MS&A, Cham, 2014), pp. 235–273 12. G. Pitton, G. Rozza, On the application of reduced basis methods to bifurcation problems in incompressible fluid dynamics. J. Sci. Comput. 73, 157 (2017)

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13. D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics (Springer, Berlin, 2013) 14. A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations (Springer, Berlin, 1994) 15. R. Wille, H. Fernholz, Report on the first European Mechanics Colloquium, on the Coanda effect. J. Fluid Mech. 23(4), 801–819 (1965) ˇ c, A computational study on the generation of the Coanda 16. A. Quaini, R. Glowinski, S. Cani´ effect in a mock heart chamber. RIMS Kôkyûroku series, No. 2009-4 (2016) 17. Y. Maday, E.M. Ronquist, A reduced-basis element method. C. R. Math. 335(2), 195–200 (2002) 18. A.E. Lovgren, Y. Maday, E.M. Ronquist, A reduced basis element method for the steady stokes problem. ESAIM: Math. Model. Numer. Anal. 40(3), 529–552 (2006)

POD for Optimal Control of the Cahn-Hilliard System Using Spatially Adapted Snapshots Carmen Gräßle, Michael Hinze, and Nicolas Scharmacher

Abstract The present work considers the optimal control of a convective CahnHilliard system, where the control enters through the velocity in the transport term. We prove the existence of a solution to the considered optimal control problem. For an efficient numerical solution, the expensive high-dimensional PDE systems are replaced by reduced-order models utilizing proper orthogonal decomposition (POD-ROM). The POD modes are computed from snapshots which are solutions of the governing equations which are discretized utilizing adaptive finite elements. The numerical tests show that the use of POD-ROM combined with spatially adapted snapshots leads to large speedup factors compared with a high-fidelity finite element optimization.

1 Introduction The optimal control of two-phase systems has been studied in various papers, see e.g. [7, 8] and [11]. In this paper, we concentrate our investigations on the diffuse interface approach, where we assume the existence of interfacial regions of small width between the phases. This has the advantage that topology changes like droplet collision or coalescence can be handled in a natural way. Many degrees of freedom are needed in the interfacial regions in order to resemble the steep gradients well, whereas in the pure phases a small number of degrees of freedom is sufficient. Thus, in order to make numerical computations feasible, we utilize adaptive finite element methods. However, the optimization of a phase field model is still a costly issue, since a sequence of large-scale systems has to be solved repeatedly. For this reason, we replace the high-dimensional systems by low-dimensional POD approximations. This has been done in e.g. [15] for uniformly discretized snapshots.

C. Gräßle () · M. Hinze · N. Scharmacher Universität Hamburg, Hamburg, Germany e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_65

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We perform POD based optimal control using spatially adapted snapshots. The combination of POD with adaptive finite elements has been investigated for timedependent problems in [14] and [6].

2 Convective Cahn-Hilliard System We consider the Cahn-Hilliard system which was introduced in [3] as a model for phase transitions in binary systems. In a bounded and open domain Ω ⊂ Rd , d ∈ {2, 3}, with Lipschitz boundary ∂Ω, we assume two substances A and B to be given. In order to describe the spatial distribution over time I = (0, T ] with fixed end time T > 0, a phase field variable ϕ is introduced which fulfills ϕ(t, x) = +1 in the pure A-phase and ϕ(t, x) = −1 in the pure B-phase. Values of ϕ between −1 and +1 represent the interfacial area between the two substances. Introducing the chemical potential μ, the Cahn-Hilliard equations can be written as a coupled system of second-order in space ⎧ ⎪ ⎪ ϕt + v · ∇ϕ − bΔμ = ⎪ ⎨ −σ εΔϕ + σε F  (ϕ) = ⎪ ∂n ϕ = ∂n μ = ⎪ ⎪ ⎩ ϕ(0, ·) =

0 μ 0 ϕ0

in I × Ω, in I × Ω, on I × ∂Ω, in Ω.

(1)

By b > 0 we denote a constant mobility, σ > 0 describes the surface tension and ε > 0 is a parameter which is related to the interface width. For the free energy F , we consider the smooth polynomial free energy (see e.g. [5]) F (ϕ) =

1 (1 − ϕ 2 )2 . 4

A possible flow of the mixture at a given velocity field v is modeled in (1) by the transport term which, in the context of multiphase flow, represents the coupling to the Navier-Stokes system, see e.g. [9] and [2]. We use the following notations and assumptions: 1 (Ω) the space of functions in H 1 (Ω) with zero Notations 2.1 We denote by H(0) mean value and by L2σ (Ω)d = {f ∈ L2 (Ω)d : divf = 0, f · nΩ |∂Ω = 0} the space of solenoidal vector fields, for which we refer to [13] for details about welldefinedness. We use as the solution space for the phase field variable W (0, T ) = 1 (Ω)) : f ∈ L2 (0, T ; H −1 (Ω))}. {f ∈ L2 (0, T ; H(0) t (0)

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Assumptions 2.2 1 (Ω) fulfills E = E(ϕ ) < ∞ with Ginzburgi) The initial phase field ϕ0 ∈ H(0) 0 0 Landau free energy

 E(ϕ) = Ω

σε σ |∇ϕ|2 + F (ϕ)dx. 2 ε

ii) The velocity v fulfills v ∈ L∞ (0, T ; L2σ (Ω)d ) ∩ L2 (0, T ; H 1(Ω)d ). Remark 2.3 It is shown in [1, Theorem 4.1.1] that there exists a unique solution (ϕ, μ) to (1) with ϕ ∈ W (0, T ) ∩ L2 (0, T ; H 2(Ω)), μ ∈ L2 (0, T ; H 1 (Ω)). This solution satisfies   (2) ≤ C E0 + v2L2 (0,T ;L2 (Ω)d ) ϕ2L2 (0,T ;H 2 (Ω)) + ϕt 2 2 −1 L (0,T ;H(0) (Ω))

where C is independent of v and ϕ0 .

3 Optimal Control of Cahn-Hilliard We investigate the minimization of the quadratic objective functional J (ϕ, u) =

β2 γ β1 ϕ − ϕd 2L2 (0,T ;L2 (Ω)) + ϕ − ϕT 2L2 (Ω) + u2U 2 2 2

where β1 , β2 ≥ 0 are given constants, ϕd ∈ L2 (0, T ; L2 (Ω)) is the desired phase field, ϕT ∈ L2 (Ω) is the target phase pattern at final time, γ > 0 is the penalty parameter and u ∈ U = L2 (0, T ; Rm ), with m ∈ N, denotes the control variable which is a time-dependent variable and in particular independent of the current spatial discretization. The goal of the optimal control problem is to steer a given initial phase distribution ϕ0 to a given desired phase pattern. This problem can also be interpreted as an optimal control of a free boundary which is encoded through the phase field variable. We consider distributed control which enters through the transport term: ⎧ ⎪ =0 ⎪ ⎪ ϕt + (Bu) · ∇ϕ σ− bΔμ ⎨  −σ εΔϕ + ε F (ϕ) = μ ⎪ ∂n ϕ = ∂n μ = 0 ⎪ ⎪ ⎩ ϕ(0, ·) = ϕ0

in I × Ω, in I × Ω, on I × ∂Ω, in Ω.

(3)

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The by B : U → L2 (0, T ; H 1(Ω)d ), (Bu)(t) = m control operator B is defined 2 d 1 d i=1 ui (t)χi where χi ∈ Lσ (Ω) ∩ H (Ω) , 1 ≤ i ≤ m, represent given control shape functions. The admissible set of controls is Uad = {u ∈ U | ua (t) ≤ u(t) ≤ ub (t) in Rm a.e. in [0, T ]} with ua , ub ∈ L∞ (0, T ; Rm ), ua (t) ≤ ub (t) almost everywhere in [0, T ]. The inequalities between vectors are understood componentwise. Then, the optimal control problem can be expressed as min Jˆ(u) = J (ϕ(u), u) u

s.t.

(ϕ(u), u) satisfies (3)

and

u ∈ Uad .

(4)

Theorem 1 (Existence of an Optimal Control) Problem (4) admits a solution u¯ ∈ Uad . Proof The infimum infu∈Uad Jˆ(u) exists due to Jˆ ≥ 0 and Uad = ∅. Let {un }n∈N ⊂ Uad be a minimizing sequence and {ϕn }n∈N the corresponding sequence of states ϕn = ϕ(un ). Since Uad is closed, convex and bounded in L2 (0, T ; Rm ) ⊃ L∞ (0, T ; Rm ), we can extract a subsequence (denoted by the same name), which converges weakly to some u¯ ∈ Uad . Weak convergence Bun 2 B u¯ in L2 (0, T ; H 1 (Ω)d ) follows from the linearity and boundedness of B. Due to the energy estimate (2) there exists a constant M > 0 such that for all n ∈ N we have ϕn 2L2 (0,T ;H 2 (Ω)) + ϕn,t 2 2

−1 (Ω)) L (0,T ;H(0)

≤ M.

Since W (0, T ) ∩ L2 (0, T ; H 2 (Ω)) is reflexive, there exists another subsequence (denoted by the same name) that converges weakly to some ϕ¯ ∈ W (0, T ) ∩ L2 (0, T ; H 2 (Ω)). It remains to show, that the pair (ϕ, ¯ u) ¯ is admissible, i.e. ϕ¯ = ϕ(u). ¯ While passing to the limit in the weak formulation is clear for the linear terms, the nonlinear ones require further investigation. Since W (0, T ) ∩ L2 (0, T ; H 2(Ω)) is compactly embedded in L2 (0, T ; H 1(Ω)) (see [12, Sect. 8, Corr. 4]), the sequence {ϕn }n∈N converges strongly to ϕ¯ in L2 (0, T ; H 1 (Ω)). For the control term we have for v ∈ H 1 (Ω) the splitting 

T 0

(Bun · ∇ϕn − B u¯ · ∇ ϕ, ¯ v)L2 (Ω) dt =  0

T

 (Bun · ∇(ϕn − ϕ), ¯ v)L2 (Ω) dt +

0

T

(B(un − u) ¯ · ∇ ϕ, ¯ v)L2 (Ω) dt.

Due to ∇ ϕ¯ ∈ L2 (0, T ; H 1 (Ω)d ), the product v∇ ϕ¯ ∈ L2 (0, T ; L2 (Ω)d ) gives rise to a continuous linear functional on L2 (0, T ; H 1(Ω)d ). Hence, the right term

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vanishes for n → ∞ by definition of weak convergence. For the left term we estimate B B B T B B B (Bun · ∇(ϕn − ϕ), ¯ v)L2 (Ω) dt B ≤ B B 0 B Bun L2 (0,T ;H 1 (Ω)d ) ϕn − ϕ ¯ L2 (0,T ;H 1 (Ω)) vH 1 (Ω) , which also vanishes for n → ∞. For the nonlinearity F  we infer from B B |F  (ϕ) − F  (ψ)| ≤ C(ϕ 2 + ψ 2 ) Bϕ − ψ B for all ϕ, ψ ∈ R and some C > 0 the estimate B B B T B B B (F  (ϕn ) − F  (ϕ), ¯ v)L2 (Ω) dt B ≤ B B 0 B ¯ L2 (0,T ;H 1 (Ω)) vH 1 (Ω) , C(ϕn2 L2 (0,T ;L2 (Ω)) + ϕ¯ 2 L2 (0,T ;L2 (Ω)) )ϕn − ϕ which gives the desired convergence due to L∞ (0, T ; H 1 (Ω)) ⊂ L4 (0, T ; L4 (Ω)). Finally, the lower semi-continuity of J yields J (ϕ, ¯ u) ¯ = inf Jˆ(u). u∈Uad

& %

Problem (4) is a non-convex programming problem, so that different minima might exist. Numerical solution methods will converge in a local minimum which is close to the initial point. In order to compute a locally optimal solution to (4), we consider the first-order necessary optimality condition given by the variational inequality Jˆ (u), ¯ u − u ¯ U  ,U ≥ 0 ∀u ∈ Uad .

(5)

Following the standard adjoint techniques, we derive that (5) is equivalent to 

T 0

  m 

γ u¯ i (t) + (χi (x) · ∇ϕ(t, x))p(t, ¯ x)dx (ui (t) − u¯ i (t))dt ≥ 0 i=1

(6)

Ω

for all u ∈ Uad where the function p¯ is a solution to the adjoint equations ⎧ ⎪ −pt − σ εΔq + σε F  (ϕ)q ¯ − Bu · ∇p = −β1 (ϕ¯ − ϕd ) in I × Ω, ⎪ ⎪ ⎨ −q − bΔp = 0 in I × Ω, ⎪ on I × ∂Ω, ∂n p = ∂n q = 0 ⎪ ⎪ ⎩ p(T , ·) = −β2 (ϕ(T ¯ , ·) − ϕT ) in Ω. (7) The variable ϕ¯ in (7) denotes the solution to (3) associated with an optimal control u. ¯

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4 POD-ROM Using Spatially Adapted Snapshots The optimal control problem (4) is discretized by adaptive finite elements and solved by a standard projected gradient method with an Armijo line search rule. In order to replace the resulting high-dimensional PDEs by low-dimensional approximations, we make use of POD-ROM, see e.g. [10] or [16]. The nonlinearity is treated using DEIM, cf [4]. In order to combine POD-ROM with spatially adapted snapshots, we follow the ideas in [14] and [6].

5 Numerical Results We consider the unit square Ω = (0, 1)×(0, 1), the end time T = 0.0125 and utilize a uniform time grid with time step size Δt = 2.5 × 10−5. The mobility is b = 2.5 × 10−5 , the surface tension is σ = 25.98 and the interface parameter is set to ε = 0.02. In the cost functional we use γ = 0.0001, β1 = 20 and β2 = 20. We use m = 1 control shape function given by χ(x) = (sin(πx1 )cos(πx2 ), −sin(πx2 )cos(πx1 ))T . The desired state is shown in Fig. 1. The initial state ϕ0 coincides with ϕd (0). In order to fulfill the Courant-Friedrichs-Lewy (CFL) condition, we impose the control constraints ua = 0, ub = 50 and demand hmin > 0.00177. The optimization is initialized with an input control u = 0 ∈ Uad . We compute the POD basis with respect to the L2 (Ω)-inner product for the snapshot ensemble formed by the desired phase field ϕd , which is discretized using adaptive finite elements. Figure 2 shows the finite element solution and the POD solution for the phase field using  = 10 and  = 20 POD modes, respectively. It turns out that a large number of POD modes is needed in order to smoothen out oscillations due to the convection.

Fig. 1 Desired phase field at t0 , t250 , t500 with adaptive meshes

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Fig. 2 Finite element (top) and POD-DEIM optimal solution with  = 10 (middle) and  = 20 (bottom) of the phase field ϕ at t = t0 , t250 , t500 with adaptive meshes

Table 1 (left) summarizes the iteration history for the finite element projected gradient method where we used the stopping criterion Jˆ (uk )Uh < 0.01 · Jˆ (u0 )Uh + 0.01. Table 1 (right) tabulates the POD-ROM optimization. Note that the value of the POD cost functional Jˆ (uk ) stagnates due to the POD error. The value of the full-order cost functional at the POD solution is Jˆ(u¯ POD ) = 7.31 × 10−4 . If  = 20 POD modes are used, the relative L2 (0, T ; L2 (Ω))-error between the finite element and the POD solution for the phase field is errϕ = 7.19 × 10−3 ; for POD-DEIM it is errϕ = 7.38 × 10−3 . In Table 2 the computational times for the uniform FE, adaptive FE, POD and POD-DEIM optimization are listed. The offline costs for POD when using spatially adapted snapshots are as follows: the interpolation of the snapshots takes 212 s, the POD basis computation costs 40 s and the computations for DEIM take 30 s. In comparison, the use of uniformly discretized snapshots leads to the computational time of 243 s for POD basis computation and 193 s for the DEIM computations.

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Table 1 Iteration history finite element optimization (left) and POD optimization (right) with  = 20 k

Jˆ(uk )

Jˆ (uk )Uh

sk

k

Jˆ (uk )

Jˆ (uk )Uh

sk

0 1 2 3 4 5

8.61 × 100 6.48 × 10−1 1.90 × 10−2 3.82 × 10−3 1.18 × 10−3 6.80 × 10−4

2.85 × 100 2.32 × 100 4.56 × 10−1 1.93 × 10−1 8.45 × 10−2 3.67 × 10−2

1.0 0.25 0.25 0.25 0.25

0 1 2 3 4 5

8.77 × 100 7.98 × 10−1 5.79 × 10−2 5.02 × 10−2 4.76 × 10−2 4.76 × 10−2

2.81 × 100 2.41 × 100 3.67 × 10−1 1.63 × 10−1 7.45 × 10−2 3.48 × 10−2

1.0 0.25 0.25 0.25 0.25

The Armijo step size is denoted by sk Table 2 Computational times for the FE, POD and POD-DEIM optimization Optimization → solve each state eq. → solve each adjoint eq.

Uniform FE 36, 868 s 1660 s 761 s

Adaptive FE 5805 s 348 s 121 s

POD 675 s 42 s 16 s

POD-DEIM 0.3 s 0.02 s 0.01 s

6 Outlook In future work, we intend to embed the optimization of Cahn-Hilliard in a trust-region framework in order to adapt the POD model accuracy within the optimization. We further want to consider a relaxed double-obstacle free energy which is a smooth approximation of the non-smooth double-obstacle free energy. We expect that more POD modes are needed in this case to get similar accuracy results as in the case of a polynomial free energy. Moreover, we intend to couple the smoothness of the model to the trust-region fidelity. Acknowledgements We like to thank Christian Kahle for providing many libraries which we could use for the coding. The first author gratefully acknowledges the financial support by the DFG through the priority program SPP 1962. The third author gratefully acknowledges the financial support by the DFG through the Collaborative Research Center SFB/TRR 181.

References 1. H. Abels, Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids, vol. 36 (Max-Planck Institute for Mathematics in the Sciences, Leipzig, 2007) 2. H. Abels, H. Garcke, G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Mod. Methods Appl. Sci. 22(3), 1150013 (2012) 3. J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958) 4. S. Chaturantabut, D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)

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5. C.M. Elliott, S. Zheng, On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339– 357 (1986) 6. C. Gräßle, M. Hinze, POD reduced-order modeling for evolution equations utilizing arbitrary finite element discretizations. Adv. Comput. Math. 1–38 (2018) 7. F. Haußer, S. Rasche, A. Voigt, The influence of electric fields on nanostructures – simulation and control. Math. Comput. Simul. 80, 1449–1457 (2010) 8. M. Hintermüller, D. Wegner, Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50(1), 388–418 (2012) 9. P.C. Hohenberg, B.I. Halperin, Theory of dynamic critical phenomena. Rev. Mod. Phys. 49(3), 435–479 (1977) 10. P. Holmes, J.L. Lumley, G. Berkooz, C.W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, vol. 2. Cambridge Monographs on Mechanics (Cambridge University Press, Cambridge, 2012) 11. E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in three dimension. SIAM J. Control Optim. 53(3), 1654–1680 (2015) 12. J. Simon, Compact sets in the space Lp (0, T ; B). Ann. Math. Pura Appl. (IV) 146, 65–96 (1987) 13. H. Sohr, The Navier-Stokes Equations. Birkhäuser Advanced Texts: Basler Lehrbücher (Birkhäuser Verlag, Basel, 2001) 14. S. Ullmann, M. Rotkvic, J. Lang, POD-Galerkin reduced-order modeling with adaptive finite element snapshots. J. Comput. Phys. 325, 244–258 (2016) 15. S. Volkwein, Optimal control of a phase-field model using proper orthogonal decomposition. ZAMM-J. Appl. Math. Mech. 81(2), 83–97 (2001) 16. S. Volkwein, Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling. Lecture Notes (University of Konstanz, Konstanz, 2013)

Part XVIII

Recent Advances on Polyhedral Discretizations

New Strategies for the Simulation of the Flow in Three Dimensional Poro-Fractured Media Stefano Berrone, Andrea Borio, Sandra Pieraccini, and Stefano Scialò

Abstract Two novel approaches are presented for dealing with three dimensional flow simulations in porous media with fractures: one method is based on the minimization of a cost functional to enforce matching conditions at the interfaces, thus allowing for non conforming grids at the interfaces; the other, instead, takes advantage of the new Virtual Elements to easily build conforming polygonal grids at the fracture-porous matrix interfaces. Both methods are designed to minimize the effort in mesh generation process for problems characterized by complex geometries. The methods are described in their simplest fashion in order to keep the notation as compact and simple as possible.

1 Introduction One of the main challenges in the simulation of the flow in porous media with fractures is the generation of a good quality mesh of the computational domain, which typically consists of a large and intricate network of fractures immersed in a polyhedral porous medium. Matching conditions at fracture-matrix and fracturefracture interfaces need to be imposed. This requires some level of conformity of the meshes at the interfaces when standard discretization approaches are used. This work presents two different approaches designed in order to tackle mesh generation difficulties from two different standpoints: a first method is based on the minimization of a cost functional to enforce the matching conditions, thus relaxing the requirement of mesh conformity at the interfaces. The other approach instead enforces matching conditions in a standard way, but relies on the new Virtual Element Method (VEM, see [1, 2]) to easily build conforming polygonal meshes at

S. Berrone · A. Borio · S. Pieraccini · S. Scialò () Politecnico di Torino, Turin, Italy e-mail: [email protected]; [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_66

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the interfaces. The two methods are described in their simplest formulation in order to keep the notation compact and highlight their key aspects.

2 The Model Problem  4 4 Let us consider the domain D = F i=1,2 Di as shown in Fig. 1, representing two blocks of a porous medium Di , i = 1, 2 separated by a fracture that is modeled according to the DFM model as a two dimensional planar interface F . The following problem is given in D: ⎧   ⎪ ⎪ −∇ · Ki ∇Hi = fi ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎨−∇π · KF ∇π HF = fF − Q H1 = H2 = HF ⎪   ⎪ ⎪ ⎪ Q · ni = K ∇H i i i ⎪ ⎪ ⎪ ⎪ ⎩H = 0

in Di , i = 1, 2 in F on F

(1)

i = 1, 2 on ∂D

where H is the hydraulic head in D (the sum of pressure head and elevation), Hi , i = 1, 2 its restriction to Di , HF the restriction to F , whereas Ki > 0 is the hydraulic transmissivity of the porous medium in Di , and KF is the transmissivity of the fracture F along the fracture plane π. The operator ∇(·) denotes, as usual, the gradient in the three dimensional space, whereas ∇π (·) is the gradient in a twodimensional reference system on F . The unit normal vector to F , pointing outward from Di is ni , i = 1, 2 and Qi , i = 1, 2 is the co-normal derivative of Hi on F , representing the flux entering in Di through F , with Q = Q1 + Q2 . The solution of problem (1) is representative of the hydraulic head distribution in a porous medium with fractures according to the DFM model, when the hydraulic transmissivity of the fracture in the direction normal to the fracture plane is higher than KF and Ki , i = 1, 2 (conductive fracture case). When this is not the case, different conditions at the interface between the porous matrix and the fracture should be considered, as Fig. 1 Computational domain

D D2 F D1 O

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discussed in [12]. The present work, however, is limited to the case of conductive fractures. Let us introduce the following functional spaces, for i = 1, 2:   Vi = v ∈ H10 (Di ) : v|F ∈ H10 (F ) , VF = H10 (F )

and W = H−1 (F )

then it is possible to write problem (1) in weak formulation: find (H1 , H2 , HF ) ∈ V1 × V2 × VF and Q1 , Q2 ∈ W, such that ⎧ H    G ⎪ ⎪ Ki ∇Hi , ∇v Di = fi , v Di + Qi , v|F F , ⎨     KF ∇π HF , ∇π v F = fF , v F − Q, vF , ⎪ H ⎪G ⎩ Hi|F − HF , λ F = 0

∀v ∈ Vi , i = 1, 2 ∀v ∈ VF

(2)

∀λ ∈ W, i = 1, 2

or, equivalently,  introducing the continuity  condition in the definition of the new space V = v ∈ H10 (D) : v|F ∈ H10 (F ) as: find H ∈ V, Hi = H|Di , i = 1, 2, HF = H|F such that: ⎧ ⎨K∇H, ∇v  = f, v  + GQ, v H , |F F D   D  ⎩ KF ∇π HF , ∇π v = fF , v − Q, vF , F F

∀v ∈ V ∀v ∈ VF

Two different approaches are here considered for the numerical resolution of (2): a first approach is based on the minimization of a cost functional to enforce the matching conditions at the interfaces without any requirement on mesh conformity, whereas the second approach uses the VEM and its robustness in dealing with polygonal elements also of low quality, to easily build conforming meshes at the interfaces.

3 The Optimization Approach The resolution of problem (2) can be seen as the constrained minimization of a properly defined cost functional. A mesh suitable for Finite Elements is introduced in each of the matrix blocks and on the fracture. These meshes can be generated independently from each other, since no conformity is required at the common interface F . For i = 1, 2, let Tδ,i be the triangulation of Di , δ being the mesh parameter, and Vδ,i ⊂ Vi a finite-element space associated to Tδ,i and spanned by basis functions {φi,j }j =1,...,Ni . Similarly Tδ,F is the triangulation on F and Vδ,F ⊂ VF the associated discrete space spanned by functions {ϕF,j }j =1,...,NF . Ni hi,j φi,j , Then hi is the discrete counterpart of Hi , i = 1, 2, given by hi = j =1 and, similarly, the discrete functions representing the fluxes Qi , i = 1, 2, on F

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NiF  F are defined as qi = j =1 qi,j (φi,j )|F , and hF = N j =1 hF,j ϕF,j , is the discrete counterpart of HF . The quadratic functional J (h) is then introduced: J :=

2

hi (qi ) − hF (q)2F

(3)

i=1

measuring the L2 (F )-norm of the error in the fulfillment of the matching condition across the interface. The discrete solution to problem (2) is then obtained solving the following PDE-constrained optimization problem: min J (h) such that, for i = 1, 2       Ki ∇hi , ∇φi,j D = f, φi,j D + qi , φi,j F , j = 1, . . . , Ni , i

i

and       KF ∇π hF , ∇π ϕF,j F = fF , ϕF,j F − q1 + q2 , ϕF,j F , j = 1, . . . , NF It is to remark that here the method is presented in a very simple domain. The generalization to more complex domains accounting for multiple intersecting fractures immersed in a polyhedral domain can be found in [11], where further details can be found on the resolution of the discrete problem when the Boundary Element Method (BEM) is used for discretization of the matrix blocks and FE-based discretizations are used on the fracture planes. As mentioned the advantages of this approach lie in the possibility of relaxing the conformity of the mesh at the interfaces in order to enforce the matching conditions. Not only matrix-fracture interfaces can be handled, but also fracturefracture matching conditions can be imposed via the same approach (see [6–10]). This is of paramount importance when dealing with intricate geometries.

4 The VEM-Based Approach The second proposed strategy relies on the flexibility and robustness of the VEM in handling polygonal meshes also with elongated elements. The BEM will be used, coupled to the VEM in order to solve the problem in the three dimensional domains. As a consequence it is only needed to introduce a discrete setting on the boundary ∂ the mesh on ∂D , of Di , ∂Di , i = 1, 2 and on the fracture F . Let us denote by Tδ,i i for i = 1, 2, and by Tδ,F the mesh on F . In this case the mesh at the interface is ∂ on F , belonging to the perfectly matching, which means that the elements in Tδ,i ∂ set denoted by (Tδ,i )|F , i = 1, 2 coincide with the elements in Tδ,F .

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The discrete VEM space is introduced on Tδ,F , defined as:   Vδ,F = v ∈ VF : v|E ∈ VE δ,F , ∀E ∈ Tδ,F being, for each element E ∈ Tδ,F 1 VE δ,F := {v ∈ H (E) : Δv ∈ Pk (E), v ∈ Pk (e) ∀e ⊂ ∂E,     v|∂E ∈ C0 (∂E), v, p E = Πk∇ v, p ∀p ∈ Pk (E)/Pk−2 (E)} E

where Pk (ω) is the space of polynomials of maximum order k on ω and the oblique projector Πk∇ : H1 (E) → Pk (E) is defined as:     = 0 , ∀p ∈ Pk (E) ∇ v − Πk∇ v , ∇p 

Πk∇ v, 1

E

 E

= (v, 1)E .

The following set of Degrees of Freedom (DOFs) will be used, on each element E ∈ Tδ,F to identify a function v ∈ Vδ,F : i) the nE values of v at the nE vertexes of E; ii) the k − 1 values of v at Gauss-Lobatto internal nodes on each edge of E; iii) the 12 k(k − 1) values of the moments of v with respect to polynomials in Pk−2 .  F Each function v ∈ Vδ,F can be expressed as: v = N j =1 vj ϕj , being > = Vδ,F = span ϕj , j = 1, . . . , NF . Further details on Virtual Elements can be found in the already mentioned references [1, 2]. Let it be k ≥ 2, and let V∂δ,i be a finite dimensional function space on ∂Di ∂ , for i = 1, 2, defined as follows: associated to the discretization Tδ,i  V∂δ,i = v ∈ L2 (∂Di ) : v|E ∈ Pk−2 (E),

∂ ∀E ⊂ Tδ,i



Ni Then, any function v ∈ V∂δ,i , i = 1, 2 can be written as v = j =1 vj ψj , with > = ∂ span ψj , j = 1, . . . , Ni = Vδ,i , being each of the ψj a piecewise discontinuous ∂ and zero polynomial of maximum degree k − 2 on a single element E ∈ Tδ,i elsewhere. Let finally Wδ be the space defined by  Wδ = v ∈ L2 (F ) : v|E ∈ Pk−2 (E),

∀E ⊂ Tδ,F



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and thus, given the conformity of the meshes on the common interface F , and the definition of the basis functions of V∂δ,i , i = 1, 2, the space Wδ can be seen as the restriction to F of any of the spaces V∂δ,i , i = 1, 2. Consequently, without loss of generality, it is possible to assume that the first NW basis functions in V∂δ,i , i = 1, 2 are referred to interface elements E ∈ Tδ,F and thus it is > = Wδ = span ψj , j = 1, . . . , NW .

(4)

For simplicity of exposition, from this point onward, let be k = 2, which simplifies the computations; the procedure however can be generalized to values of k > 2. Without loss of generality it is possible to assume that the basis functions in (4) coincide with the polynomials used for the scaled moments of the definition of the third set of DOFs for the VEM functions. Ni Let hi ∈ V∂δ,i , hi = j =1 hi,j ψj be the discrete version of Hi on ∂Di , i = 1, 2 NW and qi ∈ Wδ , qi = j =1 qi,j ψj the discrete version of Qi on F , i = 1, 2. Resorting to the BEM, it is possible to express the first equation of (2) as Ai hi = bi + Bi qi , i = 1, 2 where, with a notation overload, it is denoted by hi (resp. qi ) the column vector collecting the coefficients hi,j (resp. qi,j ) for j = 1, . . . , Ni , i = 1, 2, and Ai and Bi are influence coefficient matrices deriving from the BEM and bi is a vector collecting known source terms (and eventually boundary conditions). Similarly, the second equation in (2) is expressed as AF hF = bF + BF q, with hF the column vector of coefficients hF,j , j = 1, . . . , NF , AF the stiffness matrix deriving from the VEM, BF the matrix collecting integrals of the VEM basis functions for hF with the flux basis functions, and bF a vector collecting known source terms. The last equation in (2), expressing the weak continuity at the interface is re-written as: 

   hi , ψj F − hF , ψj F = 0,

j = 1, . . . , NW .

(5)

Recalling that basis functions in Wδ coincide with the polynomials used for the definition of the last set of VEM DOFs, it is, for each j = 1, . . . , NW : 1 |E| (hF , ψj )F = hF,ζ with hF,ζ , ζ ∈ {1, . . . , NF } the selected DOF corresponding to the scaled moment with respect to ψj in the unique element E where ψj = 0. Furthermore, being ψj a piece-wise constant polynomial of degree 1 (in this 1 simplified setting) it also results that |E| (hi , ψj )F = hi,j i = 1, 2. Thus condition (5) can be simply enforced by equating properly selected DOFs, which T  can be expressed as the action of a matrix Lh = 0, h = hT1 , hT2 , hTF , where each j -th row of L, j = 1, . . . , NW , contains all zeros except for a value +1 and a value −1 placed in correspondence of the two DOFs, one for hi , for an i ∈ {1, 2} and one for hF , selected in the sense above specified by the basis function ψj .

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γb γc

γb

γa

γa γc

Fig. 2 Conforming mesh generation process

The discrete version of problem (2) thus reads as 

    AB h b = , L 0 q 0

⎛

⎞ A1 0 0 ⎜ ⎟ A = ⎝ 0 A2 0 ⎠ , 0 0 AF

⎛

⎞ B1 0 ⎟ ⎜ B = ⎝ 0 B2 ⎠ BF

 T with q = (q1T , q2T )T and b = b1T , b2T , bFT . The key advantage of the proposed approach lies in the possibility of easily generating conforming meshes on the fractures, simply joining together the meshes on the matching block faces, as shown in Fig. 2, where three fractures immersed in a cube are represented. The mesh on fracture F3 is obtained gluing together the meshes on faces γa , γb and γc belonging to six different matrix blocks. This generates on F3 polygonal elements with flat angles at some vertexes, which however can be used in conjunction with the VEM. Further, additional nodes can be added on element edges in order to obtain a conforming mesh also across fracturefracture intersections, as for example it is done on one edge of fracture F2 (see Fig. 2, right). This allows to treat fracture intersection interfaces in a similar fashion as the one proposed here, as described in more details in [5]. Alternatively, without adding nodes, the mortar method has been used in [3, 4] to enforce continuity at fracture intersections.

5 Numerical Results A numerical test is proposed for the two different approaches, on the domain shown in Fig. 3a, where a network of eight fractures crosses a unit cube. Dirichlet boundary conditions H = 1 and H = 0 are set on face γD1 and γD0 (the whole bottom face of the domain), respectively, all other faces being insulated. The solution obtained with the optimization and VEM-based approaches is reported in Fig. 3b, c, respectively, and are in extremely good agreement. Looking at Fig. 4 the different mesh at the basis of the two approaches can be appreciated: in this picture fracture

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γD1

γD0 (a)

(b)

(c)

Fig. 3 Domain of test problem (a) and solution with optimization (b) and VEM (c) approaches

F6

(a)

(b)

(c)

Fig. 4 Meshes on a selected fracture for the different approaches

F6 is extracted, along with the matrix blocks lying on one of its sides (Fig. 4a). The computational mesh for the optimization approach is shown in Fig. 4b in blue lines for the fracture and in red lines for the block faces. It can be seen that the two meshes are completely not conforming. In contrast, the mesh for the VEM-based approach reported in Fig. 4c is perfectly conforming at the interface, but this conformity is easily obtained regardless of the geometrical complexity of the domain, according to the strategy here proposed. Acknowledgements This work has been partially supported by INdAM-GNCS and by Politecnico di Torino through project Starting Grant RTD. Computational resources were partially provided by HPC@POLITO (http://hpc.polito.it). All the authors are members of the INdAM Research group GNCS.

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References 1. B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini, A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66, 376–391 (2013) 2. L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes. Math. Mod. Methods Appl. Sci. 26(04), 729–750 (2015) 3. M. Benedetto, S. Berrone, A. Borio, S. Pieraccini, S. Scialò, A hybrid mortar virtual element method for discrete fracture network simulations. J. Comput. Phys. 306, 148–166 (2016) 4. M. Benedetto, S. Berrone, A. Borio, S. Pieraccini, S. Scialò, The virtual element method for discrete fracture network flow and transport simulations, in ECCOMAS Congress 2016 Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering, vol. 2 (2016), pp. 2953–2970 5. M. Benedetto, S. Berrone, S. Scialò, A globally conforming method for solving flow in discrete fracture networks using the virtual element method. Finite Elem. Anal. Des. 109, 23–36 (2016) 6. S. Berrone, S. Pieraccini, S. Scialò, A PDE-constrained optimization formulation for discrete fracture network flows. SIAM J. Sci. Comput. 35(2), B487–B510 (2013) 7. S. Berrone, S. Pieraccini, S. Scialò, On simulations of discrete fracture network flows with an optimization-based extended finite element method. SIAM J. Sci. Comput. 35(2), A908–A935 (2013) 8. S. Berrone, S. Pieraccini, S. Scialò, An optimization approach for large scale simulations of discrete fracture network flows. J. Comput. Phys. 256, 838–853 (2014) 9. S. Berrone, S. Pieraccini, S. Scialò, Towards effective flow simulations in realistic discrete fracture networks. J. Comput. Phys. 310, 181–201 (2016) 10. S. Berrone, S. Pieraccini, S. Scialò, Non-stationary transport phenomena in networks of fractures: effective simulations and stochastic analysis. Comput. Methods Appl. Mech. Eng. 315, 1098–1112 (2017) 11. S. Berrone, S. Pieraccini, S. Scialò, Flow simulations in porous media with immersed intersecting fractures. J. Comput. Phys. 345, 768–791 (2017) 12. V. Martin, J. Jaffré, J.E. Roberts, Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005)

The Virtual Element Method on Anisotropic Polygonal Discretizations Paola F. Antonietti, Stefano Berrone, Marco Verani, and Steffen Weißer

Abstract In recent years, the numerical treatment of boundary value problems with the help of polygonal and polyhedral discretization techniques has received a lot of attention within several disciplines. Due to the general element shapes an enormous flexibility is gained and can be exploited, for instance, in adaptive mesh refinement strategies. The Virtual Element Method (VEM) is one of the new promising approaches applicable on general meshes. Although polygonal element shapes may be highly adapted, the analysis relies on isotropic elements which must not be very stretched. But, such anisotropic element shapes have a high potential in the discretization of interior and boundary layers. Recent results on anisotropic polygonal meshes are reviewed and the Virtual Element Method is applied on layer adapted meshes containing isotropic and anisotropic polygonal elements.

1 Introduction In the numerical treatment of boundary value problems the flexibility in the discretization of the computational domain has gained more and more importance during the last years. Therefore, approximation strategies applicable on general polygonal and polyhedral meshes attracted a lot of interest. These approaches include, e.g., the BEM-based FEM [12], where BEM stands for Boundary Element Method, the Virtual Element Method (VEM) [6], Mimetic Finite Differences,

P. F. Antonietti · M. Verani MOX, Dipartimento di Matematica, Politecnico di Milano, Milano, Italy e-mail: [email protected]; [email protected] S. Berrone Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy e-mail: [email protected] S. Weißer () Universität des Saarlandes, FR Mathematik, Saarbrücken, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_67

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polygonal Discontinuous Galerkin methods, and Hybrid High-Order schemes (see [4] and the papers therein cited). Since polygonal elements may contain an arbitrary number of nodes on their boundary, the notion of “hanging nodes” is naturally included in the previously mentioned approaches. Consequently, the application in adaptive mesh refinement strategies becomes very attractive. For this reason, a posteriori error estimates have been developed and employed for the BEM-based FEM as well as for MFD and VEM in recent publications, see [1, 5, 9, 11, 17, 19, 20]. For classical Finite Element Methods (FEM), it is widely recognized that anisotropic mesh refinements have significant potential for improving the efficiency of the solution process when dealing with sharp layers in the solution. Pioneering works for anisotropic triangular and tetrahedral meshes have been done by Apel [3] as well as by Formaggia and Perotto [13, 14]. Furthermore, a posteriori error estimates for driving adaptivity with anisotropic elements have been studied by Kunert [15]. The anisotropic refinement of classical elements, however, results in certain restrictions due to the limited element shapes and the necessity to remove or handle hanging nodes in the discretization. Polygonal elements, in contrast, are much more flexible and adapt to anisotropic element shapes easily. This new topic has been addressed in [18], where anisotropic interpolation error estimates have been proved and utilized to generate highly adapted meshes. The aim of this paper is to investigate the Virtual Element Method on such anisotropically adapted meshes. In Sect. 2, some results of [18] are highlighted. After a short review of the VEM in Sect. 3, we give a numerical experiment in Sect. 4 demonstrating the applicability of the method on anisotropic meshes.

2 Preliminaries Let Ω ⊂ R2 be a bounded polygonal domain and let K 4h be a decomposition of Ω into disjoint polygonal elements K such that Ω = K∈Kh K. Each element K consists of an arbitrary (uniformly bounded) number of vertices which correspond to the nodes in the polygonal mesh Kh . The edges e of K are always located between two nodes. Several nodes may lie on a straight line and thus the notion of “hanging nodes” in classical finite element discretizations is naturally included in polygonal meshes. In order to prove convergence estimates for the later discussed approach, the meshes have to fulfil certain regularity assumptions. A classical choice for them is, cf. [6, 18], that all elements K ∈ Kh with diameter hK fulfil: – K is a star-shaped polygon with respect to a circle of radius ρK and with a uniformly bounded aspect ratio hK /ρK . – For the element K and all its edges e ⊂ ∂K the ratio hK /|e| is uniformly bounded, where |e| denotes the edge length. These regularity assumptions obviously do not allow stretched anisotropic elements, since their aspect ratio degenerates. Therefore, the regularity for meshes with

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anisotropic elements has to be adapted. The geometric information of the polygonal element K is encoded in the symmetric and positive definite covariance matrix MCov (K) =

1 |K|



(x − x¯ K )(x − x¯ K )? dx ∈ Rd×d ,

x¯ K =

K

1 |K|

 x dx. K

This matrix admits an eigenvalue decomposition MCov (K) = UK ΛK UK? with U ? = U −1 and ΛK = diag(λK,1 , λK,2 ). Exploiting this information, we define the mapping x → FK (x) = AK x

with

−1/2

AK = ΛK

UK? ,

which transforms the element K into a reference configuration FK (K). An example is given in Fig. 1. Thus, we call Kh a regular anisotropic mesh if: – The reference configuration FK (K) for all K ∈ Kh is a regular polygonal element according to the previous assumptions. – Neighbouring elements in Kh behave similarly in their anisotropy, i.e., their characteristic directions are scaled and oriented in a comparable way, see [18] for details. Under these assumptions on the mesh, an anisotropic error estimate for the Clément interpolation operator on polygonal discretizations can be derived, see [18, Sec. 4.1]. For v ∈ H 1 (Ω), we denote its interpolation by IC v ∈ Vh , where the discrete space Vh is given in the next section and the expansion coefficients are defined as usual as averages over neighbouring elements of the nodes. The Clément interpolation fulfils v − IC v2L2 (Ω) ≤ c

∗ ? ∗ λK,1 u? K,1 GK (v) uK,1 + λK,2 uK,2 GK (v) uK,2 ,

K∈Kh

20 1.5 10 0

0

−10

−1.5

−20 −20 −10

0

10

20

30

40

−3

−1.5

0

1.5

3

Fig. 1 Anisotropic polygonal element (left) and transformed reference configuration (right) with ellipse given by the eigenvectors of MCov scaled by the square root of the corresponding eigenvalues

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1

1

1

0

0

0 0

1

0

1

0

1

Fig. 2 Uniform (left) and adaptive meshes with isotropic (middle) and anisotropic elements (right) after six refinement steps starting from a hexagonal mesh

where G∗K (v) =



∂v ∂v K ∂xi ∂xj

2 dx

i,j =1

∈ R2×2 for x = (x1 , x2 )? and uK,i are

the eigenvectors of MCov (K) corresponding to the eigenvalues λK,i , i = 1, 2. This result generalizes the work of Formaggia and Perotto [14]. The generation of such general meshes, however, cannot be performed with standard tools. For the mesh refinement we use a bisection of the polygonal element through its barycentre into two new elements. The direction of the bisection might be determined by uK,2 that yields an isotropic refinement or by the eigenvector corresponding to the smallest eigenvalue of G∗K (v). The later strategy results in an anisotropic refinement, where the characteristics of the function v are incorporated. In Fig. 2, three meshes are visualized which are obtained after six refinement steps using a uniform and adaptive strategy with isotropic and anisotropic bisection, starting from a hexagonal mesh. These refinement procedures are exploited in the later numerical experiment.

3 Virtual Element Method It remains to discuss the numerical approximation of boundary value problems on polygonal meshes. We restrict ourselves for simplicity to the Poisson problem. For a given source function f ∈ L2 (Ω), we consider the following formulation: find u ∈ V = H01 (Ω) such that:  f v, ∀v ∈ V , (1) a(u, v) = Ω

where a(·, ·) = (∇·, ∇·)0,Ω . Problem (1) is well-posed. For future use, it is convenient to split the (continuous) bilinear form a(·, ·) defined in (1) into a sum of local contributions:

a(u, v) = a K (u, v) ∀u, v ∈ V , where a K (·, ·) = (∇·, ∇·)0,K . K∈Kh

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In order to construct the lowest order VEM approximation of (1), we need the following ingredients: – Finite dimensional subspaces Vh (K) of V (K) = V ∩ H 1 (K) ∀K ∈ Kh ; – Local symmetric bilinear forms ahK : Vh (K) × Vh (K) → R ∀K ∈ Kh so that ah (uh , vh ) =

ahK (uh , vh )

∀uh , vh ∈ Vh ;

(2)

K∈Kh

– A duality pairing fh , ·h , where fh ∈ Vh and Vh is the dual space of Vh . The above ingredients must be built in such a way that the discrete version of (1): find uh ∈ Vh such that: ah (uh , vh ) = fh , vh h ,

∀vh ∈ Vh ,

(3)

is well-posed and optimal a priori energy error estimates hold, cf. [6]. We begin by introducing the local space Vh (K) for K ∈ Kh   Vh (K) = vh ∈ H 1 (K) | Δvh = 0, vh |∂K ∈ B1 (∂K) ,

(4)

  where B1 (∂K) = vh ∈ C 0 (∂K) | vh |e ∈ P1 (e), ∀e ∈ ∂K . The global space is then obtained by gluing continuously the local spaces:   Vh = vh ∈ H01 (Ω) ∩ C 0 (Ω) | vh |K ∈ Vh (K), ∀K ∈ Kh .

(5)

Here, we only revised the space yielding first order approximations. But, the approach can be extended to k-th order approximation spaces Vhk for k > 1, cf. [6, 7]. We endow the space (4) with the values of vh at the vertices of K. Reasoning as in [6], it is easy to see that this is a unisolvent set of degrees of freedom. Owing the definition (4) of the VE local space and the choice of the degrees of freedom, it is possible to compute the H 1 (K) projector Π1∇ : Vh (K) → P1 (K) ⎧ ⎨a K (Π ∇ v − v , q) = 0, ∀q ∈ P (K), h 1 1 h  ∇ ⎩ ∂K (Π1 vh − vh ) = 0,

∀vh ∈ Vh (K),

(6)

see [6] for details. We observe that the last condition in (6) is needed in order to fix the constant part of the energy projector. Next, we introduce the discrete right-hand side fh ∈ Vh and the associated   duality pairing, i.e. fh , vh h = K∈Kh K Π00 f v h , where Π00 is the L2 projection  1 on constants and v h = |∂K| ∂K vh . Finally, we consider the discrete bilinear form

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and we require that the local bilinear forms ahK : Vh (K) × Vh (K) → R satisfy, for all K ∈ Kh , the following two assumptions (A1) consistency: a K (q, vh ) = ahK (q, vh ) ∀q ∈ P1 (K), ∀vh ∈ Vh (K); (A2) stability: there exist two positive constants 0 < α∗ < α ∗ < +∞ (possibly depending on the shape regularity of K), such that ∀vh ∈ Vh (K) α∗ |vh |21,K ≤ ahK (vh , vh ) ≤ α ∗ |vh |21,K . Assumption (A1) guarantees that the method is exact whenever the solution of (1) is a polynomial of degree one, whereas assumption (A2) guarantees the wellposedness of problem (3). Let now Idh be the identity operator on the space Vh (K), we set for every uh , vh ∈ Vh (K) ahK (uh , vh ) = a K (Π1∇ uh , Π1∇ vh ) + ShK ((Idh − Π1∇ )uh , (Idh − Π1∇ )vh ),

(7)

where Π1∇ is defined in (6) and the local bilinear form ShK (·, ·) ShK (uh , vh ) =

dim(V h (K)∩Vh )

dofi (uh )dofi (vh )

(8)

i=1

satisfies c∗ |vh |21,K ≤ ShK (vh , vh ) ≤ c∗ |vh |21,K for all vh ∈ ker(Π1∇ ), where c∗ and c∗ might depend on the shape regularity of the polygon, and the local discrete bilinear form (7) satisfies (A1) and (A2).

4 Numerical Experiment In this last section we report some preliminary results obtained solving the Poisson problem with non-homogeneous Dirichlet boundary condition −Δu = f

in Ω = (0, 1)2 ,

u=g

on ∂Ω,

by the Virtual Element Method on uniformly and locally refined meshes with isotropic as well as anisotropic elements. The computations are done with first and higher order approximation spaces Vhk . In the considered test problem, the exact solution is   u(x) = tanh (60x2) − tanh 60(x1 − x2 ) − 30 , x = (x1 , x2 )? ∈ R2 , chosen because it has a strong boundary layer on the bottom of the domain and a strong internal layer connecting the bottom and the right part of the boundary. The B Dirichlet data and the forcing function are chosen appropriately as g = uB∂Ω and   tanh 30 − 60(x1 − x2 ) tanh (60x2) f (x) = 14400 .   + 7200 2 cosh 30 − 60(x1 − x2 ) cosh2 (60x2)

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Fig. 3 Computed solution on uniform (left) and adaptive meshes with isotropic (middle) and anisotropic elements (right) after eight refinement steps, with k = 1 10 0 10 1 10 -1

10 -2

10 -3

10 -4

10 0 uniform (k=1) anisotropic (k=1) isotropic (k=1)

uniform (k=1) anisotropic (k=1) isotropic (k=1)

dofs -1 (h -2 ) uniform (k=3) anisotropic (k=3) isotropic (k=3)

10 -1

dofs -2 (h -4 )

10 2

dofs -1/2 (h -1 ) uniform (k=3) anisotropic (k=3) isotropic (k=3) dofs -3/2 (h -3 )

10 4

degrees of freedom

10 -2

10 2

10 4

degrees of freedom

Fig. 4 Errors (L2 left and H 1 right) on uniform and adaptive meshes with isotropic and anisotropic elements in logarithmic scale

In Fig. 2, we display the meshes and in Fig. 3 the projections of the VEM solution with order k = 1 on piecewise linear functions over each element. We can easily observe that the solution is properly described on all the meshes, although the number of degrees of freedom (dofs) on them is very different. Similar approximations can be obtained considering advection dominated advection diffusion problems. For these problems the VEM solution requires special stabilization for preventing spurious oscillations [8]. The anisotropic VEM elements generated by the approach presented in [18] have a quite large aspect ratio. For these elements the VEM construction could require the implementation of suitable polynomial basis functions in order to prevent problems due to the ill conditioning of the VEM projectors introduced in Sect. 3 as described, e.g., in [2, 10, 16]. In Fig. 4, we report the convergence histories in the approximate L2 error norm defined by the Πk∇ -projection of the VEM solution and the H 1 error semi-norm defined considering the L2 projection on the space of polynomials of order k − 1 of the gradient of the solution, cf. [7]. In the convergence graphs, we consider k = 1 and k = 3, where the continuous lines indicate the theoretical rates of convergence. The errors are given with respect to the number of degrees of freedom and, in the legend, we recall the corresponding rates with respect to the mesh-size h on quasi

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uniform meshes as well. Both, for the L2 and H 1 errors as well as for k = 1 and k = 3, the convergence histories fit very well the expected theoretical behaviours. The anisotropic meshes always perform superior providing a smaller error with respect to the uniform or the isotropic adaptive mesh. Acknowledgements P.F.A. has been partially funded by SIR starting grant n. RBSI14VT0S funded by the Italian Ministry of Education, Universities and Research (MIUR). P.F.A., S.B. and M.V. thank INdAM-GNCS.

References 1. P.F. Antonietti, L. Beirão da Veiga, C. Lovadina, M. Verani, Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems. SIAM J. Numer. Anal. 51(1), 654–675 (2013) 2. P.F. Antonietti, L. Mascotto, M. Verani, A multigrid algorithm for the p-version of the virtual element method. ESAIM Math. Model. Numer. Anal. 52(1), 337–364 (2018) 3. T. Apel, Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics (B.G. Teubner, Stuttgart, 1999) 4. L. Beirão da Veiga, A. Ern, Preface [Special issue-Polyhedral discretization for PDE]. ESAIM Math. Model. Numer. Anal. 50(3), 633–634 (2016) 5. L. Beirão da Veiga, G. Manzini, Residual a posteriori error estimation for the virtual element method for elliptic problems. ESAIM Math. Model. Numer. Anal. 49(2), 577–599 (2015) 6. L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. Marini, A. Russo, Basic principles of virtual element methods. Math. Mod. Methods Appl. Sci. 23(1), 199–214 (2013) 7. L. Beirão da Veiga, F. Brezzi, D. Marini, A. Russo. Virtual element method for general secondorder elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729– 750 (2016) 8. M. Benedetto, S. Berrone, A. Borio, S. Pieraccini, S. Scialó, Order preserving SUPG stabilization for the virtual element formulation of advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 311, 18–40 (2016) 9. S. Berrone, A. Borio, A residual a posteriori error estimate for the Virtual Element Method. Math. Mod. Methods Appl. Sci. 27(8), 1423–1458 (2017) 10. S. Berrone, A. Borio, Orthogonal polynomials in badly shaped polygonal elements for the virtual element method. Finite Elem. Anal. Des. 129, 14–31 (2017) 11. A. Cangiani, E.H. Georgoulis, T. Pryer, O.J. Sutton, A posteriori error estimates for the virtual element method. Numer. Math. 137(4), 857–893 (2017) 12. D. Copeland, U. Langer, D. Pusch, From the boundary element domain decomposition methods to local Trefftz finite element methods on polyhedral meshes, in Domain Decomposition Methods in Science and Engineering XVIII. Lecture Notes in Computational Science and Engineering, vol. 70 (Springer, Berlin, 2009), pp. 315–322 13. L. Formaggia, S. Perotto, New anisotropic a priori error estimates. Numer. Math. 89(4), 641– 667 (2001) 14. L. Formaggia, S. Perotto, Anisotropic error estimates for elliptic problems. Numer. Math. 94(1), 67–92 (2003) 15. G. Kunert, An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86(3), 471–490 (2000) 16. L. Mascotto, Ill-conditioning in the virtual element method: stabilizations and bases. Numer. Methods Partial Differ. Equ. 34(4), 1258–1281 (2018)

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17. S. Weißer, Residual error estimate for BEM-based FEM on polygonal meshes. Numer. Math. 118(4), 765–788 (2011) 18. S. Weißer, Anisotropic polygonal and polyhedral discretizations in finite element analysis (2017). arXiv e-prints, arXiv:1710.10505 19. S. Weißer, Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM. Comput. Math. Appl. 73(2), 187–202 (2017) 20. S. Weißer, T. Wick, The dual-weighted residual estimator realized on polygonal meshes. Comput. Methods Appl. Math. (2017). https://doi.org/10.1515/cmam-2017-0046

Part XIX

FEM Meshes with Guaranteed Geometric Properties

On Zlámal Minimum Angle Condition for the Longest-Edge n-Section Algorithm with n ≥ 4 Sergey Korotov, Ángel Plaza, José P. Suárez, and Tania Moreno

Abstract In this note we analyse the classical longest-edge n-section algorithm applied to the simplicial partition in Rd , and prove that an infinite sequence of simplices violating the Zlámal minimum angle condition, often required in finite element analysis and computer graphics, is unavoidably produced if n ≥ 4. This result implies the fact that the number of different simplicial shapes produced by this version of n-section algorithms is always infinite for any n ≥ 4.

1 Introduction Generation and reconstruction of computational meshes with the control over their main geometric properties (angles, sizes, shapes, etc.) is becoming more and more important topic in the finite element analysis and computer graphics. In this respect, both, ‘positive results’ with various practical algorithms guaranteeing certain properties of meshes produced (see e.g. works on bisections [1, 12, 17], on red-green refinement techniques [2, 15], and on the yellow refinement algorithm [8]), and also ‘negative results’ on impossibility of construction of meshes with

S. Korotov () Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences, Bergen, Norway e-mail: [email protected] Á. Plaza · J. P. Suárez Division of Mathematics, Graphics and Computation (MAGiC), IUMA, Information and Communication Systems, University of Las Palmas de Gran Canaria, Las Palmas, Canary Islands, Spain e-mail: [email protected]; [email protected] T. Moreno Faculty of Mathematics and Informatics, University of Holguín, Holguín, Cuba e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_68

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some characteritics in principle [14] (or by some of algorithms [9, 11, 21]), are of a great interest. Zlámal minimum angle condition [20, 22], guaranteeing that the elements in the triangulations used do not ‘shrink’ is often assumed (or required to hold) in the finite element analysis and computer graphics. (The generalization of this condition to simplicial elements is given in [5].) It is very important to know a priori if the given numerical refinement algorithm provides partitions with such a regularity property or not. The current work generalizes the following two results obtained by the authors earlier on the topic: that one from [11, 19], where we proved that the repeated longest-edge n-sections (in the classical and conforming versions) of any triangle unavoidably produce an infinite sequence of subtriangles whose minimal angles converge to zero if n ≥ 4, and the paper [18], where it was proved that the repeated longest-edge n-sections (in the classical version, for n ≥ 4) of any d-simplex produce a sequence of subsimplices whose minimal interior solid angles converge to zero. In a more detail, using the key observation of [11, 19], we prove the ‘negative result’ similar to that one of [18] (but using now somewhat different arguments) with interior solid angles replaced by d-dimensional sines of simplices defined originally in [7]. In addition, we show that the number of different simplicial shapes produced by the considered n-section algorithm is always infinite for any n ≥ 4.

2 Denotations and Definitions A d-simplex (or just simplex) S in Rd , d ∈ {1, 2, 3, . . . }, is the convex hull of d + 1 vertices A1 , A2 , . . . , Ad+1 that do not belong to the same (d − 1)-dimensional hyperplane, i.e., S = conv {A1 , A2 , . . . , Ad+1 }. We denote by hS the length of the longest edge of S. Let Fi = conv{A1 , . . . , Ai−1 , Ai+1 , . . . , Ad+1 } be the facet of S opposite to vertex Ai for i ∈ {1, . . . , d + 1}. The dihedral angle α between two such facets is defined by means of the inner product of their outward unit normals n1 and n2 , cos α = −n1 · n2 , see also [7, p. 74]. Let Ω ⊂ Rd be a bounded domain. Assume that Ω is polytopic (or just a polytope). By this we mean that Ω is the closure of a domain whose boundary ∂Ω is contained in a finite number of (d − 1)-dimensional hyperplanes. Next we define a simplicial partition of a bounded closed polytopic domain Ω ⊂ Rd as follows. We subdivide Ω into a finite number of simplices (called elements

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and denoted by S), so that their union is Ω, any two distinct simplices have disjoint interiors, and any facet of any simplex is either a facet of another simplex from the partition or belongs to the boundary ∂Ω. The set of such simplices will be called a simplicial face-to-face partition (shortly only partition) and denoted by Th , where h = max hS . S∈Th

The sequence of partitions F = {Th }h→0 of Ω is called a family of partitions if for every ε > 0 there exists Th ∈ F with h < ε. By meas p we denote the p-dimensional volume (p ≤ d). Recall a definition for the d-dimensional sine of angles in Rd from [7, p. 72]. In terms of the simplex S, for any of its vertices Ai , the d-dimensional sine of the angle of S at Ai , denoted by Aˆ i , is defined as follows: sind (Aˆ i |A1 A2 . . . Ad+1) =

d d−1 (meas d S)d−1 (d − 1)! d+1 j =1,j =i meas d−1 Fj

.

(1)

Remark 1 For d = 2, sin2 (Aˆ i |A1 A2 A3 ) is the standard sine of the angle Aˆ i in the triangle A1 A2 A3 , due to the following well-known formula, e.g. for i = 1, meas 2 (A1 A2 A3 ) =

1 |A1 A2 ||A1 A3 | sin Aˆ 1 . 2

(2)

Remark 2 For d = 3, A1 = (0, 0, 0), A2 = (1, 0, 0), A3 = (0, 1, 0), and A4 = π (0, 0, 1) the solid (spatial) angle of the first octant is 4π 8 = 2 . Its three-dimensional sine is by (1) equal to sin3 (Aˆ 1 |A1 A2 A3 A4 ) =

32 (1/6)2 = 1. 2!(1/2)3

Definition 3 We say that the Zlámal minimum angle condition holds if there exists a constant α0 such that for any triangulation Th ∈ F and any triangle T ∈ Th we have αT ≥ α0 > 0, where αT is the minimum angle of the triangle T . The next definition is a natural generalization of the Zlámal condition to any dimensions (see [5]). Definition 4 A family F is called a regular family of partitions of a polytope into simplices if there exists C > 0 such that for any Th ∈ F and any S = conv{A1 , . . . , Ad+1 } ∈ Th we have ∀ i ∈ {1, 2, . . . , d + 1}

sind (Aˆ i |A1 A2 . . . Ad+1 ) ≥ C > 0,

(3)

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where sind is defined in (1). Remark 5 Another (equivalent) definitions of the regularity for a family of partitions of a polytope into simplices are presented in [3, 4]. Remark 6 Property of regularity (or its absence) of simplicial partition is very important for error control issues (in a priori and a posteriori scenarios) in the finite element analysis and computer graphics [6, 10, 16, 20, 22].

3 Main Results First, we give a precise definition of the longest-edge n-section algorithm analysed in the work. One step of the classical longest-edge (LE) n-section algorithm is defined as follows (cf. [1, 11]): I) choose the longest edge in each simplex of a given simplicial partition and split it by n − 1 points into n subedges of equal length; II) split each simplex in the partition into n subsimplices using n − 1 points lying on its longest edge. Lemma 7 Consider a simplex S with vertices A1 , A2 , . . . , Ad+1 . Let Ai denote some of its vertices. We have the following relations between sines of angles of various dimensions based at Ai sin2 (Aˆ i |Ai1 Ai2 Ai3 ) ≥ sin3 (Aˆ i |Ai1 Ai2 Ai3 Ai4 ) ≥ · · · ≥ sind (Aˆ i |A1 . . . Ad+1 ), (4) where i1 , i2 , i3 , i4 , . . . are distinct indices from the set {1, 2, . . . , d + 1}. For the proof see [7, p. 76]. Lemma 8 Let us n-sect the triangle with edges of the length a, b, and c, where a ≤ b ≤ c. Then there exists a positive constant √ n2 − n + 1 κ= (< 1) n such that the lengths of all newly generated sub-edges are not greater than κ · c. For the proof see [11, p. 70]. Theorem 9 Classical LE n-section algorithm applied to any initial simplicial partition (conforming or nonconforming one) produces the family of simplicial partitions. Proof We observe that after one step of the algorithm in each fixed simplex S we produce n − 1 new edges inside every of n − 1 two-dimensional facet of S and n

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new edges along that longest edge  of S toward which we refine S at this step. The lengths of all new edges will not be greater than the length of κ ·  due to Lemma 8. As the number κ < 1 and it does not, in fact, depend on any simplex, only on n, for any ε > 0 after a finite number of refinement steps we will produce the simplicial partition (not necessarily face-to-face) whose all edges are not greater than ε. The exact number of steps to reach this goal is equal to the smallest number q such that κ q p ≤ ε, where p is the length of the longest edge in the initial partition. Theorem 10 Classical LE n-section algorithm applied to any initial simplicial partition (conforming or nonconforming one) always produce an infinite sequence of simplices violating the Zlámal minimum angle condition (3) if n ≥ 4. Proof Obviously, we have at least one simplex in the initial partition. Then consider some of two-dimensional facets of this simplex and denote it as F . If we apply the classical n-section algorithm just to F , we produce an infinite sequence of triangles violating the minimum angle condition in two-dimensional case, due to the result from [19] if n ≥ 4. Since the original algorithm (in dimension d) produces the family of simplicial partitions due to Theorem 9, we will unavoidably produce an infinite sequence of simplices, where each simplex in this sequence is that one whose one of two-dimensional facets coincide with one of triangles from the sequence of triangles destroying the minimum angle condition in two dimensions. Now, the result of theorem immediately follows from the relations between sines of various dimensions in Lemma 7. Theorem 11 The classical LE n-section algorithm never produces a finite number of simplicial shapes if n ≥ 4. Proof The result of the theorem follows immediately from the fact that for both algorithms we always produce the infinite sequence of two-dimensional facets with their minimal angles tending to zero, see Theorem 10. Remark 12 Construction of the initial conforming partition of any polytopic domain can be done using the idea of the construction proposed in [13] for polyhedral case. Acknowledgements The authors are indebted to Jan Brandts for valuable comments on the work.

References 1. A. Adler, On the bisection method for triangles. Math. Comput. 40, 571–574 (1983) 2. J. Bey, Simplicial grid refinement: on Freudenthal’s algorithm and the optimal number of congruence classes. Numer. Math. 85, 1–29 (2000) 3. J. Brandts, S. Korotov, M. Kˇrížek, On the equivalence of regularity criteria for triangular and tetrahedral finite element partitions. Comput. Math. Appl. 55, 2227–2233 (2008) 4. J. Brandts, S. Korotov, M. Kˇrížek, On the equivalence of ball conditions for simplicial finite elements in Rd . Appl. Math. Lett. 22, 1210–1212 (2009)

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5. J. Brandts, S. Korotov, M. Kˇrížek, Generalization of the Zlámal condition for simplicial finite elements in Rd . Appl. Math. 56, 417–424 (2011) 6. P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978) 7. F. Eriksson, The law of sines for tetrahedra and n-simplices. Geom. Dedicata 7, 71–80 (1978) 8. S. Korotov, M. Kˇrížek, Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal. 39, 724–733 (2001) 9. S. Korotov, M. Kˇrížek, Red refinements of simplices into congruent subsimplices. Comput. Math. Appl. 67, 2199–2204 (2014) 10. S. Korotov, M. Kˇrížek, A. Kropáˇc, Strong regularity of a family of face-to-face partitions generated by the longest-edge bisection algorithm. Comput. Math. Math. Phys. 48, 1687–1698 (2008) 11. S. Korotov, Á. Plaza, J.P. Suárez, On the maximum angle condition for the conforming longestedge n-section algorithm for large values of n. Comput. Aided Geom. Des. 32, 69–73 (2015) 12. S. Korotov, Á. Plaza, J.P. Suárez, Longest-edge n-section algorithms: properties and open problems. J. Comput. Appl. Math. 293, 139–146 (2016) 13. M. Kˇrížek, An equilibrium finite element method in three-dimensional elasticity. Appl. Math. 27, 46–75 (1982) 14. M. Kˇrížek, There is no face-to-face partition of R5 into acute simplices. Discrete Comput. Geom. 36, 381–390 (2006) 15. M. Kˇrížek, T. Strouboulis, How to generate local refinements of unstructured tetrahedral meshes satisfying a regularity ball condition. Numer. Methods Partial Differ. Equ. 13, 201– 214 (1997) 16. Á. Plaza, S. Falcón, J.P. Suárez, On the non-degeneracy property of the longest-edge trisection of triangles. Appl. Math. Comput. 216, 862–869 (2010) 17. I.G. Rosenberg, F. Stenger, A lower bound on the angles of triangles constructed by bisection of the longest side. Math. Comput. 29, 390–395 (1975) 18. J. P. Suárez, T. Moreno, The limit property for the interior solid angles of some refinement schemes for simplicial meshes. J. Comput. Appl. Math. 275, 135–138 (2015) 19. J.P. Suárez, T. Moreno, P. Abad, Á. Plaza. Properties of the longest-edge n-section refinement scheme for triangular meshes. Appl. Math. Lett. 25, 2037–2039 (2012) 20. A. Ženíšek, The convergence of the finite element method for boundary value problems of a system of elliptic equations (in Czech). Appl. Math. 14, 355–377 (1969) 21. S. Zhang, Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes. Houston J. Math. 21, 541–556 (1995) 22. M. Zlámal, On the finite element method. Numer. Math. 12, 394–409 (1968)

Adaptive Solution of a Singularly-Perturbed Convection-Diffusion Problem Using a Stabilized Mixed Finite Element Method María González and Magdalena Strugaru

Abstract We explore the applicability of a new adaptive stabilized dual-mixed finite element method to a singularly-perturbed convection-diffusion equation with mixed boundary conditions. We establish the rate of convergence when the flux and the concentration are approximated, respectively, by Raviart-Thomas/BrezziDouglas-Marini and continuous piecewise polynomials. We consider a simple a posteriori error indicator and provide some numerical experiments that illustrate the performance of the method.

1 Introduction The numerical solution of convection-dominated convection-diffusion problems is difficult. We are interested in the simultaneous approximation of the concentration and the flux in a singularly-perturbed convection-diffusion equation with mixed boundary conditions. This type of mixed methods was first proposed and analyzed by Thomas [11]. The use of augmented mixed finite element methods allows to use a wider set of finite element subspaces in the discretization (see [1, 2, 5, 7, 9] and the references therein). However, in [2, 7] only homogeneous boundary conditions of Dirichlet type were treated.

M. González () Universidade da Coruña, A Coruña, Spain e-mail: [email protected]; [email protected] M. Strugaru Basque Center of Applied Mathematics, Bilbao, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_69

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In this work we analyze the applicability of adaptive augmented mixed finite element methods to a singularly-perturbed convection-diffusion equation with mixed boundary conditions. In Sect. 2 we describe the augmented dual-mixed variational formulation. In Sect. 3 we present the stabilized mixed finite element method. In Sect. 4, we introduce a new a posteriori error indicator that is reliable and locally efficient. Finally, numerical experiments are reported in Sect. 5.

2 Augmented Dual-Mixed Variational Formulation Let Ω be a bounded domain of R2 with a Lipschitz-continuous boundary Γ . We assume that Γ is decomposed into two disjoint open parts, ΓD and ΓN . Let b ∈ [L∞ (Ω)]2 be solenoidal in Ω. Then, given f ∈ L2 (Ω), g ∈ H 1/2(ΓD ) and z ∈ H −1/2(ΓN ), we consider the problem: find σ : Ω → R2 and u : Ω → R such that ⎧ ⎪ ⎪ −div(σ ) + b · ∇u = f in Ω , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨  −1 σ − ∇u = 0 in Ω , (1) ⎪ ⎪ u = g on ΓD , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ ·n = z on ΓN , where  > 0 is a parameter and n is the unit outward normal vector to Γ . Given s ∈ H −1/2 (ΓN ), we define the space Hs := {τ ∈ H (div; Ω) : τ · n = s on ΓN } and decompose σ as σ = σ0 + σz ∈ H0 + Hz . We consider the bilinear forms a : H0 × H0 → R, b : H 1 (Ω) × H0 → R and c : H 1 (Ω) × H 1 (Ω) → R defined by 1 a(ζ, τ ) := 



 ζ ·τ,



b(w, τ ) :=

Ω

c(w, v) :=

w div(τ ) , Ω

b · ∇w v Ω

(2) for all ζ, τ ∈ H0 and w, v ∈ H 1 (Ω), and the linear functionals m : H0 → R and l : H 1 (Ω) → R defined by  m(τ ) :=

gτ ·n − ΓD

1 



 σz · τ , Ω

for all τ ∈ H0 and v ∈ H 1 (Ω) .

l(v) := −

(f + div(σz )) v , Ω

(3)

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We then consider the following augmented variational formulation of problem (1): find (σ0 , u) ∈ H := H0 × H 1 (Ω) such that As ((σ0 , u), (τ, v)) = Fs (τ, v) ,

∀ (τ, v) ∈ H,

(4)

where H is endowed with the product norm and the bilinear form As : H × H → R and the linear functional Fs : H → R are defined by As ((ζ, w), (τ, v)) := a(ζ, τ ) + b(w, τ ) − b(v, ζ ) + c(w, v)  + κ1 (div(ζ ) − b · ∇w) (div(τ ) + b · ∇v) Ω   + κ2 (∇w −  −1 ζ ) · (∇v +  −1 τ ) + κ3 wv Ω

(5)

ΓD

and  Fs (τ, v) := m(τ ) − l(v) − κ1 (f + div(σz )) (div(τ ) + b · ∇v) Ω   κ2 −1 + σz (∇v +  τ ) + κ3 gv  Ω ΓD

(6)

for all (ζ, w), (τ, v) ∈ H. We have the following result concerning the well-posedness of problem (4). Theorem 1 Assume that b ∈ [L∞ (Ω)]2 is solenoidal in Ω and such that b·n≥ 0

(7)

on ΓN .

Assume also that 0 < κ1


1 b · nL∞ (ΓD ) . 2

(8)

Then, problem (4) has a unique solution, (σ0 , u) ∈ H. Proof It is a consequence of the Lax-Milgram Lemma.

3 Augmented Mixed Finite Element Method In what follows, we let {Th }h>0 be a family of shape-regular meshes of Ω¯ made up of triangles; we denote by hT the diameter of an element T ∈ Th and define h := maxT ∈Th hT . Given T ∈ Th and an integer l ≥ 0, we denote by Pl (T ) the space of polynomials of total degree at most l defined on T and, given an integer

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r ≥ 0, we denote by RT r (T ) the local Raviart-Thomas space of order r (cf. [10]). B Then, we define Hh := RT r = {τh ∈ H0 : τh BT ∈ RT r (T ), ∀ T ∈ Th }, or B Hh := BDMr+1 = {τh ∈ H0 : τh BT ∈ [Pr+1 (T )]2 , ∀ T ∈ Th } (see [3]) . B On the other hand, let m ≥ 1 and define Vh := Lm = {vh ∈ C(Ω) : vh BT ∈ Pm (T ), ∀ T ∈ Th }. Then, the Galerkin scheme associated to problem (4) reads: find (σ0,h , uh ) ∈ Hh := Hh × Vh such that As ((σ0,h , uh ), (τh , vh )) = Fs (τh , vh ) ,

∀ (τh , vh ) ∈ Hh × Vh .

(9)

Under the hypotheses of Theorem 1, problem (9) has a unique solution (σ0,h , uh ) ∈ Hh × Vh . Moreover, there exists a constant C > 0, independent of h, such that ||(σ0 − σ0,h , u − uh )||H ≤ C

inf

(τh ,vh )∈Hh ×Vh

||(σ0 − τh , u − vh )||H .

(10)

The corresponding rate of convergence is given in the next theorem. Theorem 2 Assume σ0 ∈ [H t (Ω)]d , div(σ0 ) ∈ H t (Ω) and u ∈ H t +1 (Ω). Then, under the assumptions of Theorem 1, there exists a positive constant Cerr = O(b2 / 2 ), independent of h, such that ||(σ0 − σ0,h , u − uh )||H ≤   ≤ Cerr hmin{t,m,r+1} ||σ0 ||[H t (Ω)]d + ||div(σ0 )||H t (Ω) + ||u||H t+1 (Ω) .

(11)

Proof It follows straightforwardly from inequality (10) and the approximation properties of the corresponding finite element subspaces.

4 A Posteriori Error Indicator Let (σ0,h , uh ) ∈ Hh be the unique solution to problem (9). We denote by EΓD the set of all the edges induced by the mesh Th that are contained in Γ D . Moreover, for each edge e ∈ EΓD , we denote by he the length of e; we also fix a unit normal vector, ne := (n1 , n2 )t , and let te := (−n2 , n1 )t be the corresponding fixed unit tangential vector along e.

Adaptive Solution of a Singularly-Perturbed Convection-Diffusion Problem

We consider the global a posteriori error indicator θ :=



2 T ∈ Th θ T

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1/2 , where

1 (σz + σ0,h )||2[L2 (T )]2    ∂ he g − uh 2L2 (e) +  (g − uh )2L2 (e) . ∂te

θT2 := ||f + div(σz + σ0,h ) − b · ∇uh ||2L2 (T ) + ||∇uh − +

e∈EΓD ∩∂T

(12) We remark that θT consists of two terms in interior elements and elements with a side on the Neumann boundary, whereas it contains two additional terms on elements with a side on the Dirichlet boundary. In the next theorem we establish the reliability and local efficiency of θ . Theorem 3 Let (σ0 , u) ∈ H and (σ0,h , uh ) ∈ Hh be the unique solutions to problems (4) and (9), respectively. Then, there exists a positive constant Crel , independent of h, such that (σ0 − σ0,h , u − uh )H ≤ Crel θ .

(13)

Moreover, if g ∈ H 1 (ΓD ) is a piecewise polynomial on ΓD , then there exists Ceff > 0, independent of h, such that for all T ∈ Th we have  −1  u − uh 2H 1 (T ) + σ0 − σ0,h 2H (div,T ) . θT2 ≤ Ceff

(14)

Proof See [6]. In the convection-dominated regime, Crel = O(b3 /) and Ceff = O( 2 ).

5 Numerical Experiments In this section, we deal with the solution of the boundary value problem ⎧ ⎪ ⎪ −  Δu + b · ∇u = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

in Ω := (0, 1) × (0, 1) , on x = 0 ,

u=1

on x = 1 ,

 ∇u · n = 0

on y = 0,

(15) y = 1,

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where b = (1, 0)t and  ranges from 10−2 to 10−4 . The exact solution of problem (15) is (see [4]) x

u(x, y) =

e −1 1

e − 1

,

(x, y) ∈ Ω .

(16)

We remark that u has an exponential boundary layer around the line x = 1. We present numerical results for the finite element pairs (Hh , Vh ) given by (RT 0 , L1 ), (RT 1 , L2 ), (BDM1 , L1 ) and (BDM2 , L2 ) in R2 . The numerical experiments were performed using the finite element toolbox FEniCS [8]. We compare the performance of the finite element method based on uniform refinement with the adaptive method based on the a posteriori error indicator θ . For the selection of elements to be refined, we rely on the maximum strategy with a threshold γ = 0.4. We choose κ1 = 8 , κ2 = 2 and κ3 = 1. We remark that these values satisfy conditions (8). Finally, we define the total error  1/2 eh (σ0 , u) := σ0 − σ0,h H (div;Ω) + u − uh H 1 (Ω) .

(17)

In Fig. 1, we report the total error and estimator for the uniform and adaptive refinements. From these graphs, we conclude that the adaptive algorithm is more competitive than the uniform procedure. Figure 2 shows the efficiency indices for the different finite elements and the different values of  considered here. We note that their values stay in the same range for all values of  and all elements considered. In Fig. 3 we show the initial, an intermediary and final meshes for  = 10−4 . We observe that the adaptive algorithm is able to locate the boundary layer of the solution. Finally, Fig. 4 shows the final concentration and flux obtained in that case with the adaptive procedure.

