Multiscale Model Reduction: Multiscale Finite Element Methods and Their Generalizations 3031204085, 9783031204081

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Applied Mathematical Sciences

Eric Chung Yalchin Efendiev Thomas Y. Hou

Multiscale Model Reduction Multiscale Finite Element Methods and Their Generalizations

Applied Mathematical Sciences Founding Editors F. John J. P. LaSalle L. Sirovich

Volume 212

Series Editors Anthony Bloch, Department of Mathematics, University of Michigan, Ann Arbor, MI, USA C. L. Epstein, Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA Alain Goriely, Department of Mathematics, University of Oxford, Oxford, UK Leslie Greengard, New York University, New York, NY, USA Advisory Editors J. Bell, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA, USA P. Constantin, Department of Mathematics, Princeton University, Princeton, NJ, USA R. Durrett, Department of Mathematics, Duke University, Durham, CA, USA R. Kohn, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA R. Pego, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA L. Ryzhik, Department of Mathematics, Stanford University, Stanford, CA, USA A. Singer, Department of Mathematics, Princeton University, Princeton, NJ, USA A. Stevens, Department of Applied Mathematics, University of Münster, Münster, Germany S. Wright, Computer Sciences Department, University of Wisconsin, Madison, WI, USA

The mathematization of all sciences, the fading of traditional scientific boundaries, the impact of computer technology, the growing importance of computer modeling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purpose of this series is to provide such books, suitable for the user of mathematics, the mathematician interested in applications, and the student scientist. In particular, this series will provide an outlet for topics of immediate interest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. These books should be accessible to readers versed in mathematics or science and engineering, and will feature a lively tutorial style, a focus on topics of current interest, and present clear exposition of broad appeal. A compliment to the Applied Mathematical Sciences series is the Texts in Applied Mathematics series, which publishes textbooks suitable for advanced undergraduate and beginning graduate courses.

Eric Chung Yalchin Efendiev Thomas Y. Hou •

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Multiscale Model Reduction Multiscale Finite Element Methods and Their Generalizations

123

Eric Chung Department of Mathematics Chinese University of Hong Kong Shatin, New Territories, Hong Kong

Yalchin Efendiev Department of Mathematics & ISC Texas A & M University College Station, TX, USA

Thomas Y. Hou Applied and Computational Mathematics California Institute of Technology Pasadena, CA, USA

ISSN 0066-5452 ISSN 2196-968X (electronic) Applied Mathematical Sciences ISBN 978-3-031-20408-1 ISBN 978-3-031-20409-8 (eBook) https://doi.org/10.1007/978-3-031-20409-8 Mathematics Subject Classification: 65M99, 65N99, 74S99, 76S05, 35B27 © Springer Nature Switzerland AG 2023 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, expressed 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Challenges and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Multiscale problems . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Numerical challenges of solving multiscale problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Multiscale model reduction concepts . . . . . . . . . . . . . . . . . . 1.2.1 Exemplary problems . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Fine and coarse grids . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Scale separation approaches . . . . . . . . . . . . . . . . . . 1.2.4 Multiscale finite element methods and some related methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 The need for a systematic multiscale model reduction approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The basic concepts of Generalized Multiscale Finite Element Method (GMsFEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 General idea of GMsFEM. . . . . . . . . . . . . . . . . . . . 1.3.2 Multiscale basis functions and snapshot spaces . . . . 1.3.3 Reducing the degrees of freedom. . . . . . . . . . . . . . . 1.3.4 Constraint minimization concepts and mesh dependent convergence . . . . . . . . . . . . . . . . . . . . . . 1.3.5 The relation to upscaling and novel upscaled concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.7 Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.8 Methodological ingredients of GMsFEM . . . . . . . . . 1.3.9 High contrast and scale issues . . . . . . . . . . . . . . . . . 1.3.10 Modeling with multiscale methods and applications .

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1.3.11 Efficient temporal discretizations with multiscale methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.12 Learning and multiscale methods . . . . . . . . . . . . Relevant literature review . . . . . . . . . . . . . . . . . . . . . . . Overview of the content of the book . . . . . . . . . . . . . . . Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Homogenization and numerical homogenization of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Homogenization for linear problems with oscillatory coefficients. Main concepts . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Elliptic equations with heterogeneous coefficients . . . 2.1.2 Homogenization of parabolic equations . . . . . . . . . . 2.1.3 Homogenization of convection-diffusion equation . . . 2.1.4 Homogenization of convection-diffusion reaction equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Numerical homogenization for linear problems with oscillatory coefficients: Main concepts . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 A motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Local problems and macroscopic equations . . . . . . . 2.2.3 Convergence results for numerical homogenization . . 2.2.4 The choice of boundary conditions in numerical