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Fig. 1 Decay of error (left) and estimator (right) vs. number of degrees of freedom for three values of  and different elements

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Fig. 2 Efficiency indices for different finite elements

Fig. 3 Initial, intermediary (after 5) and final (after 15 iterations) mesh for  = 0.0001 when using (RT 0 , L1 )

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Fig. 4 Final concentration (left) and flux (right) for  = 0.0001 when using (RT 0 , L1 )

Acknowledgements The research of the first author was partially supported by MICINN grant MTM2016-76497-R. The research of the second author was supported by the Basque Government through the BERC 2014-2017 programme and by the Spanish Ministry of Economy and Competitivity through the BCAM Severo Ochoa excellence accreditation SEV-2013-0323.

References 1. T.P. Barrios, J.M. Cascón, M. González, A posteriori error analysis of an augmented mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Eng. 283, 909–922 (2015) 2. T.P. Barrios, J.M. Cascón, M. González, Augmented mixed finite element method for the Oseen problem: a priori and a posteriori error analyses. Comput. Methods Appl. Mech. Eng. 313, 216–238 (2017) 3. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, Berlin, 1991) 4. M. Farhloul, A.S. Mounim, A mixed-hybrid finite element method for convection-diffusion problems. Appl. Math. Comput. 171, 1037–1047 (2005) 5. M. González, Stabilized dual-mixed method for the problem of linear elasticity with mixed boundary conditions. Appl. Math. Lett. 30, 1–5 (2014) 6. M. González, M. Strugaru, Stabilization and a posteriori error analysis of a mixed FEM for convection-diffusion problems with mixed boundary conditions (submitted) 7. M. González, S. Korotov, J. Jansson, A posteriori error analysis of a stabilized mixed FEM for convection-diffusion problems. Discrete and Continuous Dynamical Systems, in Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference, Suppl., pp. 525–532 (2015) 8. A. Logg, K.-A. Mardal, G.N. Wells (eds.), Automated Solution of Differential Equations by the Finite Element Method. The FEniCS Book (Springer, Berlin, 2012) 9. A. Masud, T.J.R. Hughes, A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Eng. 191, 4341–4370 (2002) 10. J.E. Roberts, J.-M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, ed. by P.G. Ciarlet, J.L. Lions, vol. II. Finite Element Methods (Part 1) (North-Holland, Amsterdam, 1991) 11. J.-M. Thomas, Mixed finite elements methods for convection-diffusion problems, in Numerical Approximation of Partial Differential Equations (Elsevier, New York, 1987), pp. 241–250

Spaces of Simplicial Shapes Jon Eivind Vatne

Abstract When discussing shapes of simplices, e.g. in connection with mesh generation and finite element methods, it is important to have a suitable space parametrizing the shapes. In particular, degenerations of different types can appear as boundary components in various ways. For triangles, we will present two natural parametrizing sets that highlight two different types of degenerations, and then combine the properties into a new parametrizing space that allows a good basis for understanding both types of degenerations. The combined space is constructed by the process of blowing up, which in this simple case is introduction of polar coordinates. For tetrahedra, there are many different types of degenerations. In this short paper we will only give one example of what can be achieved by blowing up a natural model, namely to pull apart the two tetrahedral degenerating types known as slivers and caps.

1 Introduction The standard regularity condition, for instance used by Ciarlet [5], says that a family of simplicial meshes is well suited for finite element methods if for each simplex in a mesh in the family, the inscribed ball is not too small compared to the element diameter. An equivalent condition is that all angles are bounded away from zero, as in Ženíšek or Zlámal [4, 12, 13]. However, it is by now well known that weaker conditions can give the same rate of convergence. These conditions include semiregularity and maximal angle conditions, see e.g. work of Brandts, Hannukainen, Korotov and Kˇrížek [1–3, 7]. Compared to regularity, the advantage is that certain classes of degenerating mesh elements are allowed. For the purposes

J. E. Vatne () Western Norway University of Applied Sciences, Department of Computing, Mathematics and Physics, Bergen, Norway e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_70

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of this article, we only need to formulate these conditions for triangles (for simplices in general, angles between adjacent faces of various dimensions are used). Definition 1 (Minimal and Maximal Angle Conditions) A family of triangular meshes satisfies the minimal angle condition if there is a constant α0 > 0 such that each angle of each triangle appearing in the family of meshes is no smaller than α0 . A family of triangular meshes satisfies the maximal angle condition if there is a constant α0 > 0 such that each angle of each triangle appearing in the family of meshes is no larger than π − α0 . In order to emphasize the difference between various conditions about degenerations, it is important to have a model for simplices that includes representatives for the degenerations one is interested in. The methods introduced could also be of interest for e.g. finite volume methods, see for instance [9, 10].

2 The Set of Simplices and Symmetries In this section the sets that we consider are introduced. Definition 2 An n-simplex in Rn is given as the convex hull of n + 1 points that are in general position, i.e. that are not contained in the same affine hyperplane. A degenerate simplex is given by n + 1 points that are contained in some affine hyperplane. For emphasis, we will sometimes say that a simplex is non-degenerate. We are only interested in the shape of a simplex, not its size, position, orientation, or the ordering of vertices (this can be described using symmetry groups). Therefore we can change a given simplex as follows, preserving its shape: – Find two vertices whose distance is equal to the diameter of the simplex. – Move the simplex so that one of these vertices, say A, is placed in the origin. – Scale the simplex so that the second vertex, say B, has distance one from A (so the diameter is one). – Rotate the simplex so that B is on the positive x-axis, i.e. has coordinates B = (1, 0, . . . , 0). – Choose a third vertex C, and rotate so that it is in the xy-plane. – For each remaining vertex, rotate the simplex so that the new vertex only has nonzero coordinates in one coordinate more than the previous vertex. We can then parametrize simplicial shapes by the space of coordinates from the third vertex onwards. There is still some finite symmetry that can be considered, but the dimension of the space cannot be further reduced. One can for instance demand that the last non-zero coordinate for each point should be positive, which can be achieved by reflection. Note that the dimension is equal to the number of non-zero

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coordinates from the third vertex onwards, which is   n(n + 1) − 1. dim set of shapes of n-simplices = 2 + 3 + · · · + n = 2 For triangles, this means that the dimension is reduced from six (two coordinates for each of three points) to just two. There are other interesting possibilities for choosing normalizations, which sometimes allow easier reasoning. For instance in [11], the author chose a parameter space without rotating simplices (so of higher dimension), which was suited to compute the probability that a randomly drawn simplex is well-centered. The model from that paper is badly suited to study degenerations, however.

3 Triangles and Degenerations of Triangles We will introduce and compare different models for studying shapes of triangles. In the explicit models in this section, we have removed finite symmetries, so that there is an actual bijection between our parametrizing sets and the (abstract) set of triangular shapes.

3.1 Triangular Shapes from Normalized Positions of Vertices If we follow the recipe from the introduction, the vertices of our modified triangle will be A = (0, 0), B = (1, 0) and C = (Cx , Cy ). The dimension is two, as both coordinates of C are allowed to vary. Since AB is the diameter of the triangle, C must be in the intersection of the unit circles centered on A and B, respectively. By reflection and reordering we can assume that Cy > 0 and that C is closer to A than to B (so that Cx ≤ 1/2) (Fig. 1). With these choices, our parametrizing set has three boundary components: Fig. 1 C is restricted to lie in the region to the left of the dotted vertical line, above the x-axis and inside the circular segment centered on B

× A

1 2

× B

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Fig. 2 A degeneration where both the minimal angle and the maximal angle conditions are violated

Cx Fig. 3 The two smallest angles α and β are bounded by α = 0, α = β and β =π −α−β

β

1

isosceles, two largest angles equal ( π3 , π3 ), equilateral

isosceles, two smallest angles equal α

(a) The component Cx = 1/2 parametrizes isosceles triangles where the two smaller edges have equal length. (b) The component (Cx − 1)2 + Cy2 = 1 parametrizes isosceles triangles where the two largest edges have equal length. (c) The component Cy = 0 represents degenerations where A, B and C are collinear. Understanding degenerations and their place in the parametrizing space is important for us, so let us look at a point C = (Cx , 0) representing a degeneration. In Fig. 2 we see a concrete family degenerating to this point. Note that for 0 < Cx ≤ 1/2, the angles at A and B tend to zero, whereas the angle at C tends to π; both the maximal and the minimal angle conditions are violated.

3.2 Triangular Shapes from Angles The shape of a triangle can also be given by its angles. Since the three angles sum to π, it is sufficient to consider the two smallest angles α ≤ β, so that the largest angle γ = π − α − β ≥ β. This space is shown in Fig. 3. The set parametrizing the angles is again bounded by three components: (a) The component α = β parametrizes isosceles triangles where the two smallest angles are equal. (b) The component β = π − α − β parametrizes isosceles triangles where the two largest angles are equal.

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Fig. 4 A degenerating family with fixed middle angle β violates the minimal but not the maximal angle condition

C

A

β

B

(c) The component α = 0 represents degenerations where two vertices come together. The third case is the most interesting for us as α = 0 represents degenerate triangles. So consider a degenerate triangle represented by (0, β) in Fig. 3. A concrete family degenerating to this is shown in Fig. 4. Note that the angle at B tends to zero, but that the maximal angle condition is satisfied. Remark 3 The main difference between our two models is the degenerations that are represented by points on the boundary. In Fig. 1, a point (Cx , 0) with 0 < Cx ≤ 1/2 represents a degeneration where three different points are collinear, and where the maximal angle condition is violated. Any degeneration where the maximal angle condition is satisfied must be represented by the single point (Cx , Cy ) = (0, 0). In Fig. 3, a point (0, β) with 0 < β ≤ π/2 represents a degeneration where two points come together, and where the maximal angle condition is satisfied. Any degeneration with three different collinear points must be represented by the single point (α, β) = (0, 0). Our goal is to integrate the two good properties of these two models into a single model. We will achieve this starting from the model in Fig. 3, but changing to polar coordinates (r, θ ) (Fig. 5).

degeneration with fixed middle angle π/2 degeneration with three different collinear points

isosceles, two largest angles equal

π/4 isosceles, two smallest angles equal

Fig. 5 The set described in Theorem 4

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Theorem 4 The set of triangular shapes can be parametrized as a subset of the (r, θ )-plane with the following properties: a) It is bounded by the four curves θ = π/2, θ = π/4, r = 0 and r sin θ = π − r cos θ − r sin θ . b) The boundary components θ = π/4 and r sin θ = π − r cos θ − r sin θ parametrize isosceles triangles. c) The boundary component θ = π/2 parametrizes degeneration limits with fixed middle angle. d) The boundary component r = 0 parametrizes degeneration limits with three different collinear points. Proof We will start from the model in Fig. 3, and then introduce polar coordinates (r, θ ). The line α = β is then changed to the line θ = π/4 and the line α = 0 to the line θ = π/2. Since the line β = π − α − β does not pass through the origin, it is changed into a more complicated curve with equation r sin θ = π − r cos θ − r sin θ . Finally, the point (0, 0) is transformed into the whole line segment (0, θ ) for θ ∈ [π/4, π/2]. Since polar coordinates and cartesian coordinates are in bijection outside the locus r = 0, we get parts a), b) and c) in the theorem. To see that part d) also holds, we will need to understand what a family degenerating to a point (0, θ ) looks like. If we think of this as the limit limr→0 (r, θ ), we see that it is represented by a family where the ratio of the two smallest angles is constant. By the law of sines, we then see that this is the same information as is contained in a degeneration to the x-axis in the model represented in Fig. 1. Remark 5 We could have started from the model in Fig. 1 and introduced polar coordinates to achieve a similar result.

3.3 Mappings Between Models Starting from representing triangular shapes by the two smallest angles (α, β), we can find the coordinates of the third vertex C (A = (0, 0) and B = (1, 0) are the two first) by a simple computation, giving a map  φ : (α, β) →

tan α tan β tan α , tan α + tan β tan α + tan β

 .

In Fig. 3, the domain of definition of φ is the complement of the origin, and the image of φ in Fig. 1 is the complement of the x-axis. However, if we precompose φ with the polar coordinates map (r, θ ) → (r cos θ, r sin θ ), we get a map φ˜ : (r, θ ) →



tan(r cos θ) tan(r cos θ) tan(r sin θ) tan(r cos θ)+tan(r sin θ) , tan(r cos θ)+tan(r sin θ)

 .

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The map φ˜ as defined has domain of definition {(r, θ )| r > 0}, but can be continuously extended to r = 0 by  (0, θ ) →

 1 , 0 . 1 + tan θ

With this extension, we see that our good model in Theorem 4 maps surjectively to both our previously defined models. Furthermore, it maps bijectively onto the open subsets corresponding to non-degenerate simplices, as well as to the boundary components that represent degenerations that are adequately handled by each of the two models.

4 Tetrahedra Due to limitations of space, only a short comment about tetrahedral shapes is included. Degenerations of tetrahedra come in several flavours, see e.g. Edelsbrunner [6]. Different types of degenerations form a partially ordered set, as some can be thought of as further degenerations of others. The simplest example is that the intersection of the two families of degenerations known as caps and slivers represent the degenerations known as spades, see Fig. 6. In a tetrahedral model similar to the one shown for triangles in Fig. 1, caps and slivers would be represented by four-dimensional boundary components of a fivedimensional set representing tetrahedral shapes. Their intersection, representing spades, would be three-dimensional. If this intersection is blown up, we would get a model where spades are also represented as a four-dimensional boundary component, whereas the components representing caps and slivers would be pulled apart and intersect in a lower dimensional set. The process of blowing up replaces a lower-dimensional subset with a higher dimensional one, see e.g. the lecture notes of Melrose [8] for a good introduction to these ideas. The simplest case is blowing up the origin of a plane, which basically introduces polar coordinates. In [6], caps and slivers are handled in separate ways. Therefore it is sensible to have a model where these two types of degenerations are kept apart. ◦

◦

cap

◦

◦

◦ ◦

◦

◦

◦

spade

◦

Fig. 6 Spades as common degenerations of caps and slivers

◦

◦ sliver

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There are several types of degenerations of tetrahedra, grouped into flat (four points that almost span only a plane) and skinny (four points that almost span only a line) tetrahedra. Some of these have acceptable behaviour for finite element meshes, whereas others do not [7]. As for triangular shapes, different models for tetrahedra can highlight different degeneration types. If we wish to have the opportunity to study different types together, blowing up can be a useful tool. However, because of the greater variety of tetrahedral degenerations, the limitations of available space in the present article excludes a detailed analysis in this case. Hopefully, this can be addressed in a publication in the near future.

References 1. J. Brandts, S. Korotov, M. Kˇrížek, Dissection of the path-simplex in Rn into n pathsubsimplices. Linear Algebra Appl. 421(2–3), 382–393 (2007) 2. J. Brandts, A. Hannukainen, S. Korotov, M. Kˇrížek, On angle conditions in the finite element method. SEMA J. 56, 81–95 (2011) 3. J. Brandts, S. Korotov, M. Kˇrížek, A geometric toolbox for tetrahedral finite element partitions, in Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations, ed. by O. Axelsson, J. Karátson (Bentham Science Publishers Ltd., Sharjah, 2011), pp. 101–122 4. J. Brandts, S. Korotov, M. Kˇrížek, Generalization of the Zlámal condition for simplicial finite elements in Rd . Appl. Math. 56, 417–424 (2011) 5. P. Ciarlet, Basic Error Estimates for Elliptic Problems, vol. II. Handbook of Numerical Analysis, II (North-Holland, Amsterdam, 1991), pp. 17–351 6. H. Edelsbrunner, Geometry and Topology for Mesh Generation (Cambridge University Press, Cambridge, 2006) 7. A. Hannukainen, S. Korotov, M. Kˇrížek, Maximum angle condition for n-dimensional simplicial elements, in Numerical Mathematics and Advanced Applications - ENUMATH 2017, vol. 126, ed. by F.A. Radu, K. Kumar, I. Berre, J.M. Nordbotten, I.S. Pop (Springer, Cham, 2018). https://doi.org/10.1007/978-3-319-96415-7 8. R. Melrose, Real Blow Up. Lecture notes from a course at the session Introduction to Analysis on Singular Spaces (MSRI, 2008). Available from http://math.mit.edu/~rbm/ 9. H.M. Nilsen, J.M. Nordbotten, X. Raynaud, Comparison between cell-centered and noda-based discretization schemes for linear elasticity. Comput. Geosci. 22(1), 233–260 (2018) 10. J.M. Nordbotten, I. Aavatsmark, G.T. Eigestad, Monotonicity of control volume methods. Numer. Math. 106(2), 255–288 (2007) 11. J.E. Vatne, Simplices rarely contain their circumcenter in high dimensions. Appl. Math. 62(3), 213–223 (2017) 12. A. Ženíšek, The convergence of the finite element method for boundary value problems of a system of elliptic equations (in Czech). Appl. Math. 14, 355–377 (1969) 13. M. Zlámal, On the finite element method. Numer. Math. 12, 394–409 (1968)

Simplicial Vertex-Normal Duality with Applications to Well-Centered Simplices Jan Brandts and Michal Kˇrížek

Dedicated to Sergey Korotov on the occasion of his 50-th birthday.

Abstract We study the relation between the set of n + 1 vertices of an n-simplex S having Sn−1 as circumsphere, and the set of n + 1 unit outward normals to the facets of S. These normals can in turn be interpreted as the vertices of another simplex Sˆ that has Sn−1 as circumsphere. We consider the iterative application of the map ˆ study its convergence properties, and in particular that takes the simplex S to S, investigate its fixed points. We will also prove some statements about well-centered simplices in the above context.

1 Introduction Any bounded convex polytope in Rn can be defined in two equivalent but fundamentally different ways. In the one definition, it is the convex hull of a finite number of points. In the other, it is defined as the intersection of a finite number of closed half spaces. Now, in particular, let S be an n-simplex with vertices v0 , . . . , vn , and let j ∈ {0, . . . , n}. Write Fj for the facet of S opposite vj and νj for the outward unit

J. Brandts () Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, Netherlands e-mail: [email protected] M. Kˇrížek Institute of Mathematics, Czech Academy of Sciences, Praha 1, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_71

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normal vector to Fj . Then S can be written in the above-mentioned two ways as conv{v0 , . . . , vn } = S =

n M

Hj .

(1)

j =0

Here, Hj is the closed half space whose boundary is the affine hyperplane containing the facet Fj of S defined by Hj = {x ∈ Rn | νj? x ≤ bj }

(2)

and bj equals νj? vi for all i = j . Note that S is fully determined by its vertices v0 , . . . , vn or by its outward normals ν0 , . . . , νn and the numbers b0 , . . . , bn . In this short note we investigate the relation between the set of n + 1 vertices and of an n-simplex with Sn−1 as circumsphere and the set of its n + 1 unit outward normals. Intimately connected to this relation are two well-known types of simplices. The first is characterized in terms of its vertices, the second by its normals. Definition 1 A simplex S is well-centered if its circumcenter is an interior point of S, or equivalently, a strictly convex combination of its vertices. Definition 2 A simplex is acute if the angle between each pair of distinct unit outward normals νi , νj is larger than right, or equivalently, νi? νj < 0. Well-centered simplices were extensively studied in [6–8], whereas acute simplices, as a subclass of nonobtuse simplices, were reviewed in [3].

2 Vertex-Normal Duality for Simplices Here we describe the relation between vertices and normals of a simplex. This relation is similar to the relation between a matrix and its inverse, and hence it is nontrivial. See also [2] for an easy introduction.

2.1 Relations Between Vertices and Normals of Simplices Let n ≥ 2. Almost any set of n + 1 points on the sphere Sn−1 ⊂ Rn is the set of vertices of a nondegenerate n-simplex having Sn−1 as circumsphere and 0 as circumcenter. In contrast, ν0 , . . . , νn ∈ Sn−1 are outward normals to the facets of an n-simplex S having Sn−1 as insphere if and only if they do not lie on the same hemisphere [5]. To explain the latter, write N for the n×(n+1) matrix with columns ν0 , . . . , νn . Assume that any n columns of N are linearly independent and let V be

Simplicial Vertex-Normal Duality with Applications to Well-Centered Simplices

ν0

S1

v2 ν1

0 ν2

S

v0 ν2

v1 v1

v0

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ν1 S1 0

S

v2

ν0

Fig. 1 ν0 , ν1 , ν2 are outward normals to S if and only if 0 ∈ S

the unique n × (n + 1) matrix with columns v0 , . . . , vn satisfying ⎡ ⎤ ⎤ ν0? v0 − 1 1 ··· 1 ⎢ ⎥ ⎢. .. ⎥ .. ⎥. ⎥, D = ⎢ . where E = ⎢ . . . ⎣ ⎦ ⎣ ⎦ νn? vn − 1 1 ··· 1 (3) ⎡

N ? V = E + D,

It is easily verified that v0 , . . . , vn are the vertices of a nondegenerate n-simplex S, that ν0 , . . . , νn are normals to the facets of S, and that these facets of S are all tangent to Sn−1 . The central observation is now that these normals are all pointing outward if and only if the origin 0 is an interior point of S, which means that 0 is a strictly convex combination of the columns of V , and hence that there exists a vector u > 0 such that V u = 0. This is illustrated in Fig. 1, where on the left 0 and S1 lie outside S, and on the right inside S. As N and V have rank n and since N ? V is symmetric by (3), the equivalences Vu = 0

⇔

N ?V u = 0

⇔

V ? Nu = 0

⇔

Nu = 0

(4)

show that there exists u > 0 such that V u = 0 if and only if 0 is a strictly convex combination of the normals ν0 , . . . , νn . This, in turn, is equivalent to ν0 , . . . , νn not lying on the same hemisphere. A set with this property was called a global set by Gaddum in [5]. In view of Definition 1, this immediately leads to the following theorem. Theorem 3 The set of unit outward normals to the facets of any simplex S is the set of vertices of a well-centered simplex Sˆ with circumsphere Sn−1 . Proposition 4 Endow Sn−1 with the uniform probability measure. The probability that ν0 , . . . , νn ∈ Sn−1 are outward normals to a simplex S equals 2−n . Proof The one-dimensional null space in Rn+1 of the matrix N with columns ν0 , . . . , νn intersects the positive orthant with probability 2 · 2−n−1 . 

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As a corollary of Proposition 4 and Theorem 3, we have an alternative proof of a result by Vatne [9]. Corollary 5 The probability that n + 1 vertices v0 , . . . , vn , selected uniformly randomly from Sn−1 , are vertices of a well-centered simplex S equals 2−n .

2.2 The Vertex-to-Normal Mapping Theorem 3 motivates the investigation of the integral powers, or equivalently, of iterates by repeated application, of the mapping 6n+1 6n+1 5 5 → Sn−1 : Bn (v0 , . . . , vn ) = (ν0 , . . . , νn ) (5) Bn : Sn−1 that, given a vertex set of an n-simplex S with circumsphere Sn−1 , maps it onto the set of its n + 1 corresponding unit outward normals. Remark 6 We will also use the symbol Bn to map the n-simplex with vertices v0 , . . . , vn ∈ Sn−1 to the n-simplex with vertices ν0 , . . . , νn ∈ Sn−1 . Theorem 3 then states that Bn (S) is well-centered for any S with circumsphere Sn−1 . Proposition 7 If S is a regular simplex with vertices v0 , . . . , vn ∈ Sn−1 , then Bn (S) is also regular. Moreover, Bn2 (S) = S.

(6)

If n = 2 (and as will be shown further on, only if n = 2), regular simplices are the only fixed points of Bn2 . To see this, let Tk be a triangle with circumcircle S1 . The angles between each pair of distinct vertices of Tk , seen as vectors in R2 , correspond to subsets of S1 with measure αk , βk , γk . The outer normal vectors to the edges of Tk bisect each of these sets in two parts of equal measure, hence each application of B2 linearly redistributes these angles as depicted and described in Fig. 2. v0

ν1 Tk

ν2 v1

0

⎡

⎤ ⎡ ⎤⎡ ⎤ 0 1 1 αk αk+1 1 ⎣ βk+1 ⎦ = ⎣ 1 0 1 ⎦ ⎣ βk ⎦ 2 γk+1 γk 1 1 0

v2 ν0

Fig. 2 Linear redistribution of the angles: the angle between ν0 and ν2 is the average of the two angles between v0 and v1 , and between v1 and v2

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Here, αk+1 , βk+1 , γk+1 are the angles between the outer normals to Tk . The iterative application of B2 is therefore a linear Markov chain having a trivial unique equilibrium distribution in which all three angles are equal.

2.3 Vertex-Normal Duality in Higher Dimensions To study this process in higher dimensions, observe that the n + 1 vertices v0 , . . . , vn ∈ Sn−1 of an n-simplex triangulate the surface of Sn−1 into n + 1 spherical (n − 1)-simplices, each having n of the n + 1 points as vertices. Similar to the case n = 2, the application of Bn can be viewed as a transformation of such a triangulation, but it will turn out that this transformation is not linear anymore. To study Bn , we prove a lemma. Lemma 8 Let v1 , . . . , vn ∈ Sn−1 be vertices of an (n − 1)-simplex F . Then both unit normals ±ν to F have equal spherical distance to each vertex of F . Proof Note that ν ∈ Sn−1 is a normal to F if and only if ν ⊥ νi − νj for all i, j ∈ {1, . . . , n}, i = j , hence if and only if ν ? νi = ν ? νj . Now, ν ? νi is the cosine of the angle between ν and νi . This angle is the spherical distance between them.  Remark 9 The geometrical intuition behind Lemma 8 is as follows, see Fig. 3 for an illustration. Choose a ν ∈ Sn−1 that is equidistant to each vertex of F as north pole of Sn−1 . Then F is in horizontal position, hence ν ⊥ F . It also shows that the projection of ν onto the hyperplane H containing F equals the circumcenter of F , as each of the arcs connecting ν with a vertex of F projects onto H as a straight line, and all these lines have equal length because all the arcs have equal length. Theorem 10 Let S be an n-simplex with circumsphere Sn−1 . Then Bn2 (S) = S

(7)

if and only if all circumradii r0 , . . . , rn of the facets F0 , . . . , Fn of S are equal. Fig. 3 Illustration of Lemma 8 and Remark 9 for n = 3. As a consequence, we can now prove a generalization of Proposition 7

F S2 0

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Proof Write V = {v0 , . . . , vn } for the vertex set of S. For each j ∈ {0, . . . , n}, let Sj be for the spherical (n − 1)-simplex with vertex set V \ {vj }, and νj the unit outward normal to Fj . According to Lemma 8, νj has equal spherical distance dj to each vertex of Sj . Now, consider v0 , which is a vertex of S1 , . . . , Sn . It has spherical distance dj to νj for all j ∈ {1, . . . , n}. Again by Lemma 8, only if d1 = . . . = dn is v0 equidistant to each element of {ν1 , . . . , νn }. Repeating the same argument with v1 instead of v0 shows moreover that d0 = d1 . Hence, Bn (v0 , . . . , vn ) = (ν0 , . . . , νn ) and Bn (ν0 , . . . , νn ) = (v0 , . . . , vn ). By Remark 9, d0 = . . . = dn is equivalent to r0 = . . . = rn .



If n = 2, only for equilateral triangles the circumradii of all three facets are equal. In the next section we study the next simplest of tetrahedra.

2.4 Tetrahedra Whose Facets Have Equal Circumradius It is instructive to investigate for which tetrahedra all four triangular facets have equal circumradius. For this, consider Fig. 4, which is a folded-out impression of the triangulation of S2 into four spherical triangles induced by an inscribed tetrahedron. v0

3

v1

v2 0

T0 1 2

v3

v0

v0 Fig. 4 Fold-out of triangulation of S2 into four spherical triangles whose circumradii are all equal, hence all dashed lines have equal spherical length

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Thus, vertex v0 appears three times, and each edge between v0 and vj with j = 0 two times. Assume all dashed lines to have equal spherical length. This subdivides the spherical triangle T0 with vertices v1 , v2 , v3 into three isosceles triangles. Some of the angles of these triangles are indicated by α, β, γ in Fig. 4. Each of these then also appears in one of the isosceles triangles sharing an edge with T0 , because these are mirror images of the ones in T0 . Moreover, around each vertex v1 , v2 , v3 , all six angles, of which two are unknown but equal to one another, add up to 2π. Again using that all dashed lines have equal length, this gives the six angles around v0 , being π − α − β,

π − α − β,

π − α − γ,

π − α − γ,

π − β − γ,

π −β −γ

which add up to 2π if and only if α + β + γ = π, hence, around each vertex vj each angle α, β, γ appears two times. As a consequence, all four triangles are spherically congruent and all have surface area equal to π. Hence, as a corollary of Theorem 10, we get the following. Theorem 11 Let T be a tetrahedron with circumsphere S2 . Then B32 (T ) = T if and only if T is equifacetal, which means that all faces are congruent. Equifacetal tetrahedra were already studied in [1], and were called isosceles. There it was proved that each two opposite edges of an equifacetal tetrahedron T have equal lengths, and that any acute triangle can serve as facet of T . There do not exist equifacetal tetrahedra with triangular facets that are not acute. Equifacetal simplices were more recently studied in [4]. Final Remarks Since equifacetal tetrahedra have acute triangular facets, the facets themselves are well-centered. A well-centered tetrahedron with well-centred facets is called fully well-centered. Hence, the fixed points of the mapping B32 are in fact fully well-centered tetrahedra. It is an interesting question whether the iterative application of Bn2 always converges for any starting simplex. This will be investigated in another paper. Acknowledgements Michal Kˇrížek was supported by grant no. 18-09628S of the Grant Agency of the Czech Republic.

References 1. N. Altshiller-Court, Modern Pure Solid Geometry, 2nd edn. (Chelsea, New York, 1964); MacMillan, New York, 1935 2. J.H. Brandts, S. Korotov, M. Kˇrížek, Dissection of the path-simplex in Rn into n pathsubsimplices. Linear Algebra Appl. 421(2–3), 382–393 (2007) 3. J.H. Brandts, S. Korotov, M. Kˇrížek, J. Šolc, On nonobtuse simplicial partitions. SIAM Rev. 51(2), 317–335 (2009) 4. A.L. Edmonds, The geometry of an equifacetal simplex. Mathematika 52(1–2), 31–45 (2009)

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5. J.W. Gaddum, The sums of the dihedral and trihedral angles in a tetrahedron. Am. Math. Mon. 59(6), 370–371 (1952) 6. R. Hosek, Face-to-face partition of 3D space with identical well-centered tetrahedra. Appl. Math. 60(6), 637–651 (2015) 7. E. Vanderzee, A.N. Hirani, D. Guoy, E.A. Ramos, Well-centered triangulation. SIAM J. Sci. Comput. 31(6), 4497–4523 (2009/2010) 8. E. Vanderzee, A.N. Hirani, D. Guoy, V. Zharnitsky, E.A. Ramos, Geometric and combinatorial properties of well-centered triangulations in three and higher dimensions. Comput. Geom. 46(6), 700–724 (2013) 9. J.E. Vatne, Simplices rarely contain their circumcenter in high dimensions. Appl. Math. 62(3), 213–223 (2017)

Maximum Angle Condition for n-Dimensional Simplicial Elements Antti Hannukainen, Sergey Korotov, and Michal Kˇrížek

Abstract In this paper the Synge maximum angle condition for planar triangulations is generalized for higher-dimensional simplicial partitions. In addition, optimal interpolation properties are presented for linear simplicial elements which can degenerate in certain ways.

1 Introduction Consider a family F = {Th }h→0 of face-to-face triangulations Th of a bounded polygonal domain. In 1957, J. Synge proved that linear triangular finite elements yield the optimal interpolation order in the C-norm provided the maximum angle condition is satisfied, i.e., there exists a constant γ0 < π such that for any triangulation Th ∈ F and any triangle T ∈ Th one has (see [27]) γT ≤ γ0 ,

(1)

where γT is the maximum angle of T . In 1975/1976, Babuška and Aziz [3], Barnhill and Gregory [4], and Jamet [14] independently derived the optimal interpolation order in the energy norm of finite element approximations under the condition (1).

A. Hannukainen Department of Mathematics and Systems Analysis, Aalto University, Aalto, Finland e-mail: [email protected] S. Korotov Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences, Bergen, Norway e-mail: [email protected] M. Kˇrížek () Institute of Mathematics, Czech Academy of Sciences, Prague 1, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_72

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Later the maximum angle condition was investigated in various norms in [1, 2, 15– 18, 22, 24, 25]. In 1992, the condition (1) was generalized by Kˇrížek [19] to tetrahedral elements as follows: There exists a constant γ0 < π such that for any face-to-face tetrahedralization Th ∈ F and any tetrahedron T ∈ Th one has γD ≤ γ0

and γF ≤ γ0 ,

(2)

where γD is the maximum dihedral angles between faces of T and γF is the maximum angle in all four triangular faces of T . According to [20], the associated finite element approximations preserve the optimal interpolation order in the H 1 norm under the condition (1). Note that degenerated tetrahedral elements have a lot of real-life technical applications. For example, in calculation of physical fields in eletrical rotary machines, see [20]. Flat tetrahedral elements are also used to approximate thin slots, layers, or gaps. Moreover, they are suitable when the true solution of some problem changes more rapidly in one direction than in another direction (e.g. in anisotropic materials) [1]. From Fig. 1 we observe that the condition (2) is satisfied for a needle, splinter, and wedge tetrahedron. For the other degenerated tetrahedra from Fig. 1, the interpolation error may diverge in the H 1 -norm. On the other hand, the finite SKINNY TETRAHEDRA

spire / needle

splinter

spindle

spear

spike

FLAT TETRAHEDRA

wedge

spade

cap

Fig. 1 Classification of degenerated tetrahedra according to [7, 9]

sliver

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element method may converge in the H 1 -norm, see [12]. Hence, (2) represents only a sufficient (and not necessary) condition for the convergence of the finite element method, see [21–23, 25]. Let us point out that the two conditions in (2) are independent. For instance, for a sliver (and also cap) tetrahedron the condition γD ≤ γ0 does not hold while γF ≤ γ0 holds. On the other hand, for a spike tetrahedron γD ≤ γ0 holds and γF ≤ γ0 is violated. Since there are six dihedral angles of each tetrahedron and twelve angles between its adjacent edges, a direct generalization of (2) into higher dimensions would be technically quite complicated. Therefore, in the next section we introduce another concept which is based on the d-dimensional sine for d > 1. We will survey the main results from our previous paper [13], see also [5, 6].

2 The Maximum Angle Condition in Higher Dimensions Recall that a d-simplex S in Rd , d ∈ {1, 2, 3, . . . }, is the convex hull of d+1 vertices A0 , A1 , . . . , Ad that do not belong to the same (d − 1)-dimensional hyperplane, i.e., S = conv{A0 , A1 , . . . , Ad }. Let Fi = conv{A0 , . . . , Ai−1 , Ai+1 , . . . , Ad } be the facet of S opposite to the vertex Ai for i ∈ {0, . . . , d}. In 1978, Eriksson has introduced a generalization of the sine function to an arbitrary d-dimensional spatial angle, see [10, p. 74]. Definition 1 Let Aˆ i be the angle at the vertex Ai of the simplex S. Then d-sine of the angle Aˆ i for d > 1 is given by sind (Aˆ i |A0 A1 . . . Ad ) =

d d−1 (meas d S)d−1 (d − 1)! dj=0,j =i meas d−1 Fj

.

(3)

Remark 2 Let us show that d-sine is really a generalization of the classical sine function. Set d = 2 and consider an arbitrary triangle A0 A1 A2 . Denote by Aˆ 0 its angle at the vertex A0 . Then, obviously, meas 2 (A0 A1 A2 ) =

1 |A0 A1 ||A0 A2 | sin Aˆ 0 . 2

(4)

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Comparing this relation with (3), we find that sin Aˆ 0 = sin2 (Aˆ 0 , A0 A1 A2 ).

(5)

Definition 3 A family F = {Th }h→0 of partitions of a polytope into d-simplices is said to satisfy the generalized minimum angle condition if there exists C > 0 such that for any Th ∈ F and any S = conv{A0 , . . . , Ad } ∈ Th one has ∀ i ∈ {0, 1, . . . , d}

sind (Aˆ i |A0 A1 . . . Ad ) ≥ C > 0.

(6)

This condition is investigated in the paper [6]. It generalizes the well-known Zlámal minimum angle condition for triangles (see [8, 28, 29]), which is stronger than (1). Definition 4 A family F = {Th }h→0 of partitions of a polytope into d-simplices is said to satisfy the generalized maximum angle condition if there exists C > 0 such that for any Th ∈ F and any S = conv{A0 , . . . , Ad } ∈ Th one can always choose d edges of S, which, when considered as vectors, constitute a (higher-dimensional) angle whose d-sine is bounded from below by the constant C. Remark 5 From (4) and (5) we observe that the condition stated in Definition 4 for d = 2 is equivalent to the maximum angle condition (1). Remark 6 Let us show that in case of tetrahedra the validity of the maximum angle condition (2) implies the desired property in Definition 4, i.e. it is really a generalization. So, let (2) be valid for a given tetrahedron T . Then one can always find, see the proof of Theorem 7 in [19, pp. 517–518], three unit vectors t1 , t2 , and t3 parallel to three edges of T , so that the volume of the parallelepiped P(t1 , t2 , t3 ) generated by t1 , t2 , t3 is bounded from below by some constant c > 0. Now we use formula (3) to estimate the 3-dimensional sine of the angle formed by the vectors t1 , t2 , t3 as follows sin3 (t1 , t2 , t3 ) =

32 (meas 3 S(t1 , t2 , t3 ))2 2! 3j =0,j =i meas 2 Fj

≥

32 ( 16 meas 3 P(t1 , t2 , t3 ))2 2!( 12 )3

≥ c2 , (7)

where S(t1 , t2 , t3 ) is the tetrahedron made by t1 , t2 , t3 originating at 0, and the area of each of the three faces Fj involved is bounded from above by 1/2 due to the fact that t1 , t2 , t3 are unit vectors. The constant c can be, in fact, estimated from below by   π −γ 0 , sin γ0 , m := min sin 2 see [19, p. 518]. Now we present the main interpolation theorem of this paper using the standard Sobolev space notation.

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Theorem 7 Let F be a family of partitions of a polytope into d-simplices satisfying the generalized maximum angle condition. Then there exists a constant C > 0 such that for any Th ∈ F and any S ∈ Th we have v − πS v1,∞ ≤ ChS |v|2,∞

∀v ∈ C 2 (S),

(8)

where πS is the standard Lagrange linear interpolant and hS = diam S. For the proof see [13].

3 Examples Example 1 Denoting by A0 , A1 , A2 , A3 the vertices of spindle, spear, spike, spade, cap, or sliver tetrahedron from Fig. 1 in an arbitrary way, we find by (3) that sin3 (Aˆ 0 , A0 A1 A2 A3 ) → 0 as the discretization parameter tends to zero. The same is true for the splinter tetrahedron. However, by Definition 4 we may choose three edges which constitute a spatial angle whose 3-sine is bounded from below by a fixed constant C > 0. Choosing the two short edges of the splinter tetrahedron and one long edge, we find that the generalized maximum angle condition holds. Example 2 We show that if sin3 of two trihedral angles (cf. [11]) are the same numbers, then the magnitude of these solid angles in steradians need not be the same. Let A0 A1 A2 A3 be the √regular tetrahedron whose edges√have length 1. Then the altitude of its faces is 3/2 and the spatial altitude is 6/3. Consequently, from (3) for the trihedral angle Aˆ 0 we get sin3 (Aˆ 0 , A0 . . . A3 ) =

√ √ 3 6 2 4 3 ) √ 2!( 43 )3

32 ( 13

√ 4 3 . = 9

Consider √ now the tetrahedron with vertices B0 = (0, 0, 0), B1 = (1, 0, 0), B2 = √ ( 933 , 4 9 3 , 0), and B3 = (0, 0, 1). Then by (3) we also find that sin3 (Bˆ 0 , B0 . . . B3 ) =

32 ( 13 meas2 B0 B1 B2 )2 2! 14 meas2 B0 B1 B2

=

√ 4 3 . 9

Now by the Girard Theorem for the spherical excess we have (see [26, p. 83]) √ π 4 3 π Bˆ 0 = + + arcsin − π = 0.8785 . . . steradians, 2 2 9

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whereas 1 Aˆ 0 = 3 arccos − π = 0.5512 . . . steradians, 3 where all dihedral angles α of A0 A1 A2 A3 are ≈ 70.52◦, since cos α = 13 . Example 3 The sliver element from Fig. 1 can be narrowed as follows. For h → 0 consider the tetrahedron with vertices (h2 , 0, 0),

(0, 0, 0),

(0, h, 0),

(h2 , h, h3 ).

Example 4 A higher-dimensional example can be constructed in the following way. Consider positive numbers r1 , r2 , . . . , rd and a simplex with vertices A0 , A1 , . . . , Ad . We fix some number k so that 0 ≤ k ≤ d. The first k + 1 vertices of the simplex are defined as follows. Let A0 = (0, 0, . . . , 0, . . . , 0). Further, let A1 = (r1 , 0, . . . , 0, . . . , 0), A2 = (0, r2 , . . . , 0, . . . , 0), . . . , Ak = (0, . . . , 0, rk , 0, . . . , 0). The remaining vertices are: Ak+1 = (0, . . . , 0, rk+1 , 0, 0, . . . , 0), Ak+2 = (0, . . . , 0, rk+1 , rk+2 , 0, . . . , 0), Ak+3 = (0, . . . , 0, rk+1 , rk+2 , rk+3 , 0, . . . , 0), . . . , Ad = (0, . . . , 0, rk+1 , rk+2 , rk+3 , . . . , rd ). Therefore, for k = 0, we get the path-simplex, and for k = d the hypercube-corner simplex. Allowing some of the rk ’s to approach zero with different rates, in general, we get various degenerated simplices still satisfying the generalized maximum angle conditions. Acknowledgements The authors are indebted to Prof. Jan Brandts, Prof. Takuya Tsuchiya, and Prof. Jon Eivind Vatne for valuable suggestions. The third author was supported by RVO 67985840 of the Czech Republic and Grant no. 18-09628S of the Grant Agency of the Czech Republic.

References 1. T. Apel, Anisotropic Finite Elements: Local Estimates and Applications. Advances in Applied Mathematics (B.G. Teubner, Stuttgart, 1999) 2. T. Apel, M. Dobrowolski, Anisotropic interpolation with applications to the finite element method. Computing 47, 277–293 (1992)

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3. I. Babuška, A.K. Aziz, On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214–226 (1976) 4. R.E. Barnhill, J.A. Gregory, Sard kernel theorems on triangular domains with applications to finite element error bounds. Numer. Math. 25, 215–229 (1976) 5. J. Brandts, S. Korotov, M. Kˇrížek, On the equivalence of ball conditions for simplicial finite elements in Rd . Appl. Math. Lett. 22, 1210–1212 (2009) 6. J. Brandts, S. Korotov, M. Kˇrížek, Generalization of the Zlámal condition for simplicial finite elements in Rd . Appl. Math. 56, 417–424 (2011) 7. S.W. Cheng, T.K. Dey, H. Edelsbrunner, M.A. Facello, S.H. Teng, Sliver exudation, in Proceedings of 15-th ACM Symposium on Computational Geometry (1999), pp. 1–13 8. P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978) 9. H. Edelsbrunner, Triangulations and meshes in computational geometry. Acta Numer. 9, 133– 213 (2000) 10. F. Eriksson, The law of sines for tetrahedra and n-simplices. Geom. Dedicata 7, 71–80 (1978) 11. J.W. Gaddum, The sums of dihedral and trihedral angles in a tetrahedron. Am. Math. Mon. 59, 370–371 (1952) 12. A. Hannukainen, S. Korotov, M. Kˇrížek, The maximum angle condition is not necessary for convergence of the finite element method. Numer. Math. 120, 79–88 (2012) 13. A. Hannukainen, S. Korotov, M. Kˇrížek, Generalizations of the Synge-type condition in the finite element method. Appl. Math. 62, 1–13 (2017) 14. P. Jamet, Estimation de l’erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. Numér. 10, 43–60 (1976) 15. K. Kobayashi, T. Tsuchiya, On the circumradius condition for piecewise linear trian-gular elements. Jpn. J. Ind. Appl. Math. 32, 65–76 (2015) 16. K. Kobayashi, T. Tsuchiya, A priori error estimates for Lagrange intrepolation on triangles. Appl. Math. 60, 485–499 (2015) 17. K. Kobayashi, T. Tsuchiya, Extending Babuška-Aziz theorem to higher-order Largange interpolation. Appl. Math. 61, 121–133 (2016) 18. M. Kˇrížek, On semiregular families of triangulations and linear interpolation. Appl. Math. 36, 223–232 (1991) 19. M. Kˇrížek, On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29, 513–520 (1992) 20. M. Kˇrížek, P. Neittaanmäki, Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications (Kluwer Academic Publishers, Dordrecht, 1996) 21. V. Kuˇcera, A note on necessary and sufficient condition for convergence of the finite element method, in Proceedings of Conference on Applied Mathematics 2015, ed. by J. Brandts et al. (Institute of Mathematical, Prague, 2015), pp. 132–139 22. V. Kuˇcera, Several notes on the circumradius condition. Appl. Math. 61, 287–298 (2016) 23. V. Kuˇcera, On necessary and sufficient conditions for finite element convergence. Numer. Math (submitted). Arxiv 1601.02942 24. S. Mao, Z. Shi, Error estimates of triangular finite elements under a weak angle condition. J. Comput. Appl. Math. 230, 329–331 (2009) 25. P. Oswald, Divergence of FEM: Babuška-Aziz triangulations revisited. Appl. Math. 60, 473– 484 (2015) 26. K. Rektorys, Survey of Applicable Mathematics I (Kluwer Academic Publishers, Dordrecht, 1994) 27. J.L. Synge, The Hypercircle in Mathematical Physics (Cambridge University Press, Cambridge, 1957) 28. A. Ženíšek, The convergence of the finite element method for boundary value problems of a system of elliptic equations (in Czech). Appl. Math. 14, 355–377 (1969) 29. M. Zlámal, On the finite element method. Numer. Math. 12, 394–409 (1968)

Part XX

Discretizations and Solvers for Multi-Physics Problems

An Oscillation-Free Finite Volume Method for Poroelasticity Massimiliano Ferronato, Herminio T. Honorio, Carlo Janna, and Clovis R. Maliska

Abstract Biot’s equations of poroelasticity are numerically solved by an Elementbased Finite Volume Method (EbFVM). A stabilization technique is advanced to avoid spurious pressure modes in the vicinity of undrained conditions. Classical benchmark problems and more realistic 3D test cases are addressed. The results show that the proposed stabilization is able to eliminate the pressure instabilities preserving the solution accuracy.

1 Introduction The coupled equations of Biot’s poroelasticity, consisting of stress equilibrium and fluid mass balance in deforming porous media, still pose severe numerical challenges and recently different discretization methods have been advanced, see for instance [1–3] and references therein. In this work, they are numerically solved by an Element-based Finite Volume Method (EbFVM) [4]. This discretization technique is very flexible, as it allows for the use of unstructured grids made by elements of different types, and provides a conservative approach for both flow and mechanics. Similarly to other discretization techniques, numerical pressure instabilities can arise when undrained conditions take place. A stabilization procedure is advanced following the so-called Physical Influence Scheme (PIS), which was originally introduced for Navier-Stokes equations [5]. The numerical model is validated against classical analytical solutions and realistic three-dimensional problems, providing evidence that the proposed stabilization is able to eliminate the spurious pressure instabilities. M. Ferronato () · C. Janna University of Padova, Department of ICEA, Padova, Italy e-mail: [email protected] H. T. Honorio · C. R. Maliska Federal University of Santa Catarina, Florianopolis, Brazil © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_73

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2 Biot’s Poroelasticity Equations Let Ω ⊂ R3 and Γ denote the domain occupied by the porous medium and its 3 boundary, respectively, with x the position   vector in R . We denote time with t, belonging to an open interval I = 0, T of length T > 0. The boundary, Γ , is decomposed as Γ = Γu ∪ Γσ = Γp ∪ Γq , with Γu ∩ Γσ = Γp ∩ Γq = ∅, and n denotes its outer normal vector. A bar above a set identifies the union of the set with its boundary, e.g. Ω = Ω ∪ Γ . We assume quasi-static saturated single-phase flow of a slightly compressible fluid. The set of governing equations consists of a conservation law of linear momentum (equilibrium) and a conservation law of mass (continuity). The strong form of the initial/boundary value problem is stated as follows [6]: Given q : Ω × I → R, b : Ω × I → R3 , u¯ˆ : Γu × I → R3 , ¯t : Γσ × I → R3 , q¯ : Γq × I → R, p¯ : Γp × I → R, and p0 : Ω → R, find uˆ : Ω × 0, T → R3   and p : Ω × 0, T → R such that   ∇ s · σ − αp1 = b

on Ω × I

(equilibrium)

(1)

on Ω × I

(Hooke’s law)

(2)

on Ω × I

(continuity)

(3)

on Ω × I

(Darcy’s law)

(4)

vs = u˙ˆ

on Ω × I

(solid velocity)

(5)

uˆ = u¯ˆ

σ = C : ∇ uˆ s

  1 p˙ + ∇ · vf + αvs = q M −μκ −1 · vf = ∇p

  σ − αp1 · n = ¯t

on Γu × I

(boundary motion)

(6)

on Γσ × I

(boundary tractions)

(7)

−vf · n = q¯

on Γq × I

(boundary Darcy flux)

(8)

p = p¯

on Γp × I

(boundary pressure)

(9)

x∈Ω

(initial pressure)

p(x, 0) = p0 (x)

(10)

where uˆ and p are the displacement vector and the excess pore pressure relative to an initial reference state; σ is the rank-2 effective stress tensor; vf and vs are the fluid and solid grain velocity vectors; C is the drained rank-4 elasticity tensor, α is Biot’s coefficient, 1 is the rank-2 identity tensor, M is Biot’s modulus; κ is the rank2 intrinsic permeability tensor and μ is the viscosity; q is a volumetric source term; ∇·, ∇ and ∇ s are the divergence, the gradient and the symmetric gradient operator, ˙ , denotes a derivative with respect to time t. The respectively; the superposed dot, () subscript 0 is used to denote the reference state.

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2.1 Numerical Discretization The numerical solution of the initial boundary value problem (1)–(10) is obtained by an element-based finite volume method. Consider the following weak form developed by using = > a Petrov-Galerkin approach: ˆ Find u(t), p(t) ∈ Su (Ω) × Sp (Ω) such that ∀t ∈ [0, T ]:  Ω

 5  6 s s ˆ ω · ∇ · C : ∇ u − αp1 dΩ − ω · b dΩ = 0 Ω

∀ω ∈ Vu (Ω),



      1 k ˙ ω ∇ · α uˆ − ∇p ω ωq dΩ = 0 p˙ dΩ − dΩ + μ M Ω Ω Ω



(11)

∀ω ∈ Vp (Ω),

(12) where the trial and test spaces are given by   Su (Ω) = uˆ | uˆ ∈ [H 1 (Ω)]3, uˆ = u¯ˆ on Γu ,   Sp (Ω) = p | p ∈ H 1 (Ω), p = p¯ on Γp ,   Vu (Ω) = ω | ω ∈ [H01(Ω)]3 , ω = 0 on Γu ,   Vp (Ω) = ω | ω ∈ H01 (Ω), ω = 0 on Γp .

(13) (14) (15) (16)

Define a partition T h of Ω made of non-overlapping elements Ω e on which the corresponding discrete trial spaces Suh and Sph are introduced. The approximate pressure and displacement vector, ph and uh , read: ph (x) =

n

j =1

φj (x)pj = Np (x)p,

uh (x) =

n

φj (x)1uj = Nu (x)u,

(17)

j =1

where Np (x) and Nu (x) are the matrices of the interpolating trial functions φj (x), and p and u are the set of pressure and displacement values at the n nodes defined by T h . The test functions ωih (x) and ωhi (x) are defined over the dual partition of T h made by the cells Ωi associated to each node i (Fig. 1), such that: 9 1, if x ∈ Ωi h , ωhi (x) = ωih (x)1. (18) ωi (x) = 0, if x ∈ / Ωi The cell Ωi is also called control volume of node i and is such that the union of all control volumes is Ω and the intersection of two control volumes is the empty set. Introducing the approximations (17) into the weak forms (11)–(12) along with the test functions (18) and applying Gauss’ divergence theorem to the integrals arising

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Fig. 1 Example of test and trial functions for a 2-D unstructured grid

over each control volume yields the EbFVM differential-algebraic discretization of Biot’s poroelasticity equations: A A  C : ∇ s Nu u ds − α1Np p ds = b dΩ, i = 1, . . . , n (19) ∂Ωi

A ∂Ωi

∂Ωi

Ωi

  k 1 q dΩ, ∇Np p ds + Np p˙ dΩ = ∂Ωi μ Ωi M Ωi

A αNu u˙ ds −

i = 1, . . . , n

(20) Finally, the system (19)–(20) is integrated in time with a backward Euler scheme.

2.2 Stabilized EbFVM Formulation The use of co-located variables with the same approximation order for both displacements and pressure produces spurious oscillation modes when approaching undrained conditions [7]. In order to avoid such a drawback, several stabilization techniques were developed, especially in the field of finite element formulations, e.g., [8, 9], to allow for the stable use of equal-order interpolation pairs. The strategy proposed in this work consists of employing the so-called PIS procedure [5] for evaluating the displacements along the boundary of each control volume. The basic idea of PIS relies on mimicking an enrichment of the displacement approximation space by using information coming from the strong form of the equilibrium equation. Assume, for instance, that piecewise linear polynomials are ˆ ip ) on any point xip used as trial functions and consider the Taylor expansion of u(x located on the boundary ∂Ωi of the i-th control volume. Since linear trial functions can account for the first-order term of the Taylor expansion, the error introduced by the use of uh (xip ) obtained with (17) will be of the order of the product of the ˆ square of partition size, h2 , and the Laplacian of u:

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  ˆ ip ) = uh (xip ) + ϕ h2 ∇ 2 uˆ u(x

(21)

where ϕ is generally unknown. An estimate of ϕ can be obtained from the strong ˆ form of (1) written as a function of u:   G∇ 2 uˆ + (λ + G) ∇ ∇ · uˆ − α∇p = b

(22)

where λ and G are Lamé’s constants. Neglecting the second term in the left-hand side of (22) and the volume load b, the Laplacian of uˆ at xip can be written as a function of ∇p: ∇ 2 uˆ ip ≈

α ∇pip G

(23)

and the displacement along the boundary of the control volume Ωi can be therefore expressed as: ˆ ip ) ( Nu (xip )u + u(x

αL2d ∇Np (xip )p G

(24)

where Ld is a mesh-dependent parameter. The expression of Ld can be obtained from geometric considerations, according to the specific elements used in the partition T h . Introducing Eq. (24) in the numerical evaluation of the first boundary integral of Eq. (20) yields the stabilized version of the local mass balance: A

A αNu u˙ ds + ∂Ωi

A λ∇Np p˙ ds −

∂Ωi

∂Ωi

k ∇Np p ds + μ

 Ωi

1 Np p˙ dΩ = M

 (q + q) ˜ dΩ, Ωi

(25) where λ = α 2 L2d 1/G and q˜ is a consistency contribution required by the introduction of the stabilization term.

3 Numerical Results The proposed formulation was implemented with the aid of an in-house developed C++ object-oriented library, called EFVLib, which allows to handle 3D hybrid grids. To verify the formulation consistency, Mandel’s problem, a classical benchmark in poroelastic applications, is numerically solved and compared with the analytical solution. The geometry and boundary conditions applied to this problem are indicated in Fig. 2, while the solid and fluid parameters are those reported in [1]. A convergence analysis in the L2 -norm is performed for both tetrahedral and hexahedral elements. Simulations are carried out for different time step sizes, as indicated in Fig. 3. As expected, pressure and displacements are second-order

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Fig. 2 Mandel’s benchmark problem

Fig. 3 Mandel’s problem: convergence profiles in the L2 -norm for pressure (a, b) and displacements (c, d)

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Fig. 4 Mandel’s problem: stabilized and non-stabilized formulations at early time levels

accurate in space. However, numerical instabilities may appear in the pressure solution if no stabilization is applied. At early time levels, the silt geomechanical properties generate an undrained condition, with a sharp pressure gradient near the outer draining boundary. In this problem, such an occurrence is practically obtained using a time step size Δt = 1 s and taking the solution at t = 10 s. Figure 4 shows that the non-stabilized formulation (19)–(20) gives an oscillatory pressure profile for both tetrahedral and hexahedral grids, with spurious modes more pronounced for the latter grid. By distinction, the stabilized formulation (25) is able to eliminate such instabilities, at the cost of some numerical diffusion. The potential effectiveness of the proposed stabilized formulation is finally tested in two realistic applications with a layered heterogeneous medium, namely (i) a classical footing problem and (ii) a groundwater withdrawal case. In the first test case, the sudden application of a surface load causes an overpressure in the low permeable layers due to the instantaneous compressive volumetric strain. Figure 5 shows the classical checkerboard pattern arising in the upper silt layer if no stabilization is added. By distinction, the stabilized formulation is able to produce a much smoother solution with no spurious modes. Similarly, Fig. 6 provides the pore pressure solution obtained at early time levels due to a groundwater withdrawal from a confined sandy aquifer. The abstraction is simulated by introducing Peaceman’s well model and a hybrid discretization made by a combination of tetrahedrons, prisms and pyramids. A well-known consequence of this occurrence is the so-called Noordbergum effect, i.e., a slight overpressure arising in the low permeable layers confining the aquifer due to the medium deformation. The smooth simulation of this phenomenon is quite challenging, because of the small overpressure rise. The results in Fig. 6 shows that the stabilized EbFVM formulation is able to reproduce the Noordbergum effect quite satisfactorily.

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Fig. 5 Early pore pressure solution on a layered footing problem

Fig. 6 Noordbergum effect due to groundwater withdrawal from a confined sandy aquifer

4 Conclusions A stabilized EbFVM is developed for solving the coupled Biot’s poroelasticity equations. The stabilization term is introduced following the PIS procedure, originally developed for Navier-Stokes equations, and allows for effectively removing the spurious pressure modes arising in the proximity of undrained conditions. The proposed methodology has been validated against analytical solutions and through a convergence analysis. Realistic 3D examples are also used to test the model in challenging configurations. In all cases the physical processes are correctly captured with a smooth pressure solution throughout the layered domain. Acknowledgements This work has been developed within the international cooperation activities sponsored by the Science without Border Program of CNPq/Brazil.

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References 1. N. Castelletto, J.A. White, M. Ferronato, Scalable algorithms for three-field mixed finite element coupled poromechanics. J. Comput. Phys. 327, 894–918 (2016) 2. J.M. Nordbotten, Stable cell-centered finite volume discretization for Biot equations. SIAM J. Numer. Anal. 54, 942–968 (2016) 3. J.W. Both, M. Borregales, J.M. Nordbotten, K. Kumar, F.A. Radu, Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017) 4. A. dal Pizzol, C.R. Maliska, A finite volume method for the solution of fluid flows coupled with the mechanical behavior of compacting porous media, in Porous Media and its Applications in Science, Engineering and Industry, vol. 1453 (2012), pp. 205–210 5. G.E. Schneider, M.J. Raw, Control volume finite-element method fot heat transfer and fluid flow using co-located variables - 1. Computational procedure. Numer. Heat Trans. 11, 363–390 (1987) 6. H.F. Wang, Theory of Linear Poroelasticity (Princeton University Press, Princeton, 2000) 7. S.Y. Yi, A study of two modes of locking in poroelasticity. SIAM J. Num. Anal. 55, 1915–1936 (2017) 8. J.A. White, R.I. Borja, Stabilized low-order finite elements for coupled solid-deformation/fluiddiffusion and their application to fault zone transients. Comput. Methods Appl. Mech. Eng. 197, 4353–4366 (2008) 9. C. Rodrigo, F.J. Gaspar, X. Hu, L.T. Zikatanov, Stability and monotonicity for some discretizations of Biot’s consolidation model. Comput. Methods Appl. Mech. Eng. 298, 183–204 (2016)

Numerical Investigation on the Fixed-Stress Splitting Scheme for Biot’s Equations: Optimality of the Tuning Parameter Jakub W. Both and Uwe Köcher

Abstract We study the numerical solution of the quasi-static linear Biot equations solved iteratively by the fixed-stress splitting scheme. In each iteration the mechanical and flow problems are decoupled, where the flow problem is solved by keeping an artificial mean stress fixed. This introduces a numerical tuning parameter which can be optimized. We investigate numerically the optimality of the parameter and compare our results with physically and mathematically motivated values from the literature, which commonly only depend on mechanical material parameters. We demonstrate, that the optimal value of the tuning parameter is also affected by the boundary conditions and material parameters associated to the fluid flow problem suggesting the need for the integration of those in further mathematical analyses optimizing the tuning parameter.

1 Introduction The coupling of mechanical deformation and fluid flow in porous media is relevant in many applications ranging from environmental to biomedical engineering. In this paper, we consider the simplest possible fully coupled model given by the quasi-static Biot equations [3], coupling classical and well-studied subproblems from linear elasticity and single phase flow in fully saturated porous media. Due to the complex structure of the coupled problem, the development of monolithic solvers is not trivial and topic of current research. Hence, instead of developing new simulation tools for the coupled problem, due to their simplicity and

J. W. Both () University of Bergen, Bergen, Norway e-mail: [email protected] U. Köcher Helmut-Schmidt-University, University of the Federal Armed Forces Hamburg, Hamburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_74

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flexibility, splitting methods have been very attractive recently allowing the use of independent, tailored simulators for both subproblems. Among various iterative splitting schemes, one of the most prominent schemes is the physically motivated fixed-stress splitting scheme [8], based on solving sequentially the mechanics and flow problems while keeping an artificial mean stress fixed in the latter. From an abstract point of view, the splitting scheme is a linearization scheme employing positive pressure stabilization, a concept also applied for the linearization of other problems as, e.g., the Richards equation [6]. Addressing the physical formulation, the definition of the artificial mean stress includes a user-defined tuning parameter. It can be chosen a priori such that the resulting fixed-stress splitting scheme is unconditionally stable in the sense of a von Neumann stability analysis [5] and it is globally contractive [1, 2, 4, 7]. Suggested values for the tuning parameter from literature are either physically motivated [5] or mathematically motivated [1, 2, 4, 7]. The latter works prove theoretically global contraction of the scheme, allowing to optimize the resulting theoretical contraction rate, and hence, proposing a value for the tuning parameter with suggested, better performance than for the physically motivated parameters. In general, the suggested values for the tuning parameter given in the literature do not necessarily yield a minimal number of iterations, which for strongly coupled problems is crucial. For increasing coupling strength the problem becomes more difficult to solve, and the performance of the fixedstress splitting scheme is very sensitive to the choice of the tuning parameter. The mentioned tuning parameters depend solely on mechanical material parameters. However, practically, it is known that the physical character of the problem governed by boundary conditions also affects the performance of the scheme [5], introducing the main difficulty finding an optimal tuning parameter. We note that the fixedstress splitting scheme can also be applied as a preconditioner for Krylov subspace methods solving Biot’s equation in a monolithic fashion. In this case, performance is less sensitive with respect to the tuning parameter. In this work, we investigate numerically whether the optimal tuning parameter obtained by simple trial and error is closer related to the mathematically or the physically motivated parameters. Furthermore, we investigate whether the optimal tuning parameter is also dependent on more than only mechanical properties. For this purpose, we perform a numerical study enhancing a test case from [2] and measure performance of the fixed-stress splitting scheme for different tuning parameters. Our main results are: • Boundary conditions affect the optimality of the tuning parameter. • Fluid flow parameters affect the optimality of the tuning parameter. • Both should be included in the mathematical analysis allowing to derive theoretically an optimal tuning parameter.

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2 Linear Biot’s Equations We consider the quasi-static Biot equations [3], modeling fluid flow in a deformable, linearly elastic porous medium Ω ⊂ Rd , d ∈ {1, 2, 3}, fully saturated by a slightly compressible fluid. Using mechanical displacement u, fluid pressure p and fluid flux q as primary variables, on the space-time domain Ω × (0, T ), the governing equations written in a three-field formulation read   −∇ · 2με(u) + λ∇ · uI − αpI = ρb g,   p + α∇ · u + ∇ · q = 0, ∂t M η q + ∇p = ρf g. k

(1) (2) (3)

Equation (1) describes balance of momentum at each time, Eq. (2) describes mass  conservation and Eq. (3) describes Darcy’s law. Here, ε(u) = 12 ∇u + ∇u? is the linearized strain tensor, μ, λ are the Lamé parameters (equivalent to Young’s E Eν and λ = (1+ν)(1−2ν) ), α is the Biot modulus E and Poisson’s ratio ν via μ = 2(1+ν) coefficient, M is the Biot modulus, ρf is the fluid density, ρb is the bulk density, k is the absolute permeability, η the fluid viscosity and g is the gravity vector. In this work, we assume isotropic, homogeneous materials, i.e., all material parameters are constants. The system (1)–(3) is closed by postulating initial conditions u = u0 , p = p0 on Ω × {0}, satisfying Eq. (1), and boundary conditions u = uD on ΓD,m × (0, T ), (2με(u) + λ∇ · uI − αpI ) · n = σ N on ΓN,m × (0, T ), p = pD on ΓD,f × (0, T ), q · n = qN on ΓN,f × (0, T ) on partitions ΓD,! ∪ ΓN,! = ∂Ω, ! ∈ {m, f }, where n is the outer normal on ∂Ω. Here and in the remaining paper, we omit introducing a corresponding variational formulation and suitable function spaces, as they appear naturally. For details, we refer to our works [2, 4].

3 Fixed-Stress Splitting Scheme We solve the coupled Biot Eqs. (1)–(3) iteratively using the fixed-stress splitting scheme [8], which decouples the mechanics and fluid flow problems. Each iteration, defining the approximate solution (u, p, q)i , i ∈ N, consists of two steps. First, the flow problem is solved assuming a fixed artificial, volumetric stress σv = Kdr ∇ · u − αp, where Kdr is a tuning parameter, which will be discussed in the scope of this paper: Given (u, p, q)i−1 , find (p, q)i satisfying 

α2 1 + M Kdr

 ∂t pi + ∇ · q i =

α2 ∂t pi−1 − α∂t ∇ · ui−1 , Kdr

(4)

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η i q + ∇pi = ρf g, k

(5)

including corresponding initial and boundary conditions. Second the mechanics problem is solved with updated flow fields: Find ui satisfying corresponding boundary conditions and   −∇ · 2με(ui ) + λ∇ · ui I − αpi I = ρb g.

(6)

3.1 The Tuning Parameter Kdr The fixed stress splitting scheme can be interpreted as a two block Gauss-Seidel method with an educated predictor for mechanical displacement used in the solution of the flow problem. More precisely, the mechanics problem (1) is solved inexactly for the volumetric deformation by reduction to the one-dimensional equation Kdr ∇ · (ui − ui−1 ) − α(pi − pi−1 ) = 0

(7)

and inserted into the flow Eq. (2). In the special case of nearly incompressible materials, i.e., μ/λ → 0 or ν → 0.5, Eq. (1) yields ∇∂t (λ∇ · u − αp) ≈ 0. Hence, we expect the ansatz (7) to be nearly exact for Kdr = λ, yielding a suitable tuning parameter for nearly incompressible materials. An exact inversion of Eq. (1) for the volumetric deformation would be given by the divergence of a Green’s function and thus would be defined locally and depend on fluid pressure, geometry, material parameters and boundary conditions, both associated with the mechanical subproblem. However, due to lack of a priori knowledge and simplicity, in the literature, the considered inexact inversion includes only fluid pressure and mechanical material parameters introduced via the tuning 1D , K 2D K 3D , cf. [5], K 2×λ , cf. [1, 2, 7], and parameter Kdr . Selected values are Kdr dr dr dr 2×dD Kdr , cf. [4, 7], defined by !

d D Kdr =

2μ + λ, d ! ∈ {1, 2, 3}, d! !

2×λ Kdr = 2λ,

2×dD dD Kdr = 2 Kdr .

d D is purely physically motivated and equals the bulk modulus of The choice Kdr ! a d -dimensional material. Independent of the spatial dimension d, for uniaxial compression, biaxial compression or general deformations, we choose d ! = 1, 2, 3, respectively. Following [5], if not known better a priori, choose d ! = d. EquadD corresponds to fixing the trace of the physical, poroelastic tion (7) with Kdr = Kdr 2×λ 2×dD and Kdr have resulted from optimization of the stress tensor. The choices Kdr obtained theoretical contraction rate. Those analyses have in common that global 2×λ 2×dD convergence is guaranteed for 0 ≤ Kdr ≤ Kdr and 0 ≤ Kdr ≤ Kdr ,

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hence, the latter also covers convergence for the physical choices. Additionally, the analyses indicate that the larger Kdr the faster convergence, suggesting that the mathematically motivated parameters should yield better performance than the physically motivated parameters. In the following section we investigate this statement numerically.

4 Numerical Study: Optimal Tuning Parameter Kdr We perform a numerical parameter study analyzing the optimality of the tuning parameter Kdr in the view of the performance of the fixed-stress splitting scheme measured in terms of number of fixed-stress iterations. Inspired by a test setting from [2], we consider four test cases based on an L-shaped domain Ω = (−0.5, 0.5)2 \ [0, 0.5]2 ⊂ R2 in the time interval (0, 0.5) under two sets of boundary conditions, identified by test cases 1a/b/c, and test case 2, cf. Fig. 1. For all cases, vanishing initial conditions p0 = 0 and u0 = 0 are prescribed. A traction σ N (t) = (0, −hmax · 256 · t 2 · (t − 0.5)2 ) is applied on the top with hmax suitably chosen. Additionally, we prescribe pD = 0 on the top and qN = 0 on the remaining boundary, zero normal displacement and homogeneous tangential traction on the left and bottom side, and a homogeneous traction on the lower right side. In the test cases 1a/b/c, we also prescribe zero normal displacement on the cut, whereas in test case 2, a homogeneous traction is applied on the cut. The different sets of boundary conditions result in two different physical scenarios. Despite the twodimensional, non-symmetric geometry, the test cases 1a/b/c are closely related to a classical uniaxial compression, whereas the second test case describes a true twodimensional deformation.

(a)

(b)

Fig. 1 Geometry and boundary conditions employed in the numerical study. (a) Test cases 1a/b/c. (b) Test case 2

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For the numerical discretization of Eqs. (1)–(3) in space we use piecewise linear, piecewise constant and lowest order Raviart-Thomas finite elements for u, p and q, respectively, defined on a structured, quadrilateral mesh with 16 elements per x- and y-direction. Additionally, for the time discretization we use the backward Euler method with a fixed time step size Δt = 0.01. We solve the discretized Eqs. (1)–(3) using the fixed-stress splitting scheme (4)–(6) for a range of tuning 1D for test cases 1a/b/c and K = ωK 2D for test case 2, ω ∈ parameters Kdr = ωKdr dr dr {0.5, 0.51, . . . , 1.3}, and present the accumulated number of iterations required for ! to convergence of the fixed-stress splitting scheme. In the following, we denote Kdr be the Kdr yielding minimal number of iterations for a single test case. As stopping criterion, we employ the discrete Euclidean norm  · l 2 for the algebraic increments between two successive solution vectors for each of the unknown variables u, p and q. The numerical examples are implemented using the deal.II library (and are verified by an implementation using the DUNE library).

4.1 Test Case 1a: Effective 1d Deformation We consider a fixed Young’s modulus E = 100 [GPa] and a varying Poisson’s ratio ν ∈ {0.01, 0.1, 0.2, 0.3, 0.4, 0.49}. Moreover, we fix k = 100 [mD], η = 1 [cP], α = 0.9, M = 100 [GPa], g = 0 [m/s2 ]. On top, we apply the normal force σ N with hmax = 10 [GPa]. The number of required fixed stress iterations in relation to 1D is suitable the tuning parameter is displayed in Fig. 2a. Here, the choice Kdr = Kdr independent of the Poisson ratio, confirming that the problem is essentially driven by uniaxial compression.

iterations

600

= = = = = =

0.01 0.10 0.20 0.30 0.40 0.49

ν ν ν ν ν ν

2000 iterations

ν ν ν ν ν ν

800

= = = = = =

0.01 0.10 0.20 0.30 0.40 0.49

1000 400

200

0.6

0.8 1 1D ω = Kdr /Kdr

(a)

1.2

0.6

0.8 1 1D ω = Kdr /Kdr

1.2

(b)

Fig. 2 Total number of fixed stress iterations vs. tuning parameter. (a) Test case 1a (E = 100 GPa). (b) Test case 1b (E = 1 GPa)

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4.2 Test Case 1b: Soft Material We modify test case 1a and consider now a softer material with Young’s modulus E = 1 [GPa], yielding a stronger coupling of the mechanics and flow problem. On top, we apply the normal force hmax = 0.1 [GPa], resulting in a comparable maximal displacement of the top boundary. Apart from that, we use the same parameters as in test case 1a. The number of required fixed stress iterations in relation to the tuning parameter is displayed in Fig. 2b. Compared to the previous test case, the nature of the mechanical problem becomes more two-dimensional, ! lying between K 1D and K 2D . More precisely, K ! /K 1D depends indicated by Kdr dr dr dr dr 1D ≈ K 2D is a suitable on ν. Only for nearly incompressible materials Kdr = Kdr dr choice. Hence, we see that the mechanical character of the problem can vary with changing material parameters but fixed boundary conditions.

4.3 Test Case 1c: Influence of Flow Parameters We consider a particular example of test case 1b (E = 1 [GPa], ν = 0.01) for varying permeability k ∈ {1e-1, 1e0, . . . , 1e3} [mD]. Apart from that, we use the same parameters as in test case 1b. The number of required fixed stress iterations in relation to the tuning parameter is displayed in Fig. 3a. We observe that although all possible mechanical input data is fixed (mechanical boundary conditions and mate! is in general also dependent on rial parameters), the optimal tuning parameter Kdr material parameters associated with the flow problem. In particular, for decreasing ! increases towards K 1D . permeability, the optimal Kdr dr

k = 1e-1 mD k = 1e0 mD k = 1e1 mD k = 1e2 mD k = 1e3 mD

4000

ν = 0.01 ν = 0.10 ν = 0.20 ν = 0.30 ν = 0.40 ν = 0.49

800 iterations

iterations

6000

600

400

2000 200 0.6

0.8

1

1D ω = Kdr /Kdr

(a)

1.2

0.6

0.8

1

1.2

2D ω = Kdr /Kdr

(b)

Fig. 3 Total number of fixed stress iterations vs. tuning parameter. (a) Test case 1c (E = 1 GPa, ν = 0.01). (b) Test case 2 (E = 100 GPa)

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4.4 Test Case 2: True 2d Deformation We consider test case 2 with same parameters as in test case 1a, changing only from Dirichlet to Neumann boundary conditions on single parts of the boundary. The number of required fixed stress iterations in relation to the tuning parameter ! is is displayed in Fig. 3b. We observe that the optimal tuning parameter Kdr 2D essentially equal to Kdr , indicating that, the boundary conditions generate a true !. two-dimensional deformation and determine fully the optimal tuning parameter Kdr

5 Conclusion ! is not trivial. It In general, the a priori choice of an optimal tuning parameter Kdr is depending on the coupling strength which itself depends on various material and discretization parameters. In the above test cases we have observed: 2×λ 2×2D and Kdr have in • The mathematically motivated tuning parameters Kdr general not in the slightest shown to be optimal, which can be confirmed without ! has difficulty for e.g. nearly incompressible materials. Instead, the optimal Kdr !D d been closer related to the physically motivated parameters Kdr . • Mechanical boundary conditions are able to determine essentially the physics ! , cf. test case 1a/2, which is consistent with [5]. and define the optimal Kdr ! , cf. test However, they do not necessarily solely determine the optimal Kdr case 1b. Furthermore, although the fixed-stress approach is based on the inexact !, inversion of the mechanics Eq. (1), also fluid flow properties can influence Kdr cf. test case 1c. We note, that in general one can expect a further dependence on the domain and boundary size, which has not been studied here. • As expected, for nearly incompressible materials, a suitable tuning parameter is d ! D ≈ 1 K 2×λ ≈ 1 K 2×2D , cf. Sect. 3.1. given by Kdr = λ ≈ Kdr 2 dr 2 dr ! does not solely depend on the Lamé All in all, the optimal tuning parameter Kdr parameters, but also other physical material parameters, the physical character of the problem and numerical discretization parameters. The latter has been also studied ! should also include the effect of in [2]. All in all, future theoretical analysis of Kdr ! those additional parameters. However, in practice, we expect the dependence of Kdr on the input data of the problem to be complex, and therefore plan to investigate !. adaptive techniques for determining a locally defined approximation of Kdr

Acknowledgements The research contribution of the second author was partially supported by E.ON Stipendienfonds (Germany) under the grant T0087 29890 17 while visiting University of Bergen.

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References 1. M. Bause, Iterative coupling of mixed and discontinuous Galerkin methods for poroelasticity, in Numerical Mathematics and Advanced Applications – ENUMATH 2017 (Springer, Cham, 2018) pp. ??-?? 2. M. Bause, F. Radu, U. Köcher, Space-time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods Appl. Mech. Eng. 320, 745–768 (2017) 3. M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941) 4. J.W. Both, M. Borregales, J.M. Nordbotten, K. Kumar, F.A. Radu. Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017) 5. J. Kim, H.A. Tchelepi, R. Juanes, Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits. Comput. Methods Appl. Mech. Eng. 200, 1591–1606 (2011) 6. F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20, 341–353 (2016) 7. A. Mikeli´c, M.F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17, 451–461 (2013) 8. A. Settari, F.M. Mourits, A coupled reservoir and geomechanical simulation system. Soc. Pet. Eng. J. 3, 219–226 (1998)

Numerical Simulation of Biofilm Formation in a Microchannel David Landa-Marbán, Iuliu Sorin Pop, Kundan Kumar, and Florin A. Radu

Abstract The focus of this paper is the numerical solution of a mathematical model for the growth of a permeable biofilm in a microchannel. The model includes water flux inside the biofilm, different biofilm components, and shear stress on the biofilmwater interface. To solve the resulting highly coupled system of model equations, we propose a splitting algorithm. The Arbitrary Lagrangian Eulerian (ALE) method is used to track the biofilm-water interface. Numerical simulations are performed using physical parameters from the existing literature. Our computations show the effect of biofilm permeability on the nutrient transport and on its growth.

1 Model Equations Oil is one of the principal energy resources. Therefore it is essential to develop efficient extraction techniques with minimal environmental impact. One good candidate is the bio-plug, were bacteria are brought in the high permeable zones. The bacterial growth reduces the permeabilities of these zones, so we can reach and recover the oil in the less permeable zones. Our motivation is to develop mathematical models that have higher fidelity to the experiments and thus describe this mechanism in a better manner. A biofilm is an assemblage of surface-associated microbial cells that is enclosed in an extracellular polymeric substance matrix (EPS) [5]. The proportion of EPS in biofilms can comprise between 50–90% of the total organic matter [5, 16]. Since

D. Landa-Marbán () · K. Kumar · F. A. Radu University of Bergen, Bergen, Norway e-mail: [email protected]; [email protected]; [email protected] I. S. Pop University of Bergen, Bergen, Norway Faculty of Sciences, Hasselt University, Diepenbeek, Belgium e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_75

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Fig. 1 The strip, coordinate system and biofilm thicknesses (left). Zoom into the domain close to the strip boundary (right)

water is the largest component of the biofilm [7], it is important to consider the water flux inside the biofilm in our model. In this section, we describe the model equations for biofilm formation in a strip considering a permeable biofilm [8]. The mathematical model considered here follows ideas from [1, 14, 15]. We consider a simplified domain, a two-dimensional pore of length L and width W , denoted by Ω := (0, L)×(0, W ). The biofilm has four components: water, EPS, active, and dead bacteria (j = {w, e, a, d}). Let θj (t, x) and ρj denote the volume fraction and the density of species j . The biomass phases and water are assumed to be incompressible and that the biofilm layer is attached to the pore walls. Further the volumetric fraction of water is taken as constant ∂t θw = 0. As observed in Fig. 1, the biofilm layers has a thickness d = d(x, t) that changes in time and depends on the location at the pore wall. For simplicity we assume symmetry in the y-direction. With this we distinguish the following: Ωw (t) :={(x, y) ∈ R2 | x ∈ (0, L), y ∈ (d(x, t), W/2)} − the water domain, Ωb (t) :={(x, y) ∈ R2 | x ∈ (0, L), y ∈ (0, d(x, t))} − the biofilm layer, Γwb (t) :={(x, y) ∈ R2 | x ∈ (0, L), y = d(x, t)} − the biofilm-water interface. Γib , Γiw , Γob , and Γow are the inflow and outflow boundaries in the water and biofilm. Γs is the pore wall boundary. The water flow is described by the Stokes model μΔqw = ∇pw ,

∇ · qw = 0 in Ωw (t),

(1)

where μ is the viscosity, pw is the water pressure, and qw is the water velocity. For the water flux in the biofilm one has the Darcy law and mass conservation ∇ · qb = 0,

qb = −

kθw ∇pb μ

in Ωb (t),

(2)

where qb and pb are the velocity and pressure of the water in the biofilm respectively, and k is the permeability of the biofilm. At the interface, one has the water mass balance, the equilibrium of normal forces across Γwb (t), and the

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well-known Beavers–Joseph–Saffman boundary condition [2, 10, 13] (qw − qb ) · ν = νn (1 − θw ) on Γwb (t), −νν · (−pw 1 + μ(∇qw +

∇qTw ))

· ν = pb α −νν · (−pw 1 + μ(∇qw + ∇qTw )) · τ = √ qw · τ k

(3)

on Γwb (t),

(4)

on Γwb (t).

(5)

Here α is the Beavers–Joseph constant, νn is the normal velocity of the interface, ν is the unit normal pointing into the biofilm, and τ is the unit tangential vector, ν = (∂x d, −1)T / 1 + (∂x d),

τ = (1, ∂x d)T / 1 + (∂x d).

(6)

For each biofilm component (l ∈ {e, a, d}) one has mass conservation [1] ∂ρl θl + ∇ · (uρl θl ) = Rl ∂t

in Ωb (t)

(7)

where Rl are the rates on the biomass volume fractions, ρl is the density of the l component, and u is the velocity of the biomass. Following [6] one has u = −∇Φ

in Ωb (t),

(8)

where Φ is the growth velocity potential satisfying − ∇2Φ =

1 Rl 1 − θw ρl

in Ωb (t).

(9)

l

We adopt Monod-type reaction rates cb − kres θa ρa , K + cb cb Re = Ye μn θa ρa , K + cb

Ra = Ya μn θa ρa

(10) (11)

Rd = kres θa ρa ,

(12)

where Ye and Ya are yield coefficients, kres is the decay rate, and K is the Monod half nutrient velocity coefficient. Denoting by cw and cb the nutrient concentration in the water and biofilm respectively, one has ∂t cw + ∇ · J w = 0, J w = −D∇cw + qw cw

in Ωw (t),

(13)

∂t (θw cb ) + ∇ · J b = Rb , J b = −θw D∇cb + qb cb

in Ωb (t),

(14)

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where D, J w , J b , and Rb are the nutrient diffusion coefficient in water, the nutrient flux in the water and biofilm, respectively, and the nutrient reaction term. For the latter one has Rb = −μn θa ρa

cb K + cb

(15)

where μn is the maximum rate of nutrient. The coupling conditions are given by the Rankine-Hugoniot condition and the continuity of the concentrations: (JJ b − J w ) · ν = νn (θw cb − cw ) θ w cb = cw

on Γwb (t),

(16)

on Γwb (t).

(17)

Accounting for the biofilm dettachment due to shear stress, the normal velocity of the interface is given by [15] ⎧ ⎪ d = W, ⎪ ⎨[νν · u]+ , T T νn = ν · u + kst r μ||(1 − ν ν )(∇qw + ∇qw )νν ||, d < 0 < d < W, ⎪ ⎪ ⎩0, d = 0,

(18)

where ksrt is a constant for the shear stress and ∂t d νn = − . 1 + (∂x d)2

(19)

1.1 Boundary and Initial Conditions At the inflow we specify the pressure and nutrient concentration and we consider homogeneous Neumann condition for the growth velocity potential and volumetric fractions. At the outflow, we specify the pressure and take Neumann conditions for the concentrations, growth velocity potential, and volumetric fractions. At the substrate, we choose no-flux for the water, nutrients, and volumetric fractions and homogeneous Neumann conditions for the growth potential. We prescribe slip boundary condition for the water flux and homogeneous Neumann conditions for the nutrients. At the biofilm-water interface, we set the growth potential to zero and we consider homogeneous Neumann conditions for the volumetric fractions. The model is completed by the given initial data for the pressure, volumetric fractions, biofilm thickness, growth potential, and nutrient concentration.

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2 Implementation The challenging aspects in the numerical solution of the mathematical model are due to the coupling of the non-linearities and the existence of a free boundary (the biofilm-water interface) as it needs to be determined as part of the solution process. We use an ALE method for tracking the free boundary [4]. This method offers an accurate representation of the interface and allows to write the mass transport equations across the interface without any modification. However, the mesh will become time dependent. There are plenty of approaches for solving coupled system of partial differential equations; for example, implicit iterative methods [9, 11, 12]. We use backward Euler for the time discretization and linear Galerkin finite elements for the discretization in space. To solve the system of equations, we split the solution process into three steps: the flux (pw , q w , pb , q b ), the nutrient (nw , nb ), and the biofilm (θe , θa , θd , φ, d) variables. Firstly, a damped version of Newton’s method is used to find the solution of the flux variables considering the initial conditions of all variables. Secondly, we find the solution of the nutrient variables considering the previous solution of the flux variables and initial conditions of the remaining variables. After, we find the solution of the biofilm variables considering the previous solution of the flux and nutrient variables and initial conditions of the biofilm variables. We iterate between the three steps until the error E (the difference between successive values of the solution) drops below a given tolerance . If the error criterion (E < ) is satisfied before N iterations (n < N), we move to the next time step and solve again until a given final time T . Figure 2 shows the flowchart for the computational scheme. The infinity norm is used for the vector norm computation in the shear stress equation. The model equations are implemented in the commercial software COMSOL Multiphysics (COMSOL 5.2a, Comsol Inc, Burlington, MA, www.comsol.com).

Fig. 2 Flow tree for solving the mathematical model

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3 Numerical Simulation of Biofilm Formation in a Strip In Table 1, the values of parameters for the numerical simulations are presented. These values are taken from [1], where the authors built a model for heterogeneous biofilm development. The dimensions of the strip and the injected nutrient concentration are also taken from [1], with values L = 600 × 10−6 m and W = 600 × 10−6 m. From previous studies, we set the biofilm permeability k = 1 × 10−10 m2 [3], the Beavers–Joseph constant α = 0.1 [2], and the stress coefficient kst r = 9 × 10−11 m/(s Pa) [8]. The initial biofilm thickness is d0 = 30 × 10−6 m. The volume fraction of water in the biofilm is 50%. The left half of the biofilm (0 < x < L/2) consists of 50% active bacteria, while the other half consists of 25% active bacteria and 25% EPS. Nutrients are injected at a pressure of P = 0.5 Pa and a concentration of C = 0.001 kg/m3 . We run the numerical simulations for 300 h. In Fig. 3, the growth velocity potential and volumetric fractions are shown. The biofilm has a greater active bacterial volume fraction on the side where the nutrients are injected, leading to a greater biofilm growth in comparison with the biofilm on the right hand side. After 300 h, we observe that the lower part of the biofilm is approximately 92% formed by water, EPS, and dead bacteria while only 8% is formed by active bacteria.

Table 1 Table of model parameters for the verification study [1] Symbol μn kres Ya Ye

Value 1.1 × 10−5 /s 2 × 10−6 /s 0.22 0.4

Symbol ρw ρe ρa ρd

Value 1000 kg/m3 60 kg/m3 60 kg/m3 60 kg/m3

Symbol K D μ

Fig. 3 Growth velocity potential (left) and total volume fractions (right)

Value 1 × 10−4 kg/m3 9.6 × 10−10 m2 /s 1 × 10−3 Pa s

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3.1 Parametric Studies Figure 4 shows the sensitivity analysis for the inlet pressure and injected nutrients. We observe that as the pressure increases, the biofilm growth is less due to the shear force. Regarding the nutrient concentration, we notice that increasing the nutrient concentration by a factor of 10, the biofilm has a faster growth. On the other hand, if we decrease the nutrient concentration by a factor of 10, the biofilm grows slowly and there is almost no biofilm growth on the outflow part. In order to quantify the impact of the biofilm permeability, we perform numerical simulations with different values of k. For the high permeable biofilm, we set k = 1 × 10−8 m2 while for the less permeable biofilm we set k = 1 × 10−12 m2 . We run again the numerical simulations for 300 h. Figure 5 shows the modeled spatial distribution of biofilm height for the different permeabilities. We observe that for the less permeable biofilm, the biofilm formation is slower. The difference in the results is related to two different phenomena. First, the nutrient transport inside the less permeable biofilm is dominated by diffusion, while the transport in the high permeable biofilm is also due to convection. The second reason is that the coupling condition at the interface results in greater water velocity gradient for the impermeable biofilm, therefore the shear stress is larger.

Fig. 4 Biofilm height at various pressures (left) and nutrient concentrations (right)

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Fig. 5 Comparison of the permeable effects on the dynamic biofilm development

4 Conclusions In this work we have considered a mathematical model for permeable biofilm in a microchannel. The solution algorithm used for numerical simulations for the considered model has been presented. A sensitivity analysis is performed. We remark that if the flow rate is low, then the flux inside the biofilm can be neglected. However, for higher flow rates we must consider the effects of the flow inside the biofilm, that affects the transport of nutrients and the flux velocity value at the interface. The latter influence the biofilm thickness via the stress force. Acknowledgements The work of DLM and FAR was partially supported by the Research Council of Norway through the projects IMMENS no. 255426 and CHI no. 255510. ISP was supported by the Research Foundation-Flanders (FWO) through the Odysseus programme (G0G1316N) and Statoil through the Akademia grant.

References 1. E. Alpkvist, I. Klapper, A multidimensional multispecies continuum model for heterogeneous biofilm development. Bull. Math. Biol. 69, 765–789 (2007) 2. G. Beavers, D. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(1), 197–207 (1967) 3. W. Deng et al., Effect of permeable biofilm on micro-and macro-scale flow and transport in bioclogged pores. Environ. Sci. Technol. 47(19), 11092–11098 (2013) 4. J. Donea et al., Arbitrary Lagrangian–Eulerian methods. Encycl. Comput. Mech. 1(14), 413– 437 (2004)

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5. R.M. Donlan, Biofilms: microbial life on surfaces. Emerg. Infect. Dis. 8(9), 881–890 (2002) 6. R. Duddu, D.L. Chopp, B. Moran, A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment. Biotechnol. Bioeng. 103, 92–104 (2009) 7. H.C. Flemming, J. Wingender, The biofilm matrix. Nat. Rev. Microbiol. 8(9), 623–633 (2010) 8. D. Landa-Marbán et al., A pore-scale model for permeable biofilm: numerical simulations and laboratory experiments (2018, under review) 9. F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(341), 341–353 (2016) 10. A. Mikelic, W.Jäger, On the interface condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60(4), 1111–1127 (2000) 11. I.S. Pop, F. Radu, P. Knabner, Mixed finite elements for the Richards equation: linearization procedure. J. Comput. Appl. Math. 168(1–2), 365–373 (2004) 12. F.A. Radu et al., A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015) 13. P.G. Saffman, On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50(2), 93–101 (1971) 14. R. Schulz, P. Knabner, Derivation and analysis of an effective model for biofilm growth in evolving porous media. Math. Meth. Appl. Sci. 40, 2930–2948 (2017) 15. T.L. van Noorden et al., An upscaled model for biofilm growth in a thin strip. Water Resour. Res. 46, W06505 (2010) 16. B. Vu et al., Bacterial extracellular polysaccharides involved in biofilm formation. Molecules 14(7), 2535–2554 (2009)

Numerical Methods for Biomembranes Based on Piecewise Linear Surfaces John P. Brogan, Yilin Yang, and Thomas P.-Y. Yu

Abstract The shapes of phospholipid bilayer biomembranes are modeled by the celebrated Canham-Evans-Helfrich model as constrained Willmore minimizers. Several numerical treatments of the model have been proposed in the literature, one of which was used extensively by biophysicists over two decades ago to study real lipid bilayer membranes. While the key ingredients of this algorithm are implemented in Brakke’s well-known surface evolver software, some of its glory details were never explained by either the geometers who invented it or the biophysicists who used it. As such, most of the computational results claimed in the biophysics literature are difficult to reproduce. In this note, we give an exposition of this method, connect it with some related ideas in the literature, and propose a modification of the original method based on replacing mesh smoothing with harmonic energy regularization. We present a theoretical finding and related computational observations explaining why such a smoothing or regularization step is indispensable for the success of the algorithm. A software package called WMINCON is available for reproducing the experiments in this and related articles.

1 Introduction Lipid bilayer is arguably the most elementary and indispensable structural component of biological membranes which form the boundary of all cells. It is known since the seminal work of Canham [5], Helfrich [12] and Evans [9] in the 70s that bending elasticity, induced by curvature, plays the key role in driving the geometric configurations of such membranes.

J. P. Brogan · T. P.-Y. Yu () Department of Mathematics, Drexel University, Philadelphia, PA, USA e-mail: [email protected]; [email protected] Y. Yang Center for Computational Engineering, M.I.T., Cambridge, MA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_76

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The so-called spontaneous curvature model of Helfrich (referred simply to as the Helfrich problem) suggests that a biomembrane surface S configures itself so as to  minimize S H 2 dA subject to the area, volume and area difference (related to the bilayer characteristics) constraints, i.e. S solves the variational problem ⎧  ⎪ ⎨ (i) A[S] := S1 dA = A0 , ˆ · nˆ dA = V0 , min W [S] := H 2 dA s.t. (ii) V [S] := 13 S [x ˆi + y ˆj + zk] ⎪ S S ⎩ (iii) M[S] := −  H dA = M . 0 S 

Here H = (κ1 + κ2 )/2 is the mean curvature. In (ii), V [S] is the enclosing volume as a surface integralof S. The connection of (iii) to bilayer area difference comes 1 from the relation − S H dA = limε→0 4ε (area(S+ε ) − area(S−ε )), where S+ε and 1 S−ε are the ‘ε-offset surfaces’, and that the thickness of the lipid bilayer, 2ε, is negligible compared to the size of the vesicle. The constraint values A0 , V0 and M0 are determined by physical conditions (e.g. temperature, concentration). W [S] is called the Willmore energy of the surface S. When the area-difference constraint (iii) is omitted, the model is referred to as the Canham problem. When even the volume constraint (ii) is omitted, there is essential no constraint as W is scaleinvariant; in this case the area constraint (i) only fixes the scale, and we refer to the variational problem as the Willmore problem. The mathematical depth and physical relevance of these problems make them timeless challenges for geometric analysts and computational mathematicians. It is observed experimentally that no topological change occurs in any accessible time-scale, so we aim to solve any of the Helfrich, Canham or Willmore problems when S is assumed to be an orientable closed surface with a fixed genus g. Spherical, i.e. g = 0, vesicles are the most ubiquitous among naturally occurring biomembranes, although higher genus ones exist also [15, 18, 19]. The Helfrich and Canham models explain the large variety of shapes observed in even a closed vesicle with a spherical topology [16, 19]. Several numerical treatments of these models have been proposed in the literature, one of which was used extensively by biophysicists over two decades ago to study real lipid bilayer membranes. While the key ingredients of this algorithm are implemented in Brakke’s well-known surface evolver software [4], the overall algorithm seems to be never explained clearly by either the geometers who invented it [4, 11, 13] or the biophysicists who used it [18, 19]. As such, most of the computational results claimed in the biophysics literature are difficult to reproduce. In this note, we give an exposition on the numerical method, explain its connections to ideas developed in the applied geometry literature [17] and ideas developed in the FEM literature [3], present some theoretical and empirical findings, propose a modification of the original method based on the idea of harmonic energy regularization, and discuss open issues. 1 We assume that the normal of any closed orientable surface points outward. In particular, it means H < 0 for a sphere.

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2 Direct Minimization of Discrete Willmore Energy A standard approach to represent surfaces of arbitrary topology is piecewise linear (PL) surfaces. A PL surface can be specified by a mesh M = (V, F ) where V ∈ R#V ×3 records the 3-D coordinates of the vertices of the control mesh, #V denotes the total number of vertices, and F ∈ I #F ×3 is a list of triplets of indices from I := {1, . . . , #V } which records the bounding vertices of each of the #F triangle faces in the mesh M. We assume that the PL surfaces realized by the mesh is closed and orientable. In a numerical method, our view is usually that F is fixed and V varies. This fits the framework of our variational problems well, as fixing F also fixes the genus of the surface, and varying V means we find the embedding of F —viewed as an abstract simplicial complex—that optimizes the Willmore energy under the corresponding constraint(s). A PL surface has a well-defined area A and enclosing volume V , but no welldefined normals or mean curvatures, hence it also does not have a well-defined total mean curvature M or Willmore energy W . As such, any numerical method for the Willmore, Canham, or Helfrich problems based on PL surfaces may be classified as a “nonconforming finite element method”. One way to derive such a method is to find a definition of M and W for PL surfaces that is consistent in the sense that for any ‘reasonable’ sequence of subsequently finer triangulations of a smooth surface, the PL W or M-energies would converge to the corresponding continuous energies of the smooth surface. We then solve any version of the variational problems numerically by fixing a face list F with the presumed topology and desired resolution, then proceed to solve the corresponding finite-dimensional constrained optimization problem with V as variables. This ‘optimizing a discretization’ approach is used in [13] for solving the Willmore problem. It is mentioned in the aforementioned biophysics work that the approach is extended to solve the Canham and Helfrich problems, however almost no computational details are found in these physics papers. Some of the details can be found, however, in the manual of the Surface Evolver. Another interesting application of this approach is found in [11] for illustrating a least bending sphere eversion. There are at least half a dozen discrete mean curvatures and discrete Willmore energies proposed in the literature. One of them is the one we first learned from the paper by Hsu et al. [13], denoted here by HHKS and WHKS , based on a particular way of discretizing the area variation characterization of the mean curvature. Recall that for any smooth orientableB surface S with continuous unit normals denoted  d B by n(x), x ∈ S, we have dt Area(S ) = −2 h(x)H (x)dA, where St := t S t =0 {x +th(x)n(x) : x ∈ S} and h is any scalar field on S. To find a consistent definition of mean curvature for PL surfaces, we need to also come up with a definition of normals for PL surfaces. The idea is to define ‘discrete normals’ n(v), ‘discrete meancurvatures’ H (v), and local areas a(v) at the vertices of a PL surface so that v a(v) = A = area of the PL surface, and ∇v A · n(v) = −2H (v)a(v),

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Fig. 1 Failure of WHKS and WBobenko for solving the genus 1 Willmore problem: method converges to a PL surface unrelated to a Clifford torus, and with an energy way below 2π 2 . One can change ‘HKS’ to ‘MDSB’ or ‘EffAreaCur’ in our code to observe a similar failure. For SS methods, one can change ‘Loop’ to ‘C2g0’ [6] to observe a even higher accuracy. (a) Initial mesh. (b) WHKS = 8.85  2π 2 . (c) WBobenko = 12.57 ≈ 4π  2π 2 . (d) Control mesh of W -minimizing Loop SS. (e) W = 19.83 2π 2

∀v. Once a(v) is assigned, then n(v) and H (v) can be chosen so that H(v) := ∇v A H (v)n(v) = − 2a(v) . The local area in HHKS is a(v) = Area(star(v))/3. The discrete mean curvature introduced in [17], denoted here by HMDSB (v), corresponds to a(v) = Area(Voronoi cell around v).2 In either case, the discrete Willmore energy WHKS/MDSB and total   mean curvature MHKS/MDSB are defined as W = 2 v H (v) a(v) and M = v H (v)a(v), respectively. Yet another discrete Willmore energy is proposed by Bobenko [1], WBobenko =  e∈edge β(e) − π#V , where β(e) is the angle formed by the two circumscribed circles of the two triangles sharing the edge e. The original paper [1] puts a heavy emphasis on its invariance property under Möbius transformation, but leaves almost no trace of why this energy has anything to do with the continuous Willmore energy. However, it is not hard to check that the energy is consistent with the continuous W energy in a certain sense [2]. How good are these PL methods when used for solving any of the variation problems? One may think that any of these PL methods is simply less accurate than the higher order methods based on subdivision surfaces (SS) [6, 7, 10]. Here we present a computational experiment comparing three methods for solving the g = 1 Willmore problem, based on a direct minimization of (i) WHKS , (ii) WBobenko , and (iii) the (true) Willmore energy of Loop SS. In each case, a regularly triangulated (i.e. all vertices have valence 6) torus with 5 and 9 grid points in the minor and major circle directions is used. By the Marques-Neves theorem, the minimum Willmore energy is 2π 2 . The approximate minimizers based on our WMINCON package is shown below: In Fig. 1e, we minimize the true Willmore energy, accurately computed by a higher order numerical quadrature, of a finite dimensional family of Loop SS. In this case, the minimum cannot be smaller than 2π 2 . WHKS and WBobenko , however, would presumably be a good approximation to the true Willmore energy only when applied to a PL surface that is a reasonable discretization of a smooth surface. The 2 The paper [17] derived the same formula based on the characterization of mean curvature by the Laplace-Beltrami operator: ΔS X(x) = H(x), x ∈ S, where X : S → R3 is the position function of the surface S.

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above experiment exposes an unfortunate phenomenon: there are PL surfaces which are by no means a ‘reasonable discretization’ of any smooth torus but yet have a WHKS/Bobenko-energy way less than 2π 2 . We note that the computational result in [2, Figure 8] seems to suggest that a minimizer of WBobenko would be an accurate approximation of a Clifford torus, which is the exact opposite of what we report here. This apparent contradiction may be caused by the non-smoothness of WBobenko ; indeed we could not get the ‘spherical torus’ in Fig. 1c without the help of a nonsmooth optimization solver (we use GRANSO [8]). See also Proposition 1 below.

2.1 Addressing the Failure A fundamental shortcoming of HHKS is already addressed in [13] and the same behavior is exhibited by HMDSB : if v is a vertex over an n-gon (assumed regular for simplicity), then HHKS (v) first increases as the height of v over the n-gon increases, but then decreases after a certain height. In other words, after a certain threshold HHKS/MDSB(v) decreases despite the PL surface at v becoming sharper and sharper—a behavior that is particularly dangerous for variational problems. See Fig. 2. Two alternatives, under the name of ‘effective area curvature’ and ‘normal curvature’ in Brakke’s surface evolver manual, were introduced to alleviate this ∇v A∇v A ∇v A drawback: HEffAreaCur = − 2∇ , HNormalCur = − 2∇ , where V is the vV  v V ,∇v A enclosing volume of the PL surface. Note that this is a pointwise issue; let us mention yet another, not so well-known, pointwise issue: all the discrete mean curvatures introduced above, except HMDSB , converge to the correct value (κ1 + κ2 )/2 when applied to samples of the graph of (u, v) → 12 (κ1 u2 + κ2 v 2 ) at a small regular n-gon near (0, 0) only when n = 6. However, computational experiences suggest that this problem is forgivable as long

Fig. 2 Pointwise behavior of four discrete mean curvatures

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as we use triangulations dominated by valence 6 vertices; this can be guaranteed by repeatedly applying midpoint subdivision to a coarse triangulation. In [7], we prove the following result pertaining to the Willmore problem: Proposition 1 For every grid size (m, n), there is a regularly triangulated family of ‘almost spherical tori’ Tm,n,ε , each with a hole of diameter ∼ ε, such that as ε ↓ 0, WBobenko (Tm,n,ε ) decreases monotonically to 4π  2π 2 = W (Clifford torus).

(5, 10, .01) (10, 20, .05) (20, 40, .1)

This negative result shows that increasing the grid sizes m and n cannot resurrect the failure seen in Fig. 1c, despite the fact that WBobenko is a consistent discretization of the Willmore energy [2]. The situation is reminiscent of the “consistency without stability does not imply convergence” situation in numerical PDEs. Similar negative results has also been established for WHKS and WEffAreaCur , with the ‘almost spherical tori’ in Proposition 1 replaced by some carefully constructed ‘planar tori’ [7].

3 Harmonic Energy Regularization All failures we observed theoretically or experimentally have one thing in common: triangles with bad aspect ratios develop. This is of course a familiar issue in FEM and mesh generation, but we believe that in the moving surface context here the issue is fundamentally different. In previous works procedures going under the name of ‘vertex averaging’, ‘edge notching’ and ‘equiangulation’, implemented in the surface evolver, were introduced to somehow interfere an optimization process by fixing up ‘bad triangles’. Such a numerical method was reported to work for solving the Willmore, Canham and Helfrich problems. While these mesh smoothing procedures are probably well studied in the mesh generation community, we do not know exactly how they should be used to give a successful numerical method. Little implementation details are available in the relevant papers, and such a method seems difficult to analyze mathematically. We propose a method with a similar spirit but a cleaner formulation, based on adding to any PL W -energy WPL a harmonic energy penalization term,  i.e. to solve min WPL + λ E subject to the relevant constraint(s). Here E = M dϕ2 dA is defined w.r.t. a conformal structure [14, Ch. 8] imposed on the underlying simplicial complex where all triangles are equilateral. (This choice of conformal structure is the most natural in the genus 0 case, although we use it in the genus 1 example below

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Fig. 3 Genus 1 WEffAreaCur/NormalCur-minimizer with a (10, 16) regular triangulation, with (b and d) and without (a and c) E -regularization

Fig. 4 Genus 0 v0 -constrained WNormalCur -minimizers

anyway; see [7] for different choices of conformal structures.) Unlike the Willmore energy, a PL surface has enough regularity for E to be well-defined; here we view a PL surface as an immersion ϕ : M → R3 of the underlying simplicial complex, now thought√of as an abstract Riemann surface M, into R3 . It is not hard to check that E = 1/ 3 e∈edge length(e)2 . See Fig. 3 for the effect of such a regularization on WEffAreaCur and WNormalCur : it does not just fix up bad triangles, it completely changes the landscape of the optimization problem. The regularized minimizers are much closer to a Clifford torus. Finally, Fig. 4 shows how the regularized method performs on the genus 0 3 0 A0 − 2 in Canham problem with reduced volume/isoperimetric ratio v0 := 3V 4π ( 4π ) three different intervals known to give the shapes of a stomatocyte, discocyte (redblood cell) and prolate. The computational result is consistent with what is reported in the biophysics literature, and those produced by SS methods. Again, harmonic energy regularization is indispensable for the success.

4 Conclusion and Open Issues Our empirical observation—supported by Proposition 1 and various experiments— is that a space of PL surfaces, in contrast to SS, would be lured by any known discrete W -energies to waste its degrees of freedom on long and skinny triangles.

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Harmonic energy regularization comes to rescue and helps putting the degrees of freedom to good use, namely to optimize the accuracy of approximation to the true solution of the continuous problem. Open questions pertaining to these observations include: How to choose the penalization parameter λ? How to design a method with a provable Γ -convergence property? Is there a ‘perfect’ discrete W -energy that does not require any regularization? We believe that it is much harder to establish Γ -convergence results for PL methods than for SS methods. More details can be found in the journal version of this paper [7]. The PL and SS methods discussed here are in the spirit of “minimizing a discretization”, whereas the method proposed by Bonito et al. [3] is in the spirit of “discretizing a minimization”: it involves a tricky time- and space-discretization of a Willmore flow that bypass the lack of regularity in PL or piecewise polynomial surfaces. While we are learning about even newer methods for these fundamental geometric problems, possibly based on entirely different ways to represent surfaces, it would be interesting to understand the existing methods more deeply. Acknowledgements TY thanks Tom Duchamp, Robert Kusner, Shawn Walker and Aaron Yip for extensive discussions. This work is partially supported by NSF grants DMS 1115915 and DMS 1522337. We also thank the support of the Office of the Provost and the Steinbright Career Development Center of Drexel University.

References 1. A. I. Bobenko, A conformal energy for simplicial surfaces. Comb. Comput. Geom. 52, 135– 145 (2005) 2. A.I. Bobenko, P. Schröder, Discrete Willmore flow, in The Eurographics Symposium on Geometry Processing (2005), pp. 101–110 3. A. Bonito, R.H. Nochetto, M.S. Pauletti, Parametric FEM for geometric biomembranes. J. Comput. Phys. 229(9), 3171–3188 (2010) 4. K.A. Brakke, The surface evolver. Exp. Math. 1(2), 141–165 (1992) 5. P.B. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 26(1), 61–76 (1970) 6. J. Chen, S. Grundel, T.P.-Y. Yu, A flexible C 2 subdivision scheme on the sphere: with application to biomembrane modelling. SIAM J. Appl. Algebra Geom. 1(1), 459–483 (2017) 7. J. Chen, T.P.-Y. Yu, P. Brogan, Y. Yang, A. Zigerelli, Numerical methods for biomembranes: SS versus PL methods (2017, in preparation) 8. F.E. Curtis, T. Mitchell, M.L. Overton, A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles. Optim. Methods Softw. 32(1), 148–181 (2017) 9. E.A. Evans, Bending resistance and chemically induced moments in membrane bilayers. Biophys. J. 4(12), 923–931 (1974) 10. F. Feng, W.S. Klug, Finite element modeling of lipid bilayer membranes. J. Comput. Phys. 220(1), 394–408 (2006) 11. G. Francis, J.M. Sullivan, R.B. Kusner, K.A. Brakke, C. Hartman, G. Chappell, The minimax sphere eversion, in Visualization and Mathematics (Springer, Berlin, 1997), pp. 3–20

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12. W. Helfich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch C 28(11), 693–703 (1973) 13. L. Hsu, R. Kusner, J. Sullivan, Minimizing the squared mean curvature integral for surfaces in space forms. Exp. Math. 1(3), 191–207 (1992) 14. J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, 6th edn. (Springer, Berlin, 2011) 15. F. Jülicher, U. Seifert, R. Lipowsky, Conformal degeneracy and conformal diffusion of vesicles. Phys. Rev. Lett. 71(3), 452–455 (1993). 16. R. Lipowsky, The conformation of membranes. Nature 349, 475–481 (1991). 17. M. Meyer, M. Desbrun, P. Schröder, A.H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, in Visualization and Mathematics III (Springer, Berlin, 2003), pp. 35–57 18. X. Michalet, D. Bensimon, Observation of stable shapes and conformal diffusion in genus 2 vesicles. Science 269(5224), 666–668 (1995) 19. U. Seifert, Configurations of fluid membranes and vesicles. Adv. Phys. 46(1), 13–137 (1997)

Heavy Metals Phytoremediation: First Mathematical Modelling Results Aurea Martínez, Lino J. Alvarez-Vázquez, Carmen Rodríguez, Miguel E. Vázquez-Méndez, and Miguel A. Vilar

Abstract This work deals with the numerical modelling of the different processes related to the phytoremediation methods for remedying heavy metal-contaminated environments. Phytoremediation is a cost-effective plant-based approach of remediation that takes advantage of the ability of plants to concentrate elements and compounds from the environment and to metabolize them in their tissues (toxic heavy metals and organic pollutants are the major targets of phytoremediation). Within the framework of water pollution, biosorption has received considerable attention in recent years because of its advantages: biosorption uses cheap but efficient materials as biosorbents, such as naturally abundant algae. In order to analyse this environmental problem, we propose a two-dimensional mathematical model combining shallow water hydrodynamics with the system of coupled equations modelling the concentrations of heavy metals, algae and nutrients in large waterbodies. Within this novel framework, we present a numerical algorithm for solving the system, and several preliminary computational examples for a simple realistic case.

A. Martínez () · L. J. Alvarez-Vázquez Universidade de Vigo, Vigo, Spain e-mail: [email protected]; [email protected] C. Rodríguez Universidade de Santiago de Compostela, Santiago, Spain e-mail: [email protected] M. E. Vázquez-Méndez · M. A. Vilar Universidade de Santiago de Compostela, Lugo, Spain e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_77

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1 Introduction The term heavy metal is used for any dense metal that is classified for its potential toxicity, especially within environmental contexts. Toxic metals can be divided into two main groups: a first group consists of metals that in small quantities are essential as nutritional elements, but become toxic to all forms of life when present in greater amounts (As, Cr, Co, Cu, Ni, Se, Zn. . . ), and a second group is formed by highly poisonous metals that are not known to have any nutritional value (Pb, Hg, Cd, Ag, B. . . ). Heavy metals can be found naturally in the earth, but they usually become hazardous as a result of anthropogenic activities. Toxic metals can enter plant, animal, and human tissues via inhalation, diet, and manual handling. Most of them are toxic at high concentrations due to formation of complex compounds within the cells (the toxic effects of arsenic, mercury, and lead were known to the ancients, but methodical studies of the toxicity of some heavy metals only appear at the end of the nineteenth century). Moreover, toxic heavy metals tend to accumulate in living organisms as they are hard to metabolize [6]. Unlike organic pollutants, heavy metals cannot be biodegraded once released into the environment: They persist indefinitely and are a major cause of air, water and soil pollution. Methods for remediation of heavy metal-contaminated environments include physical removal, detoxification, bioleaching, and bioremediation. The term bioremediation is related to any treatment that uses naturally occurring organisms to break down hazardous substances into less toxic or nontoxic substances. These treatments are usually classified as in situ or ex situ: in situ bioremediation involves treating the contaminated material on site, while ex situ involves the removal of the contaminated material and treating it elsewhere. The main example of in situ bioremediation is phytoremediation (from Greek phyto, meaning plant, and Latin remedium, meaning restoring balance), which refers to the use of plants to clean up soil, air, and water contaminated with hazardous chemicals [9]. Phytoremediation is a cost-effective plant-based approach of remediation that takes advantage of the ability of plants to concentrate elements and compounds from the environment and to metabolize them in their tissues (toxic heavy metals and organic pollutants are the major targets for phytoremediation). Within the framework of water pollution, biosorption—which uses the ability of biological materials to remove and accumulate heavy metals from aqueous solutions—has received considerable attention in recent years because of its advantages compared to traditional methods. Biosorption uses cheaper materials such as naturally abundant macroalgae and microalgae (or even by-products of fermentation industries) as biosorbents. The ability of algae to absorb metals has been recognized over the last decades, mainly for the different species of Chlorella, Scenedesmus, Euglena, Phormidium, Spirogyra, and Spirullina, which often occur abundantly in typical waterbodies. Algae possess the ability to take up toxic heavy metals from the environment, resulting in higher concentrations than those in the surrounding water. They utilize the wastes as nutritional sources and enzymatically degrade the pollutants: In our case, some heavy metals are known to be detoxified or transformed

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by algal metabolism (algae also have the ability to take up various kinds of nutrients like nitrogen and phosphorus, which is an essential point in the bioremediation of eutrophication processes [3]).

2 Mathematical Modelling Let us consider a bounded domain Ω ⊂ R2 (for instance, a channel or an estuary) occupied by shallow water, in which a series of heavy metal discharges occurs, causing a contamination of the body of water above the maximum allowed thresholds. Elementary phytoremediation techniques are based on placing a mass of algae in a subdomain K ⊂ Ω, so that its bioadsorbent capacity reduces the concentration levels of heavy metals in its area of influence. Related control problems arising are e.g. the determination of the minimal amount of algae required, as well as the optimal placement of the mass of algae, and can be treated using techniques similar to those employed by the authors, for example, in [1–3, 7]. To quantify the ability of a given biomass to remove heavy metals from the aqueous solution containing them, the process is evaluated by measuring the amount of metal remaining in solution once it is checked to be invariant in time. In this way, the adsorption isotherm is obtained, which is the equilibrium relation, at a given temperature, of the amount of solute adsorbed qe , vs. the concentration of the adsorbate in the fluid phase ce . The most common model, known as the Langmuir equilibrium model, establishes the following relation: qeL = Qmax

bce , 1 + bce

(1)

where: – qeL is the quantity of metal adsorbed under equilibrium conditions, – Qmax is the maximum adsorption capacity, – b is an equilibrium adsorption constant (adjustment parameter that measures the biosorption affinity or efficiency of the biomass), and – ce is the concentration of metal in the solution at equilibrium, that is, the amount of metal remaining in the solution once verified that it does not change over time. The most efficient bioadsorbents are those presenting high Qmax and b. It is worthwhile noting here that other alternative adsorption isotherms can be also found 1/n in the scientific literature, such as the Freundlich isotherm: qeF = KF ce , or its H linear version, Henry’s law: qe = KH ce . For adsorption kinetics, the following expression is commonly used, stating that the change in time of the quantity of heavy metal q, that has been adsorbed in algae up to the time t, is driven by the difference of q with the totally absorbed quantity

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at equilibrium q ∗ :   ∂q = κq q ∗ − q , ∂t

(2)

where: – q is the concentration of metal adsorbed in time t, – q ∗ is the concentration of metal adsorbed at equilibrium, – κq is the adsorption rate (first order mass transfer coefficient). In this model (known as Lagergren’s kinetic model), which assumes that the concentration of adsorbate on the surface of the adsorbent q ∗ is in equilibrium with the concentration of adsorbate within the solution c, the concentration q ∗ is usually evaluated by applying the Langmuir equilibrium model. A first step in introducing a mathematical formulation of the problem is to choose an appropriate indicator for measuring water quality (in our case, the concentrations of heavy metals in their different phases), and to establish a mathematical model that, in function of the data of the problem, allows to simulate the concentration of that indicator in the entire domain Ω throughout the whole simulation time. So, for x ∈ Ω ⊂ R2 and t ∈ (0, T ), we consider the following concentrations (height averaged in the water column), that will be the state variables of our problem: – c(x, t) [g/m3 ]: which measures the concentration of heavy metal in water, – q(x, t) [g/g]: which measures the concentration of heavy metal deposited in algae, – a(x, t) [g/m3 ]: which measures the concentration of algae in water, – p(x, t) [g/m3 ]: which measures the concentration of nutrients (N and/or P) in water. Taking into account above considerations for the evolution of quantities of heavy metals and the interactions between algal masses and nutrients in water, these concentrations will be the solution of the following coupled system of partial differential equations: ∂c ∂q + v · ∇c − μc Δc + κc a = F in Ω × (0, T ), ∂t ∂t   ∂q bc = κq Qmax −q in Ω × (0, T ), ∂t 1 + bc ∂a p + w · ∇a − μa Δa − λ a+γ a =0 ∂t κp + p ∂p p + v · ∇p − μp Δp + β λ a=G ∂t κp + p

in Ω × (0, T ),

in Ω × (0, T ),

(3)

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with boundary conditions: ∂c = 0, ∂n

∂a = 0, ∂n

∂p = 0 on ∂Ω × (0, T ), ∂n

(4)

and initial conditions: c(0, x) = c0 (x),

q(0, x) = q 0 (x)

a(0, x) = a 0 (x) + a˜ 1K (x),

in Ω,

(5)

p(0, x) = p0 (x) in Ω,

where: – v(x, t) [m/s] = (v1 , v2 ) represents the velocity of water, averaged in height h(x, t) [m], both solutions of the classical shallow water equations (also known as Saint-Venant equations); – μc , μa , μp [m2 /s] are the diffusion coefficients of metal, algae, and nutrient, respectively; – κc is the mass transfer coefficient; – F (x, t) [g/m3 /s] represents the source term of heavy metal; – κq [1/s] is the adsorption rate constant (Lagergren constant); – Qmax [g/g] represents the maximum adsorption capacity; – b [m3 /g] is the Langmuir constant; – w(x, t) [m/s] represents the velocity of algae (for instance, in the case of immobilized algae, w = 0); – λ [1/s] represents the luminosity coefficient; – κp [g/m3 ] is the nutrient semi-saturation constant; – γ [1/s] is the mortality rate of algae; – β [g/g] is the nutrient-carbon stoichiometric coefficient; – G(x, t) [g/m3 /s] represents the source term of nutrients; – n represents the outer unit vector normal to the boundary ∂Ω; and – 1K (x) represents the indicator function of the region K ⊂ Ω where algae is initially added with a mean concentration a˜ ≥ 0 (both, K and a, ˜ are related to the control variables of our future optimal control problem). Following usual techniques [4] it can be shown that the state variables c, q, a and p are non-negative as well as bounded, and lie in the functional space L2 (0, T ; H 1 (Ω)), provided assumptions on the domain regularity, on boundedness, non-negativity and regularity of the initial data c0 , q 0 , a 0 and p0 as well as the velocities v and w and source terms F and G are met.

3 Numerical Solution In order to solve numerically the system of state equations (3)–(5), we begin by a time semidiscretization. For this, we consider a natural number N ∈ N, take the time step Δt = T /N, and define the discrete times t n = n Δt, for n = 0, . . . , N. Then,

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considering the approximations cn (x) ( c(x, t n ), n = 0, . . . , N, (analogously for q n , a n , pn , v n , wn , F n and Gn ), and defining the parameter α = 1/Δt = N/T , we have the following system of semidiscretized equations posed on Ω: α(cn+1 − cn ) + v n+1 · ∇cn+1 − μc Δcn+1 + κc a n+1 α(q n+1 − q n ) = F n+1 ,   bcn+1 n+1 n n+1 α(q , (6) − q ) = κq Qmax −q 1 + bcn+1 α(a n+1 − a n ) + wn+1 · ∇a n+1 − μa Δa n+1 − λ

pn+1 a n+1 + γ a n+1 = 0, κp + pn+1

α(pn+1 − pn ) + v n+1 · ∇pn+1 − μp Δpn+1 + β λ

pn+1 a n+1 = Gn+1 , κp + pn+1

with the corresponding homogeneous Neumann boundary conditions on ∂Ω, for cn+1 , a n+1 and pn+1 , directly obtained from (4). Next, for a space semidiscretization, we consider a Lagrange P1 finite element method (that is, test functions that are globally continuous in Ω and locally firstdegree polynomials in each element). Thus, by choosing a triangulation τh of Ω formed by triangles τ of diameter less than h, we approximate the functional space H 1 (Ω) by the space of test functions Vh = {vh ∈ C(Ω) : vh|τ ∈ P1 , ∀τ ∈ τh }. Then, by means of the direct variational formulation of the semidiscretized system of equations (6), the fully discretized approximation of the problem is obtained in a standard way. It allows to calculate the discrete approximations of the functions cn+1 , q n+1 , a n+1 , pn+1 , which will be respectively denoted by chn+1 , qhn+1 , ahn+1 , phn+1 ∈ Vh . Additionally, in order to deal with the nonlinearities appearing in the variational formulation, an iterative fixed-point algorithm is used, where, for its resolution, we begin by first iterating in phn+1 , then in ahn+1 , and then in chn+1 . Finally, qhn+1 can be obtained by the straightforward rewriting of the corresponding equation in the equivalent form: qhn+1

1 = α + κq

 α qhn

+ κq Qmax

bchn+1 1 + bchn+1

 .

(7)

4 Computational Results In this last section, we present some preliminary numerical tests for a case posed in the estuary Ría de Vigo (Galicia, NW Spain), corresponding to a wastewater discharge, presenting a high content of lead, from several shipyards located in the surrounding area.

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Numerical results have been obtained in two ways. Firstly, by using the module Heavy metals of the 2D hydrodynamic model MIKE 21 [8], developed by DHI (Danish Institute of Technology) and widely used in the study of environmental issues. As a second alternative, the problem has also been solved by using the open source software Freefem++ [5], by means of the complete programming of the algorithm presented above. In both cases the results obtained have been very similar, both qualitatively and quantitatively. From the numerous collection of numerical tests performed, only two figures are presented here, which in this case were obtained using the code Freefem++. They correspond to the study of the case in the estuary Ría de Vigo, in order to analyze the effects of the placement of a mass of algae in the vicinity of the discharge zone. In this case, green alga Ulva, also known by the common name sea lettuce, was used as a phytoremediation method. By means of a simple visual inspection of both figures, it is possible to easily appreciate the differences between the absence of a mass of algae in the discharge zone (Fig. 1) and the presence of Ulva in that zone (Fig. 2). It can be observed graphically how the concentration of lead in water is significantly reduced by placing algae in the area corresponding to the source of pollution (most highly

Fig. 1 Levels of lead concentration in water, in the absence of algae, from an industrial wastewater discharge in the estuary Ría de Vigo. The discharge zone corresponds to the peak of concentration, which also coincides with the region K

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Fig. 2 Levels of lead concentration in water, after the placement of a mass of algae a˜ = 1.25 g/m 3 in the discharge zone, in order to minimize its harmful effects

contaminated area in Fig. 1). It can be seen that the reduction of lead concentration not only appears in the area where algae have been placed, but also expands to the surrounding zones of the estuary. This occurs mainly due to the effect of tides. Finally, it is worthwhile remarking here that, after this first phase of mathematical modelling of the problem, the immediately following steps—that are already beginning to be addressed—are directed towards the formulation, analysis and resolution of some optimal control problems for this scenario. Questions related to the determination of the minimal quantity of algae to be used and the location of the optimal place for such algae mass will be addressed in a forthcoming paper, so that the best possible results of phytoremediation of water contaminated by heavy metals can be achieved, both from an ecological and an economic viewpoint. Acknowledgements This work was supported by funding from project MTM2015-65570-P of MINECO (Spain) and FEDER. The authors also thank the help and support provided by DHI with the MIKE21 modelling system.

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References 1. L.J. Alvarez-Vázquez, A. Martínez, R. Muñoz-Sola, C. Rodríguez, M.E. Vázquez-Méndez, The water conveyance problem: optimal purification of polluted waters. Math. Models Methods Appl. Sci. 15, 1393–1416 (2005) 2. L.J. Alvarez-Vázquez, F.J. Fernández, R. Muñoz-Sola, Mathematical analysis of a threedimensional eutrophication model. J. Math. Anal. Appl. 349, 135–155 (2009) 3. L.J. Alvarez-Vázquez, F.J. Fernández, A. Martínez, Optimal management of a bioreactor for eutrophicated water treatment: a numerical approach. J. Sci. Comput. 43, 67–91 (2010) 4. E. Casas, C. Ryll, F. Troltzsch, Sparse optimal control of the Schlogl and FitzHugh-Nagumo systems. Comput. Methods Appl. Math. 13, 415–442 (2013) 5. F. Hecht, New development in Freefem++. J. Numer. Math. 20, 251–265 (2012) 6. D. Mani, C. Kumar, Biotechnological advances in bioremediation of heavy metals contaminated ecosystems: an overview with special reference to phytoremediation. Int. J. Environ. Sci. Technol. 11, 843–872 (2014) 7. A. Martínez, L.J. Alvarez-Vázquez, C. Rodríguez, M.E. Vázquez-Méndez, M.A. Vilar, Optimal shape design of wastewater canals in a thermal power station, in Progress in Industrial Mathematics at ECMI 2012, ed. by M. Fontes, M. Gunther, N. Marheineke (Springer, New York, 2014), pp. 59–64 8. MIKE 21, User guide and reference manual, Danish Hydraulic Institute (DHI), Horsholm, 2001 9. H. Perales-Vela, J. Peña-Castro, R. Cañizares-Villanueva, Heavy metal detoxification in eukaryotic microalgae. Chemosphere 64, 1–10 (2006)

Urban Heat Island Effect in Metropolitan Areas: An Optimal Control Perspective Lino J. Alvarez-Vázquez, Francisco J. Fernández, Aurea Martínez, and Miguel E. Vázquez-Méndez

Abstract This work combines numerical modelling, optimization techniques, and optimal control theory of partial differential equations in order to analyze the mitigation of the urban heat island (UHI) effect, which is a very usual environmental phenomenon where the metropolitan areas present a significantly warmer temperature than their surrounding areas, mainly due to the consequences of human activities. At the present time, UHI is considered as one of the major environmental problems in the twenty-first century (undesired result of urbanization and industrialization). Mitigation of the UHI effect can be achieved by using green roofs/walls and lighter-coloured surfaces in urban areas, or (as will be addressed in this study) by setting new green zones inside the city. In order to study the problem, we introduce a well-posed mathematical formulation of the environmental problem (related to the optimal location of green zones in metropolitan areas), we give a numerical algorithm for its resolution, and finally we discuss several numerical results for several realistic 3D examples.

1 Introduction An urban heat island is a meteorological phenomenon (customarily known as UHI) related to the presence of higher temperatures in urban environments than in surrounding rural areas, especially if it is caused by anthropogenic reasons. This

L. J. Alvarez-Vázquez () · A. Martínez Universidade de Vigo, Vigo, Spain e-mail: [email protected]; [email protected] F. J. Fernández Universidade de Santiago de Compostela, Santiago, Spain e-mail: [email protected] M. E. Vázquez-Méndez Universidade de Santiago de Compostela, Lugo, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_78

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difference of temperature, as was established in past century, ranges from 2 to 12 ◦C (although most studies refer to a 5 ◦ C difference), and it is larger at night than during the day, especially in the case of absence of winds [1]. This nocturnal warming is mainly due to the fact that the shortwave radiation (absorbed during the day in cities with a decreased amount of vegetation and a large amount of materials with harmful thermal bulk and surface radiative properties) is still within concrete and asphalt, and this energy is slowly released during the night as longwave radiation, blocking the natural cooling process. At present, UHI is considered one of the major current environmental challenges, as an unwanted effect of urbanization and industrialization, and metropolises with large population and intensive economic activities are adopting strategies to mitigate UHI effect, since it poses particular risks to citizen. Mitigation of the UHI effect can be accomplished through the use of green roofs or of light-coloured surfaces in urban areas (which reflect more sunlight and absorb less heat), and also—as will be addressed in this paper—through the increasing of vegetation cover inside cities, mainly in the form of urban forests and parks, in order to maximize the multiple vegetation benefits in controlling the temperature rises [2]. In previous works of the authors [3, 4], a mathematical model for this problem has been set and analyzed, and a computational algorithm has been proposed for its numerical solution. This current study is devoted to present and discuss several numerical tests related to the mitigation of the UHI effect, posed on realistic three-dimensional scenarios. So, in Sect. 2 we recall the formulation of the control problem related to the optimal location and dimension of green zones in metropolitan areas, and in Sect. 3 a numerical algorithm for its computation is briefly summarized. Final section of the paper is devoted to show several numerical tests and discussions.

2 Analytical Formulation of the Environmental Problem In this section, we present the mathematical setting of the optimal control problem. Basically, for the state system, we consider five coupled partial differential equations (describing the velocity uA , the pressure pA and the temperature θA of air, and the temperatures of soil θS and buildings θB ) posed on three different domains whose union represents the whole domain under study (these three domains are, respectively, air ΩA , soil ΩS and buildings ΩB ). This system couples Navier-Stokes equations with several heat equations, and, for a time interval I = (0, T ), takes

Optimal Control of the Urban Heat Island Effect

the form: ⎧ ⎪ ∂uA θA ⎪ ⎪ + (uA · ∇)uA − ∇ · (νA ∇uA ) + ∇pA = ref g in ΩA × I, ⎪ ⎪ ∂t ⎪ θA ⎪ ⎪ ⎪ ⎪ = 0 in Ω × I, ∇ · u ⎪ A A ⎪ ⎨ ∂θA + uA · ∇θA − ∇ · (KA ∇θA ) = FA in ΩA × I, ⎪ ∂t ⎪ ⎪ ∂θ S ⎪ ⎪ − ∇ · (KS ∇θS ) = FS in ΩS × I, ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ∂θB ⎪ ⎪ − ∇ · (KB ∇θB ) = FB in ΩB × I, ⎩ ∂t

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(1)

completed with the boundary conditions (corresponding to wind inflow and to heat transfer by convection and/or radiation in the interfaces air/soil, air/buildings and soil/buildings) and with the usual initial conditions, both necessary in order to assure the well-posedness of the state system. A full description of this set of initial/boundary conditions, along with the definition and meaning of the different parameters appearing in the formulation, can be found in the recent paper of the authors [4]. As above commented, we are interested in finding—in an optimal way—the location of a given number M of green zones over the soil surface (obviously, avoiding the surface occupied by buildings), in such a way that the cooling effect produced by these green zones causes a temperature reduction inside a set of NB buildings under study and, consequently, minimizes the economic cost due to air conditioning. The main characteristic of green zones is the fact that they present a thermic behaviour completely different from the rest of paving materials, which is taken into account in the mathematical model given by system (1), through the specific coefficients related to the soil thermic behaviour. So, if we consider M green areas (assumed to be rectangular, for the sake of simplicity), characterized by their location over the soil, we have the control vector b = (b1 , b2 , . . . , bM ) ∈ R4M , where, for each k = 1, 2, . . . , M, bk = (p11,k , p21,k , p12,k , p22,k ) ∈ R4 represents the location of the k-th rectangular green area Gk = [p11,k , p21,k ] × [p12,k , p22,k ]. With respect to the objective function J , it corresponds to minimizing the energy cost related to the air conditioning inside each building, which depends on the mean temperature of building facade, the mean temperature of air in contact with that facade, and/or the mean temperature of soil. For this objective function, the variation of the green zones changes the coefficients in the heat equation for the soil that, by the coupling boundary conditions, also changes the temperatures of air and buildings. Taking into account above considerations, our real-world problem can be formulated as an optimal control problem, written in the form: min J (b),

b∈Bad

(2)

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where the admissible set Bad is a subset of R4M collecting all the geometrical constraints on the design variables (there exists a number L ≥ M of large, disjoint, possible regions for locating the green areas; the extent of each green area must be between a lower and an upper bound; the sum of all the areas associated to the green zones must be under a threshold given by economic budget. . .) In addition, if we introduce a new binary control variable y ∈ {0, 1}ML , representing the inclusion of one of the M green zones inside one of the possible L regions, the optimal control problem (2) can be reformulated as a problem of type MINLP (mixed integer nonlinear programming): min f (b, y) b,y

s.t. h(b, y) ≤ 0

(3)

where the new objective function f is such that f (b, y) = J (b), and the nonlinear constraints h(b, y) ≤ 0 translate above mentioned feasibility conditions. It can be proved (cf. [4]) that, under some regularity hypotheses, the nonlinear programming problem (3)—and consequently the original optimal control problem (2)—admits, at least, a solution (uniqueness is not expected due to the deeply nonlinear character of the problem).

3 Numerical Computation of the Optimal Control In this section, we show a numerical algorithm to solve the optimal control problem (2), by finding one of the solutions of the MINLP problem (3). As a first step, we need to solve the state system (1), which can be done by a semi-implicit space/time discretization based on the method of characteristics (for the time semi-discretization), and on the finite element method (for the space semidiscretization), in the spirit of that presented in previous papers of the authors (see, for instance, [5] and the references therein), and where the nonlinearities will be addressed with a suitable fixed point technique. Once able to solve the state system, the continuous optimal control problem (2) will be reformulated as a discrete version of the MINLP problem (3), in which the control variable has a real component (related to the position of green areas) and an integer component (related to the possible location of green areas within the set of given regions). This can be made by just discretizing the cost function (since control variables and constraints remain the same as in the continuous problem), employing the previously obtained numerical solution of the full space/time discretization of state system (1). Finally, this discrete nonlinear optimization problem can be solved, for instance, by any standard gradient-type algorithm, where we are only left to calculate the gradient of the discretized cost function. To compute this gradient, we have two options: either using the equations associated to the adjoint states, or using the

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linearized state systems. In our case, we have chosen the first option, since—due to the particularities of the problem—it requires to solve adjoint state system only once, which is an advantage with respect to the use of the linearized state system, that should be solved many times (once for each step of time).

4 Numerical Results We present here some numerical results obtained for three academic (although realistic) tests posed on two different 3D domains composed of a set of NB = 4 and NB = 9 equally spaced buildings. Our particular goal here will be to analyze if it is more convenient optimally locate only one large green area or more than one (assuming that they occupy the same total area). We analyze and discuss the diverse solutions we have obtained solving the problem for various scenarios. In the three scenarios, both the domain meshes and the numerical solution of the problem have been developed with Freefem++ [6]. For the time semi-discretization we have employed the method of characteristics, and for the space discretization the finite element method (P1b − P1 for the Navier-Stokes equations, and P1 for air, soil and buildings temperatures). The resolution of the nonlinear optimization problems has been performed employing the interior-point algorithm IPOPT [7] for the continuous variables, and an exhaustive search for the small number of discrete variables. (In case of a larger number of discrete variables, a branch-and-bound type method would be needed.) Scenario 1 In this first test we consider, over a soil domain of 15 m × 15 m, a symmetrically distributed set of NB = 4 buildings (each one of 3 m × 3 m) with the geometry shown in Fig. 1. As can be seen there, we have L = 9 possible regions 2D , l = 1, 2, 3), and six of 9 m2 where positioning the green areas: three of 45 m2 (ΩS,l 2D , l = 4, . . . , 9). (ΩS,l As cost function to be minimized, we consider the temperature of a thin air layer close to the soil (in search of pedestrian comfort when walking). For the different meshes we have chosen the geometric configuration given by a depth of the soil layer of HS = 2 m, a height of buildings of HB = 3 m, and a height of the air layer over the ground of HA = 6 m. With respect to the time discretization, we choose a time step size Δt = 900 s, and we solve the problem for N = 96 time steps (corresponding to a time interval of T = 24 h). Finally, we have considered an extremely weak wind entering the soil domain by the left side. In Fig. 1 we compare the results achieved for the cases of M = 1 and M = 2 green areas. We can see there both the optimal locations of the green zones and the temperatures in the air layer near the soil at final time step (for M = 0, 1 and 2 optimized green areas). Looking in detail at the last subfigure there, we can observe that the best results are achieved for M = 2 green areas, and that the worst ones—as it was expected—correspond to the absence of green zones.

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(a) (∂ Ω2D S )IN

(b) Ω2D S,3

Ω2D S,7

Ω2D S,2

Ω2D S,4

Ω2D S,1

(∂ Ω2D S )IN

Ω2D B,3

Ω2D S,8

Ω2D B,4

Ω2D S,9

Ω2D S,7

Ω2D S,5

Ω2D B,2

Ω2D B,3

Ω2D S,4

Ω2D S,6

Ω2D B,1

Ω2D S,1

(∂ Ω2D S )OU T

(c)

(d)

(e)

(f)

Ω2D S,8

Ω2D B,4

Ω2D S,9

G2 = [6.94, 14.05] × [6.50, 8.00]

Ω2D S,2

G1 = [4.5, 14.5] × [6.5, 8.5]

Ω2D B,1

Ω2D S,3

Ω2D S,5

Ω2D B,2

Ω2D S,6

(∂ Ω2D S )OU T

G1 = [4.98, 14.13] × [1.48, 250]

Surrounding soil air temperature 314

Mean temperature

312 310 308 9

306

k =1 9 9

304

k =1 9 9

302 300

k =1 9

10

20

30

40

50

Time step

60

1 (Γ A,k S )

Γ A,k S

1 (Γ A,k S )

Γ A,k S

1 (Γ A,k S ) 70

Γ A,k S

n d μ0GZ,A

(K )

n d μ1GZ,A

(K )

n θ2GZ,A d

80

(K ) 90

Fig. 1 Numerical results for the first scenario. Left to right and top to bottom, we show: (a) the optimal location for M = 1 green area; (b) the optimal location for M = 2 green areas; (c) the air temperature on the layer surrounding the soil with one optimally located green area at final time; (d) the air temperature on the layer surrounding the soil with two optimally located green areas; (e) the air temperature on the layer surrounding the soil without green areas; and (f) a comparison of the mean temperatures of the air surrounding the soil for the cases of M = 0, 1, and 2 green areas

Scenario 2 Space and time details are the same as in scenario 1. The only change here is related to the cost function, that in this case refers to minimizing the temperature of the air layer surrounding the four buildings (corresponding to optimal saving of energy cost due to air conditioning inside the buildings). In this case, we have considered a moderate to strong wind.

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(a) (∂Ω2D S )IN

(b) Ω2D S,3

Ω2D S,7

Ω2D S,2

Ω2D S,4

Ω2D S,1

(∂Ω2D S )IN

Ω2D B,3

Ω2D S,8

Ω2D B,4

Ω2D S,3

Ω2D S,9

Ω2D S,7

Ω2D S,5

Ω2D B,2

Ω2D B,3

Ω2D S,8

Ω2D B,4

Ω2D S,9

Ω2D S,4

Ω2D B,1

Ω2D S,5

Ω2D B,2

Ω2D S,6

Ω2D S,1

G1 = [5.43, 13.05] × [6.50, 8.50]

Ω2D S,2

G1 = [4.4, 14.4] × [6.5, 8.5]

Ω2D B,1

G2 = [7.90, 12.55] × [12.5, 13.52]

Ω2D S,6

(∂Ω2D S )OU T

(c)

(d)

(e)

(f)

(∂Ω2D S )OU T

Surrounding 4th building air temperature

Mean temperature

312

310

308

306 1

304

302

300

10

20

30

(Γ W,4 B ) 1

4 Γ W, B

(Γ W,4 B ) 1

W, 4 ΓB

(Γ W,4 B )

4 Γ W, B

40

n d μ0GZ,A

(K )

n μ1GZ,A d

(K )

n μ2GZ,A d

(K )

50

Time step

60

70

80

90

Fig. 2 Numerical results for the second scenario. Left to right and top to bottom, we show: (a) the optimal location for M = 1 green area; (b) the optimal location for M = 2 green areas; (c) the air temperature on the layer surrounding the four buildings with one optimally located green area at final time; (d) the air temperature on the layer surrounding the four buildings with two optimally located green areas; (e) the air temperature on the layer surrounding the four buildings without green areas; and (f) a comparison of the mean temperatures of the air surrounding the four buildings for the cases of M = 0, 1, and 2 green areas

In Fig. 2 we show the results obtained in this test: the optimal locations of the green areas for the cases M = 1 and 2, and the temperatures of the air close to the facades and roofs of the buildings (unlike previous case, cooling effect of wind can be noted now on the inflow boundary). We also incorporate a graph of the evolution of the averaged temperatures of the air surrounding the walls, giving a comparison

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Fig. 3 Numerical results for the third scenario. For the M = 4 optimized green areas, we show the temperature of the NB = 9 buildings’ facades and roofs at the last time step (left), and the temperature of soil, also at final time (right)

of the different thermal behaviours in the cases of M = 0, 1 and 2 optimally located green zones. As in previous scenario, the choice of two green areas gives better results than in the case of only one (and, evidently, both results are better than without green areas). Scenario 3 In this last test, we consider NB = 9 symmetrically distributed buildings (each one again of 3 m×3 m), with a final geometry as shown in Fig. 3. As cost function in order to minimize, we consider here the temperature of whole soil and buildings. The general details for the space/time discretizations are the same than in above scenarios. In Fig. 3 we display the results achieved for the case of M = 4 green areas. We show here, at final time step, the temperature of the nine buildings’ facades and roofs, and also the soil temperature (where locations of the four green zones can be clearly distinguished). Acknowledgements This work was supported by funding from project MTM2015-65570-P of MINECO (Spain) and FEDER.

References 1. A.J. Arnfield, Two decades of urban climate research: a review of turbulence, exchanges of energy and water, and the urban heat island. Int. J. Climatol. 23, 1–26 (2003) 2. A. Martos, R. Pacheco-Torres, J. Ordóñez, E. Jadraque-Gago, Towards successful environmental performance of sustainable cities: intervening sectors. A review. Renew. Sustain. Energy Rev. 57, 479–495 (2016) 3. F.J. Fernández, L.J. Alvarez-Vázquez, N. García-Chan, A. Martínez, M.E. Vázquez-Méndez, Optimal location of green zones in metropolitan areas to control the urban heat island. J. Comput. Appl. Math. 289, 412–425 (2015)

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4. F.J. Fernández, L.J. Alvarez-Vázquez, A. Martínez and M. E. Vázquez-Méndez, A 3D optimal control problem related to the urban heat islands. J. Math. Anal. Appl. 446, 1571–1605 (2017) 5. N. García-Chan, L.J. Alvarez-Vázquez, A. Martínez, M.E. Vázquez-Méndez, On optimal location and management of a new industrial plant: numerical simulation and control. J. Franklin Inst. 351, 1356–1371 (2014) 6. F. Hecht, New development in Freefem++. J. Numer. Math. 20, 251–265 (2012) 7. A. Wächter, L.T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)

Nitsche-Based Finite Element Method for Contact with Coulomb Friction Franz Chouly, Patrick Hild, Vanessa Lleras, and Yves Renard

Abstract The aim of this paper is to provide some mathematical results for the discrete problem associated to contact with Coulomb friction, in linear elasticity, when finite elements and Nitsche method are considered. We consider both static and dynamic situations. We establish existence and uniqueness results under appropriate assumptions on physical (friction coefficient) and numerical parameters. These results are complemented by a numerical assessment of convergence.

Many problems involve frictional contact, and are approximated numerically using the Finite Element Method. In this paper we deal with the Nitsche method originally proposed in [18] and that aims at treating the boundary or interface conditions in a weak sense, thanks to a consistent penalty term. It differs in this aspect from standard penalization techniques and from mixed methods since no Lagrange multiplier is needed and no discrete inf sup condition must be fullfilled. A Nitschebased FEM has been proposed and analyzed for static frictionless unilateral contact in [5, 8], and extended to dynamic contact in [6, 7]. Very few works deal with the adaptation of Nitsche’s method to frictional contact (see [9] where recent achievements in applying Nitsche’s method to some contact and friction problems

F. Chouly Université de Franche Comté, Besançon, France e-mail: [email protected] P. Hild Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS-IMT, Toulouse Cedex 9, France e-mail: [email protected] V. Lleras () IMAG, Univ Montpellier, CNRS, Place Eugène Bataillon, Montpellier, France e-mail: [email protected] Y. Renard INSA de Lyon, Villeurbanne, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_79

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are summarized): the Tresca’s friction problem is only considered in [4, 11] and numerical results for Coulomb friction are presented in [17, 20]. In this paper we are interested in some existence and uniqueness results at the discrete level in statics and dynamics. For the continuous static friction problem, existence of solutions hold when the friction coefficient is small enough [13]. In [19], a uniqueness result has been obtained with the assumption that a regular solution exists and that the friction coefficient is sufficiently small. At the discrete level, difficulties appear in the numerical analysis of the method [14]. Results of well-posedness for frictional contact in the dynamic case are presented in [16] for a normal compliance model, in [1, 3] for discrete systems of particles and in [12, 15] for the modified mass method.

1 Setting and Discretization We consider an elastic body Ω in Rd with d = 2, 3. Small strain assumptions are made. The boundary ∂Ω of Ω is polygonal (d = 2) or polyhedral (d = 3). The outward unit normal vector on ∂Ω is denoted n. We suppose that ∂Ω consists in three nonoverlapping parts ΓD on which the body is clamped, ΓN and the contact boundary ΓC , with meas(ΓD ) > 0 and meas(ΓC ) > 0. The contact boundary is supposed to be a straight line segment when d = 2 or a polygon when d = 3 to simplify. In the reference configuration, the body is in frictional contact on ΓC with a rigid foundation and we suppose that the unknown contact zone during deformation is included into ΓC . It is subjected to volume forces f in Ω and to surface loads g on ΓN . Static Problem We consider the unilateral contact problem with Coulomb friction in linear elastostatics. It consists in finding the displacement field u : Ω → Rd verifying the equations and conditions (1)–(2): div σ (u) + f = 0 u=0

in Ω, on ΓD ,

σ (u) = A ε(u) σ (u)n = g

in Ω, on ΓN ,

(1)

The conditions defining unilateral contact with Coulomb friction on ΓC are: un ≤ 0, ut = 0 ut = 0

σn (u) ≤ 0,

σn (u) un = 0

0⇒ |σ t (u)| ≤ −Fσn (u) ut 0⇒ σ t (u) = Fσn (u) |ut |

(i) (ii) (iii)

(2)

where F ≥ 0 stands for the friction coefficient. The notation σ = (σij ), 1 ≤ i, j ≤ T d, stands for the stress tensor field, ε(v) = (∇v + ∇v )/2 represents the linearized

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strain tensor field and A is the fourth order symmetric elasticity tensor having the usual uniform ellipticity and boundedness property. Dynamic Problem We consider the unilateral contact problem with Coulomb friction in linear elastodynamics during a time interval [0, T ) where T > 0 is the final time. We denote by ΩT := (0, T ) × Ω the time-space domain, and similarly ΓDT := (0, T ) × ΓD , ΓNT := (0, T ) × ΓN and ΓCT := (0, T ) × ΓC . We note u˙ the velocity of the elastic body and u¨ its acceleration; u0 is the initial displacement and u˙ 0 is the initial velocity. The density of the elastic material is denoted by ρ and is supposed to be a constant. The problem then consists in finding the displacement field u : [0, T ) × Ω → Rd verifying the equations and conditions (3)–(4): ρ u¨ − div σ (u) = f

σ (u) = A ε(u)

in ΩT ,

on ΓDT ,

σ (u)n = g

on ΓNT ,

in Ω,

˙ ·) = u˙ 0 u(0,

in Ω,

in ΩT ,

u=0 u(0, ·) = u0

(3)

The conditions defining unilateral contact with Coulomb friction on ΓCT are: un ≤ 0,

σn (u) ≤ 0,

σn (u) un = 0

u˙ t = 0

0⇒ |σ t (u)| ≤ −Fσn (u)

u˙ t = 0

0⇒ σ t (u) = Fσn (u)

u˙ t |u˙ t |

(i) (ii) (4)

(iii)

Additionally the initial displacement u0 should satisfy the compatibility condition u0n ≤ 0 on ΓC .

Proposition 1 Let γ be a positive function defined on ΓC . Static case: The frictional contact conditions (2) can be reformulated as follows:   σn (u) = σn (u) − γ un

R−

  , σ t (u) = σ t (u) − γ ut (−F[σ

n (u)−γ un

] R− ) .

Dynamic case: The frictional contact conditions (4) on ΓCT are equivalent to:   σn (u) = σn (u) − γ un

R−

  , σ t (u) = σ t (u) − γ u˙ t (−F[σn (u)−γ un ]

R−

)

.

1 (x − |x|) for 2 x ∈ R). Moreover, for any α ∈ R+ , we introduce the notation [·]α for the orthogonal The notation [·]R− stands for the projection onto R− ([x]R− =

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projection onto B(0, α) ⊂ Rd−1 , where B(0, α) is the closed ball centered at the 1

2 origin 0 and of radius α.  · s,D = (·, ·)s,D denotes the norm of (H s (D))d . : ;  d Let Vh ⊂ V := v ∈ H 1 (Ω) : v = 0 on ΓD , be a family of finite

dimensional vector spaces indexed by h coming from a family T h of triangulations of the domain Ω supposed to be regular and quasi-uniform. We choose a standard Lagrange finite element method of degree k with k = 1 or k = 2, i.e.:   Vh = vh ∈ (C 0 (Ω))d : vh |K ∈ (Pk (K))d , ∀K ∈ T h , vh = 0 on ΓD . We consider in what follows that γ = γh is a positive piecewise constant function on the contact interface ΓC which satisfies γ |K∩ΓC = γ0 hK , for every K that has a non-empty intersection of dimension d − 1 with ΓC , and where γ0 is a positive given constant. Let us define the discrete linear operators for a fixed parameter Θ ∈ R PnΘ,γ :

Qtγ

L2 (ΓC ) Vh → h v → Θσn (vh ) − γ vnh ,

PtΘ,γ :

Vh → (L2 (ΓC ))d−1 , vh → Θσ t (vh ) − γ vht



Vh × Vh → (L2 (ΓC ))d−1 : , (vh , v˙ h ) → σ t (vh ) − γ v˙ ht



L(vh ) =

f · vh dΩ + Ω

g · vh dΓ, ΓN

and the bilinear form: 



AΘγ (u , v ) = h

σ (u ) : ε(v ) dΩ −

h

h

h

Ω

Θ σ (uh )n · σ (vh )n dΓ. ΓC γ

Discrete Static Problem The Nitsche-based formulation for unilateral contact with Coulomb friction reads : ⎧ h h h h ⎪ ⎪ Find u ∈ V such that: ∀ v ∈ V ⎪ ⎪  ⎪ ⎪ 1 n ⎪ ⎪ ⎨ AΘγ (uh , vh ) + [P1,γ (uh )]R− PnΘ,γ (vh ) dΓ ΓC γ (5) ⎪  5 6 ⎪ ⎪ 1 ⎪ h h  · Pt ⎪ Pt1,γ (uh )  5 + ⎪ 6 Θ,γ (v ) dΓ = L(v ). ⎪ ⎪ ΓC γ −F Pn (uh ) ⎩ 1,γ

R−

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Semi Discretized Dynamic Problem Our space semi-discretized Nitsche-based method for frictional unilateral contact problems in elastodynamics then reads: ⎧ ⎪ Find uh : [0, T ] → Vh such that for t ∈ [0, T ] : ∀ vh ∈ Vh ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ 1 n ⎪ h h h h ⎪ [P1,γ (uh (t))]R− PnΘ,γ (vh ) dΓ ⎪ ⎪ ρ u¨ (t), v  + AΘγ (u (t), v ) + ⎪ ΓC γ ⎨  6 1 5 t h h h ⎪  · Pt ⎪ Qγ (u (t), u˙ h (t))  5 + 6 ⎪ Θ,γ (v ) dΓ = L(t)(v ), ⎪ n γ h ⎪ ΓC −F P1,γ (u (t )) ⎪ ⎪ ⎪ R− ⎪ ⎪ ⎪ ⎪ h h ⎩ uh (0, ·) = uh , u˙ (0, ·) = u˙ 0 , 0 (6) where uh0 (resp. u˙ h0 ) is an approximation in Vh of the initial displacement u0 (resp. the initial velocity u˙ 0 ). The notation ·, · stands for the L2 (Ω) inner product.

2 Existence and Well-Posedness Results The proofs of this section are detailed in [10].

Theorem 2 (Existence of Discrete Solutions for the Static Problem) Let us suppose that γ0 is small enough. Then for every Θ ∈ R and h > 0, the static problem (5) admits at least one solution. Moreover this solution satisfies the bound uh 1,Ω ≤ C, where the constant C > 0 depends only of the constants of V-ellipticity of a(·, ·) and of continuity of L(·), but not on the friction coefficient F and on the Nitsche’s parameter γ0 .

Sketch of the Proof We introduce the auxiliary problem involving (Tresca) friction P(g) with a fixed threshold g ∈ L2 (ΓC ), and discretized with Nitsche: ⎧ h h ⎪ ⎪ Find u ∈ V such that : ⎪ ⎪ ⎪  ⎪ ⎪ 1 n ⎨ AΘγ (uh , vh ) + [P1,γ (uh )]R− PnΘ,γ (vh ) dΓ P(g) γ ΓC ⎪ ⎪  ⎪ ⎪ 1 ⎪ ⎪ ⎪ [Pt1,γ (uh )]g · PtΘ,γ (vh ) dΓ = L(vh ), ∀ vh ∈ Vh . ⎩+ ΓC γ

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The solutions to Coulomb discrete problem (5) are the fixed point of the application h h h h h φ h: Vh → Vh defined  as follows: w → u (w ) where u (w ) is the solution to −F n [P1,γ (wh )]R− . P γ The application φ h is well defined [4]. Using standard bounds and arguments, we show that φ h is bounded and continuous in (Vh ,  · 1,Ω ). Thus we apply Brouwer’s fixed point theorem to prove the existence of, at least, one solution to Problem (5). & %

Proposition 3 (Well-Posedness) Static case: 1. If 0 ≤ F < 1, assume there exists C such that F≤C

h , γ0

((1 + Θ)2 + F(1 + Θ 2 )) ≤ C. γ0

2. or if F ≥ 1, assume there exists C such that (1 + Θ 2 )(1 + 4F2 ) ≤ C, γ0

 F≤C

h γ0

1 2

,

then Problem (5) admits one unique solution. Semi discretized dynamic case: For every value of Θ ∈ R and γ0 > 0, Problem (6) admits one unique solution uh ∈ C2 ([0, T ], Vh ).

Sketch of the Proof We introduce the following mesh- and parameter-dependent scalar product in Vh : 1

1

1

1

(vh , wh )γ = (vh , wh )1,Ω + (γ − 2 vnh , γ − 2 wnh )0,ΓC + (γ − 2 vht , γ − 2 wht )0,ΓC . – Static case: we define the (non-linear) operator Bh : Vh → Vh : 

1 n [P1,γ (vh )]R− PnΘ,γ (wh ) dΓ ΓC γ  6 1 5 t h  · Pt P1,γ (vh )  5 + 6 Θ,γ (w ) dΓ ΓC γ −F Pn (uh )

(Bh vh , wh )γ = AΘγ (vh , wh ) +

1,γ

R−

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for all vh , wh ∈ Vh . Then we prove that Bh is a one-to-one operator using Brezis’ theorem [2]. – Semi Discretized Dynamic Case Nitsche’s formulation leads to a system of (non-linear) second-order differential equations ⎧ ⎪ Find uh : [0, T ] → Vh such that for t ∈ [0, T ] : ⎪ ⎪ ⎨ Mh u¨ h (t) + Bh (uh (t), u˙ h (t)) = Łh (t), ⎪ ⎪ ⎪ ⎩ uh (0, ·) = uh , u˙ h (0, ·) = u˙ h0 . 0 with (Łh (t), wh )γ = L(t)(wh ), the mass operator Mh : Vh → Vh defined by (Mh vh , wh )γ = ρvh , wh  and with Bh : (Vh )2 → Vh , 

1 n [P1,γ (vh )]R− PnΘ,γ (wh ) dΓ γ ΓC  1 5 t h h 6 h  · Pt + Qγ (v , v˙ ) 6 5 Θ,γ (w ) dΓ, ΓC γ −F Pn (uh )

(Bh (vh , v˙ h ), wh )γ = AΘγ (vh , wh ) +

1,γ

R−

The operator Bh is Lipschitz-continuous and we conclude with the CauchyLipschitz theorem. & %

3 Numerical Results In what follows, we study an example where the three different zones characterizing friction (stick, slip and separation) exist. We consider the geometry Ωˆ = ]0, 2[×]0, 1[ and we adopt symmetry conditions (i.e., un = 0, σt (u) = 0) on ΓS = {1}×]0, 1[. We achieve the computations on the square Ω =]0, 1[×]0, 1[. We set ΓC =]0, 1[×{0} and ΓN = (]0, 1[×{1}) ∪ ({0}×]0, 1[). We suppose that the body is homogeneous isotropic material and a Poisson ratio of ν = 0.2, a Young modulus of E = 104 and a friction coefficient F = 0.5 are chosen. A density of surfaces forces F of magnitude (0.5 − y, 0) is applied on {0}×]0.5, 1[ and one of magnitude (0, x − 0.5) is applied on ]0.5, 1[×{1}. The Nitsche parameter γ0 is fixed to 100E and we consider the skew-symmetric case Θ = −1. We achieve the numerical implementation with uniform meshes with the open source finite element library GetFEM++ (see http://getfem.org/download.html). ? @ 1 1 1 1 1 , , , , The solution for mesh sizes h = are compared with a 4 8 16 32 64 reference solution on a very fine mesh (h = 1/128) and P2 Lagrange elements. Moreover, the reference solution is computed with a different discretization of the friction problem (Lagrange multipliers and Alart-Curnier augmented lagrangian).

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Figure 1 depicts the Von Mises stress and we can note a transition point on ΓC between a contact part and a separation part. Figure 2 shows the rates of convergence for the H 1 and L2 relative norms with P1 finite elements. For the H 1 norm we obtain the quasi optimality of the convergence rate whereas the suboptimality of the L2 norm may come from the lack of adjoint consistency when Θ = −1.

errors

Fig. 1 Von Mises stress with displacement amplified by 2000 10

0

10

-1

H1 norm P1(slope=0.95794) L2 norm P1 (slope=1.1206)

-2

10

-1

h

10

Fig. 2 H 1 and L2 norms on the displacement uh for P1 finite elements

Nitsche Method with Contact and Coulomb Friction

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References 1. P. Ballard, S. Basseville, Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem. M2AN Math. Model. Numer. Anal. 39, 59–77 (2005) 2. H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble), 18, 115–175 (1968), fasc. 1 3. A. Charles, P. Ballard, Existence and uniqueness of solutions to dynamical unilateral contact problems with Coulomb friction: the case of a collection of points. ESAIM Math. Model. Numer. Anal. 48, 1–25 (2014) 4. F. Chouly, An adaptation of Nitsche’s method to the Tresca friction problem. J. Math. Anal. Appl. 411, 329–339 (2014) 5. F. Chouly, P. Hild, A Nitsche-based method for unilateral contact problems: numerical analysis. SIAM J. Numer. Anal. 51, 1295–1307 (2013) 6. F. Chouly, P. Hild, Y. Renard, A Nitsche finite element method for dynamic contact: 1. Space semi-discretization and time-marching schemes. ESAIM Math. Model. Numer. Anal. 49, 481– 502 (2015) 7. F. Chouly, P. Hild, Y. Renard, A Nitsche finite element method for dynamic contact: 2. Stability of the schemes and numerical experiments. ESAIM Math. Model. Numer. Anal. 49, 503–528 (2015) 8. F. Chouly, P. Hild, Y. Renard, Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: theory and numerical experiments. Math. Comput. 84, 1089– 1112 (2015) 9. F. Chouly, R. Mlika, Y. Renard. An unbiased Nitsche’s approximation of the frictional contact between two elastic structures. Numer. Math. 139, 593–631 (2018) 10. F. Chouly, P. Hild, V. Lleras, Y. Renard, A Nitsche-based finite element method for contact with Coulomb friction (in preparation) 11. F. Chouly, M. Fabre, P. Hild, R. Mlika, J. Pousin, Y. Renard, An overview of recent results on Nitsche’s method for contact problems, in Proceedings of the UCL Workshop 2016 on Geometrically Unfitted Finite Element Methods and Applications. Lecture Notes in Computational Science and Engineering, vol. 121 (2017), pp. 93–141 12. D. Doyen, A. Ern, Analysis of the modified mass method for the dynamic Signorini problem with Coulomb friction. SIAM J. Numer. Anal. 49, 2039–2056 (2011) 13. C. Eck, J. Jarusek, Existence results for the static contact problem with coulomb friction. Math. Models Meth. Appl. Sci. 8, 445–468 (1998) 14. P. Hild, Y. Renard, Local uniqueness and continuation of solutions for the discrete Coulomb friction problem in elastostatics. Quart. Appl. Math. 63, 553–573 (2005) 15. T. Ligursky, Y. Renard, A well-posed semi-discretization of elastodynamic contact problems with friction. Quart. J. Mech. Appl. Math. 64, 215–238 (2011) 16. J.A.C. Martins, J.T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. 11, 407–428 (1987) 17. R. Mlika, Y. Renard, F. Chouly, An unbiased Nitsche’s formulation of large deformation frictional contact and self-contact. Comput. Methods Appl. Mech. Eng. 325, 265–288 (2017) 18. J. Nitsche, Uber ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36, 9–15 (1971) 19. Y. Renard, A uniqueness criterion for the Signorini problem with Coulomb friction. SIAM J. Math. Anal. 38, 452–467 (2006) 20. Y. Renard, Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity. Comput. Methods Appl. Mech. Eng. 256, 38–55 (2013)

Electrically Driven MHD Flow Between Two Parallel Slipping and Partly Conducting Infinite Plates Munevver Tezer-Sezgin and Pelin Senel

Abstract The magnetohydrodynamic (MHD) flow between two parallel slipping and conducting infinite plates containing symmetrically placed electrodes is solved by using the dual reciprocity boundary element method (DRBEM). The flow is driven by the current traveling between the plates and the external magnetic field applied perpendicular to the plates. The coupled MHD equations are solved for the velocity of the fluid and the induced magnetic field as a whole without introducing an iteration. The effects of both the slip ratio and the length of the electrodes are discussed on the flow and magnetic field behavior for increasing values of Hartmann number (H a). It is found that, an increase in the Hartmann number produces Hartmann layers of thickness 1/H a near the conducting parts and shear layers of √ order of thickness 1/ H a in front of the end points of electrodes. When the slip ratio increases Hartmann layers are weakened and the increase in the length of the electrodes retards this weakening effect of the slip on the Hartmann layers. The DRBEM discretizes only a finite portion of the plates and provides the solution inside the infinite region which is mostly concentrated in front of the electrodes. The aim of the study is to numerically simulate the MHD flow under the influence of the slipping velocity on the partly conducting plates which can not be treated theoretically.

1 Introduction Magnetohydrodynamics (MHD) combines the classical fluid dynamics with electrodynamics dealing with flows of electrically-conducting fluids subjected to an externally applied magnetic field [5]. MHD has important practical applications in engineering such as space propulsion, liquid metal mixing, MHD generators and magnetic flow meters, which mostly need the simulation of the flow between

M. Tezer-Sezgin () · P. Senel Department of Mathematics, Middle East Technical University, Ankara, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_80

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two parallel conducting plates. In most of the studies dealing with MHD flow, no-slip boundary condition is employed for the velocity of the fluid. Hunt and Williams [3] and Hunt and Malcolm [2] investigated the flow motion theoretically and experimentally, respectively, between parallel plates with electrodes subjected to an applied magnetic field perpendicular to the plates. Analytical solutions for the magnetohydrodynamic steady flow in an infinite channel and on a half-plane were developed by Sezgin [10, 11] reducing the problem to the solution of a Fredholm integral equation of the second kind which is obtained numerically. In the case of surface roughness, flow over hydrophobic surfaces and in thin film dynamics slip may occur at the fluid-solid interface. Hence, the employment of the no-slip boundary conditions are not realistic. Smolentsev [12] and Ligere [4] considered the slip at the Hartmann (perpendicular to the magnetic field) and the side (parallel to the magnetic field) walls in MHD duct flows, respectively. They presented analytical solutions in terms of infinite series for special combinations of wall conductivities. Rivero and Cuevas [8] considered the wall slip in MHD micropumps by neglecting the induced magnetic field. They simulated the flow rate with respect to the slip length for small Hartmann numbers. It is reported in [12] that, the case of completely slipping duct walls requires an asymptotic approach or a numerical solution. A numerical study based on finite difference method for the steady MHD flow through a square duct with the slip at the Hartmann walls is presented by Sarma and Deka [9]. The boundary element method (BEM) is an efficient technique for solving MHD flows in infinite regions. The idea of BEM is to transform the PDE into a boundary integral equation requiring only the discretization of the boundary, and sufficient number of interior nodes to depict the flow in the region. BEM solutions of MHD flow in an infinite region and between infinite strips were obtained by Bozkaya and Tezer-Sezgin [1, 13] taking a finite number of boundary elements on the infinite boundary. The dual reciprocity boundary element method (DRBEM) is a generalization of BEM that can tackle nonhomogeneous and nonlinear problems [6]. DRBEM considers all the terms other than the dominant PDE operator as inhomogeneity and BEM idea is applied to both sides of the equation. In this study, the MHD flow between two parallel partly conducting partly perfectly conducting plates exhibiting also the slip of the fluid is solved using the DRBEM. An external magnetic field is applied perpendicular to the plates, and an external circuit is connected to the plates at the sides of the electrodes, so a current enters the fluid from one plate and leaves from the other. The fluid starts to move with the interaction of the electric current and the imposed magnetic field. The coupled MHD equations are solved by using the DRBEM without the need of iteration for the velocity and the induced magnetic field for several values of slip ratio, electrode length and the Hartmann number. The effects of these parameters are shown by simulating the flow and the induced magnetic field in terms of equivelocity and current lines.

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2 Physical Problem and the Mathematical Model The MHD flow between two parallel slipping and partly conducting partly perfectly conducting (due to the electrodes) infinite plates is considered under the effect of vertically applied magnetic field. The fluid is viscous, incompressible, electrically conducting and the flow is laminar and steady. There is only one component of the velocity and the induced magnetic field in the z-direction having variations in the cross-section of a pipe in the xy-plane with horizontal walls tending to infinity. Figure 1 shows the geometry and the boundary conditions of the problem. The solution provides striking examples of the kind of layers which may be produced in a flow by discontinuities in the electrical boundary conditions when a strong magnetic field is applied to the fluid. As current experimental data suggest [7] the slip in MHD flow will likely occur in those fusion applications with liquid metal flows in contact with ceramics. Without the slip, the flow exhibits Hartmann layers with thickness of order 1/H a at the walls perpendicular √ to the magnetic field (Hartmann walls) and shear layers of thickness of order 1/ H a spreading along magnetic field lines where H a is the Hartmann number. In the liquid-metal blankets of a fusion reactor, if the slip behavior occurs, the slip length can be comparable or even larger than the thickness of the Hartmann layer. The degree of slip is quantified through the slip length Ls , defined as the distance from the liquid to the surface within the solid phase, where the extrapolated flow velocity vanishes. At the liquid-solid interface a linear relation between the slip velocity Us and the tangential stress as Us = Ls (dU /dn) is described. The influence of the slip at the Hartmann walls is measured by the slip ratio s = αH a where α is the dimensionless slip length. The corresponding mathematical model for the MHD flow in a channel with two parallel long enough slip walls (plates) containing electrical discontinuities due to

y=1

B=1 slip wall

∂B =0 I ∂y electrode x = −l y x=l

B0

y = −1

O

B=1 I

∂B =0 ∂y

Fig. 1 Problem geometry and the boundary conditions

V +α

∂V =0 ∂y

V −α

∂V =0 ∂y

x

x = −l x=l electrode

slip wall

B = −1

B = −1

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the electrodes placed at the middle of the walls, are dimensionless coupled equations [5] ∂B =0 ∂y ∂V =0 ∇ 2B + H a ∂y

∇ 2V + H a

in Ω

(1)

with V ±α

∂V =0 ∂y

at y = ±1, −∞ < x < ∞

B(x, ±1) = 1 x < −l, B(x, ±1) = −1 x > l,

(2)

∂B (x, ±1) = 0 − l ≤ x ≤ l ∂y (3)

where Ω = {(x, y) : −1 < y < 1, −∞ < x < ∞}, and V (x, y) and B(x, y) are the velocity and the induced magnetic field, l is the length of the electrodes. The parameter, Hartmann number is given as H a = √ dimensionless √ (B0 L σ )/( νρ) where B0 is the intensity of the applied magnetic field, L is the characteristic length (half of the distance between the plates), and σ , ν and ρ are the electrical conductivity, coefficient of viscosity and the density of the fluid, respectively.

3 The DRBEM Application The DRBEM transforms the equations in (1) into the boundary integral equations using the fundamental solution u∗ = (1/2π)ln(1/r) of the Laplace equation, [6]. Considering the convective terms as inhomogeneity, the equations are weighted by the fundamental solution u∗ and the Green’s first identity is applied two times to obtain corresponding integral equations for a source point i     ∂V ∂B ∗ dΓ = − )u dΩ = − V q ∗ dΓ − u∗ −(H a bV u∗ dΩ ci V i + ∂n ∂y Γ Γ Ω Ω (4) 

Bq ∗ dΓ −

ci Bi + Γ



u∗ Γ

∂B dΓ = − ∂n

 −(H a Ω

∂V ∗ )u dΩ = − ∂y



bB u∗ dΩ Ω

(5) where Γ = ∂Ω, q ∗ = ∂u∗ /∂n and n is the outward unit normal to the boundary Γ . ci is the constant ci = θ/2π with θ being the internal angle at the source point i measured in radians. Vi and Bi are the values at i.

Electrically Driven MHD Flow Between Two Parallel Infinite Plates

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The domain integrals on the right-hand sides of Eqs. (4) and (5) are eliminated by approximating bV and bB with the radial basis functions fj (r) = 1 + rj which are connected to the set of particular solutions as ∇ 2 uˆ j = fj , [6]. Here, rj denotes the distance between the source and the field points. Then, bV =

N+L

j =1

αj fj =

N+L

αj ∇ 2 uˆ j

j =1

bB =

N+L

βj fj =

j =1

N+L

βj ∇ 2 uˆ j

(6)

j =1

where αj and βj are the undetermined coefficients, N and L are the numbers of the boundary elements and the interior nodes, respectively. These approximations result in systems bV = Fα and bB = Fβ where F is the DRBEM coordinate matrix of size N + L constructed by taking fj ’s as columns. Substitution of approximations for bV and bB from Eq. (6) containing Laplace operator and applying the BEM procedure to both sides of Eqs. (4)–(5), the following DRBEM discretized matrix-vector equations are obtained HV − G

∂V ˆ − GQ)F ˆ −1 {H a ∂B } = −(HU ∂n ∂y

(7)

HB − G

∂B ˆ − GQ)F ˆ −1 {H a ∂V } = −(HU ∂n ∂y

(8)

where the matrices H and G of size (N + L) × (N + L) are   2 c Hij = ci δij + (ln( ) + 1) q ∗ dΓj , Hii = ci , Gij = u∗ dΓj , Gii = 2π c Γj Γj (9) c is the length of the constant element and V , B, ∂V /∂n, ∂B/∂n are the vectors with entries velocity, induced magnetic field and their normal derivative values ˆ Q ˆ are constructed by at the boundary and the interior nodes. The matrices U, taking each vector uˆ j and qˆj = ∂ uˆ j /∂n as columns, respectively. The space derivatives of V and B are approximated using the coordinate matrix F as ∂V /∂y = (∂F/∂y)F−1 V , ∂B/∂y = (∂F/∂y)F−1 B. Instead of using an iterative procedure, the coupled matrix-vector equations (7), (8) are combined and arranged to obtain a larger system and solved only once as ⎧ ⎫ ⎛⎡ ⎤ ⎡ ⎤⎞ ⎧ ⎫ ⎡ ⎤ ⎪ ∂V ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ ⎬ ⎨0⎪ ⎨ ⎬ ⎪ ⎬ ⎨V ⎪ 0 M ⎟⎪ G 0 ⎪ ⎜⎢H 0 ⎥ ∂n ⎢ ⎥ ⎢ ⎥ ⎜⎣ ⎟ . = + H a − ⎦ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠⎪ ⎪ ⎪ ⎪ ∂B ⎪ ⎭ ⎩0⎪ ⎪ ⎩B ⎭ 0 H M 0 0 G ⎪ ⎪ ⎪ ⎩ ⎭ ∂n ˆ − GQ)F ˆ −1 and M = S(∂F/∂y)F−1 . where S = (HU

(10)

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The boundary conditions for the induced magnetic field and the coupled boundary condition for the velocity slip on the walls are inserted to the system (10). Once all the unknowns are passed to the left-hand side a linear system Ax = b

(11)

is obtained and solved. This process gives the solution of the problem both in the interior and the boundary nodes in one stroke. The coefficient matrix A is a full matrix containing lots of zeros but scattered. Thus, Gaussian elimination with pivoting is used for the solution.

4 Numerical Results and Discussions The velocity V (x, y) and the induced magnetic field B(x, y) are computed by using at most N = 336 constant boundary elements on the plates, and some interior points for a finite horizontal length −3 ≤ x ≤ 3, y = ±1. The equivelocity and equivalue induced magnetic field lines are plotted for several values of perfectly conducting length l, and for slip ratio values s < 1, s = 1, s > 1 for Hartmann number values H a = 10, 50, 100.

4.1 Case 1: (s = 0, l = 0, H a Increases) When a current is injected by line electrodes set in non-conducting plates at x = 0, y = ±1, current enters the fluid at (0, 1) and leaves at (0, −1) giving conductivities B = −1 and +1 for x > 0 and x < 0, respectively, and the plates are no-slip walls. One can notice from Fig. 2 that although the MHD problem is solved for an

Fig. 2 s = 0, l = 0 from left to right H a = 10, 50, 100

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infinite strip, the flow is confined to a relatively small region near the x = 0 line on which the plates have jump in the values of magnetic field. In the rest of the region the fluid is almost stagnant. This region is reduced when H a increases. The flow emanates from the line electrodes and spreads into the channel in terms of four reversal velocity loops symmetrically located with respect to x- and y- axes. Current lines travel between the source points of line electrode in terms of closed loops. Also, in most parts of the region the value of induced magnetic field is stationary and equal to its maximum value. The velocity vanishes at the symmetry line y = 0 and shows highest values when approaching the plates. This special case of the flow induced by line electrodes has been considered by Hunt and Williams [3] and our solution for l = 0, s = 0 coincides with theirs. They examine asymptotic solution for large H a in separate regions. It is noted that as H a increases boundary layer formation starts for x > 0 and x < 0 on the plates. These are Hartmann layers of order of thickness 1/H a. There is also a thin region near x = 0 line where shear layers (parabolic √ layers) originating from the source points (0, 1) and (0, −1) of order of width 1/ H a spreading along magnetic lines for increasing values of H a.

4.2 Case 2: (l = 0, H a = 10, s Varies as s < 1, s = 1, s > 1) The effect of slip ratio is examined for a fixed Hartmann number in the absence of conducting part in the plates. When the slip ratio is less than 1 (Fig. 3), a slight slip is noticed on the plates, however Hartmann layer formation is visible on each wall. As s is increasing, slips on the plates become dominant weakening the Hartmann layers, and the magnitude of the velocity increases. Hartmann layers are diminished especially for s > 1. Also, when the slip ratio increases both the flow and induced magnetic field lines (current lines) tend to concentrate to the line electrodes (x = 0 line).

Fig. 3 l = 0, H a = 10 from left to right s < 1 (α = 0.001), s = 1 (α = 0.1), s > 1 (α = 0.2)

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Fig. 4 s = 0, H a = 50 from left to right l = 0.1, l = 0.3, l = 0.5

4.3 Case 3: (s = 0, H a = 50, l Varies as l = 0.1, 0.3, 0.5) In this case, two electrodes of length 2l are placed symmetrically with respect to the x-axis at the middle of the plates indicating the perfectly conducting parts of the noslip walls (∂B/∂n = 0). The length of the electrodes is varied as l = 0.1, 0.3, 0.5 for a fixed Hartmann number H a = 50. It is observed from Fig. 4 that, the increase in l (perfectly conducting portion) causes the formation of a stagnant region for the velocity in front of the electrodes. Induced magnetic field lines are perpendicular to y = ±1 lines for −l < x < l demonstrating the derivative condition on B (∂B/∂n = 0), and they travel from the top electrode to the bottom electrode when the electrode length increases. Parabolic layers (shear layers) in induced magnetic field lines originating from x = l and x = −l are connected at x = 0 line but are separated as l is increasing. Sezgin [10, 11] obtained the solution of MHD flow for this case by reducing the problem to dual integral equations and then to Fredholm integral equation of the second kind. In Bozkaya and Tezer-Sezgin [1, 13] the problem is solved numerically by using BEM with a fundamental solution derived for coupled MHD equations. Our DRBEM solution compares well with their solutions for line electrodes (l = 0) and electrodes with finite length (l = 0) cases.

4.4 Case 4: (H a = 10, s Varies as s < 1, s = 1, s > 1 and l Varies as l = 0.3, l = 0.5) We consider the effects of both the slipping walls and partly conducting, partly perfectly conducting walls on the behavior of the flow for a fixed Hartmann number value (H a = 10). In Fig. 5, we fix also the length of the conducting part as l = 0.3 and l = 0.5, respectively and consider the slip ratio effect when s < 1, s = 1, s > 1. The slip

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Fig. 5 H a = 10 l = 0.3 (top), l = 0.5 (bottom) and from left to right s < 1 (α = 0.001), s = 1 (α = 0.1), s > 1 (α = 0.2)

in the velocity is more observed on the plates as the slip ratio increases. Hartmann layers are only shown for s < 1 but start to diminish for s = 1 and are completely lost for s > 1. That is, the thickness of Hartmann layers and the slip length are comparable for s = 1. The increase in the electrodes length retards the weakening effect of slipping on Hartmann layers, that is, we can still see the Hartmann layers near the plates (i.e. when electrode length is l = 0.3 slip starts to weaken the Hartmann layer at s = 1, but when electrode length is l = 0.5 slip diminishes Hartmann layers when s > 1).

5 Conclusion The MHD flow between two partly conducting and partly perfectly conducting infinite plates exhibiting slip velocity is simulated by using the DRBEM. It is found that, as Hartmann number increases Hartmann layers of thickness order 1/H a are formed √ in front of the conducting parts, and also shear layers of thickness order 1/ H a, originating from the end points of the electrodes are observed. As the slip ratio is increased (s > 1) Hartmann layers are weakened and the slip velocity increases. Increase in the length of the electrodes causes the formation of a stagnant region for the velocity in front of the electrodes, and retards the weakening effect of slipping on the Hartmann layers. The effects of the increase in the Hartmann number and the increase in the slip ratio is comparable when the slip ratio is equal to one.

References 1. C. Bozkaya, M. Tezer-Sezgin, A numerical solution of the steady MHD flow through infinite strips with BEM. Eng. Anal. Bound. Elem. 36, 591–566 (2012) 2. J.C.R. Hunt, D.G. Malcolm, Some electrically driven flows in magnetohydrodynamics part 2. Theory and experiment. J. Fluid Mech. 33(4), 775–801 (1968)

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3. J.C.R. Hunt, W.E. Williams, Some electrically driven flows in magnetohydrodynamics part 1. Theory. J. Fluid Mech. 31(4), 705–722 (1968) 4. E. Ligere, I. Dzenite, A. Matvejevs, MHD flow in the duct with perfectly conducting Hartmann walls and slip condition on side walls, in 10th PAMIR International Conference-Fundamental and Applied MHD, Cagliari, Italy, 20–24 June 2016, pp. 279–283. 5. U. Muller, L. Buhler, Magnetofluiddynamics in Channels and Containers (Springer, Berlin, 2001) 6. P.W. Partridge, C.A. Brebbia, L.C. Wrobel, The Dual Reciprocity Boundary Element Method (Computational Mechanics Publications, Sauthampton, 1992) 7. B.A. Pint, K.L. More, H.M. Meyer, J.R. Distefano, Recent progress addressing compatibility issues relevant to fusion environments. Fusion Sci. Technol. 47, 851–855 (2005) 8. M. Rivero, S. Cuevas, Analysis of the slip condition in magnetohydrodynamic (MHD) micropumps. Sensors Actuators B Chem. 166–167, 884–892 (2012) 9. D. Sarma, P.N. Deka, Numerical study of liquid metal MHD duct flow under hydrodynamic “slip” condition. Int. J. Comput. Appl. 81(16), 7–10 (2013) 10. M. Sezgin, Magnetohydrodynamic flow in an infinite channel. Int J. Numer. Methods Fluids 6, 593–609 (1986) 11. M. Sezgin, Magnetohydrodynamic flow on a half-plane. Int. J. Numer. Methods Fluids 8, 743– 758 (1988) 12. S. Smolentsev, MHD duct flows under hydrodynamic ‘slip’ condition. Theor. Comput. Fluid Dyn. 23, 557–570 (2009) 13. M. Tezer-Sezgin, C. Bozkaya, The boundary element solution of the magnetohydrodynamic flow in an infinite region. J. Comput. Appl. Math. 225, 510–521 (2009)

Two Methods for the Numerical Modelling of the PM Transport and Deposition on the Vegetation ˇ Ludˇek Beneš and Hynek Rezníˇ cek

Abstract Two different methods for the simulation of particulate matter (PM) transport, dispersion and sedimentation on the vegetation are presented. A common sectional model based on a transport equation for each PM size fraction is compared to the innovative model known as moment method. It is based on solving of three transport equations for the moments of the whole PM distribution. Both methods are tested in 2D on a tree patch and in 3D on a hedgerow. The background flow field in the Atmospheric Boundary Layer (ABL) used for both methods is computed by solver based on RANS equations for viscous incompressible flow with stratification due to gravity. The two equations k −  turbulence model is used. Three effects of the vegetation are considered: slowdown or deflection of the flow, influence on the turbulence levels inside or near the vegetation and filtering of the particles present in the flow.

1 Introduction Atmospheric particulate matter (PM) is a well known risk factor to human health, especially when the concentration of PM particles is high. Near urban areas the high concentration is produced mainly by the vehicular traffic. Vegetative barriers are one of the most popular ways how to reduce the PM pollution. The investigation of vegetative barriers is difficult due to complex geometries, therefore effective and accurate method for modelling influence of the vegetation on the dustiness is essential. CFD modelling can help us to accurately assess the barrier effects despite the complex geometries. Many publications focused on the mathematical modelling of the pollutant deposition on the vegetation are available. Among the most notable the following

ˇ L. Beneš · H. Rezníˇ cek () Czech Technical University in Prague, Faculty of Mechanical Engineering, Prague, Czech Republic e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_81

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Fig. 1 Illustrative example of typical PM size distribution in ABL [6]

can be included: reviews on the topic of dry deposition on the vegetation [1, 2] and [3] or modelling studies [4] and [5] dealing with aerosols and collection efficiency. Focus of this paper can be expressed by a question: How to model a whole range of all PM particle sizes? As schematically shown in Fig. 1 the dust in ABL contains wide range of particles sizes most usually with 2-modal log-normal distribution. A classical approach divides the whole range of particle size into finite bins and solve one transport PDE for concentration at each finite bin. Other possibility is to model the evolution of the particle size distribution represented by its moments. Such approach can dramatically reduce the number of solved PDEs and consequently the computational demands. This class of methods, called the moment method, has been used earlier for the simulation of the aerosol behaviour in larger scales-models [7]. However it was rarely used in micro-scale, with the exception of [8] dealing with small scales, but the method hasn’t been used for the vegetation. Novelty of this work lies in adjusting the moment method to the vegetation settling model [1].

2 Mathematical Model of the Air Flow A baseline of the pollution modelling is of course a good determination of background flow field. Fluid Flow was computed by in-house solver described in [9]. In the formulation of Reynolds-averaged Navier-Stokes (RANS) equations the pressure p and potential temperature θ are split into background component in hydrostatic balance and fluctuations, p = p0 + p and θ = θ0 + θ  . Boussinesq approximation stating that changes in density are negligible everywhere except in the gravity term is utilized.

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Resulting set of equations is as follows: ∇ · u = 0,





∂u p + (u · ∇)u − ∇ · (νE ∇u) = −∇ + g + Tu , ∂t ρ0   νE ∂θ    + ∇ · (θ u) = ∇ ∇θ . ∂t Pr

(1) (2)

(3)

Here vector u stands for velocity, constant ρ0 represents the air density at the ground level, νE = νL + νT is the effective kinematic viscosity which is a sum of the  laminar and turbulent viscosity, g = (0, 0, −g θθ0 ) is the gravity term, Tu represent the momentum sink due to the vegetation and Pr = 0.75 stands for the Prandtl number. Turbulence is modelled by standard k −  model completed with source terms acting inside the vegetation. Equations for turbulence kinetic energy k and dissipation  are as follows:   ∂ρk μT + ∇ · (ρku) − ∇ · ∇k = Pk − ρ + ρSk , (4) ∂t σk   ∂ρ μT  2 + ∇ · (ρu) − ∇ · ∇ = C1 Pk − C2 ρ + ρS . (5) ∂t σ k k In the equations above the laminar dynamic viscosity μL is neglected and 2 the constitutive relation μT = Cμ ρ k for turbulent viscosity is used. Pk is the production of the turbulence kinetic energy, Sk and S are sources of k and  respectively. Both consist of a part due to the road traffic and a part due to the vegetation, Sk = Skr + Skv , S = Sr + Sv . Sources due to the road traffic are modelled by the model from [10] while sinks and sources due to the vegetation are described below. Following model constants are used: σk = 1.0, σ = 1.167, C1 = 1.44, C2 = 1.92 and Cμ = 0.09 [11]. Effects of Vegetation on the flow field act in two processes: firstly, a momentum sink inside the vegetation block present in the Eq. (2) is modelled by T u = −Cd LAD |u|u. Here Cd = 0.3 is the drag coefficient [12]. Secondly the influence on the turbulence is inserted. Following [11] the terms in Eqs. (4) and (5) can be written as Skv = Cd LAD(βp |u|3 − βd |u|k),

 Sv = C4 Skv . k

(6)

Constants used in the solver are βp = 1.0, βd = 5.1 and C4 = 0.9. The parameters are vertical Leaf area density (LAD ) profile which is foliage surface area per unit volume, a leaf type (broadleaf or needle) and size of the leaf.

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2.1 Numerical Methods The previous equations are computed by in-house solver with finite volume solver based on AUSM+ up scheme [13] with linear reconstruction. To prevent the overshooting the Venkatakrishnan limiter [14] is employed. The viscous terms are solved on dual (diamond type) mesh. To deal with the incompressibility the artificial compressibility method is employed [15]. Fully implicit BDF2 scheme is used to integrate the discretized system of equations in time. Each of these non-linear systems is solved by the JFNK method. Inner linear systems are solved using matrix-free GMRES solver. The linear systems are preconditioned by ILU(3) preconditioner. Necessary evaluations of the Jacobians are done via finite differences [9].

3 Pollution Transport Modelling Two methods of dust spread modelling are compared in this paper. The first one is commonly used sectional model based on the PM concentration equation and the second is a moment method which is a novelty in ABL. Sectional Model is build on simple transport equation for concentration (of passive contaminant), as follows ∂nN = −∇ · (u nN ) + ∇ · ∂t



νE ∇nN Sc

 − ∇ · (g

ρp 2 d nN ) − LAD udep nN 18μ p

(7)

where ρp denotes a particle density, μ stands for kinematic viscosity and nN (dp ) is the number concentration of the particles with given diameter dp . The term with udep represents a sink term whose calculation and approximation will be discussed later in Sect. 3.1. The whole particle sizes spectrum is divided into finite bins with thickness Δdp and the concentrations are represented with discrete value of nN . These concentration are located in the middle of each finite bin. Of course the small Δdp is counterbalanced by number of equations (for each bin one eq. is needed). Moment Method evaluates the development of the whole spectrum of particles. The concentration nN (dp ) is assumed to have log-normal distribution nN (dp ) = √

NT 2π ln σg

 exp −

(ln dp − ln dg )2 2 ln2 σg

 ,

(8)

described with three parameters: total number concentration NT , geometric mean size dg and geometric standard deviation σg . These parameters can be obtained from three moments [16] and characterize completely the distribution. That’s why three PDE only are needed to describe the whole distribution. If the k-th moment of the

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distribution is define as follows 

∞

Mk = 0

dpk nN (dp )ddp ,

(9)

a transport equation for the k-th moment can be used instead of Eq. (7), as follows ρp νE ∂Mk = −∇ · uMk + ∇ · ∇Mk − ∇ · g Mk+2 − Sdep,k (Mk ). ∂t Sc 18μ

(10)

Some moments have physical meaning, zeroth moment is equal to number concentration, second and third moments are proportional to the surface area and volume concentration respectively. These very moments are used. It has to be noticed that the choice of the moments has an non-negligible impact on the results, see discussion in [17].

3.1 Dry Deposition The deposition sink term Sdep of the Eq. (10) is equal 

∞

Sdep,k (Mk ) = −LAD 0

dpk udep (dp )nN (dp )ddp .

(11)

The deposition velocity udep acting in z-direction is computed via [18]. The implementation for the sectional model is quite straightforward [9], but some approximations are needed when the integral above is solved for the moment method. Deposition velocity is a sum of four main processes by which particles depose on the leaves: Brownian diffusion uBD = U CC Sc−2/3 Re−1/2 , sedimentation uSE = Cz gρp CC dp2 , 18μa

interception uI N = 2U Cx dp /de and impaction uI M = U Cx EI M . To evaluate the integral from Eq. (11) the deposition velocity has to be in polynomial form of dp . To achieve it the following approximations are introduced: for the Cunningham correction factor CC (in uBD and uSE ) a simplified approximation from [8] is used instead of full expression in original model [1], the impaction efficiency EI M = f (St) ≈ aSt+b is treated as piecewise linear function of Strouhal number instead of EI M = (St/(St + βd ))2 . The approximated deposition velocity is shown in the Fig. 2 where the relative difference in comparison with original model [18] is also visible.

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Fig. 2 The approximation of the deposition velocity and its relative difference in comparison with Petroff’s model

Fig. 3 Sketch of the 2D domain (left) and LAD profile (right)

4 Numerical Results Both methods were tested on 2D and 3D case. LAD profile was adopted from [19] with constants LAI=5, zm = 0.4h. Log-normal distribution NT = 106 m3 , dg = 0.86 μm, σg = 2.21 was prescribed at the inlet. The sketch of a 2D domain with tree patch and LAD profile used are displayed in Fig. 3. The intensity of point pollutant source was normalized to 1 s−1 . The volume concentration (M3 ) computed by moment method shown in the Fig. 4 was in a very good agreement with concentration computed by the sectional model, the relative difference between both methods (Fig. 4 right) is lower than 3%. However when the volume concentration is plotted in concrete points, see Fig. 5, the moment method underestimates the highest peak of concentration just after the vegetation. The slight violation of an assumption for conserving a log-normal distribution in the vegetation can explain this observation.

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Fig. 4 M3 calculated by the moment method (in μm3 /m3 ) (left) and relative difference (M3mm − M3sec )/M3mm between the sectional approach (right)

Fig. 5 Volume concentration at points [60; 15] (left), [80; 2] (middle) and [200; 2] (right). Discrete points calculated by the sectional method and the distribution calculated by the moment method are shown

Fig. 6 Sketch of the 3D domain (left) and LAD profile (right)

The 3D case represents a pollution spreading over a hedgerow near the street. The particles source was placed on a line 2 m upstream from the hedge and 0.5 m above the ground. Intensity of the source was normalized to 1 s−1 m−1 . The sketch of a domain with hedgerow and LAD profile used are displayed in the Fig. 6. The positive influence of the hedgerow which lessen the concentration behind is shown on the volume concentration in horizontal plane, see Fig. 7. The comparison between both methods displayed on the right shows quite good agreement up to 7%. The concentration computed by the moment method is underestimated mostly right after the barrier like in 2D case. The explanation can be again the slight violation of an assumption for conserving a log-normal distribution in the vegetation.

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Fig. 7 Horizontal cut at height z = 0.5 m, M3 calculated by the moment method (in μm3 m−3 ) (left). Relative difference (M3mm − M3sec )/M3mm between the sectional approach (right)

5 Conclusions The new moment method for modelling of PM concentration spread near vegetative barriers (inside ABL) was described and tested. The comparison with commonly used sectional approach showed very good agreement for simple cases (up to 7% relative difference). However the moment method slightly underestimates a volume concentration behind a barrier. The moment method is less computationally demanding, the computational time was usually six-times shorter than for the sectional method. The limited factor for the moment method lies in the strong assumption of the particle size distribution. Acknowledgements This work was supported by the grant SGS16/206/OHK2/3T/12 of the Czech Technical University in Prague. The authors are thankful for cooperation with Viktor Šíp.

References 1. A. Petroff, A. Mailliat, M. Amielh, F. Anselmet, Aerosol dry deposition on vegetative canopies. Atmos. Environ. 42, 3625–3683 (2008) 2. S. Janhäll, Review on urban vegetation and particle air pollution - deposition and dispersion. Atmos. Environ. 105, 130–137 (2015) 3. T. Litchke, W. Kuttler, On the reduction of urban particle concentration by vegetation - a review. Meteorol. Z. 17, 229–240 (2008) 4. A. Tiwary, H. Morvanb, J. Colls, Modelling the size-dependent collection efficiency of hedgerows for ambient aerosols. Aerosol Sci. 37, 990–1015 (2005) 5. J. Steffens, Y. Wang, K. Zhang, Exploration of effects of a vegetation barrier on particle size distributions in a near-road environment. Atmos. Environ. 50, 120–128 (2012) 6. F. Sand, Introduction to Particulate Matter (2017). https://publiclab.org/wiki/pm 7. C. Jung, Y. Kim, K. Lee, A moment model for simulating raindrop scavenging of aerosols. J. Aerosol Sci. 34(9), 1217–1233 (2003) 8. S. Bae, C. Jung, Y. Kim, Development of an aerosol dynamics model for dry deposition process using the moment method. Aerosol Sci. Technol. 43(6), 570–580 (2009)

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9. V. Šíp, L. Beneš, RANS solver for microscale pollution dispersion problems in areas with vegetation: development and validation (2016, Preprint). arXiv:1609.03427 10. D. Bäumer, B. Vogel, F. Fiedler, A new parameterisation of motorway-induced turbulence and its application in a numerical model. Atmos. Environ. 39(31), 5750–5759 (2005) 11. G.G. Katul, L. Mahrt, D. Poggi, C. Sanz, One- and two- equation models for canopy turbulence. Bound. Layer Meteorol. 113, 81–109 (2004) 12. V. Šíp, L. Beneš, CFD Optimization of a Vegetation Barrier. Numerical Mathematics and Advanced Applications ENUMATH 2015, ed. by B. Karasözen, M. Manguo˘glu, M. TezerSezgin, S. Goktepe, Ö. U˘gur (Springer, Berlin, 2015) 13. M.-S. Liou, A sequel to AUSM, part II: AUSM+-up for all speeds. J. Comput. Phys. 214, 137–170 (2006) 14. V. Venkatakrishnan, Convergence to steady state solutions of the euler equations on unstructured grids with limiters. J. Comput. Phys. 118, 120–130 (1995) 15. F. Muldoon, S. Acharya, A modification of the artificial compressibility algorithm with improved convergence characteristics. Int. J. Numer. Methods Fluids 55(4), 307–345 (2007) 16. E. Whitby, P. McMurry, Modal aerosol dynamics modeling. Aerosol Sci. Technol. 27(6), 673– 688 (1997) 17. V. Sip, Numerical simulations of microscale atmospheric flows and pollution dispersion, doctoral thesis (2016) FME, CTU in Prague 18. A. Petroff, L. Zhang, S. Pryor, Y. Belot, An extended dry deposition model for aerosols onto broadleaf canopies. J. Aerosol Sci. 40(3), 218–240 (2009) 19. B. Lalic, D.T. Mihailovic, An empirical relation describing leaf-area density inside the forest for environmental modeling. Appl. Meteorology. 43(4), 641–645 (2004)

On a New Mixed Formulation of Kirchhoff Plates on Curvilinear Polygonal Domains Katharina Rafetseder and Walter Zulehner

Abstract For Kirchhoff plate bending problems on domains whose boundaries are curvilinear polygons a discretization method based on the consecutive solution of three second-order problems is presented. In Rafetseder and Zulehner (SIAM J Numer Anal 56(3):1961–1986, 2018) a new mixed variational formulation of this problem is introduced using a nonstandard Sobolev space (and an associated regular decomposition) for the bending moments. In case of a polygonal domain the coupling condition for the two components in the decomposition can be interpreted as standard boundary conditions, which allows for an equivalent reformulation as a system of three (consecutively to solve) secondorder elliptic problems. The extension of this approach to curvilinear polygonal domains poses severe difficulties. Therefore, we propose in this paper an alternative approach based on Lagrange multipliers.

1 The Kirchhoff Plate Bending Problem We consider the Kirchhoff plate bending problem, where the undeformed midsurface is described by a domain Ω ⊂ R2 with a Lipschitz boundary Γ . The plate is considered to be clamped on a part Γc ⊂ Γ , simply supported on Γs ⊂ Γ , and free on Γf ⊂ Γ with Γ = Γc ∪Γs ∪Γf . Furthermore, n = (n1 , n2 )T and t = (−n2 , n1 )T represent the unit outer normal vector and the unit counterclockwise tangent vector to Γ , respectively. Then the problem reads: For given load f , find a deflection w such that   div Div C∇ 2 w = f

in Ω,

(1)

K. Rafetseder () · W. Zulehner Johannes Kepler University Linz, Institute of Computational Mathematics, Linz, Austria e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_82

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where div denotes the standard divergence of a vector-valued function, Div the rowwise divergence of a matrix-valued function, ∇ 2 the Hessian, and C a fourth-order material tensor. The boundary conditions are given by w = 0,

∂n w = 0

on Γc ,

w = 0,

Mn · n = 0

on Γs ,

Mn · n = 0, ∂t (Mn · t) + Div M · n = 0

on Γf ,

and the corner conditions M nt x = (Mn1 · t1 )(x) − (Mn2 · t2 )(x) = 0

for all x ∈ VΓ,f ,

where M denotes the bending moment tensor, given by M = −C∇ 2 w, and VΓ,f denotes the set of corner points whose two adjacent edges (with corresponding normal and tangent vectors n1 , t1 and n2 , t2 ) belong to Γf . As an example, the material tensor C for isotropic materials is given by   CN = D (1 − ν)N + ν tr(N )I ,

(2)

for matrices N , where ν is the Poisson ration, D > 0 depends on ν, Young’s modulus, and the thickness of the plate, I is the identity matrix, and tr is the trace operator for matrices. A standard (primal) variational formulation of (1) is given as follows: Find w ∈ W such that  C∇ 2 w : ∇ 2 v dx = F, v for all v ∈ W, (3) Ω

 with the Frobenius inner product A : B = i,j Aij B ij for matrices A, B, the  right-hand side F, v = Ω f v dx, and the function space W = {v ∈ H 2 (Ω) : v = 0, ∂n v = 0 on Γc ,

v = 0 on Γs }.

(4)

Here and throughout the paper L2 (Ω) and H m (Ω) denote the standard Lebesgue and Sobolev spaces of functions on Ω with corresponding norms .0 and .m 1 (Ω) denotes the set of function in H 1 (Ω) for positive integers m. Moreover, H0,Γ  which vanish on a part Γ  of Γ . The L2 -inner product on Ω and Γ  are always denoted by (., .) and (., .)Γ  , respectively, no matter whether it is used for scalar, vector-valued, or matrix-valued functions. We use H ∗ to denote the dual of a Hilbert space H and ., . for the duality product on H ∗ × H .

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2 New Mixed Formulation In our previous work [5] a new mixed variational formulation for the Kirchhoff plate bending problem with the bending moment tensor M as additional unknown is derived. The new mixed formulation satisfies Brezzi’s conditions and is equivalent to the original problem without additional convexity assumption on Ω. These important properties come at the expense of an appropriate nonstandard Sobolev space for M. In order to make this space computationally accessible, we show in [5, Theorem 4.2] a regular decomposition of it, which provides the following representation of the solution M M = pI + symCurl φ, 1 with p ∈ Q = H0,Γ (Ω) and φ ∈ (H 1 (Ω))2 satisfying the coupling condition c ∪Γs

 ∂t φ, ∇vΓ = −

p ∂n v ds

for all v ∈ W,

(5)

Γ

where ∂t φ = (Curl φ)n ∈ (H − 2 (Γ ))2 with H − 2 (Γ ) = (H 2 (Γ ))∗ . Here the symmetric Curl is defined as symCurl ψ = 12 (Curl ψ + (Curl ψ)T ) with 1

1

1



 ∂2 ψ1 −∂1 ψ1 Curl ψ = . ∂2 ψ2 −∂1 ψ2 The analogous representation for the test functions associated to M in the mixed formulation leads to the following equivalent formulation of (3): For F ∈ Q∗ , find (p, φ) ∈ V and w ∈ Q such that (pI + symCurl φ, qI + symCurl ψ)C −1 − (∇w, ∇q) = 0, − (∇p, ∇v)

= −F, v,

(6)

1 for all v ∈ Q = H0,Γ (Ω) and (q, ψ) ∈ V , where the function space V is given c ∪Γs as the subset of (q, ψ) ∈ Q × (H 1 (Ω))2 satisfying

 ∂t ψ, ∇vΓ = −

q ∂n v ds

for all v ∈ W.

Γ

Here, we use the notation (M, N )C −1 = (C −1 M, N ).

(7)

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2.1 Coupling Condition as Standard Boundary Conditions for φ In [5] we obtain for polygonal domains Ω an equivalent formulation of the Kirchhoff plate bending problem (3) in terms of three (consecutively to solve) second-order elliptic problems: 1. The p-problem: Find p ∈ Q such that (∇p, ∇v) = F, v

for all v ∈ Q.

2. The φ-problem: For given p ∈ Q, find φ ∈ Ψp = ψ[p] + Ψ0 such that (symCurl φ, symCurl ψ0 )C −1 = −(pI , symCurl ψ0 )C −1

for all ψ0 ∈ Ψ0 .

3. The w-problem: For given M = pI + symCurl φ, find w ∈ Q such that (∇w, ∇q) = (M, qI + symCurl ψ[q])C −1

for all q ∈ Q.

The second and the third problem require the construction of a particular function ψ[q] satisfying the coupling condition (7) for given q ∈ Q, for details see [5]. The space Ψ0 consists of all functions in (H 1 (Ω))2 satisfying (7) for q = 0. The approach presented in [5] is to characterize Ψ0 as space of functions ψ with standard boundary conditions available in (H 1 (Ω))2 . Originally, the boundary conditions for ψ ∈ Ψ0 are, roughly speaking, conditions for tangential derivatives of ψ of the form ∂t ψ · n = 0 ∂t2 ψ · t = 0,

∂t ψ · n = 0

on Γs ,

(8)

on Γf .

(9)

For polygonal domains we obtain from (9) a Dirichlet boundary condition for ψ. Moreover, (8) yields a Dirichlet boundary condition for the normal component ψ · n. However, the considerations heavily rely on a polygonal domain and it is not clear how to obtain standard boundary conditions in the curved case. This is our main motivation to investigate an alternative approach to incorporate the coupling condition (7) based on Lagrange multipliers, which we introduce in the next section. In [5] we propose a discretization method for the above introduced formulation using a Nitsche method to incorporate the boundary conditions in the φ-problem and present a numerical analysis of the method.

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3 Coupling Condition via Lagrange Multipliers We consider a domain Ω, whose of class 4K boundary is a curvilinear polygon ∞ curves for C ∞ . This means that Γ = E , where the edges E are C k k k=1 k = 1, 2, . . . , K and E k denotes the closure of Ek . The edges are numbered consecutively in counterclockwise direction. We denote the vertex at the startpoint of E k by ak and the interior angle at ak by ωk . Note, since we consider a closed boundary curve, the index k = 0 is in the following always identified with k = K. Furthermore, we assume that each edge Ek is contained in exactly one of the sets Γc , Γs , Γf , and the edges are maximal in the sense that two edges with the same boundary condition do not meet at an angle of π. By using the representation ∇v = (∂n v) n + (∂t v) t and incorporating the boundary conditions for v ∈ W , the coupling condition (5) reads (∂t φ · n + p, ∂n v)Γs ∪Γf + (∂t φ · t, ∂t v)Γf = 0 for all v ∈ W, provided ∂t φ ∈ (L2 (Γ ))2 . We can rewrite the condition as follows

(∂t φ · n + p, μkn )Ek +

Ek ⊂Γs ∪Γf

(∂t φ · t, μkt )Ek = 0,

(10)

Ek ⊂Γf

1 2 K for all μ = ((μ1t , μ2t , . . . , μK t ), (μn , μn , . . . , μn )) ∈ Λ where

Λ = {(∂t v, ∂n v) : for v ∈ W }, with ∂t v = (∂t v|E1 , ∂t v|E2 , . . . , ∂t v|EK ),

∂n v = (∂n v|E1 , ∂n v|E2 , . . . , ∂n v|EK ).

We view the original formulation (6) as optimality system with constraint (∇p, ∇v) = F, v and replace the space V by Q × (H 1 (Ω))2 and add (10) as additional constraint. The corresponding optimality system is the starting point for the discretization method we introduce in Sect. 3.2.

3.1 Characterization of Λ In this subsection we provide an explicit characterization of Λ. Let us consider 1 2 K k k μ = ((μ1t , μ2t , . . . , μK t ), (μn , μn , . . . , μn )), where μt and μn for k = 1, 2, . . . , K are Lipschitz continuous functions on E k . Then μ ∈ Λ if and only if the following three conditions are satisfied: 1. The boundary conditions μkt = 0 on edges Ek ⊂ Γs ∪ Γc and μkn = 0 on edges Ek ⊂ Γc have to hold.

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2. On each connected component C of Γf the compatibility condition

 Ek ⊂C Ek

μkt ds = 0

has to be satisfied. k 3. The four corner values μk−1 (ak ), μkt (ak ), μk−1 t n (ak ), μn (ak ) have to be coupled appropriately. For the case ωk = π, the conditions are given by μk−1 (ak ) + cos ωk μkt (ak ) − sin ωk μkn (ak ) = 0, t k cos ωk μk−1 (ak ) + sin ωk μk−1 t n (ak ) + μt (ak ) = 0,

(11)

for all k = 1, 2, . . . , K. These conditions follow as special case from [4, Theorem 1.5.2.8]. Remark 1 In order to describe a change of boundary condition we may also consider an interior angle ωk of π. A corresponding adaption of the conditions (11) can be found in [4]. In the following we fix a corner ak and work out the relation implied by the corresponding boundary conditions and the conditions (11) for the four involved k quantities μk−1 (ak ), μkt (ak ), μk−1 t n (ak ), μn (ak ), where we skip in the following the argument ak for better readability. We distinguish three situations: 1. Let ak be an interior corner point of Γf . Then the conditions (11) lead to μk−1 =− n

1 (cos wk μk−1 + μkt ), t sin wk

μkn =

1 (μk−1 + cos wk μkt ), sin wk t

and μkt . for arbitrary μk−1 t 2. Let ak be a corner point on the interface of Ek−1 ⊂ Γs and Ek ⊂ Γf . Then the conditions (11) provide μk−1 = 0, t

μk−1 =− n

1 μk , sin wk t

μkn =

1 cos wk μkt , sin wk

where μkt can be freely chosen. For the reverse case Ek−1 ⊂ Γf and Ek ⊂ Γs an analogous result holds. 3. In all other cases, we obtain μk−1 = μkt = μk−1 = μkn = 0. t n

3.2 The Discretization Method Let Sh (Ω) be a finite dimensional subspace of H 1 (Ω) of piecewise polynomials 1 (with respect to a subdivision of Ω) and we set Sh,0 (Ω) = Sh (Ω) ∩ H0,Γ (Ω). c ∪Γs

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The restriction of functions from Sh (Ω) to Ek is defined as Sh (Ek ) = {v|Ek : v ∈ Sh (Ω)}. 1 2 K The discrete space Λh consists of all μh = ((μ1t , μ2t , . . . , μK t ), (μn , μn , . . . , μn )), k k where μt ∈ Sh (Ek ) and μn ∈ Sh (Ek ) for k = 1, 2, . . . , K, subject to the constraints derived in Sect. 3.1. In the discrete setting the original formulation (6) is equivalent to three (consecutively to solve) second-order problems:

1. The discrete p-problem: Find ph ∈ Sh,0 (Ω) such that (∇ph , ∇vh ) = F, vh  for all vh ∈ Sh,0 (Ω). 2. The discrete (φ, λ)-problem: For given ph ∈ Sh,0 (Ω), find φh ∈ (Sh (Ω))2 /RT0 and λh ∈ Λh such that (symCurl φh , symCurl ψh )C −1 + lφ (ψh , λh ) = −(ph I , symCurl ψh )C −1 = −lp (ph , μh ),

lφ (φh , μh )

for all ψh ∈ (Sh (Ω))2 /RT0 and μh ∈ Λh , where

lφ (φ, μ) =

Ek ⊂Γs ∪Γf

lp (p, μ) =

(∂t φ · n, μkn )Ek +

(∂t φ · t, μkt )Ek ,

Ek ⊂Γf

(p, μkn )Ek ,

Ek ⊂Γf 1 2 K for μ = ((μ1t , μ2t , . . . , μK t ), (μn , μn , . . . , μn )). Here, we use the notation 2 RT0 = {ax + b : a ∈ R, b ∈ R }. 3. The discrete w-problem: For given M h = ph I + symCurl φh and λh ∈ Λh , find wh ∈ Sh,0 (Ω) such that

(∇wh , ∇qh ) = (M h , qh I )C −1 + lp (qh , λh )

for all qh ∈ Sh,0 (Ω).

In comparison with the decoupled formulation in Sect. 2.1, here the second problem, the (φ, λ)-problem, is a saddle point problem.

4 Numerical Tests As discretization space Sh (Ω) we consider B-splines of degree p ≥ 1 with maximum smoothness; see, e.g, [2, 3] for more information on this space in the context of isogeometric analysis (IGA). A sparse direct solver is used for each of

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the three sub-problems. The implementation is done in the framework of the objectoriented C++ library G+Smo (“Geometry + Simulation Modules”).1

4.1 Square Plate with Clamped, Simply Supported and Free Boundary We consider a square plate Ω = (−1, 1)2 with simply supported north and south boundaries, clamped west boundary and free east boundary. The material tensor C is given as in (2) with D = 1, ν = 0 and the load is f (x, y) = 4π 4 sin(πx) sin(πy). The exact solution is written in the form   w(x, y) = (a + bx) cosh(πx) + (c + dx) sinh(πx) + sin(πx) sin(πy), which automatically satisfies the boundary conditions on the simply supported boundary parts. The constants a, b, c and d are chosen such that the four remaining boundary conditions (on the clamped and free boundary parts) are satisfied, for details, see [6]. In Tables 1 and 2 the discretization errors for p = 1, 3 are presented. The first column shows the refinement level L, the next three pairs of columns show the respective discretization error and the error reduction relative to the previous level. rates for w and M. Table 1 Discretization errors for square plate, p = 1 L 4 5 6 7

w − wh 0 2.82 · 10−2 7.17 · 10−3 1.80 · 10−3 4.50 · 10−4

Order 1.909 1.976 1.994 1.998

w − wh 1 6.83 · 10−1 3.42 · 10−1 1.71 · 10−1 8.55 · 10−2

Order 0.992 0.998 0.999 0.999

M − M h 0 2.83 · 100 1.42 · 100 7.13 · 10−1 3.56 · 10−1

Order 0.975 0.993 0.998 0.999

Order 3.070 2.993 2.985 2.989

M − M h 0 1.10 · 10−2 1.38 · 10−3 1.75 · 10−4 2.22 · 10−5

Order 3.104 2.989 2.978 2.985

Table 2 Discretization errors for square plate, p = 3 L 4 5 6 7

w − wh 0 5.47 · 10−5 3.41 · 10−6 2.15 · 10−7 1.35 · 10−8

Order 4.147 4.001 3.984 3.989

w − wh 1 2.75 · 10−3 3.46 · 10−4 4.37 · 10−5 5.50 · 10−6

1 https://ricamsvn.ricam.oeaw.ac.at/trac/gismo/wiki/WikiStart.

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Table 3 Discretization errors for circular plate, p = 1 L 4 5 6 7

w − wh 0 3.58 · 10−4 8.98 · 10−5 2.24 · 10−5 5.62 · 10−6

Order 1.984 1.996 1.999 1.999

w − wh 1 8.37 · 10−3 4.18 · 10−3 2.09 · 10−3 1.04 · 10−3

Order 1.002 1.000 1.000 1.000

M − M h 0 8.79 · 10−3 4.38 · 10−3 2.18 · 10−3 1.09 · 10−3

Order 1.020 1.005 1.001 1.000

Order 3.163 3.034 3.004 2.999

M − M h 0 2.01 · 10−5 2.42 · 10−6 3.00 · 10−7 3.74 · 10−8

Order 3.206 3.054 3.013 3.003

Table 4 Discretization errors for circular plate, p = 3 L 4 5 6 7

w − wh 0 4.05 · 10−7 2.38 · 10−8 1.47 · 10−9 9.17 · 10−11

Order 4.319 4.083 4.019 4.004

w − wh 1 1.93 · 10−5 2.35 · 10−6 2.93 · 10−7 3.67 · 10−8

4.2 Circular Plate with Simply Supported Boundary As a second example, we consider the simply supported circular plate with radius r = 1 and uniform loading f = 1. The material tensor C is given as in (2) with D = 1 and ν = 0.3. The exact solution is given by w(x) = c1 + c2 r 2 + c3 r 4 where r 2 = x12 + x22 , c3 = 1/64 and c1 , c2 are determined from the boundary conditions. For this test reproducing the exact geometry is essential, see the discussion of the so-called Babuška paradox in [1]. Therefore, we use an exact geometry representation by means of non-uniform rational B-splines (NURBS). In Tables 3 and 4 the discretization errors for p = 1, 3 are presented. The results show optimal convergence rates for w and M. Acknowledgement The research was supported by the Austrian Science Fund (FWF): S11702N23.

References 1. I. Babuška, J. Pitkäranta, The plate paradox for hard and soft simple support. SIAM J. Math. Anal. 21(3), 551–576 (1990) 2. J.A. Cotrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis, Toward Integration of CAD and FEA (Wiley, New York, 2009) 3. L.B. da Veiga, A. Buffa, G. Sangalli, R. Vázquez, Mathematical analysis of variational isogeometric methods. Acta Numerica 23, 157–287 (2014) 4. P. Grisvard, Elliptic Problems in Nonsmooth Domains, reprint of the 1985 hardback ed. (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011) 5. K. Rafetseder, W. Zulehner, A decomposition result for Kirchhoff plate bending problems and a new discretization approach. SIAM J. Numer. Anal. 56(3), 1961–1986 (2018). https://doi.org/ 10.1137/17M1118427 6. J. Reddy, Theory and Analysis of Elastic Plates and Shells, 2nd edn. (Taylor & Francis, Philadelphia, 2007)

Part XXI

Reduced Order Models for Time-Dependent Problems

POD-Based Multiobjective Optimal Control of Time-Variant Heat Phenomena Stefan Banholzer, Eugen Makarov, and Stefan Volkwein

Abstract In the present paper, a multiobjective optimal control problem governed by a heat equation with time-dependent convection term and bilateral control constraints is considered. For computing Pareto optimal points and approximating the Pareto front, the reference point method is applied. As this method transforms the multiobjective optimal control problem into a series of scalar optimization problems, the method of proper orthogonal decomposition (POD) is introduced as an approach for model-order reduction. New strategies for efficiently updating the POD basis in the optimization process are proposed and tested numerically.

1 Introduction Many optimization problems in applications can be formulated using several objective functions, which are conflicting with each other. This leads to the notion of multiobjective or multicriterial optimization problems; cf. [4, 9, 12]. One prominent example is given by an energy efficient heating, ventilation and air-conditioning (HVAC) operation of a building with conflicting objectives such as minimal energy consumption and maximal comfort; cf. [6, 8]. In this paper we apply the reference point method [11] in order to transform a bicriterial optimal control problem into a sequence of scalar-valued optimal control problems and solve them using well-known optimal control techniques; see [13]. We build on and extend previous results obtained in [2], where a linear convectiondiffusion equation was considered. In addition, we allow the convection term to be time-dependent here. By using the a-posteriori error estimate [2, Theorem 9] we develop a new strategy for updating the POD basis while computing the Pareto front such that the error stays always below a certain predefined threshold. In our numerical examples we

S. Banholzer · E. Makarov · S. Volkwein () University of Konstanz, Department of Mathematics and Statistics, Konstanz, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_83

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compare the strategy with the simple basis extension algorithm in [2, Algorithm 3]. Moreover, we propose a method to choose an efficient initial number of POD basis functions. The paper is organized in the following manner: In Sect. 2 we present the state equation and the bicriterial optimal control problem. The reference point method and how to apply it to the problem at hand is explained in Sect. 3. Moreover, the POD method is briefly introduced to gain a speed-up in the solution process. Section 4 contains the numerical experiments and in Sect. 5 we draw a conclusion.

2 Problem Formulation The State Equation Let Ω ⊂ Rd with d ∈ {2, 3} be a bounded domain with Lipschitz-continuous boundary Γ . We choose m non-empty, pairwise disjoint subsets Ω1 , . . . , Ωm of the domain Ω. For a given end time T > 0, we set Q := (0, T ) × Ω and Σ := (0, T ) × Γ . The state equation is then given by the following diffusion-convection equation with homogeneous Neumann boundary conditions: yt (t, x) − κΔy(t, x) + β(t, x) · ∇y(t, x) =

m 

ui (t)χi (x)

in Q,

(1a)

on Σ,

(1b)

in Ω.

(1c)

i=1 ∂y ∂n (t, x)

=0

y(0, x) = y0 (x)

In (1a) the constant κ > 0 is the diffusion coefficient and the time-dependent advection β is supposed to be in L∞ (Q; Rd ). Furthermore, the function χi is given by the characteristic function of the set Ωi for all i = 1, . . . , m. For the control variable u = (u1 , . . . , um ) we assume u ∈ U = L2 (0, T ; Rm ). Finally in (1c), y0 ∈ H = L2 (Ω) is a given initial temperature. To set the framework for the weak formulation of (1), we define the Hilbert space V = H 1 (Ω) equipped with the standard inner product. The space > = Y = W (0, T ) = φ ∈ L2 (0, T ; V ) | φt ∈ L2 (0, T ; V  ) endowed with the canonical inner product is a Hilbert space; see, e.g. [3]. With similar arguments as in [1, Section 5.1] it is possible to show that for each tuple (u, y0 ) ∈ U × H there is a unique weak solution y ∈ Y of (1). Furthermore, the solution can be written as y = yˆ + Su, where yˆ ∈ Y is the weak solution of (1) for the pair (0, y0 ) and the linear operator S : U → Y is given such that Su is the weak solution of (1) to the pair (u, 0).

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The Bicriterial Optimal Control Problem For a given desired temperature yQ ∈ L2 (0, T ; H ) we introduce the cost functional J : Y × U → R2 ,

J (y, u) =

 7 72 17 7 2 y − yQ L2 (0,T ;H ) ,

1 2

u2L2 (0,T ;Rm )

?

.

Defining the set Uad = {u ∈ U | ua ≤ u ≤ ub in [0, T ]} for given ua , ub ∈ U with ua ≤ ub in [0, T ], the bicriterial optimal control problem reads min J (y, u)

s.t.

> = (y, u) ∈ (y, ˜ u) ˜ ∈ Y × Uad | y˜ = yˆ + S u˜ .

(2)

Since S is well-defined, we define the reduced cost function Jˆ : U → R2 , Jˆ(u) = J (yˆ + Su, u) and investigate the reduced formulation of (2) in this paper: min Jˆ(u)

s.t.

u ∈ Uad .

(3)

Problem (3) involves the minimization of a vector-valued function with two objectives. This is done by using the concept of Pareto optimality; cf. [4]. Definition 1 The point u¯ ∈ Uad is called Pareto optimal for (3) if there is no other control u ∈ Uad \ {u} ¯ with Jˆi (u) ≤ Jˆi (u), ¯ i = 1, 2, and Jˆ(u) = Jˆ(u). ¯

3 The Reference Point Method The theoretical and numerical aim in solving a bicriterial optimization problem is to get an approximation of the Pareto set and the Pareto front, respectively, which are given by = > Ps = u ∈ Uad | u is Pareto optimal ⊂ U and Pf = Jˆ(Ps ) ⊂ R2 . The scalarization method, in which the bicriterial function is transformed into a scalar function and then minimized using well-known techniques from scalar optimization, is one of the most popular approaches to tackle this problem, see e.g. [5, 9, 12]. The idea is that by choosing different scalarizations, both the Pareto set and the Pareto front can be approximated. One particular scalarization method is the (Euclidean) reference point method, which was previously used in [10, 11]. Given a reference point z ∈ Pf + R2≤ = {z + x | z ∈ Pf and x ∈ R2≤ } the distance function Fz : U → R,

Fz (u) =

1 2



Jˆ1 (u) − z1

2

+

1 2

 2 Jˆ2 (u) − z2

measures the Euclidean distance between Jˆ(u) and z for a given u ∈ U . The idea is that by solving the minimization problem min Fz (u) s.t.

u ∈ Uad ,

(4)

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we get a Pareto optimal point for (3). The following theorem, which is taken from [1, Theorem 3.35], guarantees this property for the problem at hand. Theorem 2 Let z ∈ Pf + R2≤ be a reference point. Then (4) has a unique solution u¯ ∈ Uad , which is Pareto optimal for (3). The algorithmic approach to approximate Pf , which we consider in this paper, first computes the two boundary points of the Pareto front. These are given by the minimizers of Jˆ1 and Jˆ2 , respectively. In our case we have to regularize the minimization of Jˆ1 because Jˆ1 is only strictly, but not strongly convex. Therefore, we minimize Jˆ1 + α Jˆ2 with a small weight 0 < α  1. We always choose the minimizer of Jˆ1 as a starting point. Given a Pareto optimal point the algorithm generates a new reference point following Pf from top to bottom, and then solves the respective reference point problem. This procedure is repeated until the end of Pf is reached. The exact scheme for computing the reference points along with a more detailed description of the algorithm can be found in [1, Section 3.4] and [2, Section 6]. When implementing this algorithm, (4) has to be solved repeatedly for different reference points z ∈ Pf + R2≤ . Each solve of (4) requires multiple solves of (1) and an adjoint equation (cf. [13, Section 3.6]) which is often computationally too costly when using a standard Finite Element (FE) method. Thus, it is reasonable to apply model-order reduction to reduce the computational effort. We use the well-known POD method; cf. [7]. In [1, 2] the procedure for our problem at hand is explained. Here, we just want to introduce some notations: Given a POD basis {ψi }i=1 of rank , we define the set V  = span {ψ1 , . . . , ψ } and the solution operator S  : U → H 1 (0, T ; V  ) of the POD solution of the u-dependent part of the state equation. The POD approximation of Jˆ is defined as Jˆ (u) = J (yˆ + S  u, u). Then, (4) is replaced by min Fz (u) =

1 2



Jˆ1 (u) − z1

2

+

1 2



Jˆ2 (u) − z2

2

s.t.

u ∈ Uad

(5)

with Jˆ2 (u) = Jˆ2 (u). For a given reference point z ∈ Pf +R2≤ we denote the optimal solutions to (4) and (5) by u¯ z and u¯ z , respectively.

4 Numerical Results In our numerical tests we consider the bicriterial optimal control problem presented in Sect. 2. We have Ω = (0, 1)2 ⊂ R2 and choose T = 1. The diffusion parameter is ˜ x) for given by κ = 0.5 and for the convection term in (1a) we use β(t, x) = cb β(t, all (t, x) ∈ Q, where β˜ is a non-stationary solution of a Navier-Stokes equation and cb ≥ 0 is a parameter to control the strength of the convection; cf. Fig. 1. We impose a floor heating of the whole room with m = 4 uniformly distributed heaters in the domains Ω1 = (0, 0.5)2 , Ω2 = (0, 0.5) × (0.5, 1), Ω3 = (0.5, 1) × (0, 0.5) and

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Ω4 = (0.5, 1)2 . The bilateral control constraints are ua = 0 and ub = 3. Finally, we choose y0 = 16 and yQ (t, x) = 16 + 2t for all (t, x) ∈ Q. All computations were carried out on a MacBook Pro 13 (middle 2012) with 2.5 GHz Intel Core i5 and 4 GB RAM. Test 1 We solve (3) for cb = 1. Then, Pf is smoothly approximated by 52 Pareto optimal points; cf. Fig. 2. Hereby, Pf ranges from P 1 = (0.0199, 4.1), which is computed with the weighted-sum method with weight α = 0.02, to P 52 = (0.6667, 0). Thus, the desired temperature can be achieved quite closely in the upper part of Pf . The four optimal controls for P 1 can be seen in right plot of Fig. 2. As in the case of a time-independent convection term all four controls adapt to the air flow, which goes from the top left corner of the room to the right bottom corner by using different heating strategies. Furthermore, one can observe a slightly wavy behaviour of the optimal controls at the beginning. This is due to the temporal changes in the dynamics of the system caused by a vortex moving over time from the top left corner to the middle of the room. Another interesting aspect is that in

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the comparison to the time-independent case (by taking average over the time) the system with time-dependent convection only needs about 10% more computation time although the latter case adds dynamics to the optimal control problem, which are more difficult to handle numerically. Test 2 Now we use an adaptive POD basis extension algorithm, which was introduced in [2, Algorithm 3], in order to investigate the influence of the timedependent convection term on the number of needed POD basis functions. As a measure for the error between uz and u¯ z induced by the POD approximation we use the a-posteriori error estimate [2, Theorem 9] and set the upper bound of the acceptable error to μ = 4 · 10−4 , as well as the initial number of POD basis functions to init = 5. Choosing the minimizer of Jˆ1 as starting point we observe that 24 POD basis functions are needed to compute the whole Pareto front in the desired approximation quality. Thereby, all 19 basis extensions are conducted on P 2 . For comparison, in [2] was shown that for a very similar problem with a timeindependent convection term only 15 POD basis functions are sufficient to achieve the same quality. This is due to the fact that a time-dependency of a convection adds more complex dynamics to the system which are hard to capture by using only one fixed POD basis. To tackle this problem we propose a POD basis improving strategy. Algorithm 1 shows the routine for computing the n-th Pareto optimal point. In this paper we investigate two strategies for determining  after each basis update. In the first case, we set  = min . In the second case, we choose  by observing the convergence rates in the control space; cf. [7, Theorem 1.49]. Namely, by choosing  ∈ [min , max ] such that  max

 λi ψi − PH ψi 2H 1 (Ω) < ε

(6)

i=+1

 denotes the H -orthogonal projection onto V  , for an ε < μ holds, where PH {ψi }i∈N is a POD basis and {λi }i∈N are the corresponding eigenvalues. The results

Algorithm 1 POD basis update algorithm Require: threshold μ > 0, 0 < min < max ; 1: Set check = 0; 2: while check = 0 do 3: Solve (5) with reference point z(n) ; 4: Compute the a-posteriori error estimate μapost for the controls; 5: if μapost < μ then 6: Set check = 1; 7: else 8: if  < max − 1 then 9: Set  =  + 2; 10: else 11: Solve (4) for u¯ z with reference point z(n) and starting point u¯ z ; 12: Compute new POD basis by using u¯ z ; 13: Choose  ∈ [min , max ] and set check = 1.

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(b)

(a)

Fig. 3 Number of used POD basis functions in Algorithm 1 using ε = 0.5 · 10−2 μ. (a)  = 10 fixed. (b) 6 ≤  ≤ 22 adaptive Table 1 Test 2: Results for ε = 0.5 · 10−2 μ Full system Basis extension with init = 5 Algorithm 1 with fixed  = 10 Algorithm 1 with adaptive  ∈ [6, 22] Algorithm 1 with adaptive  ∈ [6, 20]

CPU time 240.25 51.07 49.53 44.82 59.51

#Basis extensions – 19 11 2 0

#Basis updates – – 1 1 6

for ε = 5 · 10−3 μ and max = 22 are presented in Fig. 3 and Table 1. The value of min is set to 10 and to 6 by the first and second strategy, respectively. Using the second strategy Algorithm 1 yields the best results with respect to the CPU time. Thus, (6) estimates quite well how many POD basis functions would have been necessary in order to compute the current Pareto point. As a result only 2 basis extensions on the points P 2 and P 3 are needed. Using the first strategy, 11 basis extensions are necessary as 10 POD basis function are not enough even after a basis update. Hence, a lot of avoidable basis extensions are done. In both cases one basis update is conducted. However, the performance of the Algorithm 1 depends strongly on the choice of max . Decreasing max to 20 increases the CPU time by 33% as 6 basis updates are needed in this case. Furthermore, ε has to be chosen appropriately to avoid unnecessary basis extensions in the second case. Test 3 Now we increase the strength of the convection term cb to 2 and run the basis extension algorithm with the same settings. Surprisingly, in the basis extension algorithm [2, Algorithm 3] only 25 POD basis functions are needed to compute the whole Pareto front, although there are significant changes in the behaviour of the controls. However, as expected, increasing the strength of the convection term increases heavily the number of basis updates in Algorithm 1 for the same values of max and thus the computation time.

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5 Conclusion In the present paper we show how including a time-dependent advection term into the state equation influences the results of the bicriterial optimal control problem (2). As expected the time-dependence adds dynamics to the system which cannot be captured that easily by a single POD basis. Therefore, by introducing a new POD update strategy we are able to save about 15% of the CPU time in comparison to the basis extension algorithm in [2, Algorithm 3]. Acknowledgements S. Banholzer gratefully acknowledges support by the German DFG-Priority Program 1962 and by the Landesgraduiertenförderung of Baden-Württemberg.

References 1. S. Banholzer, POD-Based Bicriterial Optimal Control of Convection-Diffusion Equations, Master thesis, University of Konstanz, Department of Mathematics and Statistics, 2017, see http://nbn-resolving.de/ urn:nbn:de:bsz:352-0-421545 2. S. Banholzer, D. Beermann, S. Volkwein, POD-based error control for reduced-order bicriterial PDE-constrained optimization. Annu. Rev. Control 44, 226–237 (2017) 3. R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I (Springer, Berlin, 2000) 4. M. Ehrgott, Multicriteria Optimization (Springer, Berlin, 2005) 5. G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization (Springer, Berlin, 2008) 6. K.F. Fong, V.I. Hanby, T.-T. Chow, HVAC system optimization for energy management by evolutionary programming. Energy Build. 38, 220–231 (2006) 7. M. Gubisch, S. Volkwein, Proper orthogonal decomposition for linear-quadratic optimal control, in Model Reduction and Approximation: Theory and Algorithms, ed. by P. Benner, A. Cohen, M. Ohlberger, K. Willcox (SIAM, Philadelphia, 2017), pp. 5–66 8. A. Kusiak, F. Tang, G. Xu, Multi-objective optimization of HVAC system with an evolutionary computation algorithm. Energy 36, 2440–2449 (2011) 9. K. Miettinen, Nonlinear Multiobjective Optimization. International Series in Operations Research and Management Science (Springer, Berlin, 1998) 10. S. Peitz, S. Oder-Blöbaum, M. Dellnitz, Multiobjective optimal control methods for fluid flow using reduced order modeling, in 24th Congress of Theoretical and Applied Mechanics (ICTAM), Montreal, Canada, 21–26 August 2016, see http://arxiv.org/pdf/1510.05819v2.pdf 11. C. Romaus, J. Böcker, K. Witting, A. Seifried, O. Znamenshchykov, Optimal energy management for a hybrid energy storage system combining batteries and double layer capacitors, in IEEE Energy Conversion Congress and Exposition, San Jose, 2009, pp. 1640–1647 12. W. Stadler, Multicriteria Optimization in Engineering and in the Sciences (Plenum Press, New York, 1988) 13. F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, 2nd edn. (American Mathematical Society, Providence, 2010)

Greedy Kernel Methods for Accelerating Implicit Integrators for Parametric ODEs Tim Brünnette, Gabriele Santin, and Bernard Haasdonk

Abstract We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method’s features.

1 Problem Setting We consider a d-dimensional, autonomous, first order parametric initial value problem: For a given vector of parameters μ ∈ P ⊂ Rp from an admissible set P, solve 9 u(t; ˙ μ) = f (u(t; μ), μ), t ∈ [0, T ] IVP(μ) : u(0; μ) = u0 (μ) ∈ Rd .

T. Brünnette · G. Santin () · B. Haasdonk University of Stuttgart, Institute of Applied Analysis and Numerical Simulation, Stuttgart, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_84

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We assume that IVP(μ) has a unique solution u(t; μ) := u(t; μ, u0 (μ)), t ∈ [0, T ], for any value μ ∈ P and for any initial value u0 (μ) ∈ Rd . Conditions on f such that this requirement is fulfilled are well known, and we refer e.g. to [5] for details. Existence and uniqueness of solutions allow to define a parametric time evolution or flow mapping Φ(t, u0 (μ)) := u(t; μ),

(1)

which maps the initial value and the time to the corresponding solution vector in Rd , and for which it holds Φ(s, u(t, μ)) = u(t + s; μ). Although the dependency on the parameters in IVP(μ) can be quite general, we require that Φ(t, u0 (μ)) = Φ(s, u0 (ν)) for all t, s ∈ [0, T ] and for all μ, ν ∈ P s.t. (t, μ) = (s, ν), i.e., different parameters lead to non intersecting trajectories. We further assume to have an implicit time integrator which is able to numerically solve IVP(μ) with any given accuracy, provided a small enough time step is used. Although our acceleration algorithm applies to general single-step integration methods, in this paper, for the sake of presentation, we will concentrate on the Implicit Euler method (IE), and we refer again to [5] for details on its accuracy. Such integration method considers a timestep Δt > 0 and a uniform time discretization of [0, T ] in Nt := NΔt ∈ N intervals 0 = t0 < t1 < · · · < tNt ≤ T , with ti+1 − ti = Δt, 0 ≤ i ≤ Nt − 1, and computes a discrete-time approximation of u as ui (μ) ≈ u(ti ; μ), 0 ≤ i ≤ Nt . A numerical time evolution map φ : R × Rd → Rd analogous to (1) can be defined from the approximate solution as φ(Δt, ui (μ)) := ui+1 (μ), 0 ≤ i ≤ Nt − 1,

(2)

i.e., the solution vector at the current time point is mapped to the solution vector at the next time point. Observe that, under the hypotheses of arbitrary accuracy of the integration method and of non intersection of the trajectories, we assume that also the discrete trajectories are non intersecting. This means that φ is a globally defined function independent of the parameter μ ∈ P. At each discrete time point, the integrator needs to solve a generally nonlinear, ddimensional system of equations to determine the approximation ui (μ). We assume that this equation is solved with an iterative method, e.g., the Newton method, using an initialization u¯ i (μ) ∈ Rd at time ti . Common choices of this value for the IE method are, e.g., the previous approximation ui−1 (μ) or the approximation obtained by one step of the Explicit Euler method. The goal of this paper is to present a way to accelerate the computation of the numerical solution {ui (μ)}i for an arbitrary parameter vector μ ∈ P. The acceleration is realized by constructing a surrogate sφ : R × Rd → Rd of the numerical time evolution map φ such that sφ (Δt, u) ≈ φ(Δt, u) for all (Δt, u) ∈ [0, T ] × Rd , while the evaluation of sφ is much faster than the evaluation of φ. This surrogate is computed in an offline phase in a data-dependent fashion, i.e., it

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is trained using a set of precomputed numerical trajectories {ui (μj )}ij for multiple parameter values Pt r := {μ1 , . . . , μNμ } ⊂ P, Nμ ∈ N, and possibly multiple timesteps Δt. In the online phase, for a new parameter μ ∈ P the numerical solution is computed by the same time integrator and with timestep Δt, and, at each time ti , the nonlinear solver is initialized by sφ (Δt, ui−1 ), i.e., u¯ i (μ) ∈ Rd is replaced by the surrogate prediction based on the previous timestep. If the surrogate is accurate, sφ (Δt, ui−1 (μ)) is a good approximation of φ(Δt, ui−1 (μ)) = ui (μ), so the nonlinear solver will converge in possibly significantly less iterations, ideally in 0 iterations if a residual criterion is used before starting the fix-point loop. This reduction of the iterations, combined with the fast evaluation of sφ , will produce a speedup of the overall computational time. Moreover, since the same time integrator and nonlinear solver are used in the accelerated algorithm, we should expect no degradation of the accuracy, provided the surrogate prediction is accurate enough so that the initialization point is within the area of convergence of the nonlinear solver. This is in contrast to general surrogate modeling or model reduction, where the approximation typically results in an accuracy loss. The surrogate is constructed using the Vectorial Kernel Orthogonal Greedy Algorithm (VKOGA) [11], which will be discussed in Sect. 2. In particular, it is a kernel-based interpolation algorithm that constructs a nonlinear surrogate sφ . The full specification of the training data and the complete acceleration algorithm will be described in Sect. 3, but we anticipate that arbitrary unstructured trajectory data {ui (μj )}ij in possibly high dimension d can be used. We will conclude this paper with different numerical experiments in Sect. 4 to demonstrate the capabilities of our method. Moreover, similar acceleration methods have been presented in the papers [1, 2], where instead a linear surrogate is employed.

2 Kernel Based Surrogates and the VKOGA We briefly outline here the fundamentals of interpolation with kernels and of the VKOGA algorithm, and we refer to [10] and to [4, 11] for the respective details. We assume to have a function f : Ω ⊂ Rp → Rq and a training dataset composed of pairwise distinct data points X := {xi }N i=1 ⊂ Ω and data values q . We will specify in the following section the definition Y := {f (xi )}N ⊂ R i=1 of the dataset for the current algorithm. The general form of the surrogate is sf (x) :=

N

j =1

αj K(x, xj ), x ∈ Ω,

(3)

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where αj ∈ Rq are coefficient vectors and K : Ω × Ω → R is a symmetric and positive definite kernel. This means that the matrix AX,K ∈ RN×N ,   strictly AX,K ij := K(xi , xj ), is positive definite for all N ∈ N and for all sets X ⊂ Ω of N pairwise distinct points. A particular K, i.e., the Gaussian kernel K(x, y) := exp(−ε 2 x − y22 ), with a positive shape parameter ε > 0, will be used in Sect. 4. The coefficient vectors in (3) can be uniquely determined by imposing interpolation conditions (4) sf (xi ) := f (xi ), 1 ≤ i ≤ N,   which result, defining α T := [α1 , . . . , αN ], bT := f (x1 ), . . . , f (xN ) , α, b ∈ Rq×N , in the solution of the linear system AX,K α = b. This, indeed, has a unique solution as AX,K is positive definite by assumption. This interpolation method is well studied, and we just recall that convergence rates are proven for functions f in the space HK (Ω), which is a Reproducing Kernel Hilbert Space (RKHS) associated to the particular kernel K, and which is norm equivalent to a Sobolev space W2τ (Ω), τ > d/2, for certain kernels (see [10]). The goal of the VKOGA is to approximate the surrogate (3) by a sparse expansion of the same form, i.e., one where most of the αj are the zero vector. A good selection of the sparsity pattern results into an approximate surrogate which is as good as the full one, while being much faster to evaluate, since the sum involves only n  N elements. The selection of the non-zero coefficients and their computation is realized by a greedy procedure in HK (Ω), which iteratively selects nested data point sets ∅ ⊂ Xn−1 ⊂ Xn ⊂ Ω by maximizing a selection criterion at each step, and solves the corresponding interpolation problem. Possible choices in the VKOGA are the f -, P -, and f/P -greedy selection rules [3, 7, 9]. The algorithm has theoretical grounds, e.g. provable convergence rates [7, 11], which are also quasi-optimal in Sobolev spaces for P -greedy [8], and has been successfully applied in several application contexts, e.g. [6]. Moreover, the numerical computation of the surrogate can be efficiently implemented using a partial Cholesky decomposition of the kernel matrix AX,K , where only the columns appearing in the sparse surrogate need to be computed and stored.

3 The Complete Algorithm: VKOGA-IE We can now describe the complete algorithm, which we name VKOGA-IE. The target function is f := φ, which is defined on Ω := [0, T ] × Rd to Rd , i.e., p := d +1, q := d. What remains to specify is the exact definition of the training set (X, Y ) used by VKOGA to construct the surrogate sφ , as described in the previous section. As mentioned in Sect. 1, we solve IVP(μ) for Nμ ∈ N different parameters from a parameter-training set Pt r ⊂ P, each μj with a timestep Δtj . If the same parameter is used more than once with different timesteps, we just count it multiple times in Pt r . This generates trajectory data which we assign at temporary sets

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Algorithm 1 VKOGA-IE (Offline phase) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:

N

μ Input: {Δtj , μj }j =1 for j = 1, . . . , Nμ do u0 (μj ) := u(0; μj ) for i = 1, . . . , Nt do Initialize u¯ i (μj ) Compute ui (μj ) with IE end for Nt −1 Xj := {(Δtj , ui (μj ))}i=0 Nt −1 Yj := {ui+1 (μj )}i=0 end for Nμ Xj X := ∪j =1

N

μ 12: Y := ∪j =1 Yj 13: Train sφ on dataset (X, Y ) with VKOGA 14: Output: sφ

Algorithm 2 VKOGA-IE (Online phase) 1: 2: 3: 4: 5: 6: 7: 8:

Input: Δt, μ, sφ μ ∈ P , Δt > 0 u0 (μ) := u(0; μ) for i = 1, . . . , Nt do Initialize u¯ i (μ) = sφ (Δt, ui−1 (μ)) Compute ui (μ) with IE end for t Output: {ui (μ)}N i=0

Nt −1 t −1 Xj := {(Δtj , ui (μj ))}N i=0 , Yj := {ui+1 (μj )}i=0 , representing input-output pairs N

N

μ μ of φ. The dataset is defined as X := ∪j =1 Xj , Y := ∪j =1 Yj . The complete offline phase is summarized in Algorithm 1. Instead of working with a fixed kernel shape parameter ε > 0, typically step 13 implies a parameter selection procedure, e.g. via cross validation. Moreover, we assume for simplicity that T /Δt ∈ N. In the online phase, instead, we only need to run the IE method and solve at each iteration the nonlinear equation using the initialization provided by the surrogate, as described in Algorithm 2.

4 Experiments To demonstrate the features of VKOGA-IE, we consider the Burgers equation ⎧ 1 2 ⎪ (t, x) ∈ [0, T ] × [−r, r] ⎪ ⎪ ∂t θ (t, x) + 2 ∂x θ (t, x) = 0, ⎨ θ (0, x) = θ0 (x) x ∈ [−r, r] ⎪ θ (t, −r) = ul , t ∈ [0, T ] ⎪ ⎪ ⎩ θ (t, r) = ur , t ∈ [0, T ],

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which is transformed into an ODE by a semi-discrete finite volume discretization in space based on the Lax-Friedrichs flux. We consider d := 200 cells in (−r, r) with r := 5, and T := 2. This produces a d dimensional IVP(μ) depending on a twodimensional parameter vector μ := (ul , ur ). We concentrate here on shock wave solutions, i.e., ul > ur . The resulting ODE is then simulated from t = 0 to t = T , with varying time-step Δt. The nonlinear system is solved at each timestep using the Newton method, which is terminated with a maximal number of 100 iterations or when a tolerance of 10−14 on the residual is reached. The VKOGA is run with the Gaussian kernel and with a termination tolerance of 10−12. The kernel depends on a parameter ε > 0, which is chosen via five-fold cross validation from a set of 50 logarithmically equally spaced values in [10−4, 102 ]. The first experiment uses a fixed Δt = 0.01 and a single training parameter Pt r = {(3.4, 0.2)}, i.e., N = 200 = T /Δt. Observe that a fixed Δt means that the model is in practice d to d dimensional. The VKOGA selects n = 50 points, and the model is tested to solve IVP(μ) with parameters Pt e := {(3.4 + i 0.2, 0.2 + j 0.2), i, j ∈ {−1, 0, 1}} and T = Tt e := 2. The results are summarized in Table 1. The average number of iterations for the standard initialization with the previous value (‘Old value’ column) and with the VKOGA model (‘VKOGA’ column) are reported, as well as the test parameters where the minimal and maximal gain of our technique is realized. The table contains also the computational times in seconds, which are the averages over ten repetitions of the same simulation based on a Matlab implementation. It is evident that a good speedup is reached when the model is tested on the training parameter, while the quality degrades for different ones. The second experiment uses a model trained again with fixed Δt = 0.01 and the same test parameters Pt e , but instead with training parameters Pt r := {(3.2, 0), (3.2, 0.4), (3.6, 0), (3.6, 0.4)}, i.e., the corners of the square containing Pt e . The resulting training set has N = 200 × 4 = 800 points, and the VKOGA selects n = 166 points. The results are summarized in Table 2. In this case, as Table 1 Results of the first experiment with Ptr = {(3.4, 0.2)} and fixed timestep Δt = 0.01

Mean Min Max

Old value Iter 25.40 24.27 25.30

Time 2.20 2.06 2.20

VKOGA Iter 26.09 27.86 16.65

Time 2.35 2.49 1.56

Gain Iter −2.73% −14.75% 34.18%

Time −6.86% −21.11% 28.98%

μ (3.2, 0.4) (3.4, 0.2)

Table 2 Results of the second experiment, i.e., model trained with Ptr := {(3.2, 0), (3.2, 0.4), (3.6, 0), (3.6, 0.4)} and fixed timestep Δt = 0.01

Mean Min Max

Old value Iter 25.40 25.24 26.30

Time 1.35 1.29 1.36

VKOGA Iter 20.62 24.95 17.09

Time 1.19 1.39 0.95

Gain Iter 18.75% 1.15% 35.01%

Time 12.08% −8.13% 30.30%

μ (3.4, 0.4) (3.6, 0)

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Table 3 Results of the third experiment, i.e., Ptr = {(3.4, 0.2)} and multiple timesteps

Mean Min Max

Old value Iter 32.45 9.20 23.98

Time 2.22 5.36 1.50

VKOGA Iter 29.61 9.98 19.69

Time 2.37 6.53 1.24

Gain Iter 7.04% −8.51% 17.89%

Time −0.22% −21.83% 17.35%

Δt 10−3 8.7910−3

expected, we obtain a significant reduction of the number of iterations for all the test parameters. The minimal reduction is realized for the parameter μ = (3.4, 0.4), which is far from the training set, although not the farthest one. This is a further indication that the quality of the model degrades with the distance from the training set, which is a reasonable behavior but also a promising feature, since a model trained on a larger parameter training set should improve the acceleration. This reduction is reflected also in a speedup in terms of computational cost, except for μ ∈ {(3.4, 0), (3.4, 0.2), (3.4, 0.4)}. Although for the other test parameters the reduction in the number of iteration is above 15%, for these three it is respectively of 2.43%, 1.36%, 1.15%. This suggests that the additional cost required by the evaluation of the kernel model is relevant in the case of a small reduction of the number of iterations. Nevertheless, the computational time is highly dependent on the implementation, while the number of iterations is not. Finally, we test the behavior of the method with respect to a change in the timestep Δt. To this end, we use Pt r = Pt e = {(3.4, 0.2)}, but we train the model with the solutions computed for Δt ∈ {0.01, 0.005, 0.001} and test for 10 logarithmically equally spaced timesteps in [0.001, 0.05]. The algorithm selects n = 221 points. The results are reported in Table 3. In this case we achieve a reduction of the number of Newton iterations for Δt large enough, namely, except for Δt ∈ {10−3 , 1.54 · 10−3}, where the increase is of 8.51% and 2.48%. The reduction for the third smallest timestep Δt = 2.38·10−3 is 0.54%, i.e., it is positive but almost zero. In these three cases the negative or small reduction causes an increase of the simulation time. Nevertheless, the reduction of the number of iterations for large enough timestep suggests that the kernel model captures well the dependence on the timestep, so one could expect to use this technique in more general settings without the need of including in the training sets many solutions obtained with different timesteps.

5 Conclusion and Further Work In this work we described a general nonlinear forecasting method used for the acceleration of implicit ODE integrators. The method is suited for parametric problems and multi-query scenarios, and it realizes a significant acceleration possibly without accuracy degradation.

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The algorithm can be extended to non-autonomous ODEs, adaptive-timestep or multi-stage Runge-Kutta time integrators. In each case, more simulation data should be included in the training set, such as the current time or the partial solutions of the intermediate stages. Another interesting aspect that could be investigated is the analysis of the accuracy of the method. Indeed, if it is possible to prove that the surrogate has a small enough uniform error, it would be guaranteed that the initialization point is inside the convergence area of the nonlinear solver. Acknowledgements The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart.

References 1. K. Carlberg, J. Ray, B. van Bloemen Waanders, Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting. Comput. Methods Appl. Mech. Eng. 289, 79–103 (2015) 2. K. Carlberg, L. Brencher, B. Haasdonk, A. Barth, Data-driven time parallelism via forecasting. ArXiv preprint 1610.09049 3. S. De Marchi, R. Schaback, H. Wendland, Near-optimal data-independent point locations for radial basis function interpolation. Adv. Comput. Math. 23(3), 317–330 (2005) 4. B. Haasdonk, G. Santin, Greedy kernel approximation for sparse surrogate modelling, in Proceedings of the KoMSO Challenge Workshop on Reduced-Order Modeling for Simulation and Optimization, 2017 5. E. Hairer, S.P. Nø rsett, G. Wanner, Solving Ordinary Differential Equations. I: Nonstiff Problems. Springer Series in Computational Mathematics, vol. 8, 2nd edn. (Springer, Berlin, 1993) 6. T. Köppl, G. Santin, B. Haasdonk, R. Helmig, Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and machine learning techniques, Tech. report, University of Stuttgart, 2017 7. S. Müller, R. Schaback, A Newton basis for kernel spaces. J. Approx. Theory 161(2), 645–655 (2009) 8. G. Santin, B. Haasdonk, Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation. Dolomites Res. Notes Approx. 10, 68–78 (2017) 9. R. Schaback, H. Wendland, Adaptive greedy techniques for approximate solution of large RBF systems. Numer. Algorithms 24(3), 239–254 (2000) 10. H. Wendland, Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 17 (Cambridge University Press, Cambridge, 2005) 11. D. Wirtz, B. Haasdonk, A vectorial kernel orthogonal greedy algorithm. Dolomites Res. Notes Approx. 6, 83–100 (2013)

Part XXII

Limiter Techniques for Flow Problems

Third-Order Limiter Functions on Non-equidistant Grids Birte Schmidtmann and Manuel Torrilhon

Abstract We have recently developed a third-order limiter function for the reconstruction of cell interface values on equidistant grids (J Sci Comput, 68(2):624–652, 2016). This work now extends the reconstruction technique to non- uniform grids in one space dimension, making it applicable for more elaborate test cases in the context of finite volume schemes. Numerical examples show that the new limiter function maintains the optimal third-order accuracy on smooth profiles and avoids oscillations in case of discontinuous solutions.

1 Introduction We want to conduct simulations for hyperbolic conservation laws with finite volume methods. In this framework, a numerical flux function needs to be evaluated at each cell interface, taking as input values the left and right limiting value at these cell interfaces. These limit-values can simply be the cell mean values; however, to obtain second- or higher-order accurate schemes, the limit needs to be more sophisticated. This is obtained by reconstruction. We are interested in reconstructions of cell interface values which only require information of three cells, leading to second- or third-order accurate schemes. Achieving higher-order accuracy with finite volumes generally requires larger stencils. This however leads to increased communication among grid cells and is undesirable when thinking of boundary conditions and parallel codes. Furthermore, the aim of our work is to make our third-order reconstruction easily implementable in existing codes within the standard limiter framework. Therefore, we want to remain on the most compact stencil of three cells in one dimension.

B. Schmidtmann () · M. Torrilhon MathCCES, RWTH Aachen University, Aachen, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_85

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2 Reconstruction with Third-Order Limiting For the sake of simplicity, we restrict the theoretical development of the limiter function to one dimensional scalar equations. The transition from one to two dimensions can be obtained via a dimensional splitting, cf. [9]. Also, the theory easily extends to systems of conservation laws by applying it component-wise. In the one-dimensional case of Cartesian grids, we divide the domain4 of interest Ω ⊂ R in non-overlapping cells Ci = [x 1 , x 1 ) such that Ω = i Ci . We i− 2

i+ 2

denote by xi the cell centers and by Δxi = xi+ 1 − xi− 1 the size of cell Ci . Figure 1 2 2 depicts the here-introduced notation for the equidistant case, i.e. Δxi = xi+ 1 − 2 xi− 1 ≡ Δx ∀i. 2 We are interested in hyperbolic conservation law of the form ⎧ ⎨∂ u(x, t) + ∇ · f (u(x, t)) t ⎩u(x, 0)

=0 = u0 (x)

(1)

To avoid boundary effects, we impose periodic boundary conditions. Having introduced the necessary notation we can now discretize Eq. (1). The semi-discrete finite volume formulation is given by d u¯ i 1 =− dt Δxi

  (−) (+) (−) (+) ˆ ˆ f (u 1 , u 1 ) − f (u 1 , u 1 ) , i+ 2

i+ 2

i− 2

i− 2

(2)

 1 u(x, t)dx. The numerical flux functions fˆi± 1 take as input the where u¯ i ≈ Δx i Ci 2 left and right limiting values at the cell interface, see Fig. 1. As mentioned before, one could simply insert the left and right cell mean values which yields a first-order accurate scheme [5]. In order to achieve higher-order accuracy, one possible way is to reconstruct the interface values. The well-known formulation of reconstructions, found in standard literature such as [5] reads (−) i+ 12

= u¯ i +

1 2

φ(θi )δi− 1 ,

(3a)

u(+)1 = u¯ i −

1 2

φ(θi−1 )δi+ 1 .

(3b)

u

i− 2

2

2

Fig. 1 Stencil for reconstruction of interface values on equidistant grids, i.e. Δxi = xi+ 1 − 2 xi− 1 ≡ Δx ∀i 2

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Here, θi = δi− 1 /δi+ 1 denotes the ratio of consecutive gradients or undivided 2 2 differences between neighboring cells which are defined by δi− 1 = u¯ i − u¯ i−1 ,

δi+ 1 = u¯ i+1 − u¯ i .

2

2

(4)

The limiter function φ has only one variable θi which acts as a smoothness indicator. As detailed in [8], there are many advantages of changing the notation of the limiter to the two–variable formulation H (·, ·) u(−)1 = u¯ i + i+ 2

u

(+) i− 12

= u¯ i −

1 2

H (δi− 1 , δi+ 1 ),

(5a)

1 2

H (δi+ 1 , δi− 1 ).

(5b)

2

2

2

2

This form relates to Eq. (3) via φ(θi )δi− 1 = H (δi− 1 , δi+ 1 ). 2 2 2 The function H fully determines the way limiting is performed and thus the order of accuracy of the resulting scheme.

2.1 Formulation on Equidistant Meshes This section shortly summarizes the formulation of the third-order limiter functions developed in [8] for equidistant grids. Full Third-Order Reconstruction Consider a quadratic ansatz function in cell i called pi (x) that has to maintain the cell averages in the three cells Ci+ ,  ∈ {−1, 0, 1}. Evaluating this polynomial at the cell boundaries xi± 1 , we obtain the full 2 third-order (unlimited) reconstruction formulation, leading to a third-order accurate scheme. Rewriting pi (xi± 1 ) according to (5) yields 2

u

(−) i+ 12

= pi (xi+ 1 ) = u¯ i + 2

u(+)1 = pi (xi− 1 ) = u¯ i − i− 2

2

1 2δi+ 12 + δi− 12 2 3 1 δi+ 1 + 2δi− 1 2

2

2

3

(6a) (6b)

or u(∓)1 = u¯ i ± i± 2

1 δi+ 12 + δi± 12 + δi− 12 , 2 3

(7)

respectively. This leads to the reconstruction function H3 (δi− 1 , δi+ 1 ) := 2

2

 1 2δi+ 1 + δi− 1 . 2 2 3

(8)

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For solutions only containing smooth parts, this full (unlimited) third-order reconstruction shows good results. However, applying H3 to solutions containing discontinuities leads to spurious oscillations, as predicted by Godunov’s theorem [5]. Therefore, non-linear reconstruction functions need to be applied in these cases. Reconstructions with Limiter Functions In [8] we recently developed a limiter ˇ function, called H3L based on the works of Artebrant and Schroll [1] and Cada (c) and Torillhon [2]. Furthermore, we designed the combined limiter function H3L which includes a decision criterion η. This criterion evaluates the magnitude of the undivided differences between neighboring cells, see [7] for details on η. 2 η = η(δi− 1 , δi+ 1 ) = 2

2

(Δxi− 1 )2 + (Δxi+ 1 )2 2 2 2 , 5 2 α Δx 2

(9a)

α ≡ max |u0 (x)|,

(9b)

with

x∈Ω\Ωd

the maximum second derivative of the initial conditions on smooth parts of the domain. This criterion is able to distinguish between smooth extrema and discontinuities. The combined limiter applies the full third-order reconstruction H3 on parts which are classified as smooth and switches to the limited function H3L if η indicates large gradients. The combined limiter function reads (c) H3L (δi− 1 , δi+ 1 ) = 2

2

⎧ ⎨H3 (δ

i− 12 , δi+ 12 )

⎩H3L(δ

i− 12 , δi+ 12 )

if η(δi− 1 , δi+ 1 ) < 1 2

2

if η(δi− 1 , δi+ 1 ) ≥ 1. 2

(10)

2

2.2 Generalization to Non-equidistant Meshes For general grids, the size of cell Ci , denoted by Δxi is not uniform for all cells, i.e. Δxi = Δx ∀i, see Fig. 2. In this case, the definition of the undivided differences δi± 1 Eq. (4) is not meaningful anymore and new concepts need to be developed. 2

Full Third-Order Reconstruction Starting again with the full third-order reconstruction, consider a quadratic polynomial pi (x) in cell i that has to maintain the cell averages in the three cells Ci+ ,  ∈ {−1, 0, 1}. This polynomial is then evaluated

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Fig. 2 Notation for the interface reconstruction for non-equidistant grids

at the cell boundaries xi± 1 and yields the reconstructed cell interface values 2

1 ! u(−)1 = pi (xi+ 1 ) = u¯ i + H3,neq i+ 2 2 2 1 ! (+) u 1 = pi (xi− 1 ) = u¯ i − H3,neq. i− 2 2 2

(11a) (11b)

Even though this procedure is similar to the full third-order reconstruction on uniform grids, Eq. (8), the reconstruction function H3,neq differs from H3 since the different cell sizes need to be taken into account. The full (unlimited) third-order reconstruction function for non-equidistant grids reads     Δxi− 1 Δxi+1 Δxi 1 2 H3,neq δi− 1 , δi+ 1 , Δxi , Δxi−1 , Δxi+1 = Δi 3 2 Δx 1 δi+ 1 + Δx 1 δi− 1 2

2

i+

2

2

i−

2

2

(12a) with the abbreviations (see Fig. 2)

Δi =

Δxi−1 +Δxi +Δxi+1 , 3

Δxi− 1 = 2

Δxi−1 +Δxi , 2

Δxi+ 1 = 2

Δxi +Δxi+1 . 2

(12b) As in the equidistant case, comparing to Eq. (6), the reconstructed interface values can be compactly reformulated as u

(∓) i± 12

= u¯ i ±

8 8 8 Δxi Δxi−1 δi+ 12 + Δxi δi± 12 + Δxi+1 δi− 12 2 Δxi−1 + Δxi + Δxi+1

(13a)

with the slopes 8 δi− 1 = 2

δi− 1 2

Δxi− 1 2

,

8 δi+ 1 = 2

δi+ 1 2

Δxi+ 1 2

.

(13b)

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It can easily be seen that for equidistant grids, i.e. Δxi−1 = Δxi = Δxi+1 ≡ Δx, the abbreviated terms reduce to Δi = Δxi− 1 = Δxi+ 1 = Δx and therefore, the 2 2 formulations for H3,neq and H3 coincide, as expected. Equations (12) and (13) indicate that for non-equidistant meshes, the equivalent of the undivided differences δi± 1 are the scaled slopes 2

δi− 1 → Δxi+1 2

u

(−) i+ 12

δi− 1

δi− 1 = Δxi+18

2

Δxi− 1

2

2

: δi+ 1 → Δxi− 1 2

2

(14)

δi+ 1

Δxi Δxi−1 8 8 δ 1+ δi+ 1 = 2 2 i+ 2 2

2

Δxi+ 1 2

for the reconstruction of the right interface of cell Ci and δi− 1 → Δxi+ 1 2

2

u(+)1 :

δi− 1

=

2

Δxi− 1 2

i− 2

δi+ 1 → Δxi−1 2

δi+ 1 2

Δxi+ 1

Δxi Δxi+1 8 8 δ 1+ δi− 1 2 2 i− 2 2 (15)

= Δxi−18 δi+ 1 2

2

for the reconstruction of the left interface of cell Ci . These expressions resemble the δi− 1 smoothness indicators introduced by Jiang and Shu [4], which are given by Δxi8 2 and Δxi8 δ 1. i+ 2

Reconstructions with Limiter Functions In order to generalize the third-order limiter function developed in [8], we replace the undivided differences as mentioned above to obtain the reconstructions 1 (c)  u(−)1 = u¯ i + H3L Δxi+18 δi− 1 , Δxi− 1 i+ 2 2 2 2  1 (c) (+) u 1 = u¯ i − H3L Δxi−18 δi+ 1 , Δxi+ 1 i− 2 2 2 2

8 δi+ 1

 (16a)

2

 8 δi− 1 .

(16b)

2

(c)

with the limiter function H3L described in Sect. 2.1. The non-equidistant version of the limiter function can be explicitly defined by (c) H3L,neq

(c) H3L,neq





δi− 1 , δi+ 1 , Δxi , Δxi−1 , Δxi+1 = 2



2



δi+ 1 , δi− 1 , Δxi , Δxi+1 , Δxi−1 = 2

2

 (c) H3L

Δxi+1 Δx 

(c) H3L

δi− 1 2

i− 1 2

δi+ 1

Δxi−1 Δx

2

i+ 12

, Δxi− 1 2

, Δxi+ 1 2



δi+ 1

2

Δxi+ 1

2

δi− 1

2

Δxi− 1



.

2

(17)

Third-Order Limiter Functions on Non-equidistant Grids

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The decision criterion η for non-uniform meshes reads 2 η(δi− 1 , δi+ 1 ) = 2

2

δ2 1 + δ2 1 i− i+ 2 2 2 , 5 2 α dx 2

(18)

 where dx denotes the average mesh size, dx = ( i Δxi )/#cells and α is defined in Eq. (9b). The average mesh size dx could also be replaced by Δxi —the size of the central cell—for a more local description. Up to our knowledge, this formulation still reproduces reasonable results. However, if the mesh is quite irregular, consisting of big and small cells differing by a factor around 10% or larger, it might be useful to use the average mesh size dx. In these cases, numerical experiments proved to be more robust using this formulation.

3 Numerical Examples In this section we present two numerical examples validating the concepts introduced in Sect. 2. We first prove that third-order accuracy is obtained on non-equidistant onedimensional grids. Furthermore, a discontinuous solution is shown to verify the non-oscillatory property of the new limiter function. For the time update, the third-order strong stability preserving Runge-Kutta (SSP-RK3) time integrator by Gottlieb et al. [3] has been used. Smooth Solution We consider the linear advection equation with smooth initial conditions ⎧ ⎨u + u = 0 t x (19) ⎩u(x, 0) = sin(2π x) on the domain [0, 1] with periodic boundary conditions. Simulations are carried out with N = 25 × 2j , j = 0, . . . , 6 grid cells until time tend = 1.0 and CFL number 0.95. Since we are interested in non-equidistant grids, the original grid is perturbed by adding 0.02 · sin(10π xi+1/2) to each cell boundary xi+1/2 . (c) Figure 3 shows the exact solution as well as the solution obtained with H3L,neq on a grid with 25 cells. The solution is compared to WENO-JS (the third-order WENO method developed by Liu et al. [6] with smoothness measures by Jiang and Shu [4]) and the full third-order reconstruction H3,neq. It can be observed that H3 yields (c) , since only smooth structures are present. Even though as good results as H3L,neq WENO-JS also reaches third-order accuracy, its solution does not show the same quality.

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1

uex H3,neq

0.5

H(c) 3L,neq

0

WENO-JS,neq -0.5 -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 3 Smooth solution on [0, 1] with N = 25 grid cells at time tend = 1.0 and CFL number 0.95 Table 1 L1 -, L∞ -errors and EOC of the solution to (19) (c) with H3L,neq

u − uex 1 8.322E–03 1.347E–03 1.817E–04 2.323E–05 2.920E–06 3.656E–07

Grid 25 50 100 200 400 800

EOC 2.63 2.89 2.97 2.99 3.00

u − uex ∞ 1.311E–02 2.209E–03 2.921E–04 3.663E–05 4.589E–06 5.743E–07

EOC 2.57 2.92 3.00 3.00 3.00

Bold values are the empirical order of conver˚ genceUthe fact that they tend to 3.00 means that the whole paper is meaningful since we obtain the desired order of convergence not only on equidistatnt but also on non-equidistant grids uex

1 0.8

H3,neq

0.6

H

0.4

(c) 3L,neq

WENO-JS,neq

0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 4 Discontinuous solution on [0, 1] with N = 50 grid cells at time tend = 1.0 and CFL number 0.95 c Finally, Table 1 displays the L1 - and L∞ -errors of H3L,neq . The corresponding (c) empirical order of convergence (EOC) shows that H3L,neq obtains the desired accuracy already on coarse meshes.

Discontinuous Solution The same test case with the same setting as before is performed, this time with discontinuous initial condition. The exact solution of the step function (x > 0.5) is shown in Fig. 4 together with the numerical solutions.

Third-Order Limiter Functions on Non-equidistant Grids

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As expected, H3,neq creates oscillations near the discontinuous parts and WENO-JS does not oscillate but does not approach the steep gradient as nicely. The solution (c) does not show any spurious oscillations. At obtained by limiter function H3L,neq the same time it approaches the gradient around x = 0.5 as well as H3,neq. This shows that the newly developed third-order limiter function avoids oscillations while maintaining third-order accuracy—on equidistant as well as non-equidistant grids.

4 Conclusion (c)

In this work we extended the third-order limiter function H3L from equidistant to non-equidistant grids in one space dimension. We were able to show that this extension still yields third-order accurate solutions and does not create oscillations at discontinuities. Furthermore, a comparison with the well-known WENO method shows the excellent performance of the limiter.

References 1. R. Artebrant, H.J. Schroll, Limiter-free third order logarithmic reconstruction. SIAM J. Sci. Comput. 28(1), 359–381 (2006) ˇ 2. M. Cada, M. Torrilhon, Compact third order limiter functions for finite volume methods. J. Comput. Phys. 228(11), 4118–4145 (2009) 3. S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability preserving high order time discretization methods. SIAM Rev. 43(1), 89–112 (2001) 4. G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996) 5. R.J. Le Veque, Numerical Methods for Conservation Laws, 2nd edn. (Birkhäuser, Basel, 1992) 6. X.-D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994) 7. B. Schmidtmann, R. Abgrall, M. Torrilhon, On third-order limiter functions for finite volume methods. Bull. Braz. Math. Soc. 47(2), 753–764 (2016) 8. B. Schmidtmann, B. Seibold, M. Torrilhon, Relations between WENO3 and third-order limiting in finite volume methods. J. Sci. Comput. 68(2), 624–652 (2016) 9. B. Schmidtmann, P. Buchmller, M. Torrilhon, Third-order limiting for hyperbolic conservation laws applied to adaptive mesh refinement and non-uniform 2d grids (2017, preprint). arXiv:1705.10608

A Linearity Preserving Algebraic Flux Correction Scheme of Upwind Type Satisfying the Discrete Maximum Principle on Arbitrary Meshes Petr Knobloch

Abstract Various choices of limiters in the framework of algebraic flux correction (AFC) schemes applied to the numerical solution of scalar steady-state convection– diffusion–reaction equations are discussed. A new limiter of upwind type is proposed for which the AFC scheme satisfies the discrete maximum principle and linearity preservation property on arbitrary meshes.

1 Introduction This paper is devoted to the development of algebraic flux correction (AFC) schemes (in the sense of [7–9]) for stabilizing finite element discretizations of the scalar steady-state convection–diffusion–reaction equation − ε Δu + b · ∇u + c u = g

in Ω ,

u = ub

on ∂Ω .

(1)

Here Ω is a bounded d-dimensional domain (d = 2, 3) having a polygonal (resp. polyhedral) Lipschitz-continuous boundary ∂Ω. We assume that ε > 0 is constant, 1 and b ∈ W 1,∞ (Ω)d , c ∈ L∞ (Ω), g ∈ L2 (Ω), and ub ∈ H 2 (∂Ω) ∩ C(∂Ω) are given functions satisfying ∇ · b = 0 and c ≥ 0 a.e. in Ω. First rigorous analysis of a general AFC scheme was published in [1, 2] where the solvability of the scheme was investigated, an error estimate was derived, and the discrete maximum principle (DMP) for the limiter proposed in [7] was proved on Delaunay meshes. In [3], a new limiter was proposed for which the AFC scheme was proved to be linearity preserving and to satisfy the DMP on arbitrary simplicial

P. Knobloch () Charles University, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Prague, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_86

909

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P. Knobloch

meshes. However, the limiter from [3] does not possess the favourable upwind features of the limiter from [7]. Therefore, in this paper, we propose a new limiter which combines the advantages of the limiters from [7] and [3]. In Sect. 2, we introduce a finite element discretization of (1) and the corresponding AFC scheme. Then, in Sect. 3, a general result on the DMP is formulated. Finally, in Sect. 4, the properties of different limiters are discussed and a new limiter of upwind type is proposed for which the AFC scheme satisfies the discrete maximum principle and linearity preservation property on arbitrary meshes.

2 An Algebraic Flux Correction Scheme To discretize the problem (1), we introduce finite element spaces Wh = {vh ∈ C(Ω) ; vh |T ∈ P1 (T ) ∀ T ∈ Th } ,

Vh = Wh ∩ H01 (Ω) ,

where Th is a simplicial triangulation of Ω and P1 (T ) is the space of linear polynomials on T . We denote by x1 , . . . , xN the vertices of Th and assume that M of them (0 < M < N) are interior vertices which are numbered first, i.e., x1 , . . . , xM ∈ Ω and xM+1 , . . . , xN ∈ ∂Ω. We denote by ϕ1 , . . . , ϕN the standard basis functions of Wh , i.e., one has ϕi (xj ) = δij , i, j = 1, . . . , N, where δij is the Kronecker symbol. Then the functions ϕ1 , . . . , ϕM form a basis in Vh . For any vh ∈ Wh , we denote by {vi }N determined coefficients of vh with i=1 the uniquely  N respect to the above basis of Wh , i.e., vh = i=1 vi ϕi . First, we introduce the following discretization of (1): Find uh ∈ Wh such that uh (xi ) = ub (xi ), i = M + 1, . . . , N, and ah (uh , vh ) = (g, vh )

∀ vh ∈ Vh ,

(2)

where ah (uh , vh ) = ε (∇uh , ∇vh ) + (b · ∇uh , vh ) +

M

(c, ϕi ) ui vi

i=1

and (·, ·) denotes the inner product in L2 (Ω) or L2 (Ω)d . Note that the usual reaction term (c uh , vh ) is replaced by a diagonal approximation, analogous to a mass lumping in discretizations of transient problems. Due to the assumptions on the data of the problem (1), the above discretization is uniquely solvable. We denote aij = ah (ϕj , ϕi ) for i, j = 1, . . . , N, gi = (g, ϕi ) for i = 1, . . . , M, and ubi = ub (xi ) for i = M + 1, . . . , N. Then uh is a solution of (2) if and only if

A Linearity Preserving AFC Scheme of Upwind Type

911

the corresponding coefficient vector U = (u1 , . . . , uN ) satisfies the linear system N

aij uj = gi ,

i = 1, . . . , M ,

j =1

ui = ubi ,

i = M + 1, . . . , N .

Since the above discretization is unstable in the convection-dominated regime, we apply the algebraic flux correction as described in [2] to enforce the discrete maximum principle. This leads to the following system of nonlinear equations: N

aij uj +

j =1

ui = ubi ,

N

(1 − αij (U )) dij (uj − ui ) = gi ,

i = 1, . . . , M ,

(3)

j =1

i = M + 1, . . . , N ,

(4)

where αij (U ) ∈ [0, 1], i, j = 1, . . . , N, are limiters that depend on the solution U = (u1 , . . . , uN ) and form a symmetric matrix, and dij are entries of an artificial diffusion matrix defined by dij = dj i = − max{aij , 0, aj i }

∀ i = j ,

dii = −

dij .

j =i

Moreover, for any i, j ∈ {1, . . . , N}, we assume that αij (U )(uj −ui ) is a continuous function of U = (u1 , . . . , uN ). This property makes it possible to prove that the nonlinear problem (3), (4) has a solution, see [2]. The assumed symmetry of the limiters αij assures that the matrix corresponding to the additional term in (3) is positive semidefinite so that it is important not only for conservativity but also for stability and deriving error estimates. Moreover, it can be shown that a lack of symmetry of the limiters may cause that the AFC scheme is not solvable, cf. [1]. In the following section we shall state a general result on the DMP for the AFC scheme (3), (4) and then we shall discuss various choices of the limiters assuring the DMP and further favourable properties of the scheme. It can be verified (cf. [2, 3]) that the limiters presented in this paper possess the above-mentioned continuity property.

3 General Condition for the Discrete Maximum Principle In this section we formulate a general condition on the limiters αij assuring that the AFC scheme (3), (4) satisfies the DMP.

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The DMP will be formulated locally, with respect to index sets Si = {j ∈ {1, . . . , N} \ {i} : xi and xj are connected by an edge of Th } , i = 1, . . . , M, and the following property will be assumed. Assumption (A) Consider any U = (u1 , . . . , uN ) ∈ RN and any i ∈ {1, . . . , M}. If ui is a strict local extremum of U with respect to Si , i.e., ui > uj

∀ j ∈ Si

or

ui < uj

∀ j ∈ Si ,

then aij + (1 − αij (U )) dij ≤ 0

∀ j ∈ Si .

Then one can prove the following local form of the DMP. Theorem 1 Let U = (u1 , . . . , uN ) ∈ RN be a solution of (3) with limiters αij satisfying the assumption (A). Consider any i ∈ {1, . . . , M}. Then gi ≤ 0

⇒

ui ≤ max u+ j ,

gi ≥ 0

j ∈Si

ui ≥ min u− j ,

⇒

j ∈Si

− where u+ j = max{0, uj } and uj = min{0, uj }. If, in addition,

gi ≤ 0

⇒

ui ≤ max uj ,

gi ≥ 0

j ∈Si

⇒

N

j =1 aij

= 0, then

ui ≥ min uj . j ∈Si

& %

Proof See [6].

The global DMP can be proved under the same assumptions as the local DMP using the techniques of [4, Theorems 5.1 and 5.2].

4 Old and New Limiters In this section we present four examples of limiters with different properties. First we mention two limiters proposed earlier and then, combining their advantages, we introduce two new limiters. The first step in defining any of these limiters is the computation of the local quantities Pi± and Q± i for i = 1, . . . , M. Then one defines 9 Ri+

= min 1,

Q+ i Pi+

9

C ,

Ri−

= min 1,

Q− i Pi−

C ,

i = 1, . . . , M .

A Linearity Preserving AFC Scheme of Upwind Type

913

If Pi+ or Pi− vanishes, one sets Ri+ = 1 or Ri− = 1, respectively. For i = M + 1, . . . , N, one defines Ri+ = Ri− = 1. Furthermore, one introduces preliminary limiters ⎧ + ⎪ ⎨ Ri if fij > 0 , 8 αij = i, j = 1, . . . , N . 1 if fij = 0 , ⎪ ⎩ R − if f < 0 , ij i The below limiters differ in the definition of Pi± , Q± i and in the way how the symmetric limiters αij are obtained from the nonsymmetric limiters 8 αij .

4.1 Limiter by Kuzmin [7] (Limiter 1) The quantities Pi± , Q± i are defined by Pi+ =

fij+ ,

Pi− =

j ∈ Si aji ≤ aij

fij− ,

Q+ i =−

fij− ,

Q− i =−

j ∈Si

j ∈ Si aji ≤ aij

fij+ ,

j ∈Si

where fij = dij (uj − ui ), fij+ = max{0, fij }, and fij− = min{0, fij }. For any i, j ∈ {1, . . . , N} with aj i ≤ aij , one sets αij = αj i = 8 αij . This limiter is often used in computations and was thoroughly investigated in [2]. It was proved in [6] that the DMP holds provided that min{aij , aj i } ≤ 0

∀ i = 1, . . . , M , j = 1, . . . , N , i = j ,

(5)

which is always true if Th is a Delaunay triangulation. However, the condition (5) may be satisfied also on non-Delaunay meshes, particularly in the convectiondominated case, since the convection matrix is skew-symmetric. Nevertheless, on arbitrary meshes, the condition (5) is not satisfied in general and it was also shown in [6] that then the DMP generally does not hold.

4.2 Limiter by Barrenechea et al. [3] (Limiter 2) For i = 1, . . . , M, one denotes umax = max uj , i j ∈Si ∪{i}

umin = min i

j ∈Si ∪{i}

uj ,

q i = γi

j ∈Si

dij ,

(6)

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where γi > 0 are arbitrary constants, and one sets Pi+ =

fij+ ,

Pi− =

j ∈Si

fij− ,

j ∈Si

max Q+ i = qi (ui − ui ) ,

min Q− i = qi (ui − ui ) .

αij ,8 αj i }. Then, for any i, j ∈ {1, . . . , N}, one defines αij = min{8 Note that large values of the constants γi cause that more limiters αij will be equal to 1 and hence less artificial diffusion is added, which makes it possible to obtain sharp approximations of layers. On the other hand, however, large values of γi ’s also cause that the numerical solution of the nonlinear algebraic problem becomes more involved. It was proved in [3] that the DMP holds for arbitrary meshes. Moreover, the constants γi can be chosen in such a way that the AFC scheme (3), (4) is linearity preserving, i.e., the modification added to the original algebraic problem vanishes in regions where the approximate solution is a linear function. This property, which can be interpreted as a weak consistency requirement, is believed to lead to improved accuracy in regions where the solution is smooth. Moreover, it has been observed in different works that linearity preservation improves the quality of the numerical solution on distorted meshes. The linearity preservation is equivalent to the condition d αij ({u(xi )}N i=1 ) = 1 ∀ u ∈ P1 (R ) ,

i = 1, . . . , M, j = 1, . . . , N .

(7)

To assure the validity of (7), it suffices to choose the constants γi in such a way that ≤ γi (umax − ui ) ui − umin i i

∀ u ∈ P1 (Rd ) ,

(8)

see [3]. It was proved in [3] that (8) holds with γi = 1 if the patch Δi = supp ϕi is symmetric with respect to the vertex xi , and with max |xi − xj |

γi =

xj ∈∂Δi

dist(xi , ∂Δconv ) i

in general, where Δconv is the convex hull of Δi . i It was shown in [6] that the condition aj i < aij used at several places in the definition of the previous Limiter 1 often (but not always) means that xi is the upwind vertex. It turns out that this upwind feature of Limiter 1 has a positive influence on the quality of the approximate solutions and on the convergence of the iterative process for solving the nonlinear problem (3), (4). However, Limiter 1 is not linearity preserving in general and therefore we introduce the following limiter combining the ideas leading to Limiters 1 and 2.

A Linearity Preserving AFC Scheme of Upwind Type

915

4.3 A Linearity Preserving Limiter of Upwind Type (Limiter 3) For i = 1, . . . , M, one denotes

Pi+ = fij+ ,

Pi− =

j ∈ Si aji ≤ aij

fij− ,

j ∈ Si aji ≤ aij

and

q i = γi

dij ,

max Q+ i = qi (ui − ui ) ,

min Q− i = qi (ui − ui ) ,

(9)

j ∈ Si aji ≤ aij

max are defined as in (6). Then, where γi > 0 are constants satisfying (8) and umin i , ui for any i, j ∈ {1, . . . , N} with aj i ≤ aij , one sets αij = αj i = 8 αij .

Theorem 2 If (5) holds, then the AFC scheme (3), (4) with Limiter 3 satisfies the DMP. Proof It suffices to verify Assumption (A). Consider any U = (u1 , . . . , uN ) ∈ RN , i ∈ {1, . . . , M}, and j ∈ Si . Let ui be a strict local extremum of U with respect to Si . We want to prove that aij + (1 − αij (U )) dij ≤ 0 .

(10)

If aij ≤ 0, then (10) holds since (1 − αij (U )) dij ≤ 0. If aij > 0, then aj i ≤ 0 due to (5) and hence αij = 8 αij and dij = −aij < 0. If ui > uk for any k ∈ Si , then fij > 0, Pi+ > 0 and umax = ui . Consequently, 8 αij = Ri+ = 0. Similarly, if i − ui < uk for any k ∈ Si , then fij < 0, Pi < 0 and umin = ui , so that 8 αij = Ri− = 0. i Thus, (10) holds again. & % Theorem 3 The AFC scheme (3), (4) with Limiter 3 is linearity preserving. Proof Consider any i ∈ {1, . . . , M} and u ∈ P1 (Rd ). Then, in view of (8), Pi+ =

dij (uj − ui ) ≤

j ∈ Si aji ≤ aij , uj < ui

dij (umin − ui ) i

j ∈ Si aji ≤ aij

≤

+ dij γi (ui − umax i ) = Qi

j ∈ Si aji ≤ aij

and hence Ri+ = 1. Similarly, one obtains Ri− = 1. Consequently, (7) holds and the AFC scheme is linearity preserving. & %

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Thus, Limiter 3 possesses the same upwind features as Limiter 1 and leads to the validity of the DMP under the same assumption as Limiter 1. However, in addition, it provides a linearity preserving AFC scheme, which has an important impact on the convergence of the approximate solutions. In particular, for the problem (1) with dominant diffusion and a polynomial solution u considered in Example 20 of [2], resp. Example 7.1 of [3], the convergence is of optimal rate on a sequence of Friedrichs–Keller meshes, like for Limiter 2, whereas no convergence is observed for Limiter 1 in this case, see the computational results for Limiters 1 and 2 in [2, 3]. Although one usually concentrates on the convection-dominated regime, the convergence behaviour in the diffusion-dominated case is important for applications where the convection may locally vanish. However, as we mentioned in the discussion to Limiter 1, the assumption (5) does not hold on non-Delaunay meshes in general and hence the DMP cannot be guaranteed for arbitrary meshes. Therefore, combining Limiter 3 with Limiter 2, we finally introduce the following limiter, which is the main result of this paper.

4.4 A Linearity Preserving Limiter of Upwind Type Satisfying the DMP on Arbitrary Meshes (Limiter 4) For any i, j ∈ {1, . . . , N}, the limiters αij = αj i are defined as for Limiter 3 if min{aij , aj i } ≤ 0 and as for Limiter 2 otherwise. The upwind features of the limiter are important in the convection-dominated case in which the assumption (5) is typically satisfied and hence Limiter 4 coincides with Limiter 3. Therefore, Limiter 4 can be viewed as a limiter of upwind type. Since Limiter 2 satisfies Assumption (A) on arbitrary meshes without any additional assumption, it follows from the properties of Limiter 3 that the AFC scheme with Limiter 4 satisfies the DMP on arbitrary meshes. Finally, one observes that Limiter 4 satisfies (7) and hence the AFC scheme with Limiter 4 is linearity preserving on arbitrary meshes. Consequently, Limiter 4 combines all the advantages of Limiters 1 and 2, which was also verified by numerous computational tests. Finally, let us mention an additional advantage of Limiter 4 in comparison with Limiter 2. Depending on the magnitude of the factors qi defined in (6) or (9), perturbations of linear functions still lead to αij = 1 so that the stabilizing term in the AFC scheme (3), (4) vanishes. If the right-hand side g in (1) does not vanish, this often leads to approximate solutions containing wiggles which do not represent a violation of the DMP but decrease the accuracy of the results. To reduce this effect known as terracing (which also causes a bad convergence of solvers for the nonlinear AFC scheme), it is desirable to have the magnitude of qi as small as possible, i.e., to use a small value of γi satisfying (8). Comparing Limiters 2 and 3, one observes that Limiter 2 allows larger perturbations of linear functions leading to αij = 1 than Limiter 3 and hence Limiter 3 (resp. Limiter 4 under the assumption min{aij , aj i } ≤ 0) provides less pronounced terracing in comparison to Limiter 2.

A Linearity Preserving AFC Scheme of Upwind Type

917

To illustrate the effects described in the previous paragraph, we shall present numerical results for the following example. Example 1 Problem (1) is considered with Ω = (0, 1)2 , ε = 10−8 , b = (2, 3)T , and c = 0. The right-hand side g and the boundary condition ub are chosen in such a way that 

2 (x − 1) u(x, y) = x y − y exp ε 2



2



 3 (y − 1) − x exp ε   2 (x − 1) + 3 (y − 1) + exp ε

is the solution of (1). The triangulation Th of the above domain Ω used in all computations was constructed by dividing Ω into 20 × 20 equal squares and by cutting each square into two triangles by drawing the diagonal from bottom left to top right. The first row of Fig. 1 shows the approximate solutions of Example 1 obtained using the AFC scheme (3), (4) with Limiter 2 (left) and Limiter 4 (right). One can clearly observe that the above-mentioned terracing is less pronounced for Limiter 4, which also causes that the AFC scheme with Limiter 4 is more accurate. This can also be seen in the second row of Fig. 1 which shows the difference between the respective approximate solutions and the piecewise linear interpolate of the solution u. Let us mention that it was shown in [5] that the large error at the outflow boundary cannot

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2 0

0

0.2

0.4 x

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1 y

0.2 0 0

0.16

0.12

0.12

0.08

0.08

0.04

0.04

0

0

0.2

0.4 x

0.6

0.8

1 0

0.4

0.6

0.8

x

0.16

-0.04

0.2

0.2

0.4

0.6

0.8

1 y

1 0

0.2

0.4

0.6

0.8

y

0 -0.04 0

0.2

0.4

0.6 x

0.8

1 0

0.2

0.4

0.6

1

0.8

1 y

Fig. 1 Example 1: approximate solutions obtained using Limiter 2 (top left) and Limiter 4 (top right), and the corresponding errors (bottom)

918

P. Knobloch

be avoided by the choice of the limiter so that other techniques have to be applied to decrease it. Acknowledgements This work has been supported through the grant No. 16-03230S of the Czech Science Foundation.

References 1. G.R. Barrenechea, V. John, P. Knobloch, Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension. IMA J. Numer. Anal. 35(4), 1729–1756 (2015) 2. G.R. Barrenechea, V. John, P. Knobloch, Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54(4), 2427–2451 (2016) 3. G.R. Barrenechea, V. John, P. Knobloch, An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes. Math. Models Methods Appl. Sci. 27(3), 525–548 (2017) 4. P. Knobloch, Numerical solution of convection–diffusion equations using a nonlinear method of upwind type. J. Sci. Comput. 43(3), 454–470 (2010) 5. P. Knobloch, On the application of algebraic flux correction schemes to problems with nonvanishing right-hand side. Boundary and Interior Layers, Computational and Asymptotic Methods – BAIL 2014, ed. by P. Knobloch. Lect. Notes Comput. Sci. Eng., vol. 108 (Springer, Berlin, 2015), pp. 99–109 6. P. Knobloch, On the discrete maximum principle for algebraic flux correction schemes with limiters of upwind type. Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016, ed. by Z. Huang, M. Stynes, and Z. Zhang. Lect. Notes Comput. Sci. Eng., vol. 120 (Springer, Berlin, 2017), pp. 129–139 7. D. Kuzmin, Algebraic flux correction for finite element discretizations of coupled systems, in Proceedings of the International Conference on Computational Methods for Coupled Problems in Science and Engineering, ed. by M. Papadrakakis, E. Oñate, B. Schrefler (CIMNE, Barcelona, 2007), pp. 1–5 8. D. Kuzmin, Explicit and implicit FEM-FCT algorithms with flux linearization. J. Comput. Phys. 228, 2517–2534 (2009) 9. D. Kuzmin, Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J. Comput. Appl. Math. 236, 2317–2337 (2012)

Part XXIII

New Frontiers in Domain Decomposition Methods: Optimal Control, Model Reduction, and Heterogeneous Problems

Optimized Schwarz Methods for Advection Diffusion Equations in Bounded Domains Martin J. Gander and Tommaso Vanzan

Abstract Optimized Schwarz methods use better transmission conditions than the classical Dirichlet conditions that were used by Schwarz. These transmission conditions are optimized for the physical problem that needs to be solved to lead to fast convergence. The optimization is typically performed in the geometrically simplified setting of two unbounded subdomains using Fourier transforms. Recent studies for both homogeneous and heterogeneous domain decomposition methods indicate that the geometry of the physical domain has actually an influence on this optimization process. We study here this influence for an advection diffusion equation in a bounded domain using separation of variables. We provide theoretical results for the min-max problems characterizing the optimized transmission conditions. Our numerical experiments show significant improvements of the new transmission conditions which take the geometry into account, especially for strong tangential advection.

1 Introduction We study optimized Schwarz methods for the advection diffusion equation −νΔu + a · ∇u = 0 in Ω, u = g on ∂Ω,

(1)

where ν ∈ R+ , a = (a1 , a2 )T ∈ R2 and Ω is a bounded domain in two dimensions. As a model problem we consider the geometry given in Fig. 1. We decompose Ω = (−L, L) × (0, L) into two subdomains Ω1 = (−L, 0) × (0, L) and Ω2 = (0, L) × (0, L), and because of the linearity of (1), we study the optimized Schwarz methods

M. J. Gander · T. Vanzan () Section de mathématiques, Université de Genève, Genève, Switzerland e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_87

921

922

M. J. Gander and T. Vanzan

Fig. 1 Geometry of the domain Ω on the left, and decomposition into two subdomains on the right

y

y L

L Γ Ω1

Ω

Ω2

x 0

-L

x

L

0

-L

L

for the errors ejn = uj − unj , −νΔe1n + a · ∇e1n = 0 in Ω1 , (ν∂x + s1 )(e1n )(0, ·) = (ν∂x + s1 )(e2n−1 )(0, ·), −νΔe2n + a · ∇e2n = 0 in Ω2 , (ν∂x − s2 )(e2n )(0, ·) = (ν∂x − s2 )(e1n−1 )(0, ·),

(2)

where s1 , s2 ∈ R are to be determined to get fast convergence. This is typically done using Fourier transforms in the simplified setting of unbounded domains, see [3] and references therein, and in particular [2] for the case of advection diffusion problems. Advection diffusion problems have however often boundary layers, which can not be taken into account using unbounded domain analysis. The influence of geometry on the optimization for Laplace’s equation on a rectangular domain has been studied in [4], and for circular domain decomposition, see [6, 8]. We study here for the first time the influence of geometry on the optimization of transmission conditions for advection diffusion problems, looking for a separation of variables solution of the form ejn (x, y) = φjn (x)ψjn (y), j = 1, 2 in rectangular domains. Substituting the separation of variables Ansatz into (1), we obtain on Ω1 ν∂yy ψ1n − a2 ∂y ψ1n + λψ1n = 0, ψ1n (0)

=

ψ1n (L)

y ∈ (0, L),

(3)

x ∈ (−L, 0),

(4)

= 0,

ν∂xx φ1n − a1 ∂x φ1n − λφ1n = 0,

φ1n (−L) = 0 and ν∂x φ1n (0) + s1 φ1n (0) =

r2n−1 ,

where r2n−1 is the Robin data to impose on Γ . Equation (3) is a Sturm-Liouville eigenvalue problem,  Lψ = λψ, ψ(0) = ψ(L) = 0, Lψ := νe

a2 y ν

? @ a y d − 2ν dψ(y) − e . dy dy

(5)

Hence the λ are the eigenvalues of the differential operator L with the associated boundary conditions, and the eigenfunctions ψ form an orthogonal basis for the Hilbert space L2 (0, L) with respect to the weighted L2 inner product 

L

f, gw = 0

f (y)g(y)e−

a2 y ν

 dy,

f 2w =

L 0

f 2 e−

a2 y ν

dy.

OSM for Advection Diffusion Equations in Bounded Domains

923

For λ ≤ 0, Eq. (5) does not admit a solution due to the boundary conditions. For λ > 0, the general solution of (5) is given by ⎛ ψ(y) = e

a2 y 2ν

N

− a22

4λν ⎜ n ⎝C1 cos( 2ν

⎞

N

− a22

4λν 2ν

y) + C2n sin(

⎟ y)⎠ .

The boundary conditions prescribe C1n = 0 and a quantization on λ such that λ = λl =

νπ 2 l 2 L2

+

a22 4ν ,

l ∈ N. Equation (4) then has the associated solution 

φ1n (x)

=e

a1 x 2ν

√

√ D1n e

a12 +4νλl x 2ν

+

D2n e−

a12 +4νλl x 2ν

 .

Since boundary layers appearing on lateral boundaries will not reach the interface, we assume for simplicity that the domain is unbounded in the x direction and therefore get D2n = 0. With similar calculations for e2 (x, y), we obtain for the error functions on both subdomains by linearity (

e1n (x, y)

=

∞  l=1

e2n (x, y) =

∞  l=1

n eˆ1,l e n e eˆ2,l

a1 x+a2 y 2ν

a1 x+a2 y 2ν

sin( lπy L )e sin( lπy L )e

2 2 2 a12 +a22 + 4ν l π L2 2ν

x

( 2 2 2 a 2 +a 2 + 4ν l π 1 2 L2 − 2ν

, (6) x

,

n where eˆj,l are constants to be determined imposing the Robin transmission conditions on Γ . Inserting the series expansions for e1n ,e2n into the transmission conditions of the optimized Schwarz methods (2) and using the orthonormality of the eigenfunctions, we find for each l

⎛ ⎜ a1 ⎝2 + ⎛ ⎜ a1 ⎝2 −

( 2 2 2 a12 +a22 + 4ν l 2 π L

2 ( 2 2 2 a12 +a22 + 4ν l 2 π L

2

⎞

⎛

⎟ n ⎜ + s1 ⎠ eˆ1,l = ⎝ a21 − ⎞

⎛

⎟ n ⎜ − s2 ⎠ eˆ2,l = ⎝ a21 +

( 22 2 a12 +a22 + 4ν l 2 π L

2 ( 22 2 a12 +a22 + 4ν l 2 π L

2

⎞ ⎟ n−1 + s1 ⎠ eˆ2,l , ⎞

(7)

⎟ n−1 − s2 ⎠ eˆ1,l .

n = ρ(l)eˆ n−2 , eˆ n = ρ(l)eˆ n−2 , where the We thus obtain over a double step eˆ1,l 2,l 1,l 2,l convergence factor is given by

2 ρ(l) := 2

a12 + a22 +

4ν 2 l 2 π 2 L2

− a1 − 2s1

a12 + a22 +

4ν 2 l 2 π 2 L2

− a1 + 2s2

2 2

a12 + a22 +

4ν 2 l 2 π 2 L2

+ a1 − 2s2

a12 + a22 +

4ν 2 l 2 π 2 L2

+ a1 + 2s1

.

(8)

924

M. J. Gander and T. Vanzan

2 Optimization To optimize s1 , s2 for fast convergence, we need to minimize the absolute value of the convergence factor ρ(l) over all relevant l ∈ {1, . . . , N}, where N is determined by the2numerical truncation of the infinite series.1 We introduce the function f : 2 2 2

1 1 and s2 := f (p)+a , where p → a12 + a22 + 4 ν pL2π , and set first s1 := f (p)−a 2 2 + 2 p ∈ R , since f depends on p . Inserting these expressions for s1 , s2 into (8), and considering for simplicity a continuous set for l instead of the discrete one, see [9], we obtain the min-max problem

B B B f (l) − f (p) 2 B B B min max |ρ(l, p)| = min max B B. B p∈R+ l∈[1,N] p∈R+ l∈[1,N] B f (l) + f (p)

(9)

We now solve the general min-max problem (9) under the only condition that the function f is strictly monotonic, which holds for our case. Theorem 1 If the function f is strictly monotonic, then the solution √ of the min-max problem (9) is given by the unique p∗ which satisfies f (p∗ ) = f (1)f (N). Proof We prove the result when f is strictly increasing, the argument when f is strictly decreasing is analogous. Since the objective function is squared, we can omit the absolute value, and we compute ∂ρ 4(f (l) − f (p))fl (l)f (p) = , ∂l (f (l) + f (p))3

4(f (l) − f (p))fp (p)f (p) ∂ρ =− , ∂p (f (l) + f (p))3

(10)

where fl , fp are the derivatives of f with respect to l and p. If p < l for every l then, since f is strictly increasing, we know that f (l) > f (p) and fp (p) > 0, ∂ρ therefore ∂p < 0 and we can not be at the optimum since increasing p decreases maxl∈[1,N] |ρ(l, p)|. The same argument holds if p > N and we conclude that at the optimum p ∈ [1, N]. Due to the monotonicity of f , the convergence factor has a unique zero at l = p, and from the partial derivative with respect to l, we see that ρ(l, p) has only two local maxima located at l = 1 and l = N, ∀p ∈ [1, N]. Therefore, maxl∈[1,N] |ρ(l, p)| = max{ρ(1, p), ρ(N, p)}. Now since ∂ρ(1,p) > 0 ∂p ∀p ∈ (1, N] and

∂ρ(N,p) ∂p

< 0 ∀p ∈ [1, N), by continuity the optimal p∗ satisfies

ρ(1, p) = ρ(N, p). The uniqueness of p∗ follows from the strict sign of √ l = 1, N and a direct computation leads to f (p∗ ) = f (1)f (N).

∂ρ(l,p) ∂p

for

the unbounded domain analysis, one minimizes over all frequencies k := πl L ∈ [kmin , kmax ], L with kmin := πL and kmax = πh , where h = N+1 is the mesh size and N the number of mesh points on the interface Γ . From (6), we see that kmin corresponds to l = 1, and for l = N, πN π L ≈ h = kmax , like in e.g. [3].

1 In

OSM for Advection Diffusion Equations in Bounded Domains

925

Theorem 2 When N → +∞, the asymptotic behaviour of the optimized Schwarz method given by Theorem 1 is √

max |ρ(l, p∗ )| = 1 −

1≤l≤N

where

p∗

=

(2

a12 + a22 +

( 2 2 2L a12 +a22 + 4ν 2π L N( (

/ 0 0 √ 1 νπ 2

4ν 2 π 2 L L2 2πν

√

2 2 a12 +a22 + 4ν 2π L

√1 , N

(11)

ν2π 2 L2

N.

Proof We make the Ansatz p∗ = Cp N α and we solve the equation f (p∗ ) = √ f (1)f (N) asymptotically. We consider now the more general case where s1 , s2 depend on two different 1 1 parameters, s1 = f (p)−a and s2 = f (q)+a , with p, q ∈ R+ , and study 2 2 B B B f (l) − f (p)   f (l) − f (q) B B B min max |ρ(l, p, q)| = min max B B. f (l) + f (q) B p,q∈R+ l∈[1,N] p,q∈R+ l∈[1,N] B f (l) + f (p) (12) Theorem 3 If the function f is strictly monotonic, then the solution of the min-max problem (12) is given by two couples (pj∗ , qj∗ ), j = 1, 2 which satisfy |ρ(1, p, q)| = ˆ p, q)| = |ρ(N, p, q)|, where lˆ is an interior local maximum such that f (l) ˆ = |ρ(l, √ ∗ ∗ ∗ ∗ f (p)f (q), and we have p2 = q1 and q2 = p1 . Proof We observe that ρ(l, p, q) is invariant if we exchange p and q, therefore we focus our attention on the case p < q. The sign of the partial derivatives with respect to p and q satisfies 

∂|ρ| sign ∂p







= sign −f (l) + f (p) ,



∂|ρ| sign ∂q



  = sign −f (l) + f (q) .

Repeating the argument of Theorem 1, we obtain that at the optimum we must have p, q ∈ [1, N]. For the derivative with respect to l, we obtain ∂|ρ| 2fl (l)(f (p) + f (q))(f (l)2 − f (q)f (p)) = sign(ρ(l, p, q)) . ∂l (f (l) + f (p))2 (f (l) + f (q))2 We thus have three local maxima with respect to l, one located at l = 1, one at ˆ with p < lˆ < q which is the unique l = N, and an interior local maximum at l, ˆ 2 − f (q)f (p) = 0. The uniqueness zero of the partial derivative, satisfying f (l) ˆ of l follows from the strict monotonicity of f . We thus have max |ρ(l, p, q)| = l∈[1,N]

ˆ p, q)|, |ρ(N, p, q)|}. Now we observe that for 1 < p < max{|ρ(1, p, q)|, |ρ(l,

926

M. J. Gander and T. Vanzan

q < N, we have ˆ p, q)| < 0 ∂p |ρ(N, p, q)| < 0, ∂p |ρ(1, p, q)| > 0 ∂p |ρ(l, ˆ p, q)| > 0 ∂q |ρ(N, p, q)| < 0. ∂q |ρ(1, p, q)| > 0 ∂q |ρ(l,

(13)

Suppose that |ρ(1, p, q)| < |ρ(N, p, q)|. Then from (13), we see that ˆ |ρ(N)|}. In the case increasing p uniformly improves max{|ρ(1)|, |ρ(l)|, |ρ(1, p, q)| > |ρ(N, p, q)|, similarly decreasing q uniformly improves ˆ |ρ(N)|}. Thus, at the optimized parameters p∗ and q ∗ , we max{|ρ(1)|, |ρ(l)|, must have |ρ(1, p∗ , q ∗ )| = |ρ(N, p∗ , q ∗ )|, and a direct computation leads to the condition f (p∗ )f (q ∗ ) = f (1)f (N). We can thus focus on one parameter only, say p, letting q = q(p), varying such that the constraint f (p)f (q(p)) = f (1)f (N) is satisfied. Since f is strictly increasing, q(p) must be a decreasing function of p in order to satisfy the constraint. Moreover q(1) = N and q(N) = 1. We thus obtain the equivalent min-max problem ˆ p, q(p))|}. min max{|ρ(1, p, q(p))|, |ρ(l,

1≤p≤lˆ

(14)



f (p)f (q(p)) Combining (13), the implicit differentiation dq(p) dp = − f  (q(p))f (p) and the explicit expressions for the partial derivatives, we obtain

ˆ q(l))| ˆ > 0, |ρ(1, 1, q(1))| = 0, |ρ(1, l, ˆ ˆ ˆ ˆ |ρ(l, 1, q(1))| > 0, |ρ(l, l, q(l))| = 0,

d|ρ(1,p)| dq(p) = ∂|ρ(1,p,q(p))| + ∂|ρ(1,p,q(p))| dp ∂p ∂q dp ˆ ˆ ˆ d|ρ(l,p)| ∂|ρ(l,p,q(p))| ∂|ρ(l,p,q(p))| dq(p) = + dp ∂p ∂q dp

> 0, < 0.

(15) These observations are sufficient to conclude, as in the last steps in Theorem 1, that there exists a unique p∗ , solution of the min-max problem (14), so that the solution of (12) is given by p∗ , solution of (14), and q ∗ defined implicitly by f (p∗ )f (q ∗ ) = f (1)f (N). The same argument can be repeated for the case 1 < q < p < N, and since |ρ(l, p, q)| is invariant under the change p ↔ q, we obtain the desired result.

3 Numerical Experiments The transmission conditions for the advection diffusion equation (1) were analyzed in [2] using Fourier transforms assuming unbounded domains in the y direction, which led to the convergence factor ρ(k, ˜ s1 , s2 ) :=

2 2 a12 −4iνa2 k+4ν 2 k 2 −a1 −2˜s1 a12 −4iνa2 k+4ν 2 k 2 +a1 −2˜s2 2 2 . a12 −4iνa2 k+4ν 2 k 2 −a1 +2˜s2 a12 −4iνa2 k+4ν 2 k 2 +a1 +2˜s1

(16)

−p˜ p˜ This convergence factor was then optimized for s˜1 := a12ν and s˜2 := a+ 2ν , where we use the tilde to distinguish from our variables sj and p. Equations (16)

OSM for Advection Diffusion Equations in Bounded Domains

927

and (8) are similar in many aspects, but they differ profoundly in the dependence on the tangential advection a2 since Eq. (16) is a complex quantity if a2 = 0. This comes directly from the Fourier transform, as the calculations in [2] show, and makes the corresponding min-max problems much harder to solve than the ones proposed here. In addition, our numerical tests below show that our new optimized parameters perform substantially better when strong advection is present along the interface. We discretize the problem using finite differences, and use as initial guess random functions with values in the interval [−1, 1] multiplied by the factor a2 y e 2ν , so that we introduce errors in all eigenfunction components in (6). Without the weight, a projection on the orthogonal basis (6) showed that the lowest and highest frequencies are statistically less present compared to the intermediate ones, and this leads initially to an artificially faster contraction. We stop the algorithm when both e1n |Γ w and e2n |Γ w are less than  = 10−8 . This norm corresponds  n 2 to our analysis, since by Parseval, ejn |Γ w = l |eˆj,l |Γ | . For comparison, we 2 will also measure to the sine  then error in the L norm, which corresponds using n n 2 basis, ejn |Γ = e ¯ sin(j πy), since by Parseval e |  = | e ¯ | l j,l l j,l Γ | . We j Γ 2 show in Fig. 2 two iteration plots: in the left panel, for weak tangential advection, the estimates for the optimal parameters f (p∗ ) and p˜ are similar; in the right panel, for strong tangential advection, the analysis presented in this manuscript is able to provide a parameter which leads to a more efficient algorithm than p. ˜ We verified also that the discrepancy between the iteration counts obtained with the two norms disappears when we decrease the tolerance . In Fig. 3, we show the corresponding iteration plots for the two parameter case. For the unbounded domain convergence factor (16), there is no theorem available giving the optimal choice for the parameters, since the complex nature of |ρ| ˜ makes the min-max problem extremely difficult to solve, and we thus solved it numerically here. 64

75

62 70

60

65

56

Iterations

Iterations

58

54 52

60 55

50 48

50

46 44 60

80

100

120

Optimal parameters

140

160

45 70

80

90

100

110

120

130

140

150

Optimal parameters

Fig. 2 Iteration numbers for weak tangential advection, ν = 1, a1 = 20, a2 = 1, (left), and strong tangential advection, ν = 1, a1 = 1, a2 = 20 (right). The blue circle corresponds to f (p∗ ) from the new bounded analysis, and the red cross to p˜ from the unbounded analysis. The continuous black line is for the weighted norm  · w , and the triangle for the standard norm  · 2

928

M. J. Gander and T. Vanzan

55

450

500

15 200

31

32

300

350

34

35 36

34

3839

35 37 36 3839 0 4 41 42 3 4

37

250

31

32

33

33

38

400

20

31

29

30 35 36

4445 4647 484950

25

30

30

28

33

350

32

34

36

3738 3940 1 442 3 4

300

36

43 44

250

35

37 8 3 9 3 40 41 2 4 43

32 34 36

35

3940

36

35

33 35

34

34

32

31 30

4142

30 31

30

32

37 38

15 200

33

32

33

34

33

32 29

37

28

34

20

31

31

36 35

34

33

31

40

38 37

36 35

38

30

29

25

32

29

29

45

34

33

31

27

35

30

34

40 39

38 37

50

36 35

32 30

42 41

3940

5 363

31

33

35

55 38 37

39 40

33

45

43

4142 3 378

40 39

38 37 36 35

34

40

42 41

3940

3 356

50

60

43 44

4142

32

3940 3378

34

60

37

40 41 42 43

44 4546 47 48 49

400

450

500

Fig. 3 Iteration numbers for weak tangential advection, ν = 1, a1 = 20, a2 = 1 (left), and strong tangential advection, ν = 1, a1 = 1, a2 = 20 (right). The blue circle stands for the parameters from the new bounded analysis, and the red cross for the ones from the unbounded analysis

4 Conclusion The key step in the derivation of the convergence factor is that we could use separation of variables: this allowed us to expand the errors ejn in a series of orthogonal eigenfunctions along the interface, and to obtain recursive relations for each frequency, i.e. we could diagonalize the iteration operator, see also [7]. Fourier invented the Fourier transform in his famous treatise on the ‘Théorie analytique de la chaleur’ from 1822 precisely by using separation of variables, and the success of the Fourier transform is based on the fact that it permits such a diagonalization for many classical PDEs such as diffusion, reaction-diffusion, Helmholtz and, by the way, also an advection-diffusion equation with only normal advection, in the presence of Dirichlet boundary conditions. On unbounded domains, even more differential operators can be diagonalized, also the advection diffusion operator with general advection term. The presence of the Dirichlet boundary conditions and tangential advection however changes the eigenbasis, see (6), and also the inner product in which the eigenfunctions are orthogonal, it is not L2 (0, L) any more, but a weighted inner product. Since the sine functions are not eigenfunctions of the general advection diffusion operator on the bounded domain, if we started the 0 sin(πy), already after the first iteration, optimized Schwarz methods with ej0 = e¯j,1 1 sin(πy), the errors ej1 would not be proportional to the same sine function, ej1 = e¯j,1  ∞ 1 sin(lπy), but contain in general a combination of sine functions, ej1 = l=−∞ e¯j,l and therefore it would not be possible to obtain recursive relations like in (7). This insight sheds light on the emerging field of heterogeneous optimized Schwarz methods. Great care is needed in the procedure used to obtain the convergence factor. While no issues seem to be present for problems where a common eigenbasis is shared by the PDEs of interest, see [5], recent failures of the standard analysis for more complicated couplings [1] may be traced back to the lack of a shared eigenbasis which leads to an inappropriate derivation of ρ(l).

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References 1. M. Discacciati, L.G. Giorda, Optimized Schwarz methods for the Stokes-Darcy coupling. IMA J. Numer. Anal. 48, 2091–2116 (2017) 2. O. Dubois, Optimized Schwarz methods with robin conditions for the advection-diffusion equation, in Domain Decomposition Methods in Science and Engineering XVI. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2006) 3. M.J. Gander, Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006) 4. M.J. Gander, On the influence of geometry on optimized Schwarz methods. Bol. Soc. Esp. Mat. Apl. 52, 71–78 (2011) 5. M.J. Gander, T. Vanzan, Heterogeneous Optimized Schwarz methods for coupling Helmholtz and Laplace Equations, in Domain Decomposition Methods in Science and Engineering XXIV. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2018) 6. M.J. Gander, Y. Xu, Optimized Schwarz methods for circular domain decompositions with overlap. SIAM J. Numer. Anal. 52(4), 1981–2004 (2014) 7. M.J. Gander, Y. Xu, Optimized Schwarz methods for model problems with continuously variable coefficients. SIAM J. Sci. Comput. 38(5), A2964–A2986 (2016) 8. M.J. Gander, Y. Xu, Optimized Schwarz methods with nonoverlapping circular domain decomposition. Math. Comput. 86(304), 637–660 (2017) 9. S. Loisel, J. Cote, M.J. Gander, L. Laayouni, A. Qaddouri, Optimized domain decomposition methods for the spherical Laplacian. SIAM J. Numer. Anal. 48(5), 524–551 (2010)

Optimal Coarse Spaces for FETI and Their Approximation Faycal Chaouqui, Martin J. Gander, and Kévin Santugini-Repiquet

Abstract One-level iterative domain decomposition methods share only information between neighboring subdomains, and are thus not scalable in general. For scalability, a coarse space is thus needed. This coarse space can however do more than just make the method scalable: there exists an optimal coarse space in the sense that we have convergence after exactly one coarse correction, and thus the method becomes a direct solver. We introduce and analyze here a new such optimal coarse space for the FETI method for the positive definite Helmholtz equation in one and two space dimensions for strip domain decompositions. We then show how one can approximate the optimal coarse space using optimization techniques. Computational results illustrating the performance and effectiveness of this new coarse space and its approximations are also presented.

1 Introduction Domain decomposition techniques are widely used for solving algebraic systems resulting from the discretization of partial differential equations. The basic idea of these methods is to decompose the original domain Ω into subdomains Ω1 , . . . , ΩN which may or may not overlap, and then to solve local subproblems in each subdomain Ωi , which are coupled by an iteration through artificial boundary conditions at the interfaces between subdomains. Such one level methods are in general not scalable, since the communication between subdomains is local, and hence convergence deteriorates when the number of subdomains N grows. A natural remedy for this is to introduce an additional coarse correction to these methods

F. Chaouqui () · M. J. Gander Université de Genève, Section de mathématiques, Geneva, Switzerland e-mail: [email protected]; [email protected] K. Santugini-Repiquet Université de Bordeaux, IMB, Bordeaux, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_88

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which will allow global communication between subdomains, and hence improve their convergence behavior. In this paper, we study the one-level FETI (Finite Element Tearing and Interconnect) iterative method, also known as Dirichlet-Dirichlet method, see [1, 6] and references therein. Following ideas in [2–4], we show how an optimal coarse space for FETI can be developed and approximated in an optimized way. We give an error analysis for strip decompositions in 1D and 2D, and illustrate the performance of our new coarse spaces by numerical experiments. Our approximate coarse spaces are related to the ones developed in [5], which were obtained differently, namely by adding coarse space components to improve an estimate in the general convergence proof of FETI methods.

2 One- and Two-Level FETI Algorithms Let Ω be an open domain in Rd , d = 1, 2. We consider the model problem ηu − Δu = f in Ω,

u = 0 on ∂Ω,

(1)

where f ∈ L2 (Ω), and η > 0. We decompose the domain Ω into non-overlapping subdomains (Ωi )1≤i≤N with interfaces (Γi )1≤i≤N−1 as shown in Fig. 1, a so called strip decomposition. We denote by ni the outward normal corresponding to Γi as seen from subdomain Ωi , and introduce Algorithm 1 (One-level FETI algorithm) 1. Initialize λ0i,j = −λ0j,i = λ0i on each interface ∂Ωi ∩ ∂Ωj . 2. Until convergence (a) Solve the Neumann followed by the Dirichlet problems ηuni − Δuni = f in Ωi ,

ηψin − Δψin = 0 in Ωi ,

∂uni = λni,j on ∂Ωi ∩ ∂Ωj , ∂ni

ψin =

uni = 0 on ∂Ωi ∩ ∂Ω,

 1 n ui − unj on ∂Ωi ∩ ∂Ωj , 2

ψin = 0 on ∂Ωi ∩ ∂Ω.

(2)

(b) Update the Neumann traces by λn+1 i,j

=

λni,j

1 − 2



∂ψjn ∂ψin + ∂ni ∂ni

 on ∂Ωi ∩ ∂Ωj .

(3)

This algorithm has no mechanism for global communication and is therefore not scalable, as we will see in Sect. 3, so a coarse correction needs to be added. Note that when convergence is reached, the local corrections ψin are zero, meaning that

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Fig. 1 Strip decomposition of the domain Ω

Γi−1 Ωi−1

Γi Ωi

Ωi+1

∂Ω

the iterates uni have no Dirichlet jump. Therefore, a good coarse correction should be capable to reduce the jump between the iterates in order to speed-up convergence. As in [3] for optimized Schwarz methods, we define the quadratic functional q:

N .

H 1 (Ωi ) → R+ ,

(ui )1≤i≤N →

i=1



∂Ωi ∩∂Ωj =∅ ∂Ωi ∩∂Ωj

|ui − uj |2 ds,

where H 1 (Ωi ) denotes the Sobolev space that consists of square integrable functions on Ωi with square integrable gradient. Since the iterates uni of FETI have continuous normal derivatives at interfaces, we choose a coarse space with continuous normal derivatives at interfaces, namely, Xd =

⎧ ⎨ ⎩

v∈

N .

? H 1 (Ωi ) : ∀i, ηvi − Δvi = 0,

i=1

∂vi ∂ni

⎫ ⎬

@ Γi

= 0, vi |∂Ωi ∩∂Ω = 0 . ⎭ (4)

This leads to the optimal two level FETI algorithm given by

Algorithm 2 Two-level FETI algorithm with optimal coarse space 1. Initialize λ0i,j = −λ0j,i = λ0i on each interface ∂Ωi ∩ ∂Ωj . 2. Until convergence (a) Solve the Neumann followed by the Dirichlet problems ηuni − Δuni = f in Ωi ,

ηψin − Δψin = 0 in Ωi ,

∂uni = λni,j on ∂Ωi ∩ ∂Ωj , ∂ni

ψin =

uni = 0 on ∂Ωi ∩ ∂Ω,

 1 n u˜ − u˜ nj on ∂Ωi ∩ ∂Ωj , 2 i

ψin = 0 on ∂Ωi ∩ ∂Ω,

(5)



   where q (u˜ ni )1≤i≤N = min q (uni )1≤i≤N + v . v∈Xd

(b) Update the Neumann traces by λn+1 i,j

∂ u˜ ni 1 = − ∂ni 2



∂ψjn ∂ψin + ∂ni ∂ni

 on ∂Ωi ∩ ∂Ωj .

(6)

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3 Convergence Analysis Since problem (1) is linear, it suffices to set f = 0 and to analyze the convergence of (λni )1≤i≤N−1 to zero. We start in the 1D case, see Fig. 2. Using explicit solutions, we obtain 6T 5 Lemma 1 Let λn := λn1 , λn2 , . . . , λnN−1 ∈ RN−1 . Then for N ≥ 3, Algorithm 1 for f = 0 is equivalent to the iteration λn = T λn−1 , where the matrix T ∈ R(N−1)×(N−1) is given by ⎡ −1 ⎢ 4 ⎢ 1 ⎢ 4c ⎢ ⎢ 1 ⎢ 4 1 ⎢ T := 2 ⎢ 0 s ⎢ ⎢ . ⎢ . ⎢ . ⎢ . ⎢ . ⎣ .

0

1 4

0 ··· ··· 1 ... 4 1 ... 4 .. .. .. . . .

0 .. . .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ 1 0 −2 ⎥ ⎥ .. .. ⎥ . . 0 ⎥, ⎥ .. 1 ⎥ . 4 0 − 12 0 14 ⎥ ⎥ .. 1 1 ⎥ . 4 0 − 12 4c ⎦ 0 · · · · · · 0 14 0 − 14 − 12 0

√ √ with c := cosh ( ηH ) and s := sinh ( ηH ). Proof Solving for uni in (2) in one spatial dimension, we get √ √ η(x − xi−1 )) n cosh ( η(xi − x)) − λi−1 , i = 2, . . . , N − 1, = √ √ ηs ηs √ √ sinh ( η(xN − x)) n n sinh ( η(x − x0 )) n n u1 (x) = λ1 , uN (x) = −λN−1 . √ √ ηc ηc uni (x)

cosh ( λni

x0 = a

H

x1

Ω1 Fig. 2 One-dimensional geometry

xi−1

H Ωi

xi

xN −1

H ΩN

x

xN = b

Optimal Coarse Spaces for FETI and Their Approximation

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We then obtain for ψin in (2) √ √ sinh( η(x − xi−1 )) sinh( η(xi − x)) − A , i = 3, . . . , N − 2, √ √ i−1 2 ηs 2 ηs √ sinh( η(x − x0 )) ψ1n (x) = A˜ 1 , √ 2 ηs √ √ sinh( η(x − x1 )) sinh( η(x2 − x)) n ˜ ψ2 (x) = A2 − A1 , √ √ 2 ηs 2 ηs √ √ sinh( η(x − xN−2 )) sinh( η(xN−1 − x)) n ψN−1 (x) = A˜ N−1 − A , √ √ N−2 2 ηs 2 ηs √ sinh( η(xN − x)) ψNn (x) = −A˜ N−1 , √ 2 ηs ψin (x) = Ai

  1 n 1 n c s n n ˜ ˜ where Ai := − 1s λni+1 + 2c s λi − s λi−1 , A1 := − s λ2 + s + c λ1 , AN−1 :=   − 1s λnN−2 + cs + sc λnN−1 . Using then formula (3), we get the desired result. √

√ 2) , then the one-level Algorithm 1 is convergent in 1D, Theorem 2 If H > ln(1+ η and satisfies the convergence estimate

max

1≤i≤N−1

|λni | ≤

1   max |λ0i |. sinh ηH 1≤i≤N−1 2n

(7)

Proof Using Lemma 1, we have that : T ∞ = max

1 3 1 1 , + , 2s 2 4s 2 4cs 2 s 2

; =

1 , s2

since c ≥ 1, which completes the proof. Theorem 3 Algorithm 2 converges in 1D after one iteration. Proof Since it is possible with the chosen coarse space to cancel the jump, i.e. (q((u˜ ni )1≤i≤N ) = 0), the iterate u˜ n defined as u˜ n|Ωi = u˜ ni is continuous in both Dirichlet and Neumann traces, and hence it is precisely the monodomain solution, which concludes the proof. Theorem 3 shows that the coarse space Xd is optimal, i.e. it is the smallest coarse space leading to Theorem 3. In 1D, the coarse space Xd is of dimension precisely N − 1, which is needed to remove the N − 1 constraints at the interfaces.

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x0 = a

H

x1

xi−1

Ω1

L

H

xi

xN −1

Ωi

H

x

xN = b

ΩN

Fig. 3 Two-dimensional geometry

In 2D, we restrict the analysis to the case where the subdomains are rectangles as shown in Fig. 3, and expand unj and ψjn in a sine series, unj (x, y) =

∞

uˆ nj (x, m) sin(km y),

ψjn (x, y) =

m=1

where km := coefficients.

mπ L .

∞

ψˆ jn (x, m) sin(km y),

(8)

m=1

This allows us to study the convergence based on the Fourier

n Lemma 4 Let λˆ (m) :=

5 6T ∈ RN−1 . Then for λˆ n1 (m), λˆ n2 (m), . . . , λˆ nN−1 (m)

n−1 n (m), where the matrix N ≥ 3 Algorithm 1 is equivalent to λˆ (m) = Tm λˆ Tm ∈ R(N−1)×(N−1) is given by

⎡

⎤ 0 ··· ··· 0 .. ⎥ . ⎥ . ⎥ − 12 0 14 . . ⎥ . ⎥ 1 ... . ⎥ 1 . ⎥ 0 −2 4 ⎥ .. .. .. .. .. , . . . . . 0 ⎥ ⎥ ⎥ .. 1 . 4 0 − 12 0 14 ⎥ ⎥ ⎥ .. 1 1 1 . 4 0 − 2 4c ⎥ ⎦

− 14 0

1 4

⎢ ⎢ 1 ⎢ 4cm ⎢ ⎢ 1 ⎢ 4 1 ⎢ ⎢ Tm := 2 ⎢ 0 sm ⎢ ⎢ . ⎢ .. ⎢ ⎢ . ⎢ .. ⎣ 0 ··· ··· 0 with sm := sinh

1 4

(9)

m

0 − 14

  2 H and c := cosh 2H . η + km η + km m

Proof Since the Fourier coefficients uˆ nj (x, m) and ψˆ jn (x, m) satisfy a sequence of one-dimensional problems on each Ωi , 2 )u 2 )ψ ˆ n − ∂xx ψˆ n = 0, ˆ ni − ∂xx uˆ ni = 0, (η + km (η + km i i uˆ n (xi−1 ,m)−uˆ ni−1 (xi−1 ,m) ∂x uˆ ni (xi−1 , m) = λˆ ni−1 (m), ψˆ in (xi−1 , m) = i , 2 uˆ ni (xi ,m)−uˆ ni+1 (xi ,m) n n n ψˆ i (xi , m) = , ∂x uˆ i (xi , m) = λˆ i (m), 2

Optimal Coarse Spaces for FETI and Their Approximation

937

2, the iteration matrix Tm becomes the matrix T of Lemma 1 if we replace η by η +km which concludes the proof.

Theorem 5 If H >

√ ln(1+ 2) 2 , η+k12

then the one level Algorithm 1 is convergent in 2D,

and satisfies the convergence estimate max

1≤i≤N−1

λni 2 ≤

1  max λ0i 2 . 2 1≤i≤N−1 2n 2 sinh η + k1 H

  Proof We define the sequences Λni = λˆ ni (m)

m≥1

λˆ n+1 (m) = i



1 2 sm

(10)

. By Lemma 4, we have

1 n 1 1 λˆ (m) − λˆ ni (m) + λˆ ni+2 (m) 4 i−2 2 4

for each m ≥ 1. Using then Parseval’s identity λni 22 =

L 2

∞  m=1



λˆ ni (m)2 =

L n 2 2 Λi 2 ,

we have for i = 3, . . . , N − 3 √   L 1 n 1 n 1 n λn+1 Λ Λ Λ  ≤  +  +  √ 2 2 2 2 i−2 2 i 2 i 4 i+2 2s1 4 ≤

1 s12

max

1≤i≤N−1

= sinh

2

2

λni 2

1 η

 + k12 H

max

1≤i≤N−1

λni 2 ,

2 in m to bound it where we used the triangle inequality, and the monotonicity of 1/sm 2 with 1/s1 . Similarly one can show that the same bound also holds for the remaining subdomains i = 1, 2, N − 2, N − 1, and hence we get the stated result.

Algorithm 2 still converges after one iteration in two spatial dimension provided that the optimal coarse space Xd is used. However, this coarse space is now infinite dimensional, and even though it becomes finite dimensional after discretization, it is in practice too big to be used. To obtain an effective approximation, note that according to Theorem 5, low frequencies are responsible for the deterioration of the algorithm. Thus, the choice of the approximate coarse space can be optimized choosing to include the most slowly converging low frequency modes. We thus propose as approximation of the optimal coarse space Xd the optimized coarse space :  ; 8d := v ∈ Xd : ∀i, ∂x vi (xi , y) ∈ span sin (km y), m = 1, . . . , J , X (11) where J ≥ 1 can be suitably chosen to enrich the coarse space.

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8d satisfies the Theorem 6 Algorithm 2 with the approximate coarse space X convergence estimate max

1≤i≤N−1

λni 2 ≤ sinh

2n

1

2

η



+ kJ2 +1 H

Proof Since at each iteration n, we have uni (x, y) = 8d Parseval’s identity, we have for v ∈ X

max

1≤i≤N−1

∞  m=1

λ0i 2 .

(12)

uˆ ni (x, m) sin(km y), using

 LB B2 B n B B(ui+1+v)(xi+,y)−(uni+v)(xi−,y)B dy= 0

J ∞ B B2 L B2 L BB n B B B n + − n +ˆ v )(x ,m)−( u ˆ +ˆ v )(x ,m) + ( u ˆ B B i+1 i+1 i Buˆ i+1(xi+,m)−ˆuni(xi−,m)B , i i i 2 2 m=J +1

m=1

we obtain U = argmin n

8d v∈X

= argmin 8d v∈X

N−1

 LB i=1

0

N−1 J

B2 B B n B(ui+1 + vi+1 )(xi+ , y) − (uni + vi )(xi− , y)B dy B B2 B n B B(uˆ i+1 + vˆi )(xi+ , m) − (uˆ ni + vˆi )(xi− , m)B ,

i=1 m=1

and since we are now minimizing a finite dimensional quadratic, the quantity can be (m) = 0 for m ≤ J , and following the proof of Theorem 5, canceled. Hence, λˆ n+1 i we get the stated bound.

4 Numerical Experiments We start with the one dimensional problem on Ω = (−1, 1) with η = 2 and f (x) = 1. We discretize Algorithms 1 and 2 using centered finite differences with mesh size Δx = 10−4 , and run the one-level algorithm with 30 equally sized subdomains. As expected, Fig. 4 (left) shows that without coarse correction the algorithm fails to converge, and with the optimal coarse correction we get convergence after one iteration. In 2d on Ω = (−1, 1)2 divided into 10 equally sized subdomains, we run Algorithm 2 on the error equation ηe − Δe = 0 with η = 2, discretized by centered finite differences using the mesh size Δx = Δy = 5 · 10−3 . We can see in Fig. 4 (right) that approximations of the optimal coarse space are enough to

Optimal Coarse Spaces for FETI and Their Approximation 10 5

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10 5 Algorithm Algorithm Algorithm Algorithm

Algorithm 1 Algorithm 2 10 0 error

error

10 0

1 2 ( = 1) 2 ( = 2) 2 ( = 4)

10

-5

10 -10

10

5

10 iteration

15

20

-5

10 -10

5

10 iteration

15

20

Fig. 4 Convergence of Algorithms 1 and 2 in 1D (left) and 2D (right)

obtain a convergent iterative FETI method, and the more low frequency components we include in the optimized coarse space, the better the convergence becomes.

References 1. V. Dolean, P. Jolivet, P. Nataf, An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation (SIAM, Philadelphia, 2015) 2. M.J. Gander, A. Loneland, SHEM: an optimal coarse space for RAS and its multiscale approximation, in Domain Decomposition Methods in Science and Engineering XXIII (Springer, Cham, 2017), pp. 313–321 3. M.J. Gander, L. Halpern, K. Santugini, Discontinuous coarse spaces for DD-methods with discontinuous iterates, in Domain Decomposition Methods in Science and Engineering XXI (Springer, Cham, 2014), pp. 607–615 4. M.J. Gander, L. Halpern, K. Santugini, A new coarse grid correction for RAS/AS, in Domain Decomposition Methods in Science and Engineering XXI (Springer, Cham, 2014), pp. 275–283 5. A. Klawonn, M. Kühn, O. Rheinbach, Adaptive coarse spaces for FETI-DP in three dimensions. SIAM J. Sci. Comput. 38(5), A2880–A2911 (2016) 6. A. Quarteroni, A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (Clarendon Press, Gloucestershire, 1999)

Coupling MPC and HJB for the Computation of POD-Based Feedback Laws Giulia Fabrini, Maurizio Falcone, and Stefan Volkwein

Abstract In this paper we use a reference trajectory computed by a model predictive method to shrink the computational domain where we set the HamiltonJacobi Bellman (HJB) equation. Via a reduced-order approach based on proper orthogonal decomposition(POD), this procedure allows for an efficient computation of feedback laws for systems driven by parabolic equations. Some numerical examples illustrate the successful realization of the proposed strategy.

1 Introduction The numerical solution of nonlinear optimal control problems for system driven by partial differential equations is a challenging topic that has a great impact in many areas. By means of the Dynamic Programming Principle (DPP) one can obtain a Hamilton-Jacobi-Bellman equation which gives a characterization of the value function of a fully-nonlinear control problem. The value function is in fact the unique viscosity solution of a nonlinear Hamilton–Jacobi equation and solving this equation one can derive the approximation of an optimal feedback control. It is well known that the DP approach suffers from the so called curse of dimensionality and one of the main tasks is the choice of the domain where we want compute the value function. To solve this problem we apply the algorithm presented in [1] to a parabolic partial differential equation. We follow a reduced order modeling approach based on POD and we derive an a-posteriori error estimator for the optimal trajectory. The paper is organized as follows. Section 2 is devoted to the presentation of the optimal control problem. In Sect. 3 we introduce the reduced order modeling whereas in Sect. 4 we briefly explain the main features of HJB and G. Fabrini () · S. Volkwein University of Konstanz, Department of Mathematics and Statistics, Konstanz, Germany e-mail: [email protected] M. Falcone La Sapienza Università di Roma, Dipartimento di Matematica, Roma, Italy © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_89

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MPC method, how we couple them and we provide an a posteriori error estimator which give us a criterium to choose the reduced domain. Finally, Sect. 5 presents a couple of numerical tests.

2 The Optimal Control Problem Assume that Ω = (0, 1)2 ⊂ R2 with boundary Γ = ∂Ω. We set Q = (0, ∞) × Ω and Σ = (0, ∞) × Γ . Moreover, let H = L2 (Ω) and V = H 1 (Ω) with dual V  . Recall that the space B > = Y = W (0, ∞) = ϕ ∈ L2 (0, T ; V ) B ϕt ∈ L2 (0, ∞; V  ) is a Hilbert space endowed with the common inner product; cf. [3]. Let B = > Uad = u ∈ R B ua ≤ u ≤ ub ⊂ R, where ua , ub ∈ Rm are lower and upper bounds, respectively. For U = L2 (0, ∞; R) the set of admissible control is given by B = > Uad = u = u ∈ U B u(t) ∈ Uad a.e. in [0, ∞) . where ‘a.e’ stands for ‘almost everywhere’. For a given y0 ∈ H , b ∈ L2 (Γ ), u ∈ Uad and ν > 0 the state equation is yt = νΔy in Q a.e.,

∂y ν ∂n = ub on Σ a.e.,

y(0) = y0 in Ω a.e.

(1)

Introducing the continuous bilinear form  ∇ϕ · ∇ψ dx,

a(ϕ, ψ) = ν

for ϕ, ψ ∈ V

Ω

we can express (1) in the form d dt y(t), ϕH

+ a(y(t), ϕ) = u(t) b, ϕL2 (Γ ) y(0), ϕH = y0 , ϕH

∀ϕ ∈ V , f.a.a. t > 0, ∀ϕ ∈ H,

(2)

where ‘f.a.a.’ stands for ‘for almost all’. It is well-known that (2) admits a unique solution y ∈ Y; (see, e.g., [3]). We introduce the Hilbert space X = Y × U endowed with common product topology. The set of admissible points is given as B Xad = {x = (y, u) ∈ X B y solves (2) for u ∈ Uad }.

Coupling MPC and HJB for the Computation of Feedback Laws

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For a given discount factor λ > 0, positive weight σ and desired state yd ∈ H the quadratic cost is defined as J (y, u) =

1 2



∞

0

 B 2 y(t) − yd 2H + σ Bu(t)| e−λt dt,

for (y, u) ∈ X.

Now, we consider the infinite horizon, linear-quadratic optimal control problem min J (x) subject to

x ∈ Xad .

(P)

3 Reduced-Order Modeling (ROM) We discretize (P) by using proper orthogonal decomposition (POD) (see [6, 7] for 2 details). For large T > 0 and K ∈ N let {y k (t)}K k=1 ⊂ L (0, T ; V ) be some given trajectories which will be specified later. Then, we consider the linear space of snapshots B V = span {y k (t) B t ∈ [0, T ] and 1 ≤ k ≤ K} ⊂ V

with d = dim V ≤ ∞.

For any finite  ≤ d we are interested in determining a POD basis of rank  which minimizes the mean square error between the trajectories {y k (t)}K k=1 and their corresponding -th partial Fourier sums on average in [0, T ]: ⎧ K  ⎪

⎪ ⎪ ⎨ min

T 0

 7 72

7 k 7 y k (t), ψi V ψi 7 dt 7y (t) − V

k=1 i=1 ⎪ ⎪ ⎪ ⎩ s.t. {ψ } ⊂ V and ψ , ψ  = δ for 1 ≤ i, j ≤ . i i=1 i j V ij

(3)

Assuming that we have computed a POD basis {ψi }i=1 , we define V  =   span {ψ1 , . . . , ψ } ⊂ V . Then, the POD state y  (t) = i=1 yi (t)ψi ∈ V is determined by the POD Galerkin scheme for (2): d dt

y  (t), ψj H + a(y (t), ψ) = u(t) b, ψL2 (Γ ) y  (0), ψV = y0 , ψH

∀ψ ∈ V  , f.a.a. t > 0, ∀ψ ∈ V  . (4)

We define the vectors   y(t) = yi (t) ∈ R ,

  y0 = y0 , ψi H ∈ R ,

  b = b, ψi L2 (Γ ) ∈ R

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and the matrices M, A ∈ R× by Mij = ψj , ψi ,

Aij = −a(ψj , ψi ).

  Setting F(y, u) = M−1 Ay + ubu ∈ R , (y, u) ∈ R × R, we can write (4) as   y˙ (t) = F y(t), u(t) f.a.a. t > 0,

y(0) = y0

(5)

which is a -dimensional system of differential equations. Let us also define yd = yd , ψi H ∈ R ,

L(y, u) = (y? My + y? yd + σ u2 )/2

for (y, u) ∈ R × R. Then, the POD Galerkin approximation for (P) reads 

∞

min J  (y, u) =

  L y(t), u(t) e−λt dt

0

s.t.

(y, u) ∈ Xad

(P )

where the admissible set is B Xad = {(y, u) ∈ X B y solves (5) for u ∈ Uad } and X = H 1 (0, ∞; R ) × U. Remark 1 In our numerical realization of the POD Galerkin scheme we first have to introduce a high-fidelity discretization for (2). We use a discretization based on piecewise finite elements (FE) with n 3  degrees of freedom. ♦

4 Coupling MPC and HJB For the solution of (P ) we use the algorithm proposed in [1] and we provide a new a posteriori error-estimate. HJB Equation Let us briefly recall the HJB equation (see e.g. [2, 4]). For y◦ ∈ R let us introduce the value function   B   v  (y◦ ) = inf min J  (y, u) B y˙ (t) = F y(t), u(t) f.a.a. t > 0, y(0) = y◦ . u∈Uad

The value function can be characterized in terms of the Bellman equation λv  (y◦ ) + max

u∈Uad

=

 ?  − F y◦ , u ∇v  (y◦ ) − L(y◦ , u) = 0,

for all y◦ ∈ R .

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In the numerical realization we have to replace R by a sufficiently large, but bounded subset D ⊂ R . The main advantage of this approach is that once the value function v  has been computed, the optimal feedback control is  =  ? u(y ¯ ◦ ) = arg minu∈Uad − F y◦ , u ∇v  (y◦ ) − L(y◦ , u)

for all y◦ ∈ D

which can be used as a closed-loop control. The optimal state solves (5) for the feedback control u = u(y(·)). ¯ For the numerical solution of the HJB equation we consider a fully-discrete semi-Lagrangian scheme which is based on the discretization of the system dynamics with time step h and a mesh parameter k,  leading to a fully discrete approximation vhk (y◦ ):   vhk (y◦i ) = min {(1 − λh)I [vhk ](y◦i + hF(y◦i , u)) + L(y◦i , u)} u ∈ Uad

for every node y◦i ∈ D, i = 1, . . . , Np , of the discretized state space. Note that in general, the arrival point y◦i + hF(y◦i , u) is not a node of the space grid. Therefore, this value is computed by means of a linear interpolation operator, denoted by  ]. The bottleneck of this approach is related to the so-called curse of the I [vhk dimensionality, namely, the computational cost increases dramatically as soon as the dimension  does. Model Predictive Control (MPC) MPC is an optimization based method for the computation of closed-loop controls for (non-)linear dynamical systems (see [5, 8] for details). It consists in solving iteratively finite time horizon open-loop problems. Let N ∈ N, t◦ ≥ 0 and t◦N = t◦ + NΔt for a chosen time step Δt > 0. We introduce the finite time horizon cost as  J ,N (y[t◦ ,u] , u) =

t◦N t◦

L(y[t◦ ,u] (t), u(t))e−λt dt,

where the state y = y[t◦ ,u] solves y˙ (t) = F(y(t), u(t)) f.a.a. t ∈ (t◦ , t◦N ] and y(t◦ ) = y◦ . The method works as follows: For t◦ = 0 and y◦ = y0 we minimize J ,N over [t◦ , t◦N ) and store the optimal control u¯ 1 on the subinterval (t◦ , t◦ +Δt] together with the associated optimal state y[t0 ,u¯ 1 ] . Then, we initialize the next finite time horizon control problem by setting y◦ = y[t◦ ,u¯ 1 ] (t◦ + Δt), t◦ = t◦ + Δt and t◦N = t◦ + NΔt. This process is iterated. Coupling MPC and HJB The idea is to combine the advantages from both methods: HJB is global and gives the feedback law for every initial condition once the value function has been computed. On the other hand, MPC is faster, but gives an approximate feedback control just for a single initial condition. Let us assume that we are interested in the approximation of feedback controls for an optimal control problem given the initial condition y◦ . We compute via MPC a reasonable

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suboptimal trajectory y¯ N that we can use as reference trajectory for building a small domain, where we are going to compute the approximate value function. In our approach, we choose a rather short prediction horizon NΔt to obtain y¯ N quickly. Then, the trajectory y¯ N is used to build the smaller domain Dρ ⊂ D, in which we solve the HJB equation (instead of solving the HJB on the whole domain D). We construct Dρ as a tube around y¯ N defining Dρ = {y◦ ∈ R | dist(y◦ , y¯ N ) ≤ ρ}

(6)

where dist(· , ·) will be specified later. A larger ρ will allow for a better approximation of the value function, but at the same time enlarging ρ we will loose the localization around our trajectory, increase the number of nodes and the CPU time. Error Estimator Let x¯ = (y, ¯ u) ¯ be the solution to (P) and T > 0 sufficiently large. Due to the Bellman principle, u¯ is also optimal on [0, T ]. Suppose that we have computed an approximate MPC control u¯ N ∈ L2 (0, T ) satisfying u¯ N (t) ∈ Uad in [0, T ] a.e. Then, we usually have u¯ = u¯ N . Let y¯ N be the associated approximate MPC state solving (5) on [0, T ] for u = u¯ N . To evaluate the a-posteriori error estimator we have to compute the Lagrange multiplier pN solving the dual equation   −M˙pN (t) = A? pN (t) − e−λt M¯yN (t) + yd for t ∈ [0, T ),

pN (T ) = 0.

Then, it follows from [9] that u¯ − u¯ N L2 (0,T ) ≤

eλT σ

ζL2 (0,T ) ,

where the perturbation ζ ∈ U(T ) is defined as: ⎧ B N  = > −λt N ? N ⎪ ⎨ (σ e u¯ (t) − b p (t))− in Aa = =s ∈ [0, T ] BB u¯ (s) = ua >, ζ(t) = (σ e−λt u¯ N (t) − b? pN (t)) + in Ab = s ∈ [0, T ] B u¯ N (s) = ub , ⎪ ⎩ −σ e−λt u¯ N (t) − b? pN (t) in I = [0, T ] \ (A ∪ A ). a b Here, [w]− and [w]+ denote respectively the negative and positive part of w. Since y¯ solves (5) on [0, T ] for u = u, ¯ we have 2 1 d y − y¯ N )(t)M 2 dt (¯

+ (¯y − y¯ N )(t)A ≤ b2 (¯y − y¯ N )(t)2 |(u¯ − u¯ N )(t)|

f.a.a. t ∈ [0, T ], where, e.g.,  · M is the weighted norm in R induced respectively by the positive definite matrix M and  · 2 stands for the Euclidean norm. By applying Young’s inequality we derive the existence of a constant c > 0, which depends on b, M and A, so that 1 d y − y¯ N )(t)2M 2 dt (¯

≤ c |(u¯ − u¯ N )(t)|2

f.a.a. t ∈ [0, T ]

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which implies max (¯y − y¯ N )(t)M ≤

t ∈[0,T ]

√ eλT c σ

ζL2 (0,T ) .

Now, to define Dρ we set dist(y◦ , y¯ N ) = y◦ −¯yN M and ρ = eλT

(7) √ c ζL2 (0,T ) /σ .

5 Numerical Test We consider Eq. (2) with ν = 1, b = χΓ and y0 = 0.5χ[0.5,1]. For the cost functional we take σ = 0.01, λ = 1 and yd = 1. To realize the reduced-approach numerically, we have to choose a high-fidelity spatial approximation. We apply a piecewise linear finite element (FE) model with n = 2673 degrees of freedom. The snapshots for the POD method are chosen as the solution to the state and dual equation for the reference control u = 1 and for T = 1. The POD basis rank chosen for the simulations is  = 2. Test 1 The chosen discretization parameters are Δt = h = 0.02, Δτ = 0.01 (the time step to integrate the trajectories). We discretize the set Uad = [−4, 4] in 21 equidistant discrete values for the approximation of the value function and 81 for the computation of the state trajectories. In Fig. 1 we can see on the left the first approximation of the MPC solver (dotted line) and the optimal solution corresponding to the feedback control (dashed line) in Dρ , while on the right we show the same trajectories in a smaller portion of Dρ , centered around them. In Table 1, we present the CPU times and distance of the controlled trajectory from the reference trajectory in L2 -norm at the final time. We can observe that the trajectory obtained via MPC is a rough approximation; we improve the result with a localized version of the DP and the trajectory obtained is closer to the target. Moreover if we compare the trajectories computed using the information given by the value function

1

0.5

0

-0.5

-1 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 1 Test 1: domain Dρ with optimal trajectories (left panel). The latter, optimal trajectory via MPC (dotted line) and via HJB (dashed line), are detailed in the right panel

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Table 1 Test 1: CPU times and distance of the controlled trajectory from the reference trajectory at the final time T = 1 CPU time in [s] ¯y(T ) − yd M

MPC (N = 3) 11 0.07

HJB in Dρ 25 0.03

HJB in D 50 0.03

Table 2 Test 2: CPU times and distance of the controlled trajectory from the reference trajectory at the final time T = 1 CPU time in [s] ¯y(T ) − yd L2 (D)

MPC (N = 3) 10 0.04

HJB in Dρ 18 0.01

HJB in D 58 0.01

Fig. 2 Test 1: optimal trajectory via MPC (dotted line) and via HJB (dashed line) in Dρ , optimal trajectories for different initial points (right)

in Dρ and D we have the same values for ¯y(T ) − yd M . Concerning the CPU time, in the fourth column we show the global time needed to get the approximation of the value function in the whole domain and the time to obtain the optimal trajectory, whereas the third column shows the global time needed to compute all the steps of our algorithm. We can observe a speed up of factor 2. Test 2 In this test we consider the smoother initial state y◦ (x) = sin(πx), the parameters chosen are the same of Test 1. Now we have a speed up of factor 3, as shown in Table 2. Compared to Test 1 we are able to steer the solution closer to the target yd in Dρ and D. One of the advantages of computing the value function in Dρ is that we are able to reconstruct the feedback in all the points of the domain with a small computational effort. So we can change the initial state (with the constraint that the state projected in the reduced space has to be a point in Dρ ) and compute the optimal trajectory in a fast way (( 4s). To illustrate this property we consider some perturbations of the initial state y◦ , we project the system in the reduced space and we compute the optimal trajectories, in Fig. 2 (right) we show the results for four different initial conditions. Conversely, if we want to apply MPC we have to

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start the whole procedure again, and the CPU time required is two times bigger. In the left plot of Fig. 2 we show the trajectories in the reduced domain. Acknowledgements G. Fabrini gratefully acknowledges support by the German Science Fund DFG grant Reduced-Order Methods for Nonlinear Model Predictive Control.

References 1. A. Alla, G. Fabrini, M. Falcone, Coupling MPC and DP methods for an efficient solution of optimal control problems, in Conference Proceedings of IFIP (2015) 2. M. Bardi, I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-JacobiBellman Equations (Birkhauser, Basel, 1997) 3. R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I (Springer, Berlin, 2000) 4. M. Falcone, R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and HamiltonJacobi Equations (SIAM, Philadelphia, 2013) 5. L. Grüne, J. Pannek, Nonlinear Model Predictive Control (Springer, London, 2011) 6. M. Gubisch, S. Volkwein, Proper orthogonal decomposition for linear-quadratic optimal control, in Model Reduction and Approximation: Theory and Algorithms, ed. by P. Benner, A. Cohen, M. Ohlberger, K. Willcox (SIAM, Philadelphia, 2017), pp. 5–66 7. P. Holmes, J.L. Lumley, G. Berkooz, C.W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, 2nd edn. (Cambridge University Press, Cambridge, 2012) 8. J.B. Rawlings, D.Q. Mayne, Model Predictive Control: Theory and Design (Nob Hill Publishing, Madison, 2009) 9. F. Tröltzsch, S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44, 83–115 (2009)

Adaptive Multiple Shooting for Nonlinear Boundary Value Problems Thomas Carraro and Michael Ernst Geiger

Abstract Multiple shooting methods are time domain decomposition methods suitable for solving boundary value problems (BVP). They are based on a subdivision of the time interval and the integration of appropriate initial value problems on this subdivision. In certain critical cases, systematic adaptive techniques to design a proper time domain decomposition are essential. We extend an adaptive shooting points distribution developed in the 1980s for linear boundary value problems based on ordinary differential equations (ODE) to the nonlinear case.

1 Preliminaries Shooting methods are used to solve boundary value problems (BVP) by reducing them to initial value problems (IVP). Even for well-conditioned BVP this induces a strong sensitivity on the (parameterized) initial data and thus ill-conditioning of the corresponding IVP, which leads to stability problems with simple shooting. This drawback can be overcome by splitting the solution interval, i.e. by using multiple shooting. Mattheij and Staarink proposed a way to adapt the number and location of shooting intervals [5] for linear BVP. Since their method cannot be directly applied to nonlinear BVP, we present a suitable modification for the extension to the nonlinear case.

T. Carraro () Heidelberg University, Heidelberg, Germany e-mail: [email protected] M. E. Geiger Heidelberg University, Heidelberg, Germany d-fine GmbH, Frankfurt/Main, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_90

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The general BVP reads as follows: u(t) ˙ = f (t, u(t)), t ∈ I := [a, b], 0 = r(u(a), u(b)).

(1)

The first equation comprises a general ODE system, and the second equation constitutes the (nonlinear) boundary conditions. A solution to problem (1) is a function u ∈ C 1 [a, b]d , d ≥ 1, which fulfils both the differential equation and the boundary conditions. We assume familiarity with the basic simple shooting formulation for the BVP (1), where the boundary condition at the upper boundary b is replaced by a parameterized initial condition s at the lower boundary a. This marks the transition from BVP to IVP. By means of Newton’s method, the parameter is fitted such that after convergence the original BVP (1) including the boundary condition imposed at b is solved. Stability estimates with regard to the parameter s such as u(t; s1 ) − u(t; s2 )2 ≤ eL(t −a)s1 − s2 2 ,

(2)

show that the error in the initial value s affects the solution exponentially with increasing time t (L is the Lipschitz constant of the right-hand side function f (t, ·)). The decomposition I = {τ0 } ∪

M−1 3

(τj , τj +1 ],

a =: τ0 < τ1 < · · · < τM := b

(3)

j =0

of the time domain I into smaller subintervals Ij := (τj , τj +1 ] leads from simple to multiple shooting. At the time-points τj , parameters s j are imposed as initial values. This results in the following set of IVP (for j = 0, . . . , M − 1): u˙ j (t) = f (t, uj (t)), t ∈ Ij , uj (τj ) = s j

(4)

To obtain a continuous solution, a system of matching conditions is introduced: uj (τj +1 ; s j ) − s j +1 = 0, j = 0, . . . , M − 2, r(s 0 , uM−1 (τM ; s M−1 )) = 0.

(5)

By defining a vector s¯ := (s 0 , s 1 , . . . , s M−1 )? , we can abbreviate Eq. (5) and simply write F (¯s ) = 0. To apply Newton’s method, the derivatives Gj (t; s j ) := duj (t; s j ) of the interval-wise solution functions w. r. t. their respective initial ds j

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values s j are required. They are determined by solving interval-wise variational equations ˙ j (t; s j ) = fx (t, uj (t; s j )) Gj (t; s j ), t ∈ Ij , G Gj (τj ; s j ) = Ed .

(6)

Using the abbreviation Gj (τj ; s j )(τj +1 ) =: Gj (τj +1 ), the Jacobian of the system of matching conditions is given as ⎛ G0 (τ1 ) −I ⎜ ⎜ 0 G1 (τ2 ) ⎜ . Fs (¯s ) = ⎜ ⎜ .. ⎜ ⎝ 0 ··· A 0

0 −I .. .

··· ···

0 0 .. .

⎞

⎟ ⎟ ⎟ ⎟, ⎟ ⎟ 0 GM−2 (τM−1 ) −I ⎠ ··· 0 B

(7)

with the matrices A = rx (s 0 , uM−1 (τM ; s M−1 )), B = ry (s 0 , uM−1 (τM ; s M−1 ))GM−1 (τM ) in the last row. The decomposition of the time domain and the multiple shooting reformulation of the BVP (1) allow to control the exponential stability factor from (2). However, the choice of this decomposition is a difficult task, as neither the necessary number of subintervals nor the position of optimal intermediate points is known. The construction of an adaptive process is desirable, but is complicated by lacking knowledge of the problem’s stability behavior within the solution interval.

2 Adaptive Shooting Intervals for Linear Problems As the ideas of Mattheij and Staarink refer to linear BVP, we examine the general linear problem: u(t) ˙ = A(t)u(t) + b(t), t ∈ I := [a, b] Ba u(a) + Bb u(b) = g.

(8)

Here, A(·) : I → Rd×d and b(·) : I → Rd are continuous real-valued matrix and vector functions, Ba , Bb ∈ Rd×d are given constant matrices, and g ∈ Rd is a given vector. The first objective is a replacement of fixed equidistant shooting decompositions by problem-oriented decompositions of I . In equidistant shooting grids, the number

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of subintervals influences both the subinterval sensitivities Gi (t) and the shooting matrix Fs (s). Since high sensitivities play a major role in BVP bad-conditioning (see, e. g., Mattheij [4]), it is suggestive to base the shooting point distribution in such a way to bound the sensitivities Gi (t). This is the basic idea of the approach of Mattheij and Staarink [5]. It is crucial for the following that the sensitivities Gi (t) are independent of the shooting variables s i , as is the case for linear BVP. The basic proposition of Mattheij and Staarink (see [5]) is to bound the growth of the sensitivities in some matrix norm · by a constant Csens . The choice of the constant is explained in their paper by an analysis of the rounding error propagation based on the previous work of Mattheij [3]. We call this scheme the bounding approach, which can be summarized as follows. Setting τ0 := a, the simultaneous solution of the parameterized IVP (4) and the corresponding sensitivity equation (6) are performed to detect the first subinterval I1 = [τ0 , τ1 ). The time point tj on which G(tj ) exceeds the prescribed constant Csens is chosen as a new shooting point τ1 . Then the solution process is restarted with a parameter s 1 as the new initial value. This is repeated until the fine grid {tj | j = 0, · · · , N} of the numerical ODE integrator reaches the upper endpoint tN = b of the solution interval I . Then, b = τfinal . After the shooting intervals have been fixed, the shooting system is solved by Newton’s method. In this section we illustrate the optimization of the shooting point distribution in case of linear BVP. In Example 1 the system matrix entries are piecewise constant on the solution interval and thus uniformly bounded. However, their size varies strongly over time and they exhibit discontinuities. Besides testing different strategies to measuring the size of G(t) and choosing the shooting points, we observe a nonequidistant shooting point distribution. Example 1 Consider the linear BVP 

    u˙ 1 (t) 0 1 u1 (t) = , c(t) 1 u˙ 2 (t) u2 (t)

        10 u1 (0) 00 u1 (10) 1 + = 2 2 00 10 1 u (0) u (10)

on the time interval I = [0, 10]. The matrix entry c is chosen as the piecewise constant function: ⎧ ⎨110 (t ≤ 2 ∨ t ≥ 8), c(t) = ⎩1 (2 < t < 8). Table 1 presents the results obtained by testing different quantities measuring the size of the sensitivities G(t). Different upper bounds for the sensitivities are tested (here, Csens ∈ {103 , 104 , 105 , 106 }). The growth of G(t) is bounded in the matrix norms ·1 , ·2 and ·∞ , as well as in the spectral radius ρ(·), which is computed from the eigenvalue with largest modulus and is, in general, not a norm. The table illustrates that all quantities work almost equally well. Figure 1 shows the adapted

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Table 1 Example 1: Number of shooting intervals and shooting residual for different sensitivity bounds and different strategies for measuring the sensitivity size (stopping criterion for the shooting residual: F (s)2 < 10−8 ) 103 #SI 11 11 11 9

Csens Strategy G1 G2 G∞ ρ(G)

104 #SI 7 7 7 7

F 2 5.8 · 10−13 2.0 · 10−13 6.0 · 10−13 2.2 · 10−12

105 #SI 6 6 6 5

F 2 1.3 · 10−12 5.5 · 10−12 2.2 · 10−12 3.8 · 10−11

Shooting solution before convergence first component second component

1500

500 0

0

2

4

first component second component

10

1000

6

8

10

F 2 8.8 · 10−11 6.1 · 10−10 6.4 · 10−10 1.7 · 10−09

Shooting solution after convergence

Shooting solution u

Shooting solution u

2000

F 2 1.8 · 10−11 5.1 · 10−11 1.6 · 10−11 3.1 · 10−10

106 #SI 5 5 5 5

5 0 -5 -10

0

2

Time t

4

6

8

10

Time t

Fig. 1 Example 1: The shooting solution found by means of the bounding approach (left: before convergence, right: after convergence); shooting points are mainly located in regions where c is large (criterion: ρ(G(t)) ≤ 104 )

shooting grid in the case ρ(G(t)) ≤ 104 as well as the continuous solution obtained after one shooting iteration.

3 Adaptive Shooting Intervals for Nonlinear Problems The approach for the automatic shooting point distribution suggested by Mattheij cannot be expected to work for nonlinear BVP. In fact, in the linear case the sensitivity equation does not depend on the shooting parameter s and only one shooting iteration is needed, resulting in a fixed shooting grid. In contrast, the general nonlinear parameterized IVP, u(t; ˙ s) = f (t, u(t; s)),

u(a; s) = s,

leads to the sensitivity equation ˙ s) = fx (t, u(t; s)) G(t; s), G(t;

G(a; s) = Id .

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upper bound

new sensitivity

shooting variable

sensitivity

new shooting variable

Fig. 2 The adaptive multiple shooting process; upper part: shooting variables s (denoted by ‘cross symbol’) and corresponding sensitivity growth (curved dashed lines); lower part: new shooting variables (denoted by ‘red circle’) and new sensitivity norms (curved dashed lines). Problem: In the lower part, shooting variables and intervals do not fit

The Jacobian fx (t, x) depends on s, which transfers to the sensitivity solution G(t) ≡ G(t; s). Thus, the sensitivity bounding approach leads to shooting grids depending on s. Since in each shooting cycle the parameter vector s = (s 0 , s 1 , . . . , s M−1 ) is updated, we expect a different shooting grid in each iteration. Therefore, the approach from Sect. 2 must be modified in order to allow for changing grids. Figure 2 illustrates the effect of having an adapted grid changing at every shooting iteration. Assume that in the i-th shooting iteration, we have shooting variables s i = (si0 , si1 , . . . , siMi −1 ) and a corresponding shooting grid Ti (the upper part of the figure). The next shooting iteration results in an update s i+1 = 0 , s 1 , . . . , s Mi −1 ) of the shooting variables. Based on s (si+1 i+1 , a new shooting i+1 i+1 grid Ti+1 is determined, on which the subsequent multiple shooting iteration will depend. Here the following problem arises: The solution of the original IVP as well as the sensitivity equations must be computed on the new grid Ti+1 . However, the initial values s i+1 are only available on the old grid Ti (see the lower part of Fig. 2). In addition, the shooting points τ (i+1) and the shooting variables s i+1 do usually not correspond to each other, because in the new shooting grid Ti+1 , either shooting points from Ti have been removed or additional ones have been inserted. Therefore, a method is needed to transfer the shooting variables s i+1 to the new grid Ti+1 . The obvious way to achieve this matching is to interpolate the shooting values s i+1 and evaluate the interpolant at the gridpoints of Ti+1 , yielding suitable initial values s int i+1 for the solution of original IVP. Figure 3 illustrates this idea with piecewise linear interpolation. We have employed this idea in the examples below because it has turned out to work well in practice.

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interpolated shooting variable

Fig. 3 Interpolation of the shooting variables s i+1 (here: piecewise linear interpolation) and evaluation of the interpolant in the shooting points of the new grid Ti+1 provides appropriate initial values for the multiple shooting iteration (denoted by ‘red filled circle’) Table 2 Example 2: Comparison of equidistant and adaptive shooting for different initial shooting variables √ s = (0.1, −0.1)? s = (−9, 23)? Strategy #SI #Newton Time(s) #SI #Newton Time(s) Equidistant 20 –(100) (128) 20 –(4) – 50 –(100) (129) 50 –(6) – 100 –(100) (130) 100 –(6) – 200 –(100) (130) 200 –(7) – 500 –(100) (134) 500 –(8) – Adaptive 2–9 24 62 2–58 38 102 Adaptive shooting is faster and has a larger convergence domain (criteria: ρ(G(t)) ≤ 102 , F 2 < 10−8 ). The notation ‘–(x)’ means that the inner Newton method for the implicit ODE solver failed in shooting step x, needing more than 500 Newton iterates

Example 2 Consider the nonlinear BVP u(t) ¨ =

3 u(t)2 , 2

u(0) = 4, u(10) =

4 121

with linear boundary conditions on the interval [0, 10]. This problem is a modification of a BVP taken from Bulirsch and Stoer [1]. Its exact solution is given by 4 . We rewrite it as a two-component first order BVP in the form u(t) = (1+t )2      3 1 u˙ 1 (t) u (t)2 01 2 = , u˙ 2 (t) 10 u2 (t)

 10 00

       u1 (0) 00 u1 (10) 4 + = 4 . 2 2 u (0) u (10) 10 121

Table 2 presents a comparison of the equidistant multiple shooting method and the adaptive procedure for two different pairs of initial values. The criterion for adapting the shooting grid is to bound the spectral radius of the sensitivity matrix by Csens = 102 . It is generally observed that Csens has to be chosen smaller in the nonlinear framework than for linear problems. This is probably related to the convergence region of the Newton method. However, this is currently only a heuristic proposition lacking a theoretical explanation. Summarizing the results of Table 2, the adaptive process is not only faster than equidistant multiple shooting,

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Shooting solution u

2

Shooting iteration 5

first component second component

1

0

-1

0

2

4

6

8

10

Time t

Shooting solution u

Shooting iteration 38 first component second component

5 0 -5 -10

0

2

4

6

8

10

Time t

Fig. 4 Example 2: The shooting solution found by the modified adaptive procedure (upper left: 5th shooting cycle, upper right: 15th shooting cycle, lower left: 25th shooting cycle, lower right: 38th shooting cycle (convergence)); the adaptive √ process successively reduces the number of shooting intervals (initial shooting variable s = (−9, 23)? , criteria: ρ(G(t)) ≤ 102 , F (s)2 < 10−8 )

it also takes less shooting intervals, and it converges for initial√parameters s for which equidistant shooting fails completely. For s = (−9, 23)? , the inner Newton method for solving the implicit time steps needs more than maxit = 500 iterations; the inner Newton method is then stopped, and the subinterval solutions cannot be computed. Figure 4 depicts the adaptive behavior of the multiple shooting solver. Shooting intervals of different length as well as changing shooting grids are clearly recognizable. Note y axes labeling indicating the improvement of the shooting solutions. A more detailed description of the adaptive shooting point distribution, including additional examples, algorithmic procedures and an extension to optimization problems, can be found in Geiger [2].

4 Conclusions We have presented the first successful step towards adaptive multiple shooting for nonlinear BVP. We have shown that in certain cases the choice of the proper position and the adequate number of shooting points is crucial to enhance the convergence

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region of the shooting system. The considered case with ODE based models does not pose a strict limitation on the number of shooting points. In case of partial differential equations (PDE), it is desirable to keep the number of shooting points as small as possible to reduce the dimension of the shooting system. However, the presented method can hardly be extended to the PDE case, because a measure of the sensitivities would be highly computationally expensive. In the next work we will consider a promising extension to the PDE case based on the norm of the residual rather than on sensitivities. Acknowledgements T.C. was supported by the Deutsche Forschungsgemeinschaft (DFG) through the project CA 633/2-1.

References 1. R. Bulirsch, J. Stoer, Introduction to Numerical Analysis. Texts in Applied Mathematics, vol. 12, 3rd edn. (Springer, Berlin, 2002) 2. M.E. Geiger, Adaptive multiple shooting for boundary value problems and constrained parabolic optimization problems, Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg, Fakultät für Mathematik und Informatik, 2015 3. R.M.M. Mattheij, Estimates for the errors in the solutions of linear boundary value problems, due to perturbations. Computing 27(4), 299–318 (1981) 4. R.M.M. Mattheij, The conditioning of linear boundary value problems. SIAM J. Numer. Anal. 19(5), 963–978 (1982) 5. R.M.M. Mattheij, G.W.M. Staarink, On optimal shooting intervals. Math. Comput. 42(165), 25–40 (1984)

Part XXIV

Error Analysis for Finite Element Methods for PDEs

Exponential Scaling and the Time Growth of the Error of DG for Advection-Reaction Problems Václav Kuˇcera and Chi-Wang Shu

Abstract We present an overview of the results of the authors’ paper (Kuˇcera and Shu, IMA J Numer Anal, to appear) on the time growth of the error of the discontinuous Galerkin (DG) method and set them in appropriate context. The application of Gronwall’s lemma gives estimates which grow exponentially in time even for problems where such behavior does not occur. In the case of a nonstationary advection-diffusion equation we can circumvent this problem by considering a general space-time exponential scaling argument. Thus we obtain error estimates for DG which grow exponentially not in time, but in the time particles carried by the flow field spend in the spatial domain. If this is uniformly bounded, one obtains an error estimate of the form C(hp+1/2 ), where C is independent of time. We discuss the time growth of the exact solution and the exponential scaling argument and give an overview of results from Kuˇcera and Shu (IMA J Numer Anal, to appear) and the tools necessary for the analysis.

1 Continuous Problem Let Ω ⊂ Rd , d ∈ N be a bounded polygonal (polyhedral) domain with Lipschitz boundary ∂Ω. Let 0 < T ≤ +∞ and let QT = Ω × (0, T ) be the space-time domain. We consider the following nonstationary advection-reaction equation: We seek u : QT → R such that ∂u + a· ∇ u + cu = 0 ∂t

in QT ,

(1)

V. Kuˇcera () Charles University, Faculty of Mathematics and Physics, Praha, Czech Republic e-mail: [email protected] C.-W. Shu Division of Applied Mathematics, Brown University, Providence, RI, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_91

963

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along with the initial condition u(x, 0) = u0 (x) and boundary condition u = uD on ∂Ω − × (0, T ). Here a : QT → Rd and c : QT → R are the given advective field and reaction coefficient, respectively. By ∂Ω − we denote the inflow boundary {x ∈ ∂Ω; a(x, t)·n(x) < 0, ∀t ∈ (0, T )}, where n(x) is the unit outer normal to ∂Ω at x. We assume that c ∈ C([0, T ); L∞ (Ω)) ∩ L∞ (QT ) and a ∈ C([0, T ); W 1,∞ (Ω)) with a, ∇a uniformly bounded a.e. in QT .

1.1 Time Growth of the Exact Solution In order to put the results on the time growth of error of the DG method into perspective, it is useful to first gain insight into the time growth of the exact solution itself. For this purpose, we define pathlines of the flow, i.e. the family of curves S(t; x0 , t0 ), each originating at (x0 , t0 ), by S(t0 ; x0, t0 ) = x0 ∈ Ω,

dS(t; x0 , t0 ) = a(S(t; x0 , t0 ), t). dt

This means that S(·; t0 , x0 ) is the trajectory of a massless particle in the nonstationary flow field a passing through point x0 at time t0 . Equation (1) can then be rewritten along each pathline:   ∂u du(S(t; x0 , t0 ), t) + (cu)(S(t; x0, t0 ), t) = + a · ∇u + cu (S(t; x0 , t0 ), t) = 0, dt ∂t which is an ordinary differential equation with the solution 

u(S(t; x0 , t0 ), t) = u(x0 , t0 ) exp −



t

 c(S(s; x0, t0 ), s) ds .

(2)

t0

For simplicity, if we take c(x, t) ≡ c0 ∈ R, i.e. a constant, this reduces to u(S(t; x0 , t0 ), t) = u(x0 , t0 )e−c0 (t −t0) .

(3)

From this we can see that along pathlines, the exact solution of (1) exponentially grows (c0 < 0) or decays (c0 > 0) with the rate −c. In the special case of pure advection (c0 = 0) the function u is constant along each pathline. In this short note and in the paper [6] we are concerned with the case of uniformly bounded solutions and errors. One case when this can occur is when the maximal particle ‘life-time’ T is finite. By this we mean that the maximal time any massless particle carried by the flow field a spends in Ω, before exiting through the outflow boundary, is bounded by T < +∞. As such particles follow pathlines of the flow, this is equivalent to assuming that each pathline is defined only for a finite time bounded by T before exiting Ω. Therefore in (2) and (3) we have |t − t0 | < T ,

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specifically   |u(S(t; x0 , t0 ), t)| ≤ |u(x0, t0 )| exp T cL∞ (QT ) , where u(x0 , t0 ) is either the initial condition (x0 ∈ Ω, t0 = 0) or boundary condition (x0 ∈ ∂Ω − , t0 ≥ 0). Therefore the exact solution u remains uniformly bounded for all t, although it may exponentially grow along each pathline (c < 0), which exist only for a bounded time T . We therefore have uniform boundedness of u even on a potentially infinite time interval (0, T ). It is reasonable to assume that a ‘good’ numerical method will mimic the described behavior of the exact solution. Namely that whenever the exact solution remains uniformly bounded, so will the approximate solution and error of the method. As we will see, this is the case of the DG method.

2 Exponential Scaling The main tool in the analysis of [6] is a general form of the exponential scaling. Usually when dealing with problem (1) one assumes ellipticity of the resulting advection and reaction weak forms, which leads to the requirement c − 12 diva ≥ γ0 > 0

on QT

(4)

for some constant γ0 > 0. As we have seen in Sect. 1.1 this assumption is by no means necessary in the analysis of the continuous problem. One possibility how to avoid assumption (4) in the analysis of (1) is the exponential scaling ‘trick’. We write u(x, t) = eαt w(x, t) for some α ∈ R. Substituting into (1) gives eαt

∂w + αeαt w + eαt a· ∇ w + eαt cw = 0. ∂t

Since eαt > 0 we can divide the equation by this common factor, obtaining the new problem for the unknown function w: ∂w + a· ∇ w + (c + α)w = 0. ∂t

(5)

In this equation we have the new reaction term c + α. The new ellipticity condition for (5) now reads c + α − 12 diva ≥ γ0 > 0 and it can be satisfied by choosing α sufficiently large. One can then proceed to use the ellipticity to obtain estimates for w. The drawback of this approach is that in order to obtain estimates for u, one must multiply by the exponential factor eαt , the result being an estimate that depends exponentially on T .

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For the stationary version of problem (1) the authors of [1] consider general exponential scaling with respect to space and set u(x) = eμ(x) u(x). ˜ In our spacetime setting this would correspond to taking u(x, t) = eμ(x) u(x, ˜ t).

(6)

This is a generalized version of the simpler choice μ(x) = μ0 · x for some constant vector μ0 ∈ Rd considered e.g. in [4] and [7]. Substituting (6) into (1) and dividing by eμ we get a new problem for u: ˜ ∂ u˜ + a · ∇ u˜ + (a· ∇μ + c)u˜ = 0. ∂t The condition corresponding to (4) is now: There exists μ : Ω → R such that a· ∇μ + c − 12 diva ≥ γ0 > 0

on QT .

(7)

The question then is when can such a function μ be found satisfying a· ∇μ ≥ a0 > 0 so that this term can be used to dominate the other possibly negative terms of (7). The answer is in the following theorem, cf. [2] and [1]. Theorem 1 Let a : Ω → Rd be Lipschitz continuous. Then there exists a function μ ∈ W 1,∞ (Ω) such that a· ∇μ ≥ a0 > 0 if and only if the flow field a possesses neither closed curves nor stationary points. Proof The proof itself is technical, here we only sketch the main arguments. First let a· ∇μ ≥ a0 > 0. Then clearly there cannot exist a point x where a(x) = 0. Assume that a possesses a closed curve β : [0, 1] → Ω parametrized by s ∈ d [0, 1], i.e. β(0) = β(1) and ds β(s) = a(β(s)) for all s. Then we have  0 = μ(β(1)) − μ(β(0)) = 0

1

dμ(β(s)) ds = ds



1

∇μ(β(s)) · a(β(s)) ds ≥ a0 > 0,

0

which is a contradiction. Therefore a cannot contain closed curves. The opposite implication is more technical, hence we only indicate the main ideas. A function μS is constructed in a neighborhood of each streamline S which satisfies a· ∇μS > 0 on this neighborhood. This is done via the implicit function theorem on the neighborhood of each point of S separately and connecting the functions together on the neighborhood of the whole S. The set of all streamlines along with their considered neighborhoods form a covering from which a finite subcovering can be chosen and the local functions μS can then be ‘glued’ together using the related partition of unity. & %

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2.1 General Space-Time Exponential Scaling If we were to use the nonstationary version of the spatial exponential scaling (6), we would need to assume the statement of Theorem 1 holds for all t, i.e. that a(·, t) possesses neither closed curves nor stationary points for each t ∈ (0, T ). We view this as restrictive. Therefore we consider the general form u(x, t) = eμ(x,t )u(x, ˜ t),

(8)

where μ : QT → R is an appropriate function. Substituting (8) into (1) gives  ∂μ  ∂ u˜ + a · ∇ u˜ + + a· ∇μ + c u˜ = 0 ∂t ∂t

(9)

after dividing by the common positive factor eμ . The condition corresponding to (4) and (7) now reads: There exists μ : QT → R such that ∂μ + a· ∇μ + c − 12 diva ≥ γ0 > 0 ∂t

a.e. in QT .

(10)

We will construct such a function μ in the following paragraph. Up to now we worked only with the strong form of (1). The key step was to divide the whole equation by eμ . The question is how to perform this operation in a weak formulation, where all terms are under integral signs. The solution is to take test functions of the form v(x, t) = e−μ(x,t ) v(x, ˆ t) when using (8). Then the factors eμ and e−μ cancel each other and one thus obtains the weak form of (9). This is a key step in the analysis of [6].

2.2 Construction of the Scaling Function μ If c − 12 diva is negative or changes sign frequently, we can use the expression μt + a · ∇μ to dominate this term everywhere. If we choose μ1 such that ∂μ1 + a · ∇μ1 = 1 ∂t

on QT ,

(11)

then by multiplying μ1 by a sufficiently large constant, we can satisfy condition (10) for a chosen γ0 > 0. Along pathlines Eq. (11) reads  d μ1 (S(t; x0 , t0 ), t)  ∂μ1 = + a · ∇μ1 (S(t; x0 , t0 ), t) = 1, dt ∂t

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therefore μ1 (S(t; x0 , t0 ), t) = t − t0 .

(12)

At the origin of the pathline, we have μ1 (S(t0 ; x0 , t0 ), t0 ) = 0 and the value of μ1 along this pathline is simply the time elapsed since t0 . As we assumed in Sect. 1.1, the life-time of particles is bounded by T , hence 0 ≤ μ1 (x, t) ≤ T for all (x, t) ∈ QT . In the analysis we need Lipschitz continuity of μ. This can be obtained under the assumption that there are no characteristic boundary points on the inlet boundary. We note that due to its Lipschitz continuity, μ1 is differentiable with respect to x and t a.e. in QT , which justifies the previous considerations. The proof of the following theorem is rather technical, cf. [6]. Since μ1 is defined very simply along pathlines, which are solutions of ordinary differential equations, the proof follows similar ideas as in the proof of dependence of a solution of an ODE on the initial condition. Theorem 2 Let a ∈ L∞ (QT ) be continuous with respect to time and Lipschitz continuous with respect to space. Let there exist a constant amin > 0 such that − a(x, t) · n ≥ amin for all x ∈ ∂Ω − , t ∈ [0, T ). Let the time any particle carried by the flow field a(·, ·) spends in Ω be uniformly bounded by T . Then μ1 defined by (12) on Ω × [0, T ) is uniformly Lipschitz continuous with respect to x and t and satisfies 0 ≤ μ1 ≤ T .

3 Discontinuous Galerkin Method Now we introduce the DG discretization of (1). Let Th be a triangulation (partition into mutually disjoint simplices) with hanging nodes allowed. For K ∈ Th let hK = diam(K), h = maxK∈Th hK . For K ∈ Th we set ∂K − (t) = {x ∈ ∂K; a(x, t)· n(x) < 0} where n(x) is the unit outer normal to ∂K. We seek the discrete solution in the space Sh = {vh ; vh |K ∈ P p (K), ∀K ∈ Th }, where P p (K) is the set of polynomials on K of degree at most p. For K ∈ Th and vh ∈ Sh let vh− be the trace of vh on ∂K from the side of the element adjacent to K, or vh− = 0 if the face lies on ∂Ω. Finally on ∂K we define the jump of vh as [vh ] = vh − vh− , where vh is the trace from K. We seek uh ∈ C 1 ([0, T ); Sh ) such that uh (0) = u0h ≈ u0 and  ∂u

h

∂t

 , vh + bh (uh , vh ) + ch (uh , vh ) = lh (vh ),

∀vh ∈ Sh .

(13)

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Here bh , ch and lh are the advection, reaction and right-hand side forms, respectively, defined for u, v piecewise continuous on Th as follows, [3, 8]: bh (u, v) =



(a· ∇u)v dx −

K∈Th K

 ch (u, v) =

 − K∈Th ∂K

(a· n)[u]v dS,

cuv dx, Ω

lh (v) = −



− K∈Th ∂K ∩∂Ω

(a· n)uD v dx.

3.1 Error Estimates We estimate the DG error eh (t) := u(t) − uh (t) = η(tn ) + ξ(t), where η(t) = u(t) − Πh u(t) and ξ(t) = Πh u(t) − uh (t) ∈ Sh . Here Πh is the L2 (Ω)–projection onto Sh . As in Sect. 2.1, we wish to write ξ = eμ ξ˜ and test the error equation / Sh with φ = e−μ ξ˜ = e−2μ ξ to obtain estimates for ξ˜ . However, since φ(t) ∈ this is not possible. The solution is to test by Πh φ(t) ∈ Sh and estimate the difference Πh φ(t) − φ(t). A similar result is proved in the stationary case in [1] if μ ∈ W p+1,∞ (Ω). In [6] only the Lipschitz continuity of μ and standard approximation results are used in the proof of the following. Lemma 3 Let μ be globally bounded and Lipschitz continuous as in Theorem 2. Then there exists C independent of h, t, ξ, ξ˜ such that Πh φ(t) − φ(t)L2 (K) ≤ ChK max e−μ(x,t )ξ˜ (t)L2 (K), x∈K

Πh φ(t) − φ(t)L2 (∂K) ≤

1/2 ChK max e−μ(x,t ) ξ˜ (t)L2 (K) . x∈K

Now we come to the error analysis. We subtract the equations for u and uh , set vh = Πh φ(t) and rearrange the terms to get the error equation  ∂ξ ∂t

 , Πh φ + bh (ξ, φ) + bh (ξ, Πh φ − φ) + bh (η, Πh φ) + ch (ξ, φ) + ch (ξ, Πh φ − φ) + ch (η, Πh φ) +

 ∂η ∂t

 , Πh φ = 0. (14)

The terms with φ are those where the factors eμ and e−μ cancel out as in Sect. 2.1 leading to the new reaction terms as in (9). Terms containing Πh φ − φ are estimated using Lemma 3 and η is estimated by standard approximation results. Altogether we have the following, cf. [6].

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Lemma 4 Let ξ = eμ ξ˜ , φ = e−μ ξ˜ and let μ be as in Theorem 2. Then  7 1 d 1 7 7[ξ˜ ]72 − , Πh φ + bh (ξ, φ) + ch (ξ, φ) ≥ ξ˜ 2 + γ0 ξ˜ 2 + a,∂K ∂t 2 dt 2

 ∂ξ

K∈Th

√ where f a,∂K − =  |a· n|f L2 (∂K − ) . Lemma 5 Let ξ, φ and μ be as above. Then B  ∂η B B B , Πh φ B Bbh (ξ, Πh φ − φ) + bh (η, Πh φ) + ch (ξ, Πh φ − φ) + ch (η, Πh φ) + ∂t   1 7 72 7[ξ˜ ]7 ≤ Chξ˜ 2 + Ch2p+1 |u(t)|2H p+1 + |ut (t)|2H p+1 + . a,∂K − 4 K∈Th

Now we come to the main theorem of [6] on the error of the DG scheme (13). Theorem 6 Let the assumptions of Theorem 2 hold. Let the initial condition u0h satisfy u0 − u0h  ≤ Chp+1/2 |u0 |H p+1 . Then there exists a constant C depending on T but independent of h and T such that for h sufficiently small 1 √ max eh (t) + γ0 eh L2 (QT ) + t ∈[0,T ] 2



1/2

7 7 7[eh (ϑ)]72 − dϑ a,∂K

T 0

K∈Th

  ≤ Chp+1/2 |u0 |H p+1 + |u|L2 (H p+1 ) + |ut |L2 (H p+1 ) .

(15)

Proof If we apply Lemmas 4 and 5 to (14), we get 7 d 1 7 7[ξ˜ (t)]72 − ξ˜ (t)2 + 2γ0 ξ˜ (t)2 + a,∂K dt 2 K∈Th

  ≤ Chξ˜ (t) + Ch2p+1 |u(t)|2H p+1 + |ut (t)|2H p+1 . 2

Now we choose h small enough so that Ch ≤ γ0 and the first right-hand side term can then be absorbed. We integrate over (0, t), take the maximum over t ∈ [0, T ] and apply the estimate of the initial condition. Thus we get max ξ˜ (t)2 + γ0

t ∈[0,T ]

 0

T

ξ˜ (ϑ)2 dϑ +

1 2



T 0

7 7 7[ξ˜ (ϑ)]72 K∈Th

a,∂K −

dϑ

  ≤ Ch2p+1 |u0 |2H p+1 + |u|2L2 (H p+1 ) + |ut |2L2 (H p+1 ) .

(16)

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Now we reformulate (16) as an estimate of ξ instead of ξ˜ . Because ξ˜ = e−μ ξ , ξ˜ (t)2 ≥ min e−2μ(x,t )ξ(t)2 = e−2 maxQT QT

μ(x,t )

ξ(t)2 ≥ e−2T ξ(t)2

and similarly for the other norms in (16). If we multiply the resulting estimate by e2T and take the square root, we get inequality (15) for the discrete part ξ˜ of the error eh . Finally, a similar estimate for η follows from standard approximation results which gives the estimate for eh = ξ + η. & % The interpretation of Theorem 6 is this: If one proceeds in a standard way, the need to use Gronwall’s lemma arises. This leads to exponential growth in T . By using exponential scaling we effectively apply Gronwall’s lemma along pathlines, which exist only for a finite time T , resulting in bounds uniform in T . This can be interpreted as application of Gronwall in the Lagrangian framework, not in the Eulerian. We note that the obtained results would hold if Eq. (1) were in divergence form with a nonzero divergence of a. This follows from the relation div(au) = a· ∇u + u diva, which recasts the divergence form into that of (1) with the new reaction coefficient c˜ = c + diva. For future work, we plan to extend the analysis the nonlinear convective problems, by combining the presented ideas with the technique of [9] and [5]. Acknowledgements The work of V. Kuˇcera was supported by the J. William Fulbright Commission in the Czech Republic and research project No. 17-01747S of the Czech Science Foundation. The work of C.-W. Shu was supported by DOE grant DE-FG02-08ER25863 and NSF grants DMS1418750 and DMS-1719410.

References 1. B. Ayuso, L.D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47(2), 1391–1420 (2009) 2. A. Devinatz, R. Ellis, A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives, II. Indiana Univ. Math. J. 23, 991–1011 (1974) 3. M. Feistauer, K. Švadlenka, Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems. J. Numer. Math. 12(2), 97–117 (2004) 4. C. Johnson, J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46(173), 1–26 (1986) 5. V. Kuˇcera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems. IMA J. Numer. Anal. 34(2), 820–861 (2014) 6. V. Kuˇcera, C.-W. Shu, On the time growth of the error of the DG method for advective problems. IMA J. Numer. Anal. https://doi.org/10.1093/imanum/dry013 7. U. Nävert, A finite element method for convection-diffusion problems, Ph.D. Thesis, Chalmers University of Technology, 1982 8. W.H. Reed, T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73–479, 1973 9. Q. Zhang, C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42(2), 641–666 (2004)

Maximum Norm Estimates for Energy-Corrected Finite Element Method Piotr Swierczynski and Barbara Wohlmuth

Abstract Nonsmoothness of the boundary of polygonal domains limits the regularity of the solutions of elliptic problems. This leads to the presence of the so-called pollution effect in the finite element approximation, which results in a reduced convergence order of the scheme measured in the L2 and L∞ -norms, compared to the best-approximation order. We show that the energy-correction method, which is known to eliminate the pollution effect in the L2 -norm, yields the same convergence order of the finite element error as the best approximation also in the L∞ -norm. We confirm the theoretical results with numerical experiments.

1 Introduction Let Ω ⊂ R2 be an open and simply connected polygonal domain. Consider a model Poisson problem − Δu = f

in Ω,

and u = 0

on ∂Ω.

(P)

The regularity of the solution of such a problem, when considered on a smooth domain, depends only on the regularity of the given data f . This, however, is not true anymore, when polygonal domains are considered, since then singular functions of the form si (r, φ) = η(r)r λi sin(λi φ),

λi = iπ/Θ,

(1)

enter the solution [7, Section 2.1.1]. Here, Θ is the angle of the considered corner and (r, φ) are polar coordinates defined around it. Also, η(r) is a smooth cut-off function equal to 1 for r < r and equal to 0, when r > r, for some positive

P. Swierczynski () · B. Wohlmuth Technical University of Munich, Institute for Numerical Mathematics, München, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_92

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constants r, r. Note that r can be chosen so that the cut-off functions around different corners have disjoint supports. For the sake of notational simplicity, we focus our attention on one of the corners only. This is justified since all the results presented here have a local nature and naturally extend to the case of multiple corners. Maximum norm error estimates for the standard finite element approximation of elliptic problems are well-studied, e.g., in [8, 10, 12] in multiple contexts. It was shown in [11] that the presence of the non-smooth corners in the computational domain reduces the convergence order of the finite element method in the maximum norm. A similar phenomenon, known as a pollution effect, see, e.g., [2, 3], is also known when the error is measured in the L2 -norm. The standard approach for mitigating the pollution effect in the finite element solution includes mesh grading, for which maximum error estimates exist, see, e.g., [1]. Recently an alternative approach was developed—the energy-correction based on a local modification of the bilinear form governing the problem [4]. This work was inspired by similar considerations in the context of finite difference method [13] and has been later extended to piecewise polynomial approximation spaces in [5]. We build on the analysis presented there and show that the energy-correction method converges optimally in the sense of the best approximation property when measured in the weighted L∞ -norm. The forthcoming analysis of the energy correction scheme is carried out in weighted Lebesgue and weighted Sobolev spaces, which constitute a very convenient framework for describing the regularity of elliptic problems on polygonal domains [7]. We define for β ∈ R, l ∈ Z+ ∪ {0} and 1 ≤ p ≤ ∞ the weighted Sobolev spaces l,p Wβ (Ω), as spaces of measurable functions v : Ω → R, for which the following norms are finite

r β−l+|m| D m vL∞ (Ω) , and vW l,∞ (Ω) := β

p

v

l,p Wβ (Ω)

|m|≤l

:=

p

r β−l+|m| D m vLp (Ω) ,

for 1 ≤ p < ∞,

|m|≤l

2 m represents the m-th where m is a multi-index with |m| = i=1 mi , and D 0,p generalized derivative, see, e.g., [6]. We also use the standard notation Wβ (Ω) = p

Lβ (Ω) and Wβl,2 (Ω) = Hβl (Ω). Furthermore, we define Nβl,σ (Ω) spaces of weighted Hölder continuous functions, where σ ∈ (0, 1), consisting of all l-times continuously differentiable

Maximum Norm Estimates for EC FEM

975

functions, for which the following norm is finite vN l,σ (Ω) = β

r β−σ −l+|m| D m vC 0 (Ω)

|m|≤l

+

sup

|m|=l x1 ,x2 ∈Ω

|r(x1 )β D m v(x1 ) − r(x2 )β D m v(x2 )| . |x1 − x2 |σ

Note that for l ∈ Z+ such an embedding holds Nβl,σ (Ω) 3→ Wβl,∞ (Ω). The following regularity result is a consequence of [7, Theorem 2.6.5]. Theorem 1 Let us for any  > 0 take β ≥ max(0, k + 1 − λ1 + ), k − λ1 < α < k and α˜ = α − k + 1. Assume that f ∈ Nβk−1,σ (Ω) ∩ H−k−1 α˜ (Ω) for some σ > 0. Then

u ∈ Wβk+1,∞ (Ω)∩Hαk+1 (Ω)∩H01 (Ω) is continuous. Moreover, in the neighborhood of the corner, the solution admits the following expansion

u=U+

ki si ,

i: λi 0

uW k+1,∞ (Ω) ≤ cf N k−1,σ (Ω) , β

uHαk+1 (Ω) ≤ cf H k−1 (Ω) .

and

−α˜

β

2 Energy Corrected Finite Element In this section, we give a brief overview of the energy-correction techniques used for improving the convergence order in the finite element approximations of elliptic problems on non-convex domains. Let T be a given uniform triangulation of the computational domain Ω. We define Vhk , to be a space of globally continuous, piecewise polynomial functions of order k ∈ Z+ , which have values 0 on the boundary ∂Ω. In order to remove the pollution effect from the standard finite element approximation, we introduce a modification of the bilinear form a(·, ·), which mitigates the stiffness of the problem in the vicinity of the singularity. The modified finite element k approximation of (P) reads then: find um h ∈ Vh such that ah (um h , vh ) = f, vh Ω

for all vh ∈ Vhk ,

(2)

where the bilinear form is defined as ah (u, v) := a(u, v) − ch (u, v) and a(u, v) =  ∇u · ∇v. We assume that ah (·, ·) is bilinear, continuous and elliptic, and that Ω ch (·, ·) is symmetric. In [4], the modification was defined on a one-element patch

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around the re-entrant corner. We use an extension of this idea, so that the correction of the bilinear form is supported in K element layers from the corner. Let Sh1 = {T ∈ Th : 0 ∈ ∂T },

Shi = {T ∈ T : ∂T ∩ ∂Shi−1 = ∅}, i = 1, . . . , K.

Moreover, we assume that the one element patch around the corner consists of identical isosceles triangles. We can now define the modification as ch (u, v) :=

K

 γi

i=1

Shi

∇u · ∇v dx.

(3)

Note that with each layer of elements there is an associated correction parameter γi , which still needs to be determined. Asymptotically a unique optimal sequence of parameters γ = {γ }K i=1 on a given correction patch exist. Moreover, the sequence of correction parameters depends on the number and shape of the elements T of the correction patch and on the angle Θ of the re-entrant corner only. Several effective procedures for finding it, based on nested Newton strategies, were proposed in [9] for piecewise linear finite element. This modification, together with an alternative version in case of higher order spaces, is discussed in [5]. Due to the choice of the ch (·, ·), we preserve the sparsity structure of the stiffness matrix, as only a fixed number of its entries needs to be suitably scaled. The following theorem, providing sufficient conditions for the optimal convergence of the energy-corrected method in the weighted L2 -norms, was proven in [5, Theorem 2]. Theorem 2 Let k denote the polynomial degree of the finite element space, k−λ1 < α < k, α˜ = α − k + 1 and f ∈ H−k−1 α˜ (Ω). Let the modification ch (·, ·) be defined as above and satisfy m m m m a(si − si,h , si − si,h ) − ch (si,h , si,h ) = O(hk+1 ),

for all i ≤ K.

Then for the energy-corrected finite element solution, we obtain the following optimal error estimates k ∇(u − um h )L2α (Ω) ≤ ch f H k−1 (Ω) , −α˜

k+1 u − um f H k−1 (Ω) . h L2α (Ω) ≤ ch −α˜

The following result describing the convergence of the energy-corrected finite element in negatively weighted norms can be found in [5, Lemma 4] k Remark 3 Let um h ∈ Vh be the energy-corrected approximation (2). Suppose that k+1 (Ω) ∩ H01 (Ω) the modification ch (·, ·) is chosen as in Theorem 2 and let u ∈ H−α for some 0 < α < 1. Then for some c > 0

u − um h L2

−α (Ω)

≤ chk+1 uH k+1 (Ω) . −α

Maximum Norm Estimates for EC FEM

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3 Maximum Norm Error Estimates In this section, we investigate the maximum norm error estimates of the energycorrected finite element scheme introduced above. We begin by stating some auxiliary results. Lemma 4 (Inverse Inequality) For all vh ∈ Vhk and all α ≥ −1 the following estimate holds. Moreover, when the triangulation is uniform, the constant is independent of the choice of the element. vh L∞ ≤ ch−1 vh L2α (T ) α (T )

for all T ∈ T .

Proof The proof follows from the standard scaling argument and the equivalence of finite dimensional norms. Lemma 5 (Interpolation Error) Let Ih : C(Ω) → Vhk denote the standard nodal ¯ and β > 0 interpolation operator. Then for any function u ∈ Wβk+1,∞ (Ω) ∩ C(Ω) the following estimate holds for some constant c > 0 independent of h ≤ chk+1 uW k+1,∞ (Ω) . u − Ih uL∞ β (Ω) β

Lemma 6 Let α > −1 and T ∗ ∈ T be a single element of the triangulation that lies close to the corner located at the origin, namely maxx∈T ∗ |x| < ch ˜ for constant c˜ > 0. Then the following estimate holds for all vh ∈ Vhk hα vh L2 (T ∗ ) ≤ cvh L2α (T ∗ ) . Proof Let first α ≥ 0. If the element T ∗ is not attached to the corner, then it lies in the distance of at least h from it and hence the estimate is obvious. Suppose then that the triangle T ∗ shares a vertex with the corner. Note that since vh (0) = 0, then the norm vh L∞ (T ∗ ) is attained at some point Pmax , which is one of the remaining nodal points of the finite element basis. Notice also that for some constant ρk depending only on the order of the finite element space, we have ρk h ≤ rmax , where rmax is the distance of the point Pmax from the origin. Thus, we can write α |vh (Pmax )| hα vh L2 (T ∗ ) ≤ hα |T ∗ |1/2vh L∞ (T ∗ ) ≤ ρk−α |T ∗ |1/2rmax ∗ ≤ cvh  2 ≤ ρk−α |T ∗ |1/2vh L∞ Lα (T ∗ ) . α (T )

In the last step we used the inverse inequality from Lemma 4. Suppose now that α ∈ (−1, 0). Since r ≤ ch, ˜ we have hα ≤ hα vh L2 (T ∗ ) ≤ cvh L2α (T ∗ ) Finally, we are in a position to state main result.

1 α c˜α r

and

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Theorem 7 Let β ≥ max(0, k + 1 − λ1 + ), with  > 0 and also k − λ1 < α < k, α˜ = α − k + 1. Assume that f ∈ Nβk−1,σ (Ω) ∩ H−k−1 α˜ (Ω) for some σ > 0. Then the energy corrected finite element approximation (2) of (P) admits the following convergence property k+1 ∞ u − um | log h|s (f N k−1,σ (Ω) + f H k−1 (Ω) ), h Lβ (Ω) ≤ ch −α˜

β

where s = 1, when k = 1 and s = 0 otherwise. Proof Without loss of generality we assume that the corner lies at the origin. We π since otherwise standard methods provide the desired also assume that Θ > k+1 convergence order of the scheme [11]. Following the line of proof presented in [1, 10, 11] we introduce a dyadic decomposition around this corner of the domain Ω. Let R > 0 and for j = 0, . . . , I we set ΩJ = {x ∈ Ω : dJ +1 < |x| < dJ }, where dJ = 2−J R and dI ≤ c∗ h, dI +1 = 0. Moreover, dI is chosen so4 that the correction patch of (3) is contained in ΩI . We also define Ω−1 = Ω \ IJ =0 ΩJ . Finally, we set ΩJ = ΩJ +1 ∪ ΩJ ∪ ΩJ −1 . In the proof we consider cases J = I − 1, I and the case J < I − 1 separately. For J < I − 1 we can rely on results proven in [10], namely   −1 s m ∞ (Ω ) ≤ c | log h| ) +d ) . u − um  inf u − χ u − u  ∞ 2 L L (Ω L (Ω J h h J J J

χ∈Vhk

This result was proven under the Galerkin orthogonality assumption a(u−um h , vh ) = 0. Although this does not hold globally for the energy-corrected finite element scheme, it is true for functions vh ∈ Vhk with support in ΩJ . This is sufficient for this result to hold and there is no need to use a more general version provided in [12]. As a consequence, we immediately obtain β

u − um ≤ dJ u − um h L∞ h L∞ (ΩJ ) β (ΩJ )  m  + u − u  2 ≤ c | log h|s inf u − χL∞ h L β (ΩJ ) χ∈Vhk

 β−1 (ΩJ )

 .

(4)

Now, we can move our investigations to subregions, which are close to the corners of the domain, namely, we consider J = I − 1, I . Let T ∗ ∈ T denote the element in the domain’s triangulation, in which the maximum error of the scheme, when measured on ΩJ only, is attained. Note also that T ∗ ⊂ ΩJ . Then, for any χ ∈ Vhk m ∗ ≤ u − χL∞ (Ω  ) + χ − u L∞ (T ∗ ) u − um ≤ u − um h L∞ h L∞ h β (ΩJ ) β (T ) β β J

(5)

Maximum Norm Estimates for EC FEM

979

We now focus our attention on the second term in this estimate. Applications of the inverse inequality between L∞ and L2 norms of discrete functions and Lemma 6 give for some constants cJ , cJ > 0 β m ∗ ≤ cJ h χ − u L∞ (T ∗ ) χ − um h L∞ h β (T )  m ≤ cJ hβ−1 χ − um h L2 (T ∗ ) ≤ cJ χ − uh L2

β−1 (T

∗)

.

A simple application of the Hölder inequality for β  satisfying β > β  > k + 1 − λ1 leads to u − χL2

 β−1 (ΩJ )

≤ cu − χL∞ (ΩJ ) . β

Hence, for cJ > 0 we have cJ χ − um h L2

β−1 (T

∗)

≤ cJ u − um h L2

+ cJ u − χL2

≤ cJ u − um h L2

+ cJ u − χL∞ (ΩJ )

 β−1 (ΩJ )  β−1 (ΩJ )

 β−1 (ΩJ )

β

Therefore, we obtain for all χ ∈ Vhk using (5)  ∞ u − um  ≤ c u − χL∞ (ΩJ ) + u − um h Lβ (ΩJ ) h L2 β

 β−1 (ΩJ )

 .

(6)

Combining (4) and (6) we see that for some c > 0 and for any β > k + 1 − λ1  s m ∞ ∞ u − um h Lβ (Ω) ≤ c | log h| inf u − χLβ (Ω) + u − uh L2 χ∈Vhk

β−1 (Ω)

 .

Application of the interpolation error estimate from Lemma 5 and existing energy corrected finite element estimates in weighted L2 norm stated in Theorem 2 completes the proof. Note that the weight α = β−1 is exactly the one required there. π However, when the considered angle satisfies k+1 < Θ < πk , then −1 < β − 1 < 0 and the results from Remark 3 need to be used instead. Remark 8 The energy-correction needs to be applied to all corners, for which the singular functions (1) influence the regularity stated in Theorem 1. This are exactly π the corners, for which Θ > k+1 , where k is the order of polynomials used in the finite element discretization. A similar condition appears also in schemes employing mesh grading on domains with corners, see [12].

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4 Numerical Results We consider two examples with known analytical solutions. First, we choose the  L-shape domain Ω = (−1, 1)2 \ [0, 1] × [−1, 0] with the largest interior angle of size Θ = 3π/2. In the second example, we focus our attention on the domain Ω = (−1, 1)2 \{−x ≤ y ≤ 0}, which has the angle Θ = 7π/4 located at the origin. We choose a known exact solution u = s1 + s2 + s3 , where si are known singular functions (1) corresponding to the considered angles. For the sake of presentation, it is chosen so that the energy-correction needs to be applied only around the origin, see Remark 8 in a more general case. The parameters γ in the modification (3) are computed using a version of the Newton algorithm described in [5] and in the experiments we choose the weight β = k + 1 − λ1 , where k = 1, 2, 3 is the order of polynomials used in the discretizations and λ1 = π/Θ. This choice induces a slightly stronger norm than assumed in Sect. 3 but the optimal convergence order of the energy-corrected scheme can be observed regardless of this. These numerical tests (Fig. 1) confirm the theoretical results of Theorem 7. Weighted L∞ error of the ECFEM

Weighted L∞ error of the ECFEM

10−1

10−1

10−2

10−2

10−3

10−3 10−4

10−5 10−6

P1 P2 P3 P1 with EC P2 with EC P3 with EC

10−7 10−8 10−9 10−10 −2 10

Error

Error

10−4

10−1

Mesh size h

10−5

P1 P2 P3 P1 with EC P2 with EC P3 with EC

10−6 10−7 10−8 100

10−9 −2 10

10−1

100

Mesh size h

Fig. 1 Comparison of weighted L∞ errors of piecewise polynomial finite element schemes with and without energy-correction (EC) for Θ = 3π/2 (left) and Θ = 7π/4 (right). Doted lines present the errors obtained with the standard finite element resulting in 2π/Θ orders of convergence. In the energy-corrected scheme, we set for corresponding discretization orders γ1 = 0.11753, γ2 = (0.03152, −0.00553), γ3 = (0.01289, −0.00237) for Θ = 3π/2, and γ1 = 0.18618, γ2 = (0.0737, −0.01968), γ3 = (0.04065, −0.01948, 0.0045) for Θ = 7π/4. Energy-correction improves the convergence properties of the scheme and yields optimal convergence orders in the weighted norms, namely 2, 3 and 4 for respective discretization orders

Maximum Norm Estimates for EC FEM

981

Acknowledgements We gratefully acknowledge the support of the German Research Foundation (DFG) through the grant WO 671/11-1 and, together with the Austrian Science Fund, through the IGDK1754 Training Group. We would also like to thank Dr Johannes Pfefferer for many fruitful and helpful discussions.

References 1. T. Apel, J. Pfefferer, A. Rösch, Finite element error estimates on the boundary with application to optimal control. Math. Comput. 84(291), 33–70 (2014) 2. H. Blum, M. Dobrowolski, On finite element methods for elliptic equations on domains with corners. Computing 28(1), 53–63 (1982) 3. P.G. Ciarlet, J.L. Lions, Finite Element Methods (Part 1). Handbook of Numerical Analysis, vol. II (North Holland, Amsterdam, 1991) 4. H. Egger, U. Rüde, B. Wohlmuth, Energy-corrected finite element methods for corner singularities. SIAM J. Numer. Anal. 52(1), 171–193 (2014) 5. T. Horger, P. Pustejovska, B.Wohlmuth, Higher order energy-corrected finite element methods. ArXiv e-prints (2017). http://adsabs.harvard.edu/abs/2017arXiv170405638H 6. V.A. Kondratiev, Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227–313 (1967) 7. V. Kozlov, V.G. Maz’ya, J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs, vol. 85 (American Mathematical Society, Providence, 2001) 8. J. Nitsche, A.H. Schatz, Interior estimates for Ritz-Galerkin methods. Math. Comput. 28, 973– 958 (1974) 9. U. Rüde, C. Waluga, B. Wohlmuth, Nested Newton strategies for energy-corrected finite element methods. SIAM J. Sci. Comput. 36(4), A1359–A1383 (2014) 10. A.H. Schatz, L.B. Wahlbin, Interior maximum norm estimates for finite element methods. Math. Comput. 31(138), 414–442 (1977) 11. A.H. Schatz, L.B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. Part 1. Math. Comput. 32(141), 73–109 (1978) 12. A.H. Schatz, L.B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. Part 2, Refinements. Math. Comput. 33(146), 465–492 (1979) 13. C. Zenger, H. Gietl, Improved difference schemes for the Dirichlet problem of Poisson’s equation in the neighbourhood of corners. Numer. Math. 30(3), 315–332 (1978)

Digital Operators, Discrete Equations and Error Estimates Alexander Vasilyev and Vladimir Vasilyev

Abstract We consider some correlations between theories of discrete and continuous pseudo-differential equations. The discrete theory is very useful to construct good finite approximations for continuous solutions, and solvability theory for discrete pseudo-differential equations is very similar to the theory of continuous ones. We show certain elements of such a theory, and for simplest cases give comparison estimates.

1 Introduction The theory of singular integral (and more general pseudo-differential) equations and related boundary value problems is very studied for smooth manifolds and domains with a smooth boundary [1–4]. At the same time there are some theories related to difference approximations for boundary value problems and numerical analysis for special operator equations (see, for example, [5, 6]). We would like to develop such a theory of discrete equations which might inherit all important properties of continuous theory and at the same time it will help us constructing good finite approximations. We will start from simplest operators and consider special m  canonical domains like Rm , the half-space Rm + = {x ∈ R : x = (x , xm ), xm > 0}, a m  and the cone C+ = {x ∈ R : xm > a|x |, a > 0}.

A. Vasilyev · V. Vasilyev () National Research Belgorod State University, Belgorod, Russia © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_93

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2 Digital Operators 2.1 Digital Calderon–Zygmund Operators We consider simplest Calderon–Zygmund operators of convolution type [3]   v.p. K(x − y)u(y)dy = lim K(x − y)u(y)dy, ε→0

N→+∞ε 0}. The number æ ∈ R is called an index of periodic factorization.

4 Solvability As we will see the index of factorization very influences on the solvability picture of Eq. (3). Theorem 6 If the elliptic symbol A˜ d (ξ ) ∈ Eα admits periodic factorization with index æ so that |æ − s| < 1/2 then Eq. (3) has unique solution in the space H s (Dd ) for arbitrary right-hand side vd ∈ H+s−α (Dd ).

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The solution is constructed explicitly with the help of periodic Hilbert transform with the parameter ξ  ∈ h¯ Tm−1 [14] per (Hξ  u˜ d )(ξ  , ξm )

h v.p. = 2πi

h¯ π cot

−h¯ π

h(ξm − ηm ) u˜ d (ξ  , ηm )dηm , ξm ∈ h¯ [−π, π]. 2

Remark 7 For the more simple equation (1) we have an analogous result in the space L2 (Dd ) ≡ H 0 (Dd ).

5 Discrete Equations and Comparison Here we can give a comparison between operators K and Kd in appropriate subspace of continuous functions. We consider the weight function of the form  ω(x) ˜ = (1 + |x|) ˜ α

x˜m 1 + x˜m

β ,

α,β

and let the discrete space Cγ (Dd ) be the space of functions ud (x) ˜ of discrete variable with the norm ||ud ||α,β,γ ≡ ||ω · ud ||γ , 0 < γ < 1, 0 < α + γ < m, γ < β < γ + 1, where ||ud ||γ ≡ max |ud (x)| ˜ + max x∈D ˜ d

x, ˜ y∈D ˜ d

|ud (x) ˜ − ud (y)| ˜ . |x˜ − y| ˜γ α,β

Theorem 8 The operator Kd is linear bounded operator in the space Cγ (Dd ), and its norm doesn’t depend on h. α,β

Let us note that such space Cγ (Dd ) is a discrete analogue of continuous space α,β Cγ (Rm + ). α,β α,β If we denote lh the restriction operator lh : Cγ (Rm + ) −→ Cγ (Dd ), and take α,β m the function u ∈ Cγ (R+ ), then we have Theorem 9 |[(lh K − Kdh lh )u](x)| ˜ ≤ chγ ln(1/ h), and constant doesn’t depend on h. Let us note the analogous result for the whole space D = Rm in corresponding Hölder space and some numerical experiments are given by the authors in [13, 18]. For comparison K and Kd we need to know their invertibility conditions. Now, the ellipticity condition σ (ξ ) = 0, ∀ξ ∈ S m−1 , isn’t sufficient for invertibility of K.

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Additionally, the following topological condition has been satisfied. We’ll assume σ (0, . . . , 0, −1) = σ (0, . . . , 0, +1), and +∞ d arg σ (·, ξm ) = 0. −∞

Further, for the operator Kd one can construct the analogous theory (see also [14]), and taking into account main properties of σ, σd we will obtain the following Theorem 10 Operators K and Kd are both simultaneously invertible or nonα,β α,β invertible in the spaces Cγ (Rm + ) and Cγ (Dd ) respectively. Now we will introduce a special discrete periodic kernel Kd,N (x) ˜ which is defined as follows. We take a restriction of the discrete kernel Kd (x) ˜ on the set QN ∩ Zm ≡ QdN and periodically continue it to a whole Zm . Further we consider discrete periodic functions ud,N with discrete cube of periods QdN . We can define a cyclic convolution for a pair of such functions ud,N , vd,N by the formula (ud,N ∗ vd,N )(x) ˜ =

ud,N (x˜ − y)v ˜ d,N (y)h ˜ m.

(4)

d y∈Q ˜ N

Further we introduce finite discrete Fourier transform by the formula (Fd,N ud,N )(ξ˜ ) =

˜ ξ˜ m d ud,N (x)e ˜ i x· h , ξ˜ ∈ RN ,

d x∈Q ˜ N

d = hTm ∩ hZm . Let us note that here ξ˜ is a discrete variable. where RN ¯ ¯ According to the formula (4) one can introduce the operator

(Kd,N ud,N )(x) ˜ =

Kd,N (x˜ − y)u ˜ d,N (y)h ˜ m

d y∈Q ˜ N

defined on periodic discrete functions ud,N and a finite discrete Fourier transform for its kernel

˜ ξ˜ m d σd,N (ξ˜ ) = Kd,N (x)e ˜ i x· h , ξ˜ ∈ RN . d x∈Q ˜ N

d Definition 11 A function σd,N (ξ˜ ), ξ˜ ∈ RN , is called s symbol of the operator d. Kd,N . This symbol is called an elliptic symbol if σd,N (ξ˜ ) = 0, ∀ξ˜ ∈ RN

Theorem 12 Let σd (ξ ) be an elliptic symbol. Then for enough large N the symbol σd,N (ξ˜ ) is elliptic symbol also.

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As before an elliptic symbol σd,N (ξ˜ ) corresponds to the invertible operator Kd,N in the space L2 (QdN ). Moreover using the operator is convenient for calculations.

6 Conclusion Here we have given a very small piece of our considerations devoted to digital operators and related boundary value problems. We have obtained error estimates for the whole space Rm [13, 15, 18], comparison of solvability for discrete and continue multidimensional singular integral equations in a half-space [14, 16, 17], comparison result for finite discrete and continuous equations [18, 19]. For more general case of digital pseudo-differential equations we have obtained some solvability results both for a half-space [20, 21] and a cone [8–12] which are based on studies [7]. The authors hope it will be possible to obtain error estimates for general pseudo-differential equations and related boundary value problems and their digital analogues in appropriate discrete spaces. Acknowledgements The author was supported by the State contract of the Russian Ministry of Education and Science (contract No 1.7311.2017/8.9).

References 1. G. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations (AMS, Providence, 1981) 2. F.D. Gakhov, Boundary Value Problems (Dover Publications, Mineola, 1981) 3. S.G. Mikhlin, S. Prössdorf, Singular Integral Operators (Akademie–Verlag, Berlin, 1986) 4. N.I. Muskhelishvili, Singular Integral Equations (North Holland, Amsterdam, 1976) 5. S. Roch, P.A. Santos, B. Silbermann, Non-commutative Gelfand Theories. A Tool-kit for Operator Theorists and Numerical Analysts (Springer, Berlin, 2011) 6. V.S. Ryaben’kii, Method of Difference Potentials and Its Applications (Springer, Berlin, 2002) 7. V.B. Vasilyev, Wave Factorization of Elliptic Symbols: Theory and Applications. Introduction to the Theory of Boundary Value Problems in Non-Smooth Domains (Kluwer Academic Publishers, Dordrecht, 2000) 8. V.B. Vasilyev, Discrete equations and periodic wave factorization, in AIP Conference Proceedings, vol. 1759 (2016), 5 pp. 9. V.B. Vasilyev, The periodic Cauchy kernel, the periodic Bochner kernel, and discrete pseudodifferential operators, in AIP Conference Proceedings, vol. 1863 (2017), 4 pp. 10. V.B. Vasilyev, On discrete boundary value problems, in AIP Conference Proceedings, vol. 1880 (2017), 4 pp. 11. V.B. Vasilyev, Discrete pseudo-differential operators and boundary value problems in a halfspace and a cone. Lobachevskii J. Math. 39(2), 289–296 (2018) 12. V.B. Vasilyev, On discrete pseudo-differential operators and equations. Filomat 32(3), 975–984 (2018) 13. A. Vasilyev, V. Vasilyev, Numerical analysis for some singular integral equations. Neural Parallel Sci. Comput. 20, 313–326 (2012)

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14. A. Vasilyev, V. Vasilyev, Discrete singular operators and equations in a half-space. Azerb. J. Math. 3, 84–93 (2013) 15. A. Vasilyev, V. Vasilyev, On the solvability of certain discrete equations and related estimates of discrete operators. Dokl. Math. 92, 585–589 (2015) 16. A. Vasilyev, V. Vasilyev, Periodic Riemann problem and discrete convolution equations. Differ. Equ. 51, 652–660 (2015) 17. A. Vasilyev, V. Vasilyev, Discrete singular integrals in a half-space, in Current Trends in Analysis and its Applications, ed. by V. Mityushev, M. Ruzhansky (Birkhäuser, Basel, 2015), pp. 663–670 18. A. Vasilyev, V. Vasilyev, Two-scale estimates for special finite discrete operators. Math. Model. Anal. 22, 300–310 (2017) 19. A. Vasilyev, V. Vasilyev, Discrete approximations for multidimensional singular integral operators. Lect. Notes Comput. Sci. 10187, 706–712 (2017) 20. A. Vasilyev, V. Vasilyev, Pseudo-differential operators and equations in a discrete half-space. Math. Model. Anal. 23(3), 492–506 (2018) 21. A. Vasilyev, V. Vasilyev, On some discrete potential like operators. Tatra Mt. Math. Publ. 71, (2018)

A Simple Boundary Approximation for the Non-symmetric Coupling of the Finite Element Method and the Boundary Element Method for Parabolic-Elliptic Interface Problems Christoph Erath and Robert Schorr

Abstract The non-symmetric coupling for parabolic-elliptic interface problems on Lipschitz domains was recently analysed in Egger et al. (On the nonsymmetric coupling method for parabolic-elliptic interface problems, preprint, 2017, arXiv:1711.08487). In Egger et al. (2017, Section 5) a classical FEM-BEM discretisation analysis was provided, but only with polygonal boundaries. In this short paper we will look at the case where the boundary is smooth. We introduce a polygonal approximation of the domain and compute the FEM-BEM coupling on this approximation. Note that the original quasi-optimality cannot be achieved. However, we are able to show a first order convergence result for lowest order FEM-BEM.

1 Introduction and Model Problem In [6] a semi-discretisation of a non-symmetric FEM-BEM coupling for smooth boundaries was analysed. Recently, [2] extended the analysis to general Lipschitz domains and provided the first complete analysis for the full discretisation. The FEM-BEM error estimates in [2, Section 5] only hold for polygonal boundaries which might be a restriction for real life applications. Hence we look at problems with smooth boundaries and a polygonal approximation thereof. We will see that the novel quasi-optimality result of [2] cannot be restored. Nevertheless, a first order convergence result (for lowest order FEM-BEM) can be established. The model problem reads as follows: Let Ω ⊂ R2 be a bounded domain with diam(Ω) < 1 and boundary Γ := ∂Ω, and let Ωe = R2 \ Ω be its complement.

C. Erath · R. Schorr () Graduate School of Computational Engineering, TU Darmstadt, Darmstadt, Germany Department of Mathematics, TU Darmstadt, Darmstadt, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_94

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Then find u and ue such that ∂t u − Δu = f

in Ω × (0, T ),

(1)

in Ωe × (0, T ),

(2)

u = ue

on Γ × (0, T ),

(3)

∂n u = ∂n ue

on Γ × (0, T ),

(4)

in Ω,

(5)

|x| → ∞,

(6)

−Δue = 0

u(·, 0) = 0 ue (x, t) = a(t) log |x| + O(|x|−1 )

where T > 0 is a fixed time. The co-normal derivative ∂n u = ∇u · n|Γ is taken in direction of the unit normal vector n on Γ pointing outwards with respect to Ω. Equations (3)–(4) are transmission conditions across the coupling interface Γ , Eq. (5) is an initial condition, whereas Eq. (6) is a so-called radiation condition with an unknown a(t) ∈ R for fixed t. In this paper, we will use the notation L2 (0, T ; X) for the Bochner-Sobolev T spaces with some Banach space X and write vL2 (0,T ;X) = 0 vX dx for the corresponding norm. For the L2 -scalar product we write (·, ·)ω for ω ⊆ Ω and ·, ·ω for the duality product. Furthermore, we will reformulate the exterior problem given by Eqs. (2)–(6) with the aid of the single layer operator V and the double layer operator K. For smooth enough input they read   (Vψ)(x) = G(x, y)ψ(y) dsy , (Kv)(x) = ∂ny G(x, y)v(y) dsy , Γ

Γ

1 where G(x, y) = − 2π log |x − y| denotes the fundamental solution of the Laplace operator in two dimensions. They can be extended to continuous operators V : H −1/2 (Γ ) → H 1/2(Γ ) and K : H 1/2(Γ ) → H 1/2(Γ ). The weak coupling formulation of the above problem looks as follows ([2]):

Problem 1 (Variational Problem) Given f ∈ L2 (0, T ; (H 1(Ω)) ), find u ∈ QT = {u ∈ L2 (0, T ; H 1 (Ω)) : ∂t u ∈ L2 (0, T ; (H 1(Ω)) ) and u(0) = 0} and φ ∈ BT = L2 (0, T ; H −1/2(Γ )) such that ∂t u(t), vΩ + B((u(t), φ(t)), (v, ψ)) = f (t), vΩ ,

(7)

for all test functions v ∈ H 1 (Ω) and ψ ∈ H −1/2(Γ ) and for a.e. t ∈ [0, T ]. Here, B : (QT × BT ) × (H 1 (Ω) × H −1/2 (Γ )) → R is defined by B((u(t), φ(t)), (v, ψ)) = (∇u(t), ∇v)Ω − φ(t), vΓ + (1/2 − K)u(t)|Γ , ψΓ + Vφ(t), ψΓ .

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Remark 2 The problem we look at here is a simplified model problem; extensions to three dimensions, more general parabolic equations or problems with jumps at the interface are possible.

2 A Simple Domain Approximation for Domains with Smooth Boundaries The following analysis is based on [1, 3] for the finite element part. The boundary element and coupling part is addressed in [4, 5]. Now we assume that the boundary Γ of Ω is in C 3 . Let Ωh be a polygonal approximation of Ω with a quasi-uniform triangulation T = {T } such that the vertices of Γh = ∂Ωh also lie on Γ . The induced boundary triangulation will be denoted by EΓ = {E}. Furthermore, we will need an exact triangulation T8 = {T8} 4 of Ω consisting additionally of curved triangles such that Ω¯ = T8∈T8 T8. Then we can define several mappings that transform the elements of the triangulation. Let AT ∈ R2×2 and bT ∈ R2 , then FT : Tˆ → T ,

FT (x) ˆ = AT xˆ + bT

8T := FT + transforms the reference triangle Tˆ to a triangle T . The transformation F 8 ΦT maps the reference triangle onto a curved triangle T , where ΦT ∈ C 3 (Tˆ , R2 ), a construction can be found in [1, Section 6.1]. By composing the two functions we can define a mapping from T ∈ T to T8 ∈ T8 by: Gh : Ωh → Ω,

8T ◦ F −1 , Gh |T := F T

with Gh (x) = x for all x ∈ T with Gh (T ) ∩ Γ = ∅ and Jacobi matrix DGh . The mapping has the property that for all T ∈ T (see also [3]): 7 7 7det(DGh ) − 17 ∞ ≤ chT , L (T )

(8)

with hT = diam(T ) (and h = maxT ∈T hT ). The constant c is independent of hT . By setting gh := Gh |Γh we obtain gh : Γh → Γ . For the determinants we introduce abbreviations adopted from [3]: Jh := det(DGh ),

μh := det(Dgh ).

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2.1 Analysis of the Semi-discrete System For the discretisation in space we will use a Galerkin ansatz with the space of continuous piecewise linear functions S 1 (T ) and the space of piecewise constants P 0 (EΓ ). Then our semi-discrete problem reads: Problem 3 (Semi-discrete Problem) Find uh ∈ QhT := H 1 (0, T ; S 1 (T )) and φh ∈ BTh := L2 (0, T ; P 0 (EΓ )) such that (∂t uh (t), vh )Ωh + Bh ((uh (t), φh (t)); (vh , ψh )) = (fh (t), vh )Ωh ,

(9)

for all v = (vh , ψh ) ∈ S 1 (T ) × P 0 (EΓ ) and a.e. t ∈ (0, T ). The function fh is defined by fh = (f ◦ Gh ) · Jh and Bh is defined by Bh ((uh (t), φh (t)); (vh , ψh )) =(∇uh (t), ∇vh )Ωh − φh (t), vh Γh ψh , (1/2 − K)uh (t)Γh + ψh , Vφh (t)Γh . Important here is that the bilinear form Bh is elliptic, see [2, Lemma 3]. For comparison of Problems 1 and 3 we introduce the notion of so-called lifted functions: 8 vh : Ω → R2 , 8h : Γ → R, ψ

2 8 vh := vh ◦ G−1 h for vh : Ωh → R ,

8h := ψh ◦ g −1 for ψh : Γh → R. ψ h

(10) (11)

By using the transformation formula and the definition of fh we can see that (fh , vh )Ωh = (f,8 vh )Ω . for all vh ∈ S 1 (T ). That means that the altered right-hand side of Eq. (9) is consistent with the weak formulation Eq. (7). To analyse the error, we transform our discrete system to Ω. Firstly, we reformulate the term with the time derivative (∂t uh , vh )Ωh = (

1 ∂t 8 uh ,8 vh )Ω . Jh

To reformulate the bilinear form, we use the identity: −T O −T −1 ∇(8 wh ) = ∇(wh ◦ G−1 h ) = DGh (∇wh ) = DGh (∇wh ) ◦ Gh ,

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−1 T h h where DG−T h = (DGh ) . So we obtain for (uh , φh ) ∈ QT × BT and (vh , ψh ) ∈ S 1 (T ) × P 0 (EΓ ):   Bh ((uh , φh ); (vh , ψh )) = ∇uh ∇vh dx − φh vh ds Ωh





Γh

 1 uh ψh ds − ∂n Guh dsy ψh dsx 2 Γh Γh Γh   + Gφh dsy ψh dsx +

Γh



Γh

 1 1 8h8 φ dx − vh ds Jh μh Ω Γ    1 1 1 8uh 1 dsy ψ 8h 8h + 8 uh ψ ds − ∂n G8 dsx 2 Γ μh μ μ h h Γ Γ   1 1 8h ((8 8φ 8h 8h )), 8h + dsy ψ dsx =: B uh , 8 φh ); (8 vh , ψ G μh μh Γ Γ =

DGTh ∇8 uh DGTh ∇8 vh

8 y) = G(g −1 (x), g −1 (y)). with G(x, h h Using the results from [3, Lemma 6.2] and [4, Example 1] (which is based on [5, Proposition 3.2]), we obtain: Lemma 4 (Error in Perturbed Bilinear Form) Let (uh , φh ) ∈ S 1 (T ) × P 0 (EΓ ) and (vh , ψh ) ∈ S 1 (T ) × P 0 (EΓ ), then 8h ((8 8h ); (8 8h )) − B 8h ))| vh , ψ uh , 8 φh ); (8 vh , ψ |B((8 uh , φ 7 7 7 7 7 7 7 8h )7 φ h )7 1 , ψ ≤ ch 7(8 uh , 8 v 7 7(8 h −1/2 H (Ω)×H

(Γ )

H 1 (Ω)×H −1/2 (Γ )

,

with a constant c > 0 independent of h. For the analysis we introduce the operator (Rh , rh ), often called elliptic projection. For (w, χ) ∈ H 1 (Ω) × H −1/2 (Γ ) we define it by: 8h ((Rh w, rh χ); (8 8h )) = B((w, χ); (8 8h )). B vh , ψ vh , ψ for all (vh , ψh ) ∈ S 1 (T ) × P 0 (EΓ ). This is well-defined due to the ellipticity of the bilinear form. Using [3, Lemma 6.4] we can derive an approximation result. Lemma 5 (Approximation Properties of (Rh , rh )) H 1/2(Γ ) we obtain

For (u, φ) ∈ H 2 (Ω) ×

  7 7 7 7 u − Rh uH 1 (Ω) + 7φ − rh φ 7H −1/2 (Γ ) ≤ ch uH 2 (Ω) + 7φ 7H 1/2 (Γ ) . Finally, we can show the convergence of the semi-discrete scheme.

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Theorem 6 (Convergence of the Semi-discrete Scheme) Let (u, φ) be the solution to Problem 1 with u ∈ H 1 (0, T ; H 2 (Ω)), φ ∈ L2 (0, T ; H 1/2(Γ )) and (uh , vh ) the solution of Problem 3. Then we get (using the lifting Eqs. (10) and (11)) 7 7 78 7 8 uh − uL2 (0,T ;H 1 (Ω)) + 7φ h − φ7

L2 (0,T ;H −1/2 (Γ ))

  7 7 ≤ ch uL2 (0,T ;H 2 (Ω)) + ∂t uL2 (0,T ;H 2 (Ω)) + 7φ 7L2 (0,T ;H 1/2 (Γ )) Proof The error can be split into two parts by 8 uh − Rh uL2 (0,T ;H 1 (Ω)) + Rh u − uL2 (0,T ;H 1 (Ω)) uh − uL2 (0,T ;H 1 (Ω)) ≤ 8 8h − rh φ. Furthermore and similarly for φ. Thus we set θ1 := 8 uh − Rh u, θ2 := φ 1/2 1 1 we introduce wc = ( Jh w, w)Ω . Then we can use ( Jh ∂t θ1 , θ1 )Ω = 12 ∂t θ1 2c , 8h and plugging in (θ1 , θ2 ) as trial and test functions into Eq. (9) to the ellipticity of B obtain:   1 ∂t θ1 2c + C θ1 2H 1 (Ω) + θ2 2H −1/2 (Γ ) 2 1 8h ((θ1 , θ2 ); (θ1 , θ2 )) ≤ ( ∂t θ1 , θ1 )Ω + B Jh =(

1 1 ∂t 8 uh , θ1 )Ω − ( ∂t Rh u, θ1 )Ω Jh Jh

8h ((8 8h ((Rh u, rh φ); (θ1 , θ2 )) +B uh , 8 φh ); (θ1 , θ2 )) − B = (∂t u, θ1 )Ω − (

1 ∂t Rh u, θ1 )Ω Jh

8h ((Rh u, rh φ); (θ1 , θ2 )) + B((u, φ); (θ1, θ2 )) − B = (∂t u, θ1 )Ω − (

1 ∂t Rh u, θ1 )Ω . Jh

In the last lines we first used the Galerkin orthogonality and afterwards the definition of (Rh , rh ). Inserting ( J1h ∂t u − J1h ∂t u, θ1 )Ω and using Lemma 5 and Eq. (8) yields 1 1 (∂t u − ∂t Rh u), θ1 )Ω + (∂t u − ∂t u, θ1 )Ω Jh Jh 7 7 717 7 ∂t uH 2 (Ω) θ1 L2 (Ω) + ch ∂t uH 2 (Ω) θ1 L2 (Ω) ≤ ch 7 7J 7 ∞ =(

h L (Ω)

≤ Ch ∂t uH 2 (Ω) θ1 L2 (Ω) . Because

1 Jh

is bounded in L∞ (Ω), we can treat it as a constant.

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Using Young’s inequality, the error splitting, the bound for the elliptic projection and integrating over t from 0 to T lead to the final estimate. & %

2.2 Analysis of the Fully-Discrete System For the full discretisation we use the implicit Euler method, so we introduce the discrete derivative ∂Δt un+1 := (un+1 − un )/Δt. Then the system reads: Problem 7 (Fully Discrete Formulation) Find unh ∈ S 1 (T ), φhn ∈ P 0 (EΓ ) for n = 0, · · · , N := T /Δt, such that (unh , φhn ) fulfil for all (vh , ψh ) ∈ S 1 (T )×P 0 (EΓ ) (∂Δt unh , vh )Ωh + Bh



  unh , φhn ; (vh , ψh ) = (fhn , vh )Ωh ,

(12)

for 1 ≤ n ≤ N and u0h = 0, where fhn (x) := fh (x, nΔt). Similar as in [2], the result holds in the natural energy norm where we do not need any duality arguments which are not available for the non-symmetric coupling. Theorem 8 (Convergence of the Fully-Discrete Scheme) Let (u, φ) be the solution to Problem 1 with u ∈ H 1 (0, T ; H 2(Ω)), ∂t t u ∈ L2 (0, T ; (H 1(Ω)) ) and φ ∈ H 1 (0, T ; H 1/2(Γ )) and let (unh , φhn ) be the solution of Problem 7. Then we obtain (again using the lifting Eqs. (10) and (11)) ⎛ ⎝

N

 7 72 72 7 n 7 n 7 Δt 78 φh − φ(t n )7 uh − u(t n )7H 1 (Ω) + 78

H −1/2 (Γ )

n=1



⎞1/2 ⎠

= O(h + Δt)

Proof Similar as before, write 8 unh − u(t n ) = (8 unh − Rh u(t n )) + (Rh u(t n ) − u(t n )) =: θ1n + ρ1n , 8 φhn − φ(t n ) = (8 φhn − rh φ(t n )) + (rh φ(t n ) − φ(t n )) =: θ2n + ρ2n . Plugging (θ1n , θ2n ) into Eq. (12) yields (

1 8h ((θ n , θ n ); (θ n , θ n )) ∂Δt θ1n , θ1n )Ω + B 1 2 1 2 Jh

=(

1 1 ∂Δt 8 unh , θ1n )Ω − ( ∂Δt Rh u(t n ), θ1n )Ω Jh Jh

8h ((8 8h ((Rh u(t n ), rh φ(t n )); (θ n , θ n )) +B unh , 8 φhn ); (θ1n , θ2n )) − B 1 2 = (∂t u, θ1n )Ω − (

1 ∂Δt Rh u, θ1n )Ω Jh

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= −(

1 (∂Δt u(t n ) − ∂Δt Rh u(t n )), θ1n )Ω Jh

+ (∂t u(t n ) −

1 1 ∂t u(t n ), θ1n )Ω + ( (∂t u(t n ) − ∂Δt u(t n )), θ1n )Ω . Jh Jh

Here we first used Galerkin orthogonality and the definition of the elliptic projection, then we inserted some terms and rearranged everything. Now most of the terms can be bounded as before. The term with the discrete derivative can be estimated using Taylor expansion. By summing over n from 0 to N and bounding the resulting terms we obtain the convergence result. & %

3 Numerical Experiment As numerical experiment, we look at the approximation of the unit circle by a polygonal curve for the model problem Eqs. (1)–(6) with f = 0. The initial condition will be chosen to be u(x1 , x2 , 0) = sin(2πx1 x2 ). The starting mesh and a refined mesh are shown in Fig. 1. As we do not know the analytical solution of the example, we computed a reference solution uref h on a mesh with 1,048,576 elements and then estimate the error via: ⎛ N

7 n 72 ⎝ uh − u(t n )7 Δt 78

⎞1/2

H 1 (Ω)

⎠

⎛ N 72 7

7 7 ≈⎝ Δt 7unh − uref h 7

n=1

H 1 (Ωh )

n=1

⎞1/2 ⎠

=: eh1 .

For the exterior solution we proceed analogously but replace the norm in the exterior ·2H −1/2 (Γ ) by the equivalent norm ·2V = V·, ·Γ . This error will be denoted by eh2 .

a

1

b

0.5 x2

x2

0.5 0

−0.5 −1

1

0

−0.5 −1

0 x1

1

−1

−1

0 x1

1

Fig. 1 The initial mesh and therefore initial approximation of the circle is shown in (a) and the third uniform refinement in (b). The coupling boundary is bold

Simple Boundary Approximation for Non-Symmetric FEM-BEM Table 1 The estimated errors of the numerical example with rate of convergence

1001

h

eh1

eh2

Added error

Rate

0.4203 0.2219 0.1137 0.0575 0.0289 0.0145

1.65e−01 1.59e−01 1.23e−01 8.13e−02 4.82e−02 2.53e−02

7.73e−02 6.63e−02 5.36e−02 3.68e−02 2.18e−02 1.10e−02

1.82e−01 1.72e−01 1.35e−01 8.93e−02 5.28e−02 2.76e−02

0.087 0.36 0.6 0.76 0.94

Because the input data is smooth, we would expect the full order of convergence. In Table 1 we see that we arrive at this order. Acknowledgements This work is supported by the Excellence Initiative of the German Federal and State Governments and the Graduate School of Computational Engineering at TU Darmstadt. The authors would also like to thank Herbert Egger (TU Darmstadt) for pointing out this topic.

References 1. C. Bernardi, Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26(5), 1212–1240 (1989) 2. H. Egger, C. Erath, R. Schorr, On the non-symmetric coupling method for parabolic-elliptic interface problems. Preprint (2017). arXiv:1711.08487 3. C. Elliot, T. Ranner, Finite element analysis for a coupled bulk-surface partial differential equation. IMA J. Numer. Anal. 33(2), 377–402 (2013) 4. C. Johnson, J.-C. Nédélec, On the coupling of boundary integral and finite element methods. Math. Comput. 35(152), 1063–1079 (1987) 5. M. Le Roux, Méthode d’éléments finis pour la résolution numérique de problèmes extérieurs en dimension 2. R.A.I.R.O. Analyse numérique 11(1), 27–60 (1977) 6. R.C. MacCamy, M. Suri, A time-dependent interface problem for two-dimensional eddy currents. Quart. Appl. Math. 44(4), 675–690 (1987)

Part XXV

Fluid Dynamics

Galerkin Projection and Numerical Integration for a Stochastic Investigation of the Viscous Burgers’ Equation: An Initial Attempt Markus Wahlsten and Jan Nordström

Abstract We consider a stochastic analysis of the non-linear viscous Burgers’ equation and focus on the comparison between intrusive and non-intrusive uncertainty quantification methods. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are applied to a provably stable formulation of the viscous Burgers’ equation, and compared. As measures of comparison: variance size, computational efficiency and accuracy are used.

1 Introduction The two main approaches for solving partial differential equations with random inputs can be categorized in intrusive and non-intrusive methods. Semi-intrusive methods do exist, combining intrusive and non-intrusive methods [1], but are rare. Non-intrusive methods solves the original problem multiple times using fix stochastic inputs [2]. Numerical integration and interpolation techniques are then used to compute statistics of the solution. Intrusive methods based on polynomial chaos expansions results in a system of equations for the expansion coefficients [8, 9]. The non-intrusive method unlike the intrusive one, takes advantage of an already existing deterministic solver. The aim of this article is to compare numerical integration (NI) and polynomial chaos (PC) with the stochastic Galerkin approach in terms of performance. The methods are applied to the viscous Burgers’ equation and compared in terms of computational time, variance size and accuracy. The rest of the paper proceeds as follows. Section 2 introduces the non-linear viscous Burgers’ equation. The stochastic Galerkin procedure is introduced in

M. Wahlsten () · J. Nordström Linköping University, Linköping, Sweden e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_95

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Sect. 3. In Sect. 4, we derive boundary conditions and an energy estimate for the continuous problem. Section 5 presents a stable and accurate semi-discrete finite difference formulation based on summation-by-parts (SBP) with simultaneous approximation terms (SAT). In Sect. 6, numerical results are presented comparing NI and PC. Finally, conclusions are drawn in Sect. 7.

2 The Continuous Problem We consider the viscous Burgers’ equation in one space dimension x ∈ Ω, t > 0, ut + uux = uxx , Lu = g(x, t, ξ ), x ∈ ∂Ω, t > 0, u = f (x, ξ ), x ∈ Ω, t = 0.

(1)

The solution is denoted u = u(x, t, ξ ), where, ξ = (ξ1 , ξ2 , . . . , ξL ) is the vector of variables representing the uncertainty in the solution. The viscosity  is a positive constant. The boundary operator defined on the boundary ∂Ω is denoted by L, while f (x, ξ ) and g(x, t, ξ ) are the stochastic initial and boundary data.

3 The Stochastic Galerkin Projection To perform a stochastic Galerkin projection of (1) we insert the truncated expansion u(x, t, ξ ) =

M

ui (x, t)ψi (ξ ),

(2)

i=0

where the ψi ’s are a set of orthogonal basis functions. We find M M M M

(ui )t ψi + ui (uj )x ψi ψj =  (ui )xx ψi , i=0

i=0 j =0

i=0 M

Lui ψi = g(x, t, ξ ),

i=0 M

i=0

ui ψi = f (x, ξ ).

(3)

PC and NI for Stochastic Investigations of the Viscous Burgers’ Equation

1007

By multiplying (3) with ψk , for k = 0, 1, . . . , M and integrating over the stochastic domain Ωξ , we obtain (uk )t +

M M

ui (uj )x ψi ψj , ψk  = (uk )xx ,

i=0 j =0

Luk = g(x, t, ξ ), ψk , uk = f (x, ξ ), ψk ,

(4)

for k = 0, 1, . . . , M. Hence, a deterministic system of dimension M + 1 times the size of the original system is obtained. From (4), the deterministic coefficients u0 (x, t), u1 (x, t), . . . , uM (x, t) are computed. For details involving the basics in polynomial chaos expansions and numerical integration see for example [5] and [6].

4 The Energy Estimate We now consider the system (4) written on the general form ut + Aux = uxx , x ∈ Ω, t > 0, Lu = g, x ∈ ∂Ω, t > 0, u = f, x ∈ Ω, t = 0,

(5)

where we denote u = [u0 , . . . , uM ]T , Aj k = g = [g0 , . . . , gN ], gk = g(x, ξ ), ψk ,

M

ui ψi ψj , ψk ,

i=0

f = [f0 , . . . , fN ], fk = f (x, ξ ), ψk .

(6)

The continuous problem (5) can be written in the split form [3] 1 1 ut + (Au)x + Aux = uxx , 3 3

(7)

where the matrix A is symmetric and hence can be diagonalized as A(u) = XΛXT . Following the path in [4], gives B1 B u2t + 2 ux 2 = −W T ΛM W B 0

(8)

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− + + − − where ΛM = diag(Λ+ M , ΛM ), W = V u, W = V u, W = V u, and

V+

Λ+ M

⎤ ⎤ ⎡ ⎡ T − ( 2 Λ+ )−1 X T ∂ T − ( 2 Λ− )−1 X T ∂ X+ X + ∂x − ∂x 3 3 ⎥ ⎥ ⎢ ⎢ − T ∂ T ∂ =⎣ X− X+ ⎦, V− = ⎣ ⎦, ∂x ∂x ∂ ∂ I0 − I0 ∂x I0 + I0 ∂x ⎡ ⎡ ⎤ ⎤ 2 + 2 − Λ 0 0 Λ 0 0 3 3 ⎢ ⎢ ⎥ ⎥ 2 + −1 = ⎣ 0 −( 23 Λ− )−1 0 ⎦ , Λ− 0 ⎦. M = ⎣ 0 −( 3 Λ ) 0 0 I0 /2 0 0 −I0 /2

(9)

Note that the right-hand side of (8) is cubic in u, and hence a non-linear boundary conditions is required. To bound the left-hand side (LHS) of (8), we impose the non-linear boundary conditions 2

− |Λ− M |0,1 (W )0,1 =

2

− |Λ− M |0,1 (V )0,1 u = g0,1 ,

(10)

where the subscript 0 and 1 denotes the left and right boundary, respectively. From (10) we can extract the boundary operators L0,1 as L0,1 =

2

− |Λ− M |0,1 (V )0,1 .

(11)

By imposing the boundary condition (10) in (8), gives u2t + 2 ux 2 + W0 2Γ + W1 2Γ = g0T g0 + g1T g1 .

(12)

The result can be summarized in Proposition 1 The problem (5) augmented with the boundary conditions (10) has a bounded solution. Proof By integrating (12) in time we obtain 7 7 7u(T )72 + =



T

0 T 0

2 ux 2 + W0 2Γ + W1 2Γ dt g0T g0

+ g1T g1 dt

7 72 + 7f 7 .

(13)

Hence, an energy estimate is obtained. Remark 2 In NI, the above analysis is analogous, the only differences is that u = u(x, t, ξ ), g = g(x, ξ ), f = f (x, ξ ) and A = u.

PC and NI for Stochastic Investigations of the Viscous Burgers’ Equation

1009

5 The Semi-discrete Formulation The problem (4) or equivalently (5) is solved using a finite difference formulation based on the SBP–SAT technique [7]. A stable and accurate semi-discrete SBP-SAT formulation of (5) using the split form (7) is ˜ vt + 13 (D ⊗ IM )Av + 13 A(D ⊗ IM )v − (D ⊗ IM )Dv −1 + (P E0 ⊗ IM ) 0 (L0 v − e0 ⊗ g0 ) + (P −1 EN ⊗ IM ) 1 (L1 v − eNx ⊗ g1 ) v(0) = f.

(14)

˜ = (D ⊗ IM )v where ⊗ denotes the Kronecker product. In (14), v where Dv represents u numerically, g0 , g1 and f are the numerical approximations of < g0 (x, t, ξ ), ψl >, < g1 (x, t, ξ ), ψl >, and < f (x, t, ξ ), ψl > for l = 0, 1, . . . , M, respectively. Remark 3 When using NI, the vectors g0 , g1 and f instead denote g0,1 = [g¯0,1 (ξ0 ), . . . , g¯0,1 (ξM )]T and f = [f¯(ξ0 ), . . . , f¯(ξM )]T . That is, the boundary and initial data g¯0 (ξ ), g¯1 (ξ ) and f¯(ξ ) are grid functions in ξ . The matrices  0 and  1 will be determined to ensure stability. The numerical solution is arranged in the following way ⎡ ⎤ ⎡ ⎤ v0 v0 ⎢v ⎥ ⎢v ⎥ ⎢ 1⎥ ⎢ 1⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ v = ⎢ ⎥, [vi ] = ⎢ ⎥, ⎢[vi ]⎥ ⎢ vm ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎣ . ⎦ ⎣ . ⎦ vM i vNx where vim approximates the polynomial chaos coefficient um (xi , t). Remark 4 The vector vij approximates u(xi , t, ξj ) when using NI. The derivative in the x−direction is approximated by the SBP operator D = P −1 Q. The matrix P is a positive definite diagonal matrix and Q is almost skewsymmetric satisfying Q + QT = EN − E0 = B = diag[−1, 0, . . . , 0, 1]. We denote the identity matrix of dimension M + 1 by IM . The matrices E0 and EN are zero except for the element (1, 1) and (Nx + 1, Nx + 1) respectively, which is 1. Finally, e0 and eNx denote a zero vector with the exception of the first and last element respectively, which is 1. The matrix A in (14) is given by A = diag(A¯ 0 , . . . , A¯ Nx ),

(A¯ i )j k =

M

l=0

vil < ψl , ψj , ψk > .

(15)

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Remark 5 In the NI framework (A¯ i ) = diag(vi0 , . . . , viM ). The discrete boundary operators L˜ 0 and L˜ 1 in (11) are decomposed as 2 2 − − L0 = [( |− |V ) , 0], L = [0, ( |− 0 1 M M |V )1 ]. where

⎤ XT− ⎢ ⎥ V− 0 = ⎣ 0 ⎦, I0

⎤ − 3 (− )−1 XT− 2 ⎥ ⎢ V− XT− ⎦, D =⎣ I0 ⎤ ⎡ 2 −  0 0 ⎥ ⎢3 − ˜ − 3 + −1 V− = V− 0 ⎦. 0 + VD D, M = ⎣ 0 − 2 ( ) 0 0 −I0 /2 ⎡

(16)

⎡

(17)

In (16), the matrices X+ and X− contain the eigenvectors of their corresponding positive and negative eigenvalues of A respectively.

5.1 Stability To prove stability, we use the discrete energy method and the SBP-properties described above. The result is Proposition 6 The numerical approximation (14) with the penalty matrices 2 2 − T − T Σ1 = ( |− Σ0 = ( |− M |V ) , M |V )

(18)

is strongly stable. Proof Applying the discrete energy method to (14) and integrating in time from 0 to T we obtain 7 7 7v(T )72 + P +



T

0 T 

0 T

− 

0 T

− 0

7 72 7 7 7 72 7˜ 7 2 2 7Dv 7 + 7W0 7 + 7W1 7 dt = P

7 72 gT0 g0 + gT1 g1 dt + 7f7P T    W− W− dt (C ⊗ I ) g g 0 0 T    W− W− dt (C ⊗ I ) g g 1

1

(19)

PC and NI for Stochastic Investigations of the Viscous Burgers’ Equation

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where (0, 2) are the eigenvalues to 

 1 −1 C= . −1 1

(20)

The LHS of (19) is bounded by data, leading to strong stability.

6 Numerical Experiments The initial and boundary data as well as a forcing function will be provided by the manufactured solution u(x, t, ξ ) = sin(2π(x − t)) +

1 ξ

1 + e− 10

.

(21)

As a measure of comparison, the following norm is used 

T

V ar[e] = 0

V ar[U ] − V ar[V ]2 dt,

(22)

where U denotes the exact solution projected on the grid and V denotes the numerical solution. The comparison is done using  = 0.1 and  = 0.01. A small  is generates a less smooth solution. In the calculations below a 3rd-order SBP-operator with 40 grid points in space, and a 4th-order Runge–Kutta scheme in time is used. A PC computation using 29 basis functions is used as reference solution. The uncertainty imposed is uniformly distributed between −1 and 1. Figure 1a, b illustrate the normed error of the variance as a function of number of coefficients/evaluations (M) for PC and NI using  = 0.1 and  = 0.01, respectively. Figure 1c, d show the CPU time as a function of number of M for PC and NI for the same two cases. Finally, Figure 1e, f depict the normed error of the variance as a function of CPU time for PC and NI using again  = 0.1 and  = 0.01. As can be seen from Fig. 1a–f, PC performs better than NI for smooth problems, while NI seems to be more efficient for non-smooth problems.

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-2

PC NI

PC NI

10

10 -4

-2

-6

10

10 -4

10 -8

10

10 -10 10 -12 10

-14

10

-16

10

-6

-8

10 -10

10 0

M

10 1

10

0

(a) PC NI

3

PC NI

CPU time (s)

10

10 2

2

10

0

M

10

1

10

(c)

10

1

PC NI

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-2

10 -4

-8

10 -6

-10

10 -12 10

10

M

PC NI

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10 -6 10

0

(d)

10 -2

10

1

(b)

CPU time (s)

10

10

M

10 -8

-14

10 -10

10 -16 10

2

CPU time (s)

(e)

10

3

10

2

CPU time (s)

(f)

Fig. 1 The uncertainty in all figures is uniformly distributed in [−1, 1]. (a) The normed error of the variance as a function of M, using  = 0.1. (b) The normed error of the variance as a function of M, using  = 0.01. (c) The CPU time as a function of M, using  = 0.1. (d) The CPU time as a function of M, using  = 0.01. (e) The normed error of the variance as a function of the CPU time, using  = 0.1. (f) The normed error of the variance as a function of the CPU time, using  = 0.01

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7 Conclusions We have analyzed and compared the efficiency of PC and NI. The study has been carried out using the non-linear viscous Burgers’ equation. The PC framework is applied to the continuous problem, and a stable high-order finite difference formulation on SBP-SAT form was constructed. A similar numerical scheme is constructed for NI. The normed difference of the variance was used as a measure of comparison. Numerical results suggest that the PC procedure outperforms NI for smooth problems, while NI seems to be more efficient for less smooth problems.

References 1. R. Abgrall, P.M. Congedo, A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems. J. Comput. Phys. 235, 828–845 (2013) 2. S. Hosder, R.W. Walters, R. Perez, A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. AIAA paper 891 (2006) 3. J. Nordström, Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation. J. Sci. Comput. 29, 375–404 (2006) 4. J. Nordström, A roadmap to well posed and stable problems in computational physics. J. Sci. Comput. 71(1), 365–385 (2017) 5. M.P. Pettersson, G. Iaccarino, J. Nordström, Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties (Springer, Berlin, 2015) 6. R.C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, vol. 12 (SIAM, Philadelphia, 2013) 7. M. Svärd, J. Nordström, Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014) 8. X. Wan, G.E. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209, 617–642 (2005) 9. D. Xiu, G.E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)

Nonlinear Flux Approximation Scheme for Burgers Equation Derived from a Local BVP J. H. M. ten Thije Boonkkamp, N. Kumar, B. Koren, D. A. M. van der Woude, and A. Linke

Abstract We present a novel flux approximation scheme for the viscous Burgers equation. The numerical flux is computed from a local two-point boundary value problem for the stationary equation and requires the iterative solution of a nonlinear equation depending on the local boundary values and the viscosity. In the inviscid limit the scheme reduces to the Godunov numerical flux.

1 Introduction The viscous Burgers equation is a well-known model problem in fluid dynamics, describing the nonlinear convection-diffusion balance for incompressible flow. It was introduced by Burgers in [1] to study turbulence in incompressible flow, and since then it was investigated in many publications. Solutions of the Burgers equation can exhibit steep interior/boundary layers when the viscosity is small. In the inviscid limit, discontinuous solutions (shock waves) can develop, even when the initial condition is smooth. This puts severe restrictions on the (spatial) discretisation. We pursue to construct a scheme that has the following properties. First, for positive viscosity (bounded below by a positive constant), the scheme should be second order accurate, second, it should not produce spurious oscillations in the vicinity of steep layers, and third, it has a three-point coupling.

J. H. M. ten Thije Boonkkamp () · N. Kumar · B. Koren · D. A. M. van der Woude Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands e-mail: [email protected]; [email protected]; [email protected]; [email protected] A. Linke Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_96

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For space discretisation we adopt the finite volume method, hence we need flux approximation schemes. For the inviscid equation, Godunov’s flux approximation scheme is very well known. The expression for this flux is derived from a local initial value problem, assuming a piecewise constant initial condition, the so-called Riemann problem. The Godunov scheme is a basic scheme which is only first order accurate. It can be combined with, for example, (W)ENO reconstruction and higher order Runge-Kutta time integration methods to achieve higher order. The viscous term is usually approximated by central differences. In this contribution we follow a different approach. We compute the numerical flux from a local two-point boundary value problem (BVP) for the steady, viscous Burgers equation, taking into account the fully nonlinear convection-diffusion balance. Our motivation is to derive an extension of the Godunov numerical flux for the viscous equation. This way we anticipate that the numerical solution inherits properties of the exact solution. For example, in Sect. 3 we will show, under certain conditions, that the resulting numerical scheme is monotone, implying that no spurious oscillations are generated. Moreover, we will show that our numerical flux reduces to the Godunov flux for the inviscid equation. The nonlinear scheme is inspired by the complete flux scheme presented in [6]. The complete flux scheme is a flux approximation scheme for (linear) conservation laws of advection-diffusion-reaction type. The basic idea of the complete flux scheme is to compute the numerical flux from the entire equation, including the source term. Therefore, the numerical flux is the superposition of a homogeneous flux, corresponding to the advection-diffusion operator, and an inhomogeneous flux, taking into account the effect of the source. We like to emphasize that our flux approximation scheme corresponds to the fully nonlinear convection-diffusion operator, and as such can be considered as a modification of the homogeneous flux scheme. We have organized the paper as follows. In Sect. 2 we present the local BVP defining the numerical flux. Next, in Sect. 3 we give the derivation of the numerical flux. A numerical example is given in Sect. 4 and concluding remarks are given in Sect. 5.

2 Local BVP for the Numerical Flux In this section we present the fully nonlinear BVP from which we derive the numerical flux (function). Consider the one-dimensional viscous Burgers equation and corresponding flux f (u, ux ), i.e., ut + f (u, ux )x = 0,

f (u, ux ) = 12 u2 − νux ,

(1)

Nonlinear Flux Approximation Scheme for the Burgers Equation

1017

where ν ≥ 0 is the viscosity. Equation (1) for u = u(x, t) is defined on, say, (0, 1)× [0, ∞). For space discretisation we employ the finite volume method [2]. Let Δx = 1/(N − 1) be the grid size, and let xj = (j − 1)Δx (j = 1, 2, . . . , N) and xj +1/2 = 1 2 (xj +xj +1 ) (j = 1, 2, . . . , N −1) denote the grid and interface points, respectively. Integrating (1) over a control volume Ωj = (xj −1/2 , xj +1/2 ), we obtain the semidiscretisation Δx u˙ j (t) + Fj +1/2 (t) − Fj −1/2 (t) = 0,

(2)

where u˙ j (t) ≈ ut (xj , t) and Fj +1/2 (t) ≈ f (u, ux )(xj +1/2 , t) is the numerical flux at x = xj +1/2. In the following we suppress the dependency on t. The derivation of Fj +1/2 is based on the following local two-point BVP: fx =

1

2u

2

− νux

 x

u(xj ) = uj = uL ,

= 0,

xj < x < xj +1 ,

(3a)

u(xj +1 ) = uj +1 = uR ,

(3b)

which we obtain from (1) ignoring the time derivative. In the derivations that follow, it is convenient to normalize the spatial coordinate x in (3). Therefore, we introduce the variables w(σ ) = u(x),

σ =

x − xj , Δx

ε=

ν , Δx

(4)

and can rewrite the BVP (3) as follows 1

2w

2

 − εw = 0,

w(0) = uL ,

0 < σ < 1,

w(1) = uR ,

(5a) (5b)

where the prime ( ) denotes differentiation with respect to σ . From this BVP we will derive expressions for the numerical flux (function) Fj +1/2 = F (uL , uR ; ε). Integrating the ODE (5a) twice and applying the boundary conditions (5b) we can derive the following (implicit) representation of the solution [3] Λ(σ ) λ(σ ) , w (σ ) = (uR − uL ) , Λ(1) Λ(1)  σ  w(η) dη > 0, Λ(σ ) = λ(ξ ) dξ.

w(σ ) = uL + (uR − uL ) λ(σ ) = exp

1  ε

σ 0

(6a) (6b)

0

From this representation it is evident that w is monotone, more precisely, w is monotonically decreasing if uL > uR and monotonically increasing if uL < uR . We will employ this property in the next section.

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3 Numerical Flux Function In this section we derive expressions for the numerical flux from the BVP (5). A similar derivation is given in [5], however, in this contribution we give a different representation of the numerical flux, which provides more insight. Further, we include a brief discussion on monotonicity and order of convergence. Because of the monotonicity property of w, we distinguish the two cases uL ≥ uR and uL < uR .

3.1 The Case uL ≥ uR The solution of BVP (5) satisfies w (σ ) ≤ 0, hence w is monotonically decreasing. We define the numerical flux Fj +1/2 by the relation Fj +1/2 = 12 w2 − εw = 12 c2 > 0,

(7)

where w is the solution of BVP (5). Obviously, Fj +1/2 ≥ 0. Therefore, we can write   Fj +1/2 = 12 c2 for some unknown c satisfying |c| > c+ = max |uL |, |uR | . The latter inequality readily follows if we substitute σ = 0 or σ = 1 in (7). Relation (7) is equivalent to the following first order ODE for w:  1 2 dw = w − c2 . dσ 2ε

(8)

Integrating (8) across (0, 1) and applying the boundary conditions (5b) we find for c the algebraic relation  1 + z(c) 

c H (c) = log = , 1 − z(c) ε +



 uL − uR c z(c) = 2 , c − uL uR

(9)

where 0 < z(c) < 1; see Fig. 1. H + (c) and c/ε are odd functions of c, consequently Eq. (9) has the trivial root c = 0, which we discard. Since Fj +1/2 = 12 c2 is even, we restrict ourselves to c > c+ . We can prove that H + (c) = c/ε has a unique solution c > c+ , which we compute by Newton iteration. If we choose an initial guess c0 > c+ the iteration will always converge. Note that Eq. (9) implicitly defines c = c(uL , uR ). Invoking the implicit function theorem, we conclude that ∂c/∂uL > 0 and ∂c/∂uR < 0 for c > c+ , implying that ∂Fj +1/2 > 0, ∂uL

∂Fj +1/2 < 0. ∂uR

Nonlinear Flux Approximation Scheme for the Burgers Equation 5

+

H (c) c/ c=c+

4 3

+

H (c)

1019

2 1 0 0

2

4

6

8

10

c

Fig. 1 Intersection points of the function H + (c) with c/ε, restricted to c > c+ . Parameter values are: uL = 2, uR = 1 and ε = 0.2, 1, 10 5 4

c=c

-

2

1

0.5

1 0 0

-

H (c) c/(2 )

1.5 H (c)

3

+

H (c)

2

+

H (c) c/

2

4

c

6

8

10

0

0

5 c

10

Fig. 2 Intersection points of H + (c) and c/ε (left), restricted to 0 < c < c− , and H − (c) and c/(2ε) (right). Parameter values are: uL = 1, uR = 3 and ε = 0.2, 1, 10

When combined with the explicit Euler time integration method, with a time step small enough to satisfy the stability condition, the resulting generalization of Godunov’s scheme remains monotone.

3.2 The Case uL < uR In this case w (σ ) > 0 and w is monotonically increasing. Again, the numerical flux is defined by Fj +1/2 = 12 w2 − εw with w the solution of BVP (5), however, this time we either have Fj +1/2 ≥ 0 or Fj +1/2 < 0. For a positive numerical flux we have Fj +1/2 = 12 c2 where c is the solution of (9) satisfying |c| < c− = min |uL |, |uR | ; see Fig. 2. This inequality is again a consequence of Eq. (7) if we

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substitute σ = 0 and σ = 1. On the other hand, for a negative flux we have Fj +1/2 = 12 w2 − εw = − 12 c2 < 0,

(10)

for the unknown c. From (10) we can readily derive the ODE:  1 2 dw = w + c2 . dσ 2ε

(11)

Integrating this equation across (0, 1) and applying the boundary conditions (5b) we obtain the relation  c  π  c   c − arctan + sgn(uR ) − sgn(uL ) = , (12) H − (c) = arctan uL uR 2 2ε see Fig. 2. In the derivation of (12) we used the identity arctan(z) + arctan(1/z) = π − 2 sgn(z) for z = 0. Analogous to the previous case, H (c) and c/(2ε) are odd and 1 2 Fj +1/2 = − 2 c is even, therefore we restrict ourselves to c > 0. For positive c Eq. (12) has a unique solution, which we compute with Newton iteration.

3.3 Choice of the Numerical Flux In the case uL < uR we can have either a positive or a negative flux, so we have to determine its sign. To that purpose we investigate the zero-flux condition, i.e., we solve Eq. (7) for c = 0 subject to the boundary conditions (5b). This way we find the condition 1 1 1 . − = uL uR 2ε

(13)

In [4] it is proven that, in case uL uR > 0, the numerical flux Fj +1/2 > 0 for 1/uL − 1/uR < 1/(2ε), and Fj +1/2 < 0 for 1/uL − 1/uR > 1/(2ε). This situation is displayed in Fig. 2. For ε = 0.2 Eq. (9) has a unique root 0 < c < c− and consequently Fj +1/2 > 0. In the other two cases Eq. (12) has a unique solution c > 0 and Fj +1/2 < 0. The black lines are tangent to either H + (c) or H − (c) at c = 0 and correspond to the zero-flux condition (13). Otherwise, for uL uR < 0, the numerical flux Fj +1/2 < 0. This situation is shown in Fig. 3. Clearly, in this case only Eq. (12) has a solution. Putting everything together, we obtain the following expressions for the numerical flux: • if uL ≥ uR Fj +1/2 = 12 c2 ,

H + (c) =

c ε

(c > c+ ).

10

4

8

3.5

6

3

4

2.5

H (c)

2 0

1021

-

H (c) c/(2 )

2

-

+ H (c)

Nonlinear Flux Approximation Scheme for the Burgers Equation

1.5

-2

1

+

H (c) c/

-4 -6

c=c 0

2

4

6

8

0.5

-

0

10

0

5 c

c

10

Fig. 3 (Possible) intersection points of H + (c) and c/ε (left) and H − (c) and c/(2ε) (right). Parameter values are: uL = −2, uR = 1 and ε = 0.2, 1, 10

• if uL < uR and uL uR > 0 ⎧ ⎪ ⎪− 12 c2 if ⎪ ⎪ ⎨ 1 2 Fj +1/2 = if 2c ⎪ ⎪ ⎪ ⎪ ⎩ 0 if

1 uL

−

1 uR

>

1 2ε ,

H − (c) =

c 2ε ,

1 uL

−

1 uR


0. Note however, that in the limit ν = 0 Eq. (12) has the solution c = 0, giving Fj +1/2 = 0, which is in agreement with Godunov’s flux. Second, we consider the diffusive limit (ν → ∞), restricting ourselves to uL ≥ uR . Note that z(c) → 0 for ν → ∞. Applying the approximation H + (c) ≈ 2z(c) for |z(c)|  1, we obtain from Eq. (9) that c2 − uL uR ≈ 2ε(uL − uR ). For the numerical flux we then find Fj +1/2 ≈ 12 uj uj +1 −

 ν  uj +1 − uj , Δx

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0.8

=0.1

0.6 0.4

=1 F0

-0.4 -0.6 -0.8 -1 -1

-0.5

0 u

0.5

1

L

Fig. 4 Sign of the numerical flux function. The contours denote the zero-flux condition for several values of ε

which is a second order accurate approximation of f (u, ux ). These limit cases are in agreement with the observation that the scheme exhibits first order convergence for small ε, gradually increasing to second order for larger ε [5].

4 Numerical Example In this section we present an example of our scheme; convergence tests have been reported in [5]. We apply our numerical scheme to Eq. (1) subject to the initial condition u(x, 0) = 1 + sin(2πx) for 0 < x < 1. We compute numerical solutions for ν = 10−3 . The initial smooth profile steepens and a typical solution is displayed in Fig. 5. We compute the numerical solutions for Δx = 1/20, 1/80, 1/320 and compare these with a very accurate reference solution. For time integration we use a third order stability preserving Runge-Kutta method with time step Δt = 10−5 , hence the temporal discretisation error is negligible. Newton iteration for Eq. (9) or (12) is terminated if both the residual and the update (in absolute value) drop below the tolerance value 10−12 . Clearly, the steep layer is nicely approximated for decreasing grid size.

Nonlinear Flux Approximation Scheme for the Burgers Equation

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Fig. 5 Snapshot of a numerical solution of the Burgers equation at t = 0.41

5 Concluding Remarks We have presented a novel flux approximation scheme for the viscous Burgers equation, derived from a local two-point boundary value problem for the stationary equation. The scheme is fully nonlinear, and requires the iterative solution of a nonlinear equation, for which we use Newton iteration. In the limit of zero viscosity, the numerical flux reduces to the Godunov flux. First numerical results are encouraging, however, more numerical tests are required. As possible extensions, we mention, first, inclusion of the time derivative in the flux approximation, and second, extension to two-dimensional equations.

References 1. J.M. Burgers, A Mathematical Model Illustrating the Theory of Turbulence (Academic Press, New York, 1948) 2. R. Eymard, T. Gallouët, R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, ed. by P.G. Ciarlet, J.L. Lions, vol. VII (North-Holland, Amsterdam, 2000), pp. 713–1020 3. R. Eymard, J. Fuhrmann, K. Gärtner, A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems. Numer. Math. 102, 463–495 (2006) 4. N. Kumar, Flux approximation schemes for flow problems using local boundary value problems, PhD Thesis, Eindhoven University of Technology, 2017 5. N. Kumar, J.H.M. ten Thije Boonkkamp, B. Koren, A. Linke, A nonlinear flux approximation scheme for the viscous Burgers equation, in Finite Volumes for Complex Applications VIII – Hyperbolic, Elliptic and Parabolic Problems, ed. by C. Cances, P. Omnes (Springer, Switzerland, 2017), pp. 457–465 6. J.H.M. ten Thije Boonkkamp, M.J.H. Anthonissen, The finite volume-complete flux scheme for advection-diffusion-reaction equations. J. Sci. Comput. 46, 47–70 (2011)

A Spectral Solenoidal-Galerkin Method for Flow Past a Circular Cylinder Hakan I. Tarman

Abstract Flow past a circular cylinder embodies many interesting features of fluid dynamics as a challenging fluid phenomenon. In this preliminary study, flow past a cylinder is simulated numerically using a Galerkin procedure based on solenoidal bases. The advantages of using solenoidal bases are twofold: first, the incompressibility condition is exactly satisfied due to the expansion of the flow field in terms of the solenoidal bases and second, the pressure term is eliminated in the process of Galerkin projection onto solenoidal dual bases. The formulation is carried out using a mapped nodal Fourier expansion in the angular variable while a modal polynomial expansion is used in the radial variable. A variational approach to recover the pressure variable is also presented. Some numerical tests are performed.

1 Introduction In the numerical modeling of the incompressible flow phenomena, the divergencefree condition stands as an important source of difficulty. Yet, another related issue is the numerical handling of the pressure variable appearing in the governing NavierStokes equations that usually comes without any boundary conditions. There are schemes developed solely to satisfy the continuity equation such as the fractional step scheme in [1] and the influence-matrix method in [2]. However, this can only be achieved to a limited degree of accuracy. In this work, a solenoidal spectral representation for the flow field is used in a Galerkin approach. As a consequence, the incompressibility and boundary conditions on the flow field are strictly enforced. The pressure term is then

H. I. Tarman () Middle East Technical University, Mechanical Engineering Department, Ankara, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_97

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eliminated in the Galerkin projection procedure onto a solenoidal dual space. There have been various works utilizing solenoidal spectral expansions. Moser et al. [3] presented a spectral method to automatically satisfy the continuity equation and boundary conditions and tested their method on the channel flow and the flow between concentric cylinders. They expanded the vertical and horizontal extents with Chebyshev polynomials and Fourier series, respectively. Kessler [4] studied steady and oscillatory regimes of Rayleigh-Benard convection with explicitly constructed solenoidal bases based on poloidal-toroidal decomposition. Trigonometric polynomials and the beam functions were used in the construction of the solenoidal bases satisfying the boundary conditions in a rectangular container. Clever and Busse [5] used toroidal-poloidal expansion in their numerical approach satisfying the solenoidal condition exactly; however, the procedure for eliminating the pressure leads to higher order derivatives. A related Galerkin approach to study three dimensional flow around a circular cylinder is presented in [6] where modal expansions are used in all three directions with Laguerre polynomials to resolve the infinite extent in the radial direction. Most recently Meseguer and Trefethen [7] proposed a spectral Petrov-Galerkin formulation based on divergence-free bases in terms of Chebyshev polynomials to study stability of pipe flow. Another recent study utilizing solenoidal bases is also the study of pipe flow in [8]. A mathematical analysis of the Solenoidal-Galerkin approach for the Stokes problem can be found in [9]. In this study, uniform flow past a circular cylinder is numerically simulated by using a Solenoidal-Galerkin procedure. The flow configuration consists of a simple geometry of cylinder with simple no-slip boundary conditions on the cylinder. However, the resulting flow field shows strong inhomogeneities in the wake region developing behind the cylinder with many different mechanisms at play. It is also physically relevant because this type of flow is an everyday phenomenon that can be observed when objects move in a fluid medium such as air and water. There is a vast literature on this subject since it is one of the classical problems of fluid mechanics. The text [10] on the flow around circular cylinders may be a good starting point. In this work, a concise numerical approach is presented to provide high accuracy for a subsequent study of stability and transition. For this purpose, a mapped nodal Fourier representation is used in the angular direction in order to concentrate the otherwise homogeneously distributed Fourier nodes in the wake region while modal polynomial representation based on Legendre polynomials is used in the radial direction to ensure resolution of the boundary layer around the cylinder [11].The use of Legendre polynomials with their natural interval of definition −1, 1 , of course, requires the truncation of the computational domain. This is an issue that we would like to explore further in a future study. A variational approach to recover the pressure that is vanished in the process of Galerkin projection onto dual solenoidal basis is also presented.

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2 Governing Equations The steady Navier-Stokes (N-S) equations in polar coordinates are ∇ · v = 0,   1 2 ∂p 2 ∂vθ 1 1 2 + ∇ vr − 2 − 2 vr , v · ∇vr − vθ = − r ∂r Re r ∂θ r   1 1 1 1 ∂p 2 ∂vr 2 + − 2 vθ . v · ∇vθ + vr vθ = − ∇ vθ + 2 r r ∂θ Re r ∂θ r

(1)

where Re is the Reynolds number based on the cylinder radius and the free-stream velocity U∞ that drives the flow v = (vr , vθ ), in the form v = u0 + u with the flow field u superimposed over the basic mode u0 subject to the boundary conditions, u(r = 1, θ ) = 0, limr→∞ u(r, θ ) = 0, u(r, θ + 2π) = u(r, θ ).

(2)

The basic mode u0 is given by  u0 (x) = ∇ × ψ0 ez =

 ∂ψ0 1 ∂ψ0 ,− ,0 , r ∂θ ∂r

(3)

with     r −1 1 ψ0 (x) = r − 1 − exp − sin (θ ), r δ

(4)

√ where δ = α/ Re and α = 4 following [6]. Here x = (r, θ ). It can be shown the basic mode satisfies the free-stream flow condition limr→∞ u0 (r, θ ) = (cos θ, sin θ ) at infinity.

3 Numerical Formulation This system is approximated using a Solenoidal-Galerkin approach. The SolenoidalGalerkin projection procedure starts with the expansion of the flow field u in terms of solenoidal expansion functions U q (x) which are generated from scalar expansion functions ψq (x) by U q (x) = ∇ × ψq ez

(5)

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that form the scalar field in the form

ψ(x) = ψˆ q ψq (x) = ψˆ q Pm (r)Φn (θ ) q

(6)

q

for the index vector q = (m, n), so that they satisfy ∇ · U q = 0 and the boundary conditions (Eq. (2)). The projection procedure proceeds with the weighted inner product    U p, U q = ω

0



2π

dθ

∞

U p · U q ω r dr

(7)

1

onto the dual space spanned by U p. It can be  shown that the pressure term in Eq. (1) vanishes under the projection U p , ∇p = 0 provided that dual expansion ω   functions U p satisfy ∇ · ω U p = 0 and the boundary conditions U p (r = 1, θ ) = 0, limr→∞ r ω U p (r, θ ) = 0.

4 Numerical Procedure The infinite domain is truncated at r = R and boundary condition at infinity is approximated by U q (r = R, θ ) = 0. The radial bases in the expansion (Eq. (6)), Pm (r) are then selected as polynomials with m associated with its degree and are required to satisfy the following boundary conditions Pm (1) = Pm (1) = 0 and Pm (R) = Pm (R) = 0. A suitable choice is Pm (r) = (x 2 − 1)2 Lm (x)

(8)

2 where Lm (x) denotes Legendre polynomial of order m in x(r) = R−1 (r − 1) − 1 for −1 ≤ x ≤ 1 associated with high accuracy Gauss quadrature. The angular basis in the expansion, Φn (θ ), on the other hand, are the cardinal functions in the Fourier space of 2π-periodic functions. The uniform nodal distribution associated with the cardinal functions, however, is not suitable for the physical problem where most of the activity in the computational domain occurs in the sector behind the cylinder

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according to the flow direction. For this purpose we use modified cardinal functions   Φn (θ ) = Sn F −1 (θ ) − F −1 (θn ) where  − α ) n 1   . Sn (α − αn ) = N + 1 sin 1 (α − αn ) 2 

sin

N+1 2 (α

(9)

and the map θ = F (α) is based on a Fractional Linear transformation [12] exp (iθ ) =

exp (i(α − γ )) − β . 1 − β exp (i(α − γ ))

The choice of γ = 0 helps concentrating on the wake region behind the cylinder while |β| < 1 controls the density of the nodes in that region. The numerical evaluation of the inner product integrals in the projection procedure is performed using quadrature integration, namely, Legendre-Gauss quadrature 

1 −1

f (x)dx =

K

4k f (xk )

k=0

with the quadrature nodes xk and weights 4k , and the Fourier quadrature formula 1 2π



2π 0

1 1 g(F (α)) dα = g(θn ) = N +1 2π N

n=0

 0

2π

  g(θ ) F −1 (θ ) dθ ) *+ , ω(θ)

with the mapped quadrature nodes θn = F (αn ) and αn = 2πn/(N + 1). Thus, the inner product integrals    U p, U q = ω



2π 0

U p · U q r dr

(10)

1

R − 1

4k U p (rk , θn ) · U q (rk , θn )rk 2 N

=

∞

ω(θ ) dθ K

n=0 k=0

where rk = r(xk ) = R−1 2 (xk + 1) + 1, can be evaluated exactly for the functions U p and U q associated with the expansion in Eq. (6) provided that the degree m in Eq. (8) satisfies m < K − 4. The nonuniform spatial distribution of the radial nodes rk and the mapped Fourier nodes θn is providing added spatial resolution where it is needed the most as shown in quadrature node distribution over the domain in Fig. 1.

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Fig. 1 The quadrature node distribution for β = −0.4 and γ = 0 with R = 10

The truncated representation of the stream function ψ(x) =

M N

ψˆ q Pm (r)Φn (θ )

m=0 n=0

is used to construct the solenoidal velocity field u(x) = ∇ × ψez superimposed over the basic mode u0 to construct the velocity field v = u0 + u and, in turn, is substituted into the N-S equations that are linearized around the basic mode and then projected   1 L (u) U p , N (u, u0 ) + N (u0 , u) − = Re ω   1 L (u0 ) − N (u0 , u0 ) U p, Re ω where 

v · ∇vr − 1r vθ2 N (v, v) = v · ∇vθ + 1r vr vθ



(11)

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and 

∇ 2 vr − L (v) = ∇ 2 vθ +

2 r2 2 r2

∂vθ ∂θ ∂vr ∂θ

− −

1 v r2 r 1 v r2 θ

 .

The procedure in Eq. (11) can be used iteratively to obtain an improved solution v = u0 + u by repeatedly updating u0 .

5 Pressure Calculation The pressure field, where it vanished in the Galerkin projection operation onto solenoidal bases in the previous section, can be recovered by using the velocity field v = (vr , vθ ) computed above. In order construct a variational formulation, the pressure field p(r, θ ) is expanded in the form p(r, θ ) =

pˆ q Lm (x)Φn (θ )

(12)

q

with q = (m, n) where Lm (x) denotes Legendre polynomial of order m in x(r) = 2 R−1 (r − 1) − 1 for −1 ≤ x ≤ 1. The linearized equations are then projected 

V , −∇p

 ω

  1 L(v) = V , N (v, v) − Re ω

(13)

onto the space V ∈ V×V spanned by V = span{Πm (r)×Φn (θ )} with Πm (r(x)) = (x 2 − 1)Lm (x).

6 Results In order to demonstrate the effectiveness of the approximation over the mapped Fourier nodes θn in comparison tothe uniform nodes αn we used a test function   f (θ ) = sin 2 tan−1 8 tan (θ/2) exhibiting steep behavior near θ = 0. The comparison of the truncation error f − fN ∞ in the truncated representations fN (θ ) =

N

  f (θn )SN F −1 (θ ) − F −1 (θn )

n=0

and fN (α) =

N

n=0

f (αn )SN (α − αn )

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100

10−2

10−4

10−6 norm( f (α) − fN (α) , inf )

10

norm( f (θ) − fN (θ) , inf )

−8

10−10

0

10

20

30

40

50

60

70

N

Fig. 2 The inf-norm of the truncated representation error for the uniform and the mapped nonuniform Fourier representations

Fig. 3 The computed flow field at Re = 1 (left) and Re = 5 (right) for a resolution of N = 24 and K = 20 with R = 10

in Fig. 2 shows several orders of magnitude improvement in the representation using the mapped cardinal functions. Some limited number of numerical experiments are performed for Re = 1 and Re = 5 with R = 10 to test the numerical formulation, the implementation and to develop a preliminary code. The resulting flow fields to the first approximation are shown in Fig. 3. The flow is developing a wake region behind the cylinder downstream. In this preliminary work, our objective is being to formulate and implement the numerical procedure presented. For this purpose the linearized and steady

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form of the governing Navier-Stokes equations are used. This limits the numerical experiments to small Re values at higher values of which nonlinearity becomes important. Our future objective is to use the numerical approach that is developed in this work, to perform a parametric study of transition where the accuracy and the effectiveness of the numerical approach become crucial. Another issue to be studied and to be improved upon as a future study is the truncation of the infinite domain.

References 1. S.A. Orszag, L.C. Kells, Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96(1), 159–205 (1980) 2. L. Kleiser, U. Schumann, Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows, in Proceedings of the 3rd Conference on Numerical Methods in Fluid Mechanics, ed. by E.H. Hirschel (1980), pp. 165–173 3. R.D. Moser, P. Moin, A. Leonard, A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow. J. Comput. Phys. 52(3), 524–544 (1983) 4. R. Kessler, Nonlinear transition in three-dimensional convection. J. Fluid Mech. 174, 357–379 (1987) 5. R.M. Clever, F.H. Busse, Nonlinear oscillatory convection. J. Fluid Mech. 176, 403–417 (1987) 6. B.R. Noack, H. Eckelmann, A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder. Phys. Fluids 6(1), 124 (1994) 7. A. Meseguer, L.N. Trefethen, Linearized pipe flow to Reynolds number 107 . J. Comput. Phys. 186(1), 178–197 (2003) 8. O. Tugluk, H.I. Tarman, Direct numerical simulation of pipe flow using a solenoidal spectral method. Acta Mech. 223(5), 923–935 (2012) 9. A.F. Pasquarelli, A. Quarteroni, G. Sacchi-Landriani, Spectral approximations of the Stokes problem by divergence-free functions. J. Sci. Comput. 2(3), 195–226 (1987) 10. M.M. Zdravkovich, Flow Around Circular Cylinders 1: Fundamentals (Oxford Science Publications, Oxford, 1997) 11. J.S. Hesthaven, D. Gottlieb, Spectral Methods for Time-Dependent Problems (Cambridge University Press, Cambridge, 2007) 12. J.M. Augenbaum, An adaptive pseudospectral method for discontinuous problems. Appl. Numer. Math. 5, 459–480 (1989)

Conservative Mimetic Cut-Cell Method for Incompressible Navier-Stokes Equations René Beltman, Martijn Anthonissen, and Barry Koren

Abstract We introduce a mimetic Cartesian cut-cell method for incompressible viscous flow that conserves mass, momentum, and kinetic energy in the inviscid limit, and determines the vorticity such that the global vorticity is consistent with the boundary conditions. In particular we discuss how the no-slip boundary conditions should be applied in a conservative way on objects immersed in the Cartesian mesh. We use the method to compute the flow around a cylinder. We find a good comparison between our results and benchmark results for both a steady and an unsteady test case.

To compute fluid flow in complicated geometries often curvilinear or unstructured meshes are used. The generation of these meshes is difficult and time-consuming. When the geometry depends on time, the mesh has to be updated after every time step and the cost of mesh generation will take a significant part of the total computing time. Immersed boundary methods form an increasingly popular alternative. Immersed boundary methods are methods in which one Cartesian mesh is used for the complete flow domain, with the boundaries of objects immersed in this Cartesian mesh. Near the immersed boundary the Cartesian method is adapted for the no-slip boundary condition. Within the class of immersed boundary methods essentially two approaches for modeling the boundary conditions exist. In the first approach the influence of the boundary on the fluid is modeled by an extra force term in the Navier-Stokes equations. In the second approach the sharp interface of the boundary is maintained and the boundary condition is taken into account by adjusting the discretized NavierStokes equations.

R. Beltman () · M. Anthonissen · B. Koren Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7_98

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Cut-cell methods follow the second approach. Close to the object non-Cartesian cells occur, the so-called cut cells. These cells demand a special treatment because of their more difficult shape and the no-slip boundary condition on one of their faces. One of the most popular discretization methods for the incompressible NavierStokes equations on Cartesian meshes is the MAC-method [1]. It uses a staggered mesh, which means that the velocity variables are located on the faces of the Cartesian cells and the pressure variables are located in the centers of the cells. From this variable positioning, through central difference approximations a method follows that has a compact stencil, no spurious pressure oscillations, conservation of both mass and momentum as well as conservation of secondary quantities as vorticity and energy (in the inviscid limit). Most cut-cell methods use a colocated mesh, where the velocity and pressure variables are all located in the centers, which makes the treatment of cut cells simpler. However, methods on colocated meshes do not have all the favorable properties that the MAC method has. For example, spurious pressure oscillations have to be suppressed by introducing artificial diffusion, making these methods less suitable for turbulent flow computations. A few extensions of the MAC method to cut-cell meshes have been presented in the literature [2, 3]. In these methods different cut-cell configurations are treated case by case and a mass, momentum and energy conserving extension is derived using, in 2D, a 5-point stencil and a finite-volume rationale. However, achieving a fully conservative method using the 5-point stencil is impossible. Moreover, an extension of the method to 3D is problematic due to the many possible cut-cell configurations. To our knowledge only a quasi-3D extension, where the immersed boundary is parallel to one of the Cartesian coordinate axes, has been published so far [4]. We will use the recent developments in mimetic discretization methods [5], that allow for a conservative treatment of non-Cartesian cells. Using these methods on a primal-dual mesh structure, upon completing the dual mesh near the boundary, we derive a mass, momentum and energy conserving cut-cell method that calculates vorticity in such a way that the global vorticity is consistent with no-slip boundary conditions. We first introduce the method and then present results for the flow around a cylinder.

1 The Mimetic Cut-Cell Method 1.1 The Cut-Cell Primal-Dual Cell-Complex We discuss the cut-cell mesh for the specific case of a cylinder in a rectangular domain. From this discussion it will be clear how it can be generalized to other domains. We give the cylinder its own discrete representation, independent of the

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Fig. 1 On the left the Cartesian mesh (blue) and the discretized cylinder (green) are shown. In the middle one sees the cylinder with extra intersection vertices added. Finally, on the right the resulting primal mesh is shown

Cartesian mesh, because in a future extension to general time-dependent geometries this will be needed for mass conservation. We cover the rectangular domain by a Cartesian mesh with Nx × Ny cells. For simplicity we take the mesh to be uniform. We discretize the boundary of the cylinder by uniformly taking Nθ points on the circle and connecting these by straight line segments. Subsequently, we immerse the discretized cylinder in the Cartesian mesh, add the intersections between edges of the Cartesian mesh and the cylinder as extra vertices, and remove all the edges and vertices in the interior of the cylinder. This process has been depicted in Fig. 1. The resulting computational mesh consists of two-dimensional cells, onedimensional edges and zero-dimensional vertices. We denote the mesh by G := {C(0), C(1) , C(2) }, where C(k) , for k = 0, 1, 2, is the set of k-dimensional cells. This mesh is a so-called cell-complex. This means that for each k-dimensional cell σ(k) ∈ C(k) in the mesh its boundary ∂σ(k) is made up of lower dimensional cells that are also part of the mesh. The cell-complex G covers the flow domain Ω. Our cut-cell method will also need a dual mesh. The dual mesh is a second mesh for the flow domain that is geometrically dual to the primal mesh in the sense that for each k-dimensional cell σ(k) ∈ C(k) of the primal mesh, there exists a (2 − k)dimensional dual cell σ˜ (k) ∈ C˜(k) in the dual mesh, where we denote by C˜(k) the set of (2 − k)-dimensional dual cells. We use a barycentric dual mesh. In contrast to the primal mesh, the dual mesh G˜ = {C˜(2), C˜(1) , C˜(0) } is not a cell-complex. However, to formulate a fully conservative cut-cell method it is important to extend the dual mesh to a cell-complex. This can be done as follows. We consider the restriction of the b , C b }, and then take primal mesh to the boundary ∂Ω of Ω, denoted by G b = {C(0) (1) the dual mesh to the primal mesh within ∂Ω, which we denote by G˜ b = {C˜b , C˜b }. (1)

(0)

The union of the original dual mesh and the boundary dual mesh, G¯ := G˜ ∪ G˜ b , constitutes a dual cell-complex. This construction works in arbitrary dimension. The construction of the dual cell-complex has been depicted in Fig. 2. More details concerning the primal and dual cell-complex can be found in [6].

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Fig. 2 On the left the primal (blue) and dual (red) mesh are depicted and in the middle the primal mesh and boundary dual mesh are shown. On the right one sees the dual cell-complex which is the union of the interior and boundary dual mesh

1.2 The Incidence Matrices We discretize the velocity field u as the fluxes through the edges of the primal mesh. For each edge σ(1) ∈ C(1) we set  uσ(1) :=

u · n dL,

(1)

σ(1) (1)

where n is the normal on σ(1) . Next, let u(1) = (uσ(1) ) be the vector of all these fluxes. We denote the finite linear space corresponding to unknowns on the edges by C (1), so u(1) ∈ C (1), and define C (0) (vertices) and C (2) (cells) analogously. Using the integral values as variables we can discretize the continuity equation ∇ · u = 0 without introducing a discretization error. To see this, we consider a twodimensional cell σ(2) ∈ C(2), integrate the continuity equation over σ(2) and apply the divergence theorem to obtain 0=

σ(1) ∈∂σ(2)

oσ(2) σ(1) u(1) σ(1) ,

(2)

where oσ(2)σ(1) = 1 if the orientations of σ(2) and σ(1) agree and −1 otherwise. For example, if we take σ(2) to be oriented such that the normal on ∂σ(2) points outward and the normal on σ(1) given by the orientation of σ(1) points into σ(2) , then oσ(2)σ(1) = −1. We can write Eq. (2) for all cells in C(2) together as 0(2) = D(2,1)u(1) , where 0(2) is the zero element of C (2) and D(2,1) : C (1) → C (2) is defined by Dσ(2,1) (2) σ(1) = oσ(2) σ(1) , where we extend the definition of oσ(2)σ(1) by 0 to the instances when σ(1) ∈ / ∂σ(2) . (1,0) (0) (1) (0) Similarly, we define an incidence matrix D : C → C . Let ω ∈ C (0) be a discretization of the 2D scalar vorticity field ω by evaluation on the vertices of the mesh, then the value of D(1,0)ω(0) for some edge σ(1) is, as a result of the

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fundamental theorem of calculus, given by  [D(1,0)ω(0) ]σ(1) =

rot ω · n dL = σ(1)

σ(0) ∈∂σ(1)

oσ(1)σ(0) ωσ(0) , (0)

(3)

with rot ω := (∂y ω, −∂x ω). We choose the orientation of the dual cells in relation to the orientation of their corresponding primal cells. From this it follows [6] that the incidence matrices on ˜ (1,2) : C˜(2) → C˜(1) and D ˜ (0,1) : C˜(1) → C˜(0) are given by ˜ i.e., D the dual mesh G, T T   ˜ (1,2) = D(2,1) and D ˜ (0,1) = D(1,0) . D For Eq. (3) to hold for all σ(1) ∈ C(1) it is crucial that the two boundary points that make up ∂σ(1) are in C(0). The dual mesh G˜ is not a cell-complex and therefore if p˜ (2) ∈ C˜(2) is, for example, a discretization of the continuous pressure field p on the vertices of the dual mesh, then  ˜ (1,2)p˜ (2) ]σ˜ = [D ∇p · dL (1) σ˜ (1)

only holds for an edge σ˜ (1) for which its end points are again part of the mesh, i.e., for an edge that is dual to a primal edge σ(1) that is not part of the boundary b . mesh C(1) To get a consistent discretization of the pressure gradient on all the dual edges ˜ (1,2) to G. ¯ This extension maps from C¯(2) to C¯(1) and we denote it by we extend D (1,2) (1,2) ¯ ¯ D . The part of D that maps from C¯(2) to C˜(1) will be used separately in the ¯ (1,2). It consists of an interior and boundary part, discretization and denoted by D i ˜ (1,2) I˜(1,2) ]. For more details on the construction of D ¯ (1,2) and, ¯ (1,2) = [D i.e., D i b ¯ (0,1), see [6]. analogously, D

1.3 The Discrete Hodge Operators So far, we only discussed the exact discretization of differential operators on both the primal and dual cell-complex in terms of the incidence matrices. These primal and dual incidence matrices map from C (k) to C (k+1) and from C¯(k+1) to C¯(k) , respectively. To be able to discretize higher order differential operators, like for example the vector Laplacian, and to end up with a square solvable system of equations we need an interpolation map between the primal and dual meshes. We introduce interpolation operators H(k) : C (k) → C˜(k) that interpolate a field discretized on the k-cells in C(k) to a consistent discretization of the same field on C˜(k) . For example, if u(1) ∈ C (1) is a discretization of the velocity field on primal edges as in (1), then u˜ (1) := H(1) u(1) is an approximation of the velocity field discretized on the dual edges in C˜(1) .

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We use the mimetic scalar product matrices [5], which can also be interpreted as interpolation operators [6]. These operators are constructed locally for each 2-cell and the global operator is constructed by an assembly process as in finite element methods. The operators are symmetric positive definite and diagonal in the Cartesian part of the mesh.

1.4 The Numerical Scheme We use the incidence matrices and discrete Hodge matrices to discretize the incompressible Navier-Stokes equations in space. The discrete variables are velocity, vorticity and pressure. As velocity variables we use the fluxes on the edges, i.e., u(1) ∈ C (1). The pressure variables are located on the vertices of the dual cellcomplex, i.e., p¯ (2) ∈ C¯(2). The vorticity variables will be located in the vertices of the primal mesh, i.e., ω(0) ∈ C (0) . However, as we will see shortly, we only need vorticity variables in the vertices of the non-Cartesian cut cells. We discretize the incompressible Navier-Stokes equations in the form ∂t u + ∇ · (u ⊗ u) + ν∇ × ω + ∇p = 0, with additionally ω = ∇ × u and the continuity equation ∇ · u = 0. We discretize the momentum equation on all edges of the dual mesh, i.e., we approximate the integral of the momentum equation over every dual edge. In summary, we discretize the spatial part as ⎤ ⎤ ⎡ ⎤⎡ (1) ¯ (1,2) ˜ (1) u H(1) ∂t + C[u(1) ] H(1)D(1,0) D 0 i ⎢ ⎥ ⎢ ˜(0,1) (0) ⎥ ⎥⎢ ⎥ ˜ (0,1) H(1) −ν −1 H(0) 0 ⎦⎣ ω(0) ⎦=⎢ D ⎣ ⎣ −Ib v˜ b ⎦. (1,2) (1) ¯ )T 0 0 (D p¯ (2) I(1,1) v b i ⎡

(4)

The Dirichlet boundary conditions are incorporated in the last line. The matrix ¯ (1,2))T = I˜(1,1) contains only entries equal to 1, 0 or, −1, and consists, just like (D i [(D(2,1))T I˜b(1,2) ]T , of two parts. The matrix I˜(1,1) is defined such that the equation ¯ (1,2))T u(1) = I˜(1,1)v (1) brings forth D(2,1)u(1) = 0(2) and (I˜(1,2))T u(1) = (D i b b (1,2) (1) (I˜b )T v b , i.e., the discrete incompressibility constraint and the definition of the Dirichlet boundary conditions, imposing that the fluxes for the boundary edges are (1) (1) equal to the prescribed values v b ∈ Cb . The no-slip boundary condition for the diffusive term is incorporated in the ˜(0) equation that defines the vorticity. The vector v˜ (0) b ∈ Cb contains the integrals of the tangential velocity over the boundary dual edges. The boundary term −I˜b(0,1)v˜ (0) b gives the contribution of boundary edges in the dual cell-complex to the vorticity integral over the dual cells. The entries of ω(0) are actually the vorticity variables

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multiplied with the viscosity ν. This definition of the variables increases the symmetry of the discrete system. ˜ (0,1) H(1)u(1) + The equation for the vorticity variables, i.e., H(0) ω(0) = ν(D (0,1) (0) I˜b v˜ b ), is explicit for variables in the Cartesian parts of the mesh, because the restriction of H(0) to these parts of the mesh is diagonal. Consequently, these variables become superfluous and are not used. As a result, the vorticity variables are only necessary in the cut cells. For convenience we left the corresponding adjustment of the discrete equations out of (4). For the convection term we use the central approximation introduced in [7], which conserves both momentum and kinetic energy. This discretization of the convection term fits well in the framework presented above [6].

2 Numerical Results: The Flow Around a Cylinder We consider the benchmark case [8] of the flow around an asymmetrically placed cylinder in a channel. On the inflow boundary a parabolic profile is prescribed, on the top and bottom of the channel no-slip boundary conditions apply and on the outflow boundary we impose zero stress. We test for two meshes that are uniform in the y-direction and non-uniform in the x-direction such that mesh lines are concentrated near the cylinder. We use the standard four-stage explicit Runge-Kutta method for the time integration. Results are shown for two test cases. The first case is a steady case with average inflow velocity corresponding to Re = 20. We compute the drag coefficient cD , the lift coefficient cL , the length of the recirculation zone La and the pressure difference between the front and back of the cylinder. The results are shown in Table 1. The second case is an unsteady periodic case with inflow corresponding to Re = 100. In this case we compute for one period the maximum drag coefficient cDmax , the maximum lift coefficient cLmax , the Strouhal number St, and the pressure difference between front and back of the cylinder halfway the period, where the start of the period coincides with the moment that cL (t) = cLmax . The results can be found in Table 2. In Fig. 3 the vorticity field is shown. Despite the fact that the meshes used are not body-conforming and have relatively few cells near the cylinder, characteristic values close to the benchmark values are found. Especially in the unsteady case the drag and lift coefficients are somewhat

Table 1 Results for the steady test case Nx × N y × N θ 160 × 130 × 30 300 × 255 × 60 [8] (lower bound) [8] (upper bound)

#u(1) 40,656 149,584 – –

#ω(1) 226 444 – –

#p˜ (2) 20,732 75,539 – –

cD 5.5876 5.5835 5.57 5.59

cL 0.0109 0.0122 0.0104 0.0110

La 0.0858 0.0849 0.0842 0.0852

Δp 0.1171 0.1178 0.1172 0.1176

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R. Beltman et al.

Table 2 Results for the unsteady test case Nx × N y × N θ 160 × 130 × 30 300 × 255 × 60 [8] (lower bound) [8] (upper bound)

30

0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

Fig. 3 The vorticity on the finer mesh in the unsteady case for cL (t) = cLmax

overestimated, however, the values corresponding to the finer mesh are already close to the benchmark interval. The fact that we find relatively accurate results despite the use of a non-optimal mesh and the low order of accuracy of the method in the cut cells, is to be attributed to the physical accuracy of the method. The method conserves mass, momentum and energy, even in cut cells, and determines a vorticity corresponding to a physically correct global vorticity. For proofs and numerical verifications of these properties see [6]. Acknowledgements This research is part of the EUROS program, which is supported by NWO domain Applied and Engineering Sciences and partly funded by the Ministry of Economic Affairs.

References 1. F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182–2189 (1965) 2. M. Dröge, R. Verstappen, A new symmetry-preserving Cartesian-grid method for computing flow past arbitrarily shaped objects. Int. J. Numer. Methods Fluids 47, 979–985 (2005) 3. Y. Cheny, O. Botella, The LS-STAG method: a new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties. J. Comput. Phys. 229, 1043–1076 (2010) 4. Y. Cheny, F. Nikfarjam, O. Botella, Towards a fully 3D version of the LS-STAG immersed boundary/cut-cell method, in Eighth International Conference on Computational Fluid Dynamics, paper ICCFD8-2014-0371 (2014)

Conservative Mimetic Cut-Cell method

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5. K. Lipnikov, G. Manzini, M. Shaskov, Mimetic finite difference method. J. Comput. Phys. 257, 1163–1227 (2014) 6. R. Beltman, M.J.H. Anthonissen, B. Koren, Conservative polytopal mimetic discretization of the incompressible Navier-Stokes equations. J. Comput. Appl. Math. 340, 443–473 (2018). https:// doi.org/10.1016/j.cam.2018.02.007 7. B. Perot, Conservation properties of unstructured staggered mesh schemes. J. Comput. Phys. 159, 58–89 (2000) 8. M. Schäfer, S. Turek, F. Durst, E. Krause, R. Rannacher, Benchmark computations of laminar flow around a cylinder, in Flow Simulation with High-Performance Computers II (Vieweg+ Teubner Verlag, Braunschweig, 1996), pp. 547–566

Index

A Aagaard, B.T., 654 Aanonsen, S.I., 73, 75 Aarnes, J.E., 77 Aavatsmark, I., 754 Abad, A., 738 Abdi, E.A., 381, 382 Abdulle, A., 510 Abels, H., 704, 705 Abgrall, R., 902, 1005 Abramowitz, M., 91, 207, 209 Absil, P.-A., 685, 686 Acharya, S., 862 Ackerer, P., 50, 52 Adams, R.A., 85 Adelsberger, J., 593, 595, 598, 600 Adler, A., 737 Adler, J.H., 546 Afif, M., 617, 622 Agrachev, A., 445 Aguilar, G., 646, 649 Ahmad, B., 715 Ahmed, E., 604 Alaraj, A., 36 Alboin, C., 167, 170 Ali Hassan, S., 604 Alla, A., 664, 941, 944 Allaire, G., 336, 510 Allasia, G., 94, 98, 99 Allen, S., 280, 339 Almani, T., 541 Almroth, B., 5 Alonso, J.J., 684 Alpkvist, E., 804

Alsaedi, A., 715 Alt, H.W., 50, 52 Altshiller-Court, N., 767 Alvarez, M., 113 Alvarez-Vázquez, L.J., 821, 832 Amaziane, B., 617, 622, 631 Ambrósio, J., 269, 270, 273 Amestoy, P.R., 31 Amielh, M., 860, 863 Amsallem, D., 683 Andrä, H., 295 Angermann, L., 626 Anselmet, F., 860, 863 Anthonissen, M.J.H., 1016, 1037, 1039–1042 Antonietti, P.F., 726, 731 Apel, T., 726, 770, 974, 978 Arara, A.A., 419–427 Arbenz, P., 226 Arbogast, T., 53 Arias, T.A., 683, 686 Aricó, A., 226 Arnaudon, M., 442 Arndt, D., 582, 588 Arnfield, A.J., 830 Arnold, D.N., 216, 260, 323, 467 Arnold, V.I., 312 Artebrant, R., 902 Attene, M., 160 Auckenthaler, T., 123 Augenbaum, J.M., 1029 Avril, S., 239 Axelsson, O., 230 Ayuso, B., 966, 969 Aziz, A.K., 769

© Springer Nature Switzerland AG 2019 F. A. Radu et al. (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2017, Lecture Notes in Computational Science and Engineering 126, https://doi.org/10.1007/978-3-319-96415-7

1045

1046 B Babuška, I., 5, 482, 483, 769, 877 Badics, Z., 146 Bae, S., 860, 863 Bai, C., 432 Balay, S., 31 Balázsová, M., 561–569 Ballani, J., 3–32 Ballard, P., 821, 840 Bampton, M., 5 Bangerth, W., 143, 217, 400, 547, 582, 588 Banholzer, S., 881, 884, 886, 887 Banjai, L., 146 Bänsch, E., 482, 487 Bardi, M., 944 Barenblatt, G.I., 70 Barnhill, R.E., 769 Barr, A.H., 810 Barrault, M., 13 Barrenechea, G.R., 909–911, 913, 914, 916 Barrett, J.W., 174 Barrios, T.P., 743 Barth, A., 891 Barucq, H., 145–151 Basseville, S., 840 Bassi, F., 498 Bastian, P., 193, 370 Batchelor, T., 69 Bathe, K.J., 519 Bäumer, D., 861 Bäumler, K., 593–600 Bause, M., 215, 541–544, 546–548, 551–559, 562, 566, 790 Bazilevs, Y., 875 Beamer, N., 36 Beavers, G., 804 Becker, R., 184, 400, 407 Becker, S., 420 Beermann, D., 881, 884, 886 Beirão da Veiga, L., 159, 502, 504, 519, 520, 523, 578, 715, 725, 726, 731 Belot, Y., 862, 863 Beltman, R., 302, 303, 305, 1037, 1039–1042 Belytschko, T., 193, 197 Bendsøe, M., 335 Benedetto, M., 721, 731 Benedetto, M.F., 501–508 Beneš, L., 860–863 Beneš, M., 279–286, 637 Benner, P., 270, 667, 685 Bensimon, D., 810 Benzi, M., 212 Bergamashi, N., 50, 52, 53 Berge, R., 573–580, 657

Index Berge, R.L., 653–660 Berkooz, G., 663, 708, 943 Berland, H., 420, 425 Bernardi, C., 995 Berninger, H., 604 Berre, I., 574, 653–660 Berrone, S., 501–508, 718, 721, 726, 731 Bertoluzza, S., 157–164 Bertout, J.A., 238 Besenghi, R., 94, 98, 99 Bey, J., 737 Biane, P., 441 Biegler, L.T., 833 Binev, P., 5, 19, 361, 362, 364, 472, 479 Biot, M.A., 215, 541, 646, 791 Bjørnerud, A., 68, 69 Black, F., 103, 688 Blackstock, D.T., 5, 8 Blanes, S., 439 Blum, H., 974 Blum, V., 123 Boas, D.A., 36 Böcker, J., 881, 883 Boffi, D., 184, 323, 325, 332, 698 Bokanowski, O., 390, 391 Bolis, A., 695 Bonnet, G., 290, 294 Boon, W.M., 248, 253, 574, 575, 655 Borau, C., 248 Borio, A., 501–508, 721, 726, 731 Borja, R.I., 782 Borregales, M., 541–548, 552, 562, 604, 779, 790 Bossavit, A., 260, 265 Botella, O., 1036 Both, J.W., 49–62, 541, 552, 562, 604, 779, 790 Botti, L., 174, 498 Bouloutas, E., 18, 53 Boutin, C., 289, 291 Bozkaya, C., 850 Bracewell, R., 85 Brakke, K.A., 811 Brandts, J., 738, 740, 753, 771, 772 Brandts, J.H., 762 Brebbia, C.A., 850 Brencher, L., 891 Brenner, K., 50, 575 Brenner, S., 408, 409, 412 Brezis, H., 845 Brezzi, F., 51, 159, 184, 216, 323, 325, 332, 502, 504, 519, 520, 523, 578, 698, 715, 725, 731, 746 Brisard, P., 289

Index Brix, G., 68 Brix, K., 301 Brizzi, R., 510 Brogan, F.A., 5 Brogan, P., 815, 816 Brook, R.H., 368 Brouder, Ch., 429 Brown, E., 252 Brown, J.S., 238 Brown, R., 247 Bruhn, D., 653 Bruns, T., 336 Buchmller, P., 900 Buckley, D.L., 69 Buffa, A., 493, 495, 875 Buhler, L., 849 Bui, M.M., 238 Bui-Thanh, T., 684 Bulirsch, R., 957 Bungartz, H.-J., 123 Burger, M., 695 Bürger, R., 616, 620, 621 Burgers, J.M., 1015 Burman, E., 166, 173–180, 183–191, 193, 194, 407, 593, 594, 600 Burrage, K., 423 Burrage, P.M., 423 Busse, F.H., 1026 Butcher, J.C., 419 Byrne, H., 248

C ˇ Cada, M., 902 ˇ Cesenek, J., 535 Caffarelli, L.A., 409 Cafiero, S., 520, 521, 523 Cahn, J., 280, 339 Cahn, J.W., 280, 704 Cai, Z., 329 Calandra, H., 145–151 Camacho, F., 361 Cances, C., 50 Cangiani, A., 159, 519, 520, 725, 726 Canham, P.B., 809 Cañizares-Villanueva, R., 820 Cantwell, C.D., 694 Canuto, C., 693 Cao, Y., 375 Capuzzo Dolcetta, I., 944 Carey, G.F., 31 Carlberg, K., 683, 891 Carstensen, C., 471 Cartier, P., 444

1047 Casas, E., 823 Casas, F., 439 Cascón, J.M., 743 Castelletto, N., 548, 783 Cattaneo, L., 37, 45 Cavoretto, R., 93–100 Celia, M., 18, 53 Celia, M.A., 52, 626 Cessenat, O., 493, 495 Chan, T., 905 Chang, K.-C., 647 Chaplain, M.A., 238–240, 242 Chaplain, M.A.J., 237–244 Chapman, S.J., 36 Chaponnier, C., 247 Chapoton, F., 431, 436 Chappell, G., 811 Charles, A., 840 Chartier, Ph., 429 Chaturantabut, S., 684, 708 Chave, F.A., 575 Chavent, G., 640 Chaves, J., 311 Chen, Y., 73 Chen, Z., 53, 137, 626 Cheng, K.W., 194 Cheny, Y., 1036 Cherfils, L., 53 Chernov, A., 159 Chernyshenko, A., 165–172 Chernyshenko, A.Y., 166, 169 Chevaugeon, N., 194 Chinosi, C., 519–526 Chleboun, J., 482, 483 Chojecki, M., 36 Chopp, D.L., 801 Choquet, C., 53 Chouly, F., 407, 839–846 Chow, T.-T., 881 Christiansen, S.H., 266 Chu, K.F., 673 Chu, M.T., 431 Ciarlet, P., 452, 453, 455, 461, 468, 753 Ciarlet, P.G., 533, 740, 772, 974 Cicuttin, M., 509–517 Clairambault, J., 238 Claus, P., 183 Claus, S., 166, 184, 193 Clever, R.M., 1026 Cliffe, K.A., 651 Clift, S.S., 104, 110 Cockburn, B., 216, 510, 581–584 Cohen, A., 5, 19, 474, 667 Cohen, D., 420

1048 Colls, J., 860 Colombo, A., 498 Comerford, A., 695 Congedo, P.M., 1005 Congreve, S., 493–500 Copeland, D., 725 Corey, A.T., 368 Coronel, A., 615–622 Cortial, J., 683 Cote, J., 924 Cotrell, J.A., 875 Coull, B., 36 Craig, R.J., 5 Crandall, M.G., 381, 394 Cuevas, S., 850 Cunningham, J.J., 238 Curfman McInnes, L., 31 Curry, C., 429–436, 439–446 Curtis, F.E., 813

D da Veiga, L.B., 875 Dahmen, R., 5, 19 Dahmen, W., 364, 472 dal Pizzol, A., 779 Dale, A.M., 36, 42 Dalmont, J.-P., 5, 30 Damodaran, M., 684 D’Angelo, C., 36, 37, 575 Dautray, R., 664, 882, 942 Davidson, S., 675 Davies, R.J., 653 Davydov, D., 582, 588 De Basabe, J.D., 216 De Marchi, S., 116, 892 De Neef, M., 370 De Rossi, A., 93–100 Deaton, J., 335 Debrabant, K., 419–427 Deckelnick, K., 280 Decuzzi, P., 36 Deichmann, N., 653 Deift, P., 431, 432 Deka, P.N., 850 Dellnitz, M., 883 Demlow, A., 361 Deng, W., 804 Denkena, B., 486, 487 Desbrun, M., 810 Després, B., 493, 495 Deuflhard, P., 349 Devinatz, A., 966 Devor, A., 36

Index DeVore, R., 361, 362, 364, 472, 474, 479 Dewhirst, M.W., 36 Di Pietro, D.A., 138, 139, 173, 174, 371, 452, 498, 510, 512, 513, 515, 516, 575 Diaz, J., 145–151 Diedhiou, M.M., 53 Diehl, J.A., 238 Diening, L., 471 Dietrich, P., 654 Discacciati, M., 928 Distefano, J.R., 850 Dobrowolski, M., 974 Dogru, A.H., 541 Dolcetta, I.C., 280 Dolean, V., 932 Dolejší, V., 533 Döll, P., 646 Donat, R., 619 Donatelli, M., 226, 229 Donea, J., 803 Donlan, R.M., 799 Dormieux, L., 289 Dorostkar, A., 225–232 Douglas, J., 51, 323 Doyen, D., 840 Dragt, A.J., 312 Drasdo, D., 248 Dréau, K., 194 Dröge, M., 1036 Droniou, J., 451–459 Dubois, O., 922, 926, 927 Duddu, R., 801 Dunkl, C.F., 461, 468 Dupont, T., 362 Dupuy, D.E., 673 Durst, F., 1041, 1042 Düster, A., 193 Dzenite, I., 850

E Ebrahimi-Fard, K., 429–436, 439–446 Eck, C., 840 Eckelmann, H., 1026, 1027 Edelman, A., 683, 686 Edelsbrunner, H., 759 Edmonds, A.L., 767 Efendiev, Y., 510 Eftang, J.L., 5 Egger, H., 146, 974, 975, 993, 994, 999 Ehrgott, M., 881, 883 Eichfelder, G., 883 Eigestad, G.T., 754 Eikermann-Haerter, K., 42

Index Eisenlohr, P., 289 Eisenstat, S.C., 498 Ekeland, I., 410 Elliot, C., 995 Elliott, C.M., 174, 704 Ellis, R., 966 Elman, H.C., 498 Emblem, K.E., 68, 69 Emery, M., 439, 440 Engquist, W.E.B., 510 Engwer, C., 193, 197 Epshteyn, Y., 372 Erath, C., 993, 994, 999 Eriksson, F., 740, 771 Ern, A., 138, 139, 173–180, 329, 509–517, 726, 840 Escargueil, A., 238 Esedo¯glu, S., 280 Eskin, G., 983 Esser, P., 593, 595, 598, 600 Evans, E.A., 809 Evensen, G., 75 Eymard, R., 451–459, 1017

F Fabre, M., 407, 840 Fabrini, G., 941, 944 Fahrig, R., 69 Falcone, M., 390–392, 941, 944 Falk, R.S., 260, 323 Famiglietti, J.S., 645 Fang, Q., 36 Farhat, C., 146, 683 Farhloul, M., 746 Farthing, M.W., 50 Fasshauer, G., 104 Fasshauer, G.E., 93–100 Feischl, M., 400, 471 Feistauer, M., 531–539, 969 Félix, S., 5, 30 Feng, F., 812 Feng, K., 408 Feng, Y., 673 Fereshtian, A., 104 Fernández, F.J., 821, 830, 831 Fernholz, H., 698 Ferretti, R., 390, 944 Ferronato, M., 783 Fessler, J.A., 280 Fick, L., 693 Fiedler, F., 861 Fieselmann, A., 69 Finn, J.M., 312

1049 Fiorentino, G., 226 Flemisch, B., 375, 575 Flemming, H.C., 800 Fløystad, G., 429, 434 Fong, K.F., 881 Formaggia, L., 575, 726, 728 Forrester, J.W., 67 Forsyth, P.A., 104, 110 Fortin, M., 51, 184, 323, 325, 332, 519, 698, 746 Foulger, G.R., 653 Foulon, S., 109 Franca, L.P., 407 Francis, G., 811 Frederiksen, J.L., 68 Frehse, J., 409 Frepoli, C., 626 Friedman, A., 70, 409, 966 Fries, T.-P., 193, 194, 197, 482 Frind, E., 626 Fryknäs, M., 238 Fuchs, A., 347 Fuˇcíík, R., 635–638, 640 Fuˇcík, R.L, 635–643 Fuentes, D., 673 Fuhrmann, J., 1017 Fumagalli, A., 573–580, 657

G Gabard, G., 146 Gabbiani, G., 247 Gaddum, J.W., 762, 773 Gage, M., 280 Gagnon, L., 42 Gakhov, F.D., 983, 986 Gallouët, T., 451–459, 1017 Gamkrelidze, R., 445 Gander, M.J., 922, 924, 928, 932, 933 Ganesan, S., 593, 594, 600 Ganguly, A., 69 Ganis, B., 626 Garcia-Aznar, J.M., 247, 248 García-Chan, N., 832 Garcion, E., 239 Garcke, H., 704 Garoni, C., 226–228 Gärtner, K., 1017 Gaspar, F., 646, 649 Gaspar, F.J., 541, 546, 646, 782 Gaspoz, F.D., 472 Gatenby, R.A., 238 Gedicke, J., 408, 412, 493–500 Gefen, A., 248–250, 253

1050 Geiger, M., 217 Geiger, M.E., 958 Georgoulis, E., 146 Georgoulis, E.H., 84, 726 Gersbacher, C., 367 Gerstmayr, J., 269, 270, 273 Gerstner, E.R., 69 Geuzaine, C., 636 Ghiglieri, J., 664 Giaquinta, M., 314 Gietl, H., 974 Giles, M.B., 400, 651 Gillies, R.J., 238 Gimse, T., 77 Ginting, V., 70 Giorda, L.G., 928 Girault, V., 174, 467 Gittelson, C., 495 Gittelson, C.J., 493, 495 Glassner, A.S., 312, 313 Glowinski, R., 174, 698 Gluyas, J.G., 653 Goldman, D., 36 Golub, G.H., 124 Gomez-Benito, M.K., 247 González, M., 743, 749 Gopalakrishnan, J., 324, 325, 581 Gordan, J.D., 238 Gottlieb, D., 1026 Gottlieb, S., 905 Gould, I., 36 Gralla, P., 486, 487 Grande, J., 169, 193–195 Grandhi, R., 335 Gräßle, C., 704, 708 Green, J.E.F., 70 Gregory, J.A., 769 Grepl, M., 674 Griebel, M., 593, 595, 598, 600 Grimm, E., 664 Grisvard, P., 874 Gritto, R., 653 Gropp, W.D., 31 Groß, S., 593, 595, 598, 600 Grossman, R., 435 Grote, M.J., 216 Grün, G., 704 Grüne, L., 666, 945 Gubisch, M., 664, 667, 884, 943 Gudi, T., 407, 408 Guermouche, A., 31 Guerrero, F., 619 Gugercin, S., 685 Guichard, C., 451–459

Index Guillet, C., 239 Gullbo, J., 238 Guo, D., 336 Guo, L., 432 Guoy, D., 762 Gustafsson, T., 407–413 Guzmán, J., 184, 324, 325

H Ha, C., 336 Haasdonk, B., 113–121, 274, 275, 891, 892 Hadrava, M., 531–539 Hafizogullari, Y., 301 Hairer, E., 109, 429, 890 Hakim-Dowek, M., 439, 440, 442 Halperin, B.I., 704 Halpern, L., 932, 933 Hanby, V.I., 881 Haneuse, S.J.-P.A., 252 Hannukainen, A., 753, 760, 773 Hansbo, A., 174 Hansbo, M.G., 183 Hansbo, P., 166, 174, 183–191, 193, 194, 407, 468, 469 Hansen, C.V., 381, 382 Hanson, E.A., 65–79 Harari, I., 146 Harlow, F.H., 1036 Hartman, C., 811 Hartmann, R., 143 Hashin, Z., 295 Haußer, F., 703 Havu, V., 123 Hecht, F., 825, 833 Heimann, F., 193–201 Heinecke, A., 123 Heister, T., 547, 582, 588 Helfich, W., 809 Helmig, R., 375, 575, 605, 638–640, 654, 892 Heltai, L., 547, 582, 588 Henjes, J., 486 Henkelman, R.M., 67, 69 Hennicker, J., 575 Henriksen, O., 68 Herbin, R., 451–459, 1017 Herrero, H., 693 Heryudono, A., 104, 107, 109, 110 Hesthaven, J.S., 138, 139, 148, 693, 1026 Hetmaniuk, U., 5, 146 Hibler, W.D., 346 Hild, P., 407, 825, 839, 840 Hildebrandt, S., 314 Hiley, C.T., 240

Index Hill, T., 969 Hilliard, J.E., 280, 704 Hintermüller, M., 663, 666, 703 Hinz, B., 247 Hinze, M., 704, 708 Hiptmair, R., 146, 493, 495 Hirani, A.N., 762 Hixson, E.L., 5, 15 Hochbruck, M., 135–143, 420, 425 Hochstenbach, M.E., 206 Hodneland, E., 65–79 Hohenberg, P.C., 704 Höhme, S., 248 Holmes, P., 663, 708, 943 Holter, K.E., 35–46 Hoppe, R.H.W., 216, 219 Horáˇcek, J., 531–539 Horger, T., 974, 976, 980 Horne, R.N., 654 Horneggerand, J., 69 Hosder, S., 1005 Hosek, R., 762 Hötzl, H., 654 Hou, B.L., 45 Hou, T.Y., 510 Hsu, C.-Y., 36 Hsu, K.-C., 647 Hsu, L., 811 Hsu, R., 36 Hu, C.-J., 238 Hu, X., 546, 646, 782 Huan, G., 626 Huang, L.-H., 647 Hubbert, S., 83–91 Huckle, T., 123, 230 Hudson, R.L., 441 Hughes, T.J.R., 743, 875 Hunt, J.C.R, 850, 855 Hurty, W.C., 5 Hussaini, M.Y., 693 Huttunen, T., 494, 496, 497 Huynh, D.B.P., 5, 6, 678 Huynh, P., 3–32 Hysing, S., 593, 594, 596, 600

I Iaccarino, G., 1007 Ihlenburg, F., 5 IJzerman, W.L., 302, 303, 307, 313 Illangasekare, T.H., 637 In’t Hout, K.J., 109 Iserles, A., 212, 429, 432 Ito, K., 663, 666

1051 J Jack, C., 35 Jackson, F.H., 206 Jacobsen, J., 649 Jadraque-Gago, E., 830 Jafari-Khouzani, K., 69 Jaffré, J., 167, 170, 575, 717 Jäger, W., 50, 52, 801 Jahn, M., 487 Jahnke, T., 138 James, F., 616, 618 Jamet, P., 769 Jamshidian, F., 440 Janhäll, S., 860 Jankov, D., 209 Japhet, C., 604 Jarlebring, E., 205–213 Jarusek, J., 840 Javierre, E., 247 Jawad, R., 36 Jensen, M., 68 Jentzen, A., 420 Ji, Z.L., 5, 23 Jiang, G., 390 Jiang, G.-S., 904, 905 Jingxue, Y., 617 Jirousek, J., 146 Johanni, R., 123 Johansson, A., 174 John, V., 909–911, 913, 914, 916 Johnson, C., 966, 995 Jolivet, P., 932 Jones, C., 649 Joseph, D., 804 Jost, J., 814 Jousset, P., 653 Jouve, F., 336 Juanes, R., 541 Julian, B.R., 653 Jülicher, F., 810 Jung, C., 860, 863 Juntunen, M., 407 Jurak, M., 631

K Kabel, M., 295 Kaˇcur, J., 50, 52 Kaipio, J.P., 494, 497 Kallischko, A., 230 Kalman, R.E., 73 Kalpathy-Cramer, J., 69 Kamm, R.D., 248 Kane, B., 367, 370

1052 Kanschat, G., 143, 216, 219, 547 Kao, C.-Y., 70 Karandikar, R.L., 439, 440, 442 Kärcher, M., 674 Kargin, V., 441 Karniadakis, G., 693 Karniadakis, G.E., 1005 Karpinski, S., 604, 626, 631 Katul, G.G., 861 Keilegavlen, E., 573–580, 653–660 Kells, L.C., 1025 Kermode, A.G., 68 Kern, M., 604 Kessler, R., 1026 Khaliq, A.Q.M., 104 Kharchenko, S., 230 Kikuchi, N., 335, 654 Kim, J., 541, 790 Kim, Y., 860, 863 Kimura, M., 280, 282 Kirby, R.M., 582, 584 Kirk, B.S., 31 Klapper, I, 804 Klawonn, A., 932 Kleiser, L., 1025 Klieber, W., 368, 369 Klinkovský, J., 636, 638, 640 Klitz, M., 593, 595, 598, 600 Kloeden, P.E., 420 Klöfkorn, R., 367 Klug, W.S., 812 Knabner, P., 50–53, 56, 57, 59, 541, 604, 607, 626, 628, 800, 803 Knepley, M.G., 654 Knezevic, D., 3–32 Knezevic, D.J., 5, 6 Knobloch, P., 909–914, 916, 917 Kobayashi, K., 770 Köcher, U., 215–222, 541–544, 546–548, 552–554, 556, 559, 562, 566, 790 Kochina, I.N., 70 Koehler, E., 252 Köhler, J., 135–143 Koláˇr, M., 279–286 Kolotilina, L.Y., 230 Kondratiev, V.A., 974 Köngeter, J., 654 Koppenol, D.C., 248, 253 Köppl, T., 37, 892 Koren, B., 1018, 1022, 1037, 1039–1042 Kormann, K., 581–588 Kornhuber, R., 604 Korotov, S., 737, 738, 740, 743, 753, 760, 762, 771–773

Index Kosík, A., 531–539 Koskela, A., 205–213 Kou, J., 626, 628 Kowarschick, M., 69 Kozlov, V., 974 Kratochvíl, J., 282 Krause, E., 1041, 1042 Kreiss, H.O., 384 Kretzschmar, F., 146, 149 Kreuzer, C., 471–479 Kˇrížek, M., 737, 738, 740, 741, 753, 760, 762, 770–773 Kˇrišt’an, J., 282 Krmer, L., 123 Kronbichler, M., 581–588 Kropá, A., 740 Krumbiegel, K., 663, 665 Kublik, C., 280 Kucera, V., 771 Kuˇcera, V., 964, 965, 967–971 Kuchta, M., 35–46 Kueper, B.H., 626 Kühn, M., 932 Kumar, C., 820 Kumar, K., 49–62, 541, 544, 547, 548, 604, 607, 626, 628, 779, 790 Kumar, N., 1018, 1020, 1022 Kummer, F., 194 Kundan, K., 552, 562 Kundzewicz, Z.W., 646 Kunert, G., 726 Kunisch, K., 663, 666 Kurganov, A., 395 ◦ P., 123–130 Kus, Kusiak, A., 881 Kusner, R., 811 Kusner, R.B., 811 Kuttler, W., 860 Kuzmin, D., 593, 594, 596, 600, 909, 910, 913 Kværnø, A., 419–427 Kvashchuk, A., 625–632

L Laayouni, L., 924 Lacroix, S., 626 Ladyzenskaja, O.A., 619 Lagarias, J.C., 97, 98 Lagos, R., 615–622 Lahellec, N., 289 Lalic, B., 864 Lanczos, C., 271 Landa-Marbán, D., 800 Lang, B., 123

Index Lang, J., 704 Langelaar, M., 335, 336 Langer, U., 725 Larson, A., 183 Larson, M., 194 Larson, M.G., 166, 174, 183–191, 193, 407, 468, 469 Larson, R., 435 Larsson, E., 104, 107, 109, 110 Larsson, H., 68 Larsson, R., 238 Larsson, S., 420 Lassila, T., 696, 697 Lawrence, N.D., 113 Lazarov, R., 510, 581 Le, C., 336 Le Bris, C., 510, 516 Le Roux, M., 995 Le Tallec, P., 146, 147 Le Veque, R.J., 900, 902 Lederer, H., 123–130 Lederer, P., 184, 194 Lee, K., 860 Lee, T., 626 Lefebvre, J., 42 Legendre, C., 239 Legoll, F., 510, 516 LeGresley, P.A., 684 Lehmann, F., 50, 52 Lehoucq, R., 5 Lehrenfeld, C., 184, 193–201 Lemaire, S., 173, 509–517 Lemieux, J.F., 347 Leok, M., 626 Leonard, A., 1026 Lesage, F., 42 LeVeque, R.J., 381, 383 Levesley, J., 83–91 Lewis, R.W., 646 L’Excellent, J.Y., 31 Li, B., 626 Li, H., 104 Li, L.C., 431, 432 Lie, K.-A., 77 Ligere, E., 850 Lijoka, O., 146 Linke, A., 1018, 1022 Linninger, A., 36 Lions, J.-L., 409, 664, 882, 942, 974 Lions, P.-L., 390, 394, 604 Liou, M.-S., 862 Lipnikov, K., 166, 1036, 1040 Lipowsky, R., 810 Lipton, R., 5

1053 Lisbona, F., 646, 649 List, F., 51–53, 56, 57, 541, 544, 548, 604, 628, 631, 790, 803 Litchke, T., 860 Liu, X.-D., 905 Livernet, M., 431 Livesu, M., 157–164 Lleras, V., 840 Lloyd, M.C., 238 Lo, M.-H., 645 Loan, C.F.V., 124 Loan, C.V., 206, 209 Logg, A., 184, 188, 748 Loisel, S., 924 Loneland, A., 932 Lorentzen, R.J., 65–79 Lorenz, J., 384 Lorenz, W.J., 68 Lorenzi, T., 237–244 Lorz, A., 237–244 Losch, M., 347 Lovadina, C., 726 Lovgren, A.E., 698 Lozinski, A., 510, 516 Lu, B., 626 Lu, X., 104 Lubich, C., 429 Luckhaus, S., 50, 52 Lumley, J.L., 663, 708, 943 Lundervold, A., 429, 431, 435, 436 Luostari, T., 494, 496 Luttmann, A., 487 Lynch, R.E., 584

M Maaß, P., 486, 487 MacCamy, R.C., 993 Maciag, A., 150 Maday, Y., 5, 13, 693, 698, 700 Magnus, W., 439, 445 Mahaffy, J.H., 626 Mahony, R., 685, 686 Mahrt, L., 861 Maier, M., 582, 588 Mailliat, A., 860, 863 Majda, A., 381 Majer, E., 653 Malcolm, D.G., 850 Maley, C.C., 238, 240 Malham, S.J.A., 443 Maliska, C.R., 779 Manchon, D., 431, 439, 443–445 Mandal, S., 345, 346, 348, 353

1054 Mandeville, E.T., 42 Mani, D., 820 Manni, C., 226 Manzini, G., 159, 452, 519, 520, 725, 726, 1036, 1040 Manzoni, A., 696 Mao, S., 770 March, R., 280 Marcusson, G., 104, 109 Mardal, K.-A., 35–46 Marek, A., 123–130 Marek, I., 290, 291 Marini, D., 731 Marini, L., 725 Marini, L.D., 159, 216, 502, 504, 519, 520, 523, 578, 715, 966, 969 Marinnan, T., 36 Markov, I.V., 280 Martin, V., 575, 717 Martínez, A., 821, 832 Martos, A., 830 Mascotto, L., 159, 498, 731 Massing, A., 166, 183, 184, 188, 193 Masson, R., 575 Masters, I., 646 Masud, A., 743 Mattheij, R.M.M., 951, 954 Matvejevs, A., 850 Matzavinos, A., 70 Maute, K., 335, 336 Mayne, D.Q., 666, 945 Maz’ya, V.G., 974 Mazza, M., 226 McCauley, A., 649 McClamroch, N.H., 626 McClure, M.W., 654 McCourt, M.J., 93–100 McCoy, J., 280 McGarr, A., 653 McLachlan, R., 429 McMahon, K.J., 5 McMurry, P., 862 McWhorter, D.B., 637 Mechelli, L., 663, 667 Meerbergen, K., 206 Mehlmann, C., 345–349 Meier, P., 68 Melenk, J., 494 Melrose, R., 759 Melzer, H., 400, 404, 412 Mencattini, I., 432, 435, 436 Meng, L., 45 Mengaldo, G., 695 Merkert, D., 295

Index Merlo, L.M., 238 Merton, R.C., 105 Meseguer, A., 1026 Meyer, H.M., 850 Meyer, M., 810 Micchelli, C.A., 113 Michalet, X., 810 Michiels, W., 206 Mielnik, B., 439 Mierka, O., 593–600 Miettinen, K., 881, 883 Mihailovic, D.T., 864 Mikelic, A., 215, 541, 542, 552, 553, 561, 801 Mikhlin, S.G., 983, 984, 986 Mikula, K., 281, 282 Mikyška, J., 636–638, 640 Millerand, M., 70 Milstein, G.N., 423 Minárik, V., 282 Mirrahimi, S., 241 Mitchell, T., 813 Mitra, K., 604, 607, 608 Mlika, R., 839 Modin, K., 429 Moës, N., 194 Moiola, A., 146, 149, 493, 495 Molenaar, J., 370 Moler, C., 206, 209 Mollapourasl, R., 104 Molteni, M., 226 Monchiet, V., 289, 290, 294 Monk, P., 493–497 Moran, B., 801 More, K.L., 850 Moreno, T., 738 Morin, P., 472, 476, 479 Mortensen, M., 37, 39–41 Morvanb, H., 860 Moser, R.D., 1026 Moulinec, H., 289 Mounim, A.S., 746 Mourits, F.M., 791 Moxey, D., 582, 584, 695 Mu, L., 510 Muldoon, F., 862 Mulet, P., 615–622 Müller, B., 194 Müller, C., 582, 586, 587 Müller, S., 892 Muller, U., 849 Münch, I., 339 Munjal, J.L., 4, 5, 19 Munk, D., 335 Muñoz-Sola, R., 821

Index Munthe-Kaas, H., 429–436 Musacchia, J.J., 42 Muskhelishvili, N.I., 983, 986

N Nabil, M., 36 Namiki, T., 137 Napov, A., 226 Nataf, P., 932 Nævdal, G., 65–79 Nävert, U., 966, 971 Nédélec, J.-C., 260, 995 Nederveen, C.J., 5, 30 Neitzel, I., 667 Neytcheva, M., 226 Ngnotchouye, J.M.T., 420 Nguyen, L., 3–32 Nguyen, N., 13 Nguyen, N.C., 581–584 Ni, X., 432 Niebuhr, C., 481–488 Niederwestberg, D., 486, 487 Nikfarjam, F., 1036 Nikishin, A., 230 Nilsen, H.M., 754 Nitsche, J., 407, 974 Noack, B.R., 1026, 1027 Nochetto, R.H., 51, 471, 473, 474, 476 Noelle, S., 395 Noor, A.K., 5 Norato, J., 336 Nordaas, M., 37, 39, 41 Nordbotten, J.M., 49–62, 65–79, 541, 544, 547, 548, 552, 562, 574, 575, 604, 607, 626, 628, 655, 657, 754, 779, 790 Nordström, J., 1007, 1009 Norris, L.K., 431 Nørsett, S.P., 109, 429, 432, 890 Notay, Y., 226 Nüßing, A., 197

O Oberhuber, T., 636, 638, 640 Oberlack, M., 194 Oberman, A.M., 390 Oden, J., 654 Oden, J.T., 400 Oder-Blöbaum, S., 883 Ogden, F.L., 50 Ohkawa, K., 626

1055 Ohlberger, M., 667 Ohm, P., 546 Oliker, V., 301, 303 Oliver, D.S., 73, 75 Olshanskii, M., 165–172, 184 Olshanskii, M.A., 166, 169, 193 Omerovi, S., 194 Ordóñez, J., 830 Orszag, S.A., 585, 1025 Osher, S., 381, 905 Ostermann, A., 420, 425 Oswald, P., 771 Oteo, J.A., 439 Otto, F., 52 Ouazzi, A., 345, 346, 353 Overton, M.L., 813 Owren, B., 420, 425 Oye, V., 653

P Pacheco-Torres, R., 830 Paniconi, C., 50 Pannek, J., 666, 945 Pao, W.K.S., 646 Paolucci, G., 392 Paraschivoiu, M., 400 Paredes, D., 510 Park, E.J., 50, 52, 53 Park, E.Y., 36 Parolini, N., 593, 594, 600 Parthasarathy, K.R., 441 Partridge, P.W., 850 Parvizian, J., 193 Pasquarelli, A.F., 1026 Patankar, S., 77 Patera, A.T., 3–32, 400, 674, 678, 693 Patras, F., 436, 439–446 Paulson, O.B., 68 Pauš, P., 282 Paynabar, K., 69 Peaceman, D.W., 138 Pechstein, A., 270 Peitz, S., 883 Pelteret, J.-P., 582, 588 Peña-Castro, J., 820 Pencheva, G., 626 Peng, D.-P., 390 Pennacchio, M., 157–164 Pennati, V., 226 Pennes, H.H., 674 Pepper, J.W., 238 Peraire, J., 400, 581–584

1056 Perales-Vela, H., 820 Perez, R., 1005 Perot, B., 1041 Perotto, S., 726 Perracchione, E., 93–100 Persson, J., 104, 109 Perthame, B., 238, 241 Perugia, I., 146, 149, 493–500 Peters, J.M., 5 Petersen, S., 146 Peterson, J.W., 6, 31 Petroff, A., 860, 862, 863 Petrova, G., 5, 19, 395 Petrushev, P., 472 Pettersson, M.P., 1007 Pettersson, U., 104, 109 Pfefferer, J., 974, 978 Pfeiler, C.-M., 184, 194 Philips, P., 552, 554, 555, 558, 561, 568 Pieraccini, S., 501–508, 718, 721, 731 Pinkus, A, 5 Pinnau, R., 684 Pint, B.A., 850 Pitkäranta, J., 877, 966 Pitton, G., 695, 697 Platen, A., 301 Plaza, Á., 737, 738, 740 Pleba´nski, J., 439 Pogány, T.K., 209 Poggi, D., 861 Polacheck, W.J., 248 Pontil, M., 113 Pop, I.S., 49–62, 541, 604, 607, 608, 626, 628, 631, 803 Popel, A.S., 36 Port, R., 68 Porwal, K., 408 Post, J.T., 5, 15 Pousin, J., 407 Prada, D., 157–164 Praetorius, D., 471 Pralet, S., 31 Presho, M., 70 Prins, C.R., 302, 305 Prössdorf, S., 983, 984, 986 Protter, P.E., 439, 440, 442, 443 Prud’homme, C., 5, 19 Prudhomme, S., 400 Pryer, T., 726 Pryor, S., 862, 863 Pusch, D., 725 Pustejovska, P., 974, 976, 980 Putti, M., 50, 52, 53

Index Q Qaddouri, A., 924 Qiu, B., 45 Quaini, A., 695, 698 Quarteroni, A., 36, 37, 693, 696, 698, 932, 1026

R Rachford, H.H., Jr., 138 Radavich, P.M., 5 Radu, F.A., 49–62, 541–548, 552–554, 556, 559, 562, 566, 603–613, 625–632, 779, 790, 803 Rafetseder, K., 871, 872 Rahrah, M., 645–652 Ramos, E.A., 762 Rank, E., 193 Rannacher, R., 217, 400, 412, 1041, 1042 Ranner, T., 995 Rapetti, F., 260, 265, 266 Rasche, S., 703 Raviart, P.-A., 260, 521 Raw, M.J., 782 Rawlings, J.B., 666, 945 Ray, J., 891 Raynaud, X., 754 Razouk, N., 212 Reager, J.T., 645 Reddy, J., 876 Reed, W.H., 969 Reeds, J.A., 97, 98 Reid, B.J., 238 Reischl, D., 270 Rektorys, K., 772 Remacle, J.F., 636 Renard, Y., 407, 840 Rentmeesters, Q., 685 Reusken, A., 169, 193–195, 200, 201 Reynolds, A.C., 73, 75 Rheinbach, O., 932 Rice, J.R., 584, 654 Richey, A.S., 645 Richter, T., 345–349, 404, 414 Rippa, S., 94, 97 Rivero, M., 850 Rivière, B., 368, 369, 372, 467 Roberts, J., 51, 640 Roberts, J.E., 167, 170, 575, 717, 746 Rocca, E., 703 Rocco, G., 695 Roch, S., 983 Rodell, M., 645 Rodrigo, C., 541, 546, 646, 649, 782

Index Rodríguez, C., 821 Rognes, M.E., 184, 188 Rohde, C., 603–613 Romaus, C., 881, 883 Rønquist, E.M., 5, 698, 700 Ros, J., 439 Rosasco, L., 113 Rösch, A., 663, 665, 974, 978 Rosen, B., 69 Rosenberg, I.G., 737 Rossmann, J., 974 Rotkvic, M., 704 Rovas, D.V., 5, 19 Rowley, C.W., 663, 708, 943 Rozvany, G., 336 Rozza, G., 5, 674, 678, 693, 695–697 Rttgers, A., 593, 595, 598, 600 Rubinstein, J., 280 Rüde, U., 974–976 Rusanov, V.V., 381 Russo, A., 159, 502, 504, 519, 520, 578, 715, 725, 728, 731 Ruvinskaya, L., 36 Ryaben’kii, V.S., 983 Ryll, C., 823

S Saad, Y., 229 Sacchi-Landriani, G., 1026 Safdari-Vaighani, A., 103–112 Saffman, P.G., 801 Sahu, S., 390, 391 Sakadži´c, S., 36, 42 Salvador, T., 390 Samier, P., 575 Sand, F., 860 Sander, O., 604 Sangalli, G., 875 Santin, G., 114–116, 891, 892 Santos, P.A., 983 Santugini, K., 932, 933 Sanz, C., 861 Sarma, D., 850 Sauter, M., 654 Sauter, S., 461, 468 Sævareid, O., 65–79 Saye, R., 197 Schaback, R., 116, 117, 892 Schad, L.R., 68 Schäfer, M., 1041, 1042 Schatz, A.H., 974, 978, 979 Scheichl, R., 651 Schmainda, K.M., 69

1057 Schmidt, A., 481–488 Schmidtmann, B., 900–902, 904 Schnaubelt, R., 138 Schneebeli, A., 216 Schneider, G.E., 782 Schnepp, S., 146, 149 Schöberl, J., 270, 273 Schoeder, S., 582, 586, 587 Scholes, M., 103, 688 Scholz, R., 408 Schorr, R., 993, 994, 999 Schötzau, D., 216 Schrefler, B., 646 Schröder, P., 810 Schroll, H.J., 381, 382, 902 Schultz, M.H., 498 Schulz, R., 800 Schumann, U., 1025 Schweitzer, M.A., 498 Schwenck, N., 575 Scialò, S., 501–508, 717, 718, 721, 731 Scott, L.R., 361 Scott, R., 362 Scotti, A., 575 Secomb, T., 36 Secomb, T.W., 36 Segal, A., 649 Seibold, B., 901, 902, 904 Seifert, U., 810 Seifried, A., 881, 883 Selamet, A., 5, 23 Semmler, W., 68 Sen, M.K., 216 Sen, S., 678 Sepulchre, R., 685, 686 Sepúlveda, M., 615–622 Serra-Capizzano, S., 226–229 Serres, C., 167, 170 Sesana, D., 229 Settari, A., 791 Seus, D., 603–613 Ševˇcoviˇc, D., 279–286 Sezgin, M., 850 Shabana, A.A., 270 Shaskov, M., 1036, 1040 Shcherbakov, V., 104, 107 Sherar, M., 675 Sherwin, S., 693 Sherwin, S.J., 582, 584 Shewchuk, J.R., 161 Shi, K., 510 Shi, Z., 770 Shi, Z.-C., 408 Shibata, D., 240

1058 Shipley, R.J., 36 Shishenina, E., 145–151 Shu, C.-W., 904, 905, 964, 965, 967–971 Siebert, K.G., 472, 476, 479, 487 Sigg, M., 420 Sigmund, O., 335 Silbermann, B., 983 Simon, J., 706 Simon, M.C., 238 Singh, G., 541 Šíp, V., 860–863 Skaflestad, B., 420, 425 Slodicka, M., 51–53 Slodiˇcka, M., 604 Smetana, K., 5, 6, 19, 26 Smith, B.F., 31 Smith, R.C., 1007 Smith, S.T., 683, 686 Smolentsev, S., 850 Soh, H., 704 Šolc, J., 762 Solonnikov, V.A., 619 Solovský, J., 635–643 Sorensen, D., 275, 684, 708 Sottocasa, F., 575 Souganidis, P., 390 Sourbron, S., 67, 69 Sourbron, S.P., 69 Spahn, J., 296 Speicher, R., 441 Speleers, H., 226 Sprekels, J., 313, 703 Staarink, G.W.M., 951, 954 Stadler, W., 881, 883 Stamm, S., 693 Stampacchia, G., 409 Starke, G., 329 Stefansson, I., 573–580, 657 Steffens, J., 860 Stegun, I.A., 91, 207, 209 Stenberg, R., 329, 407–413 Stenger, F., 737 Stern, P., 5 Sternberg, P., 280 Steven, G, 335 Stevenson, R., 471 Stewart, G.W., 497 Stoer, J., 957 Stogner, R.M., 31 Strang, G., 455 Strese, S., 238 Strichartz, R.S., 439 Strouboulis, T., 737 Stubgaard, M., 68

Index Stufflebeam, S.M., 69 Stykel, T., 270 Suárez, J.P., 738 Subhan, F., 84 Süli, E., 209 Sullivan, J., 811 Sullivan, J.M., 811 Sun, S., 370, 626, 628 Sunada, D.K., 637 Sung, L.-Y., 408, 409, 412 Suquet, P., 289 Suri, M., 993 Sutradhar, A., 70 Sutton, O.J., 726 Švadlenka, K., 969 Svärd, M., 1009 Svyatskiy, D., 166 Swanton, C., 240 Swenson, S., 645 Synge, J.L., 769

T Tadmor, E., 379, 905 Tambue, A., 420 Tang, F., 881 Tantardini, F., 362–364 Tarman, H.I., 1026 Tchelepi, H.A., 541, 548 Teckentrup, A.T., 651 Temam, R., 53, 410 ten Thije Boonkkamp, J.H.M., 302, 303, 307, 313, 1016, 1018, 1022 Tesini, P., 498 Teutsch, G., 654 Tezaur, R., 146 Tezer-Sezgin, M., 850 Thomas, B.F., 645 Thomas, D.H., 584 Thomas, J.M., 260, 521 Thomass, J.-M, 746 Tiesler, H., 673 Tiwary, A., 860 Tobiska, L., 593, 594, 600 Tofts, P., 68 Tofts, P.S., 42, 44 Tokoutsi, Z., 674 Tomasek, J., 247 Tomei, C., 431, 432 Torrilhon, M., 900–902, 904 Toselli, A., 160 Tozza, S., 392 Tran, T., 290, 294 Trefethen, L.N., 1026

Index Trefftz, E., 146 Tröltzsch, F., 313, 663, 664, 676, 823, 881, 946 Tsai, T.-L., 647 Tsuchiya, T., 770 Tsujikawa, T., 282 Tsukerman, I., 146 Tugluk, O., 1026 Tukker, T.W., 302, 303, 313 Turan, E., 226 Turcksin, B., 582, 588 Turek, S., 345, 346, 353, 593–600, 1041, 1042 Turner, P.E., 240 Tyrtyshnikov, E.E., 227 U Ucar, E., 654, 657 Ulbrich, S., 664 Ullmann, S., 704 Uralceva, N.N., 619 Usta, F., 84 V Valentin, F., 510 Valero, C., 247 Vallès, B., 73, 75 Vallette, B., 430 Valli, A., 932 van Bloemen Waanders, B., 891 van Dijk, N., 335, 336 Van Duijn, C., 370 van Genuchten, M.T., 631 van Keulen, F., 335, 336 van Lent, J., 229 van Lith, B.S., 313 van Noorden, T.L., 802 Vanden-Eijnden, E., 510 Vanderzee, E., 762 Vandewalle, S., 229 Vangel, M.G., 69 Vanselow, A., 347 Vanzan, T., 928 Vasilyev, A., 985, 987–990 Vasilyev, V., 985, 987–990 Vasilyev, V.B., 990 Vassilevski, Y., 166, 168, 626 Vatne, J.E., 574, 575, 764 Vatned, J.E., 755 Vázquez, R., 875 Vázquez-Méndez, M.E., 821, 832 Veeser, A., 360–364, 461–469, 471–479 Vehmeyer, J., 486, 487 Venkatakrishnan, V., 862 Venkataraman, C., 237–244

1059 Verani, M., 731 Verdi, C., 51 Verdier, O., 429 Verfürth, R., 363, 364 Verfüth, R., 362–364 Vermolen, F., 645–652 Vermolen, F.J., 248–250, 253 Veroy, K., 5, 19, 674 Verschaeve, J.C., 37, 39, 41 Versieux, H.M., 510 Verstappen, R., 1036 Videman, J., 407–413 Vihharev, J., 400 Villani, C., 303 Vilmart, G., 429 Vio, G., 335 Vita, S.F., 280 Vlasák, M., 561–569 Vogel, B., 861 Vohralík, M., 329, 368, 371, 604 Voigt, A., 703 Volkwein, S., 663, 664, 667, 703, 708, 881, 884, 886, 943, 946 Vondˇrejc, J., 290, 291 Voss, D.A., 104 Voss, K., 645 Vu, B., 799 W Wächter, A., 833 Wahlbin, L.B., 974, 978, 979 Wall, W.A., 581–588 Walters, R.W., 1005 Walther, V., 240 Waluga, C., 974, 976 Wan, X., 1005 Wang, D., 146 Wang, H.F., 646, 780 Wang, J., 510 Wang, M., 336 Wang, S.-J., 647 Wang, X., 45 Wang, Y., 860 Wanger, W., 339 Wanner, G., 109, 429, 890 Warburton, T., 138, 139, 148, 216, 219 Watson, G., 209 Wauer, J., 150 Weiland, T., 146 Weißer, S., 726, 731 Welch, J.E., 1036 Wells, D., 582, 588 Wen, P.Y., 69 Wendland, H., 94, 96, 98, 116, 891, 892

1060 Werner, J., 117 Wheeler, J., 626 Wheeler, M., 552–555, 558, 561, 568 Wheeler, M.F., 215, 216, 368, 370, 467, 541, 542, 604, 626 Whitby, E., 862 White, J.A., 548, 782, 783 Whitney, H., 260 Wi, R., 37, 39 Wick, T., 345–347, 404, 726 Widlund, O., 160 Wiese, A., 443 Willcox, K., 667, 684, 685 Wille, R., 698 Willems, P.R., 123 Williams, C.A., 654 Williams, W.E., 850, 855 Wilson, E.L., 5, 16 Wilson, M.P., 653 Wingender, J., 800 Wintersteiger, C., 184, 194 Winther, R., 39, 260 Winthers, R., 323 Wirtz, D., 275, 891, 892 Witting, K., 881, 883 Wittwar, D., 113–121 Wo, S., 70 Wohlmuth, B., 37, 375, 575, 974–976, 980 Wojtaszczyk, P., 5, 19 Wolf, K.B., 312 Wolff, M., 375 Wright, M.H., 97, 98 Wright, P.E., 97, 98 Wright, W., 429, 430 Wrobel, L.C., 850 Wu, O., 69

X Xiu, D., 1005 Xu, G., 881 Xu, Y., 922, 928

Y Yadav, N.K., 301, 302, 307

Index Yakovlev, S., 582, 584 Yang, A., 815, 816 Yaseen, M.A., 42 Yazaki, S., 280, 282 Ye, X., 510 Yeremin, A.Y., 230 Yi, S.-Y., 552, 563, 782 Yong, W.A., 51, 52 Yotov, I., 574, 575, 604, 626, 655 Yu, T.P.-Y, 812 Yucel, M.A., 42

Z Zahedi, S., 184 Zang, A., 653 Zanna, A., 65–79, 429, 432 Zanotti, P., 360, 461–469 Zarba, R., 18, 53 Zdravkovich, M.M., 1026 Zeman, J., 290, 291 Zenger, C., 974 Ženíšek, A., 738, 753, 772 Žgalji´c Keko, A., 631 Zhang, H., 408, 409 Zhang, J., 137 Zhang, K., 860 Zhang, L., 45, 862, 863 Zhang, N., 45 Zhang, Q., 971 Zhang, S., 361, 738 Zhang, Y., 408, 409, 412 Zharnitsky, V., 762 Zheltov, I.P., 70 Zhen, F., 137 Zheng, S., 704 Zienkiewic, O.C., 146 Zierler, K.L., 68 Zigerelli, A., 815, 816 Zikatanov, L.T., 546, 646, 782 Zilian, A., 482 Zimmermann, R., 687, 689 Zlámal, M., 740, 753, 772 Znamenshchykov, O., 881, 883 Zulehner, W., 871, 872 Zunino, P., 36, 37, 45

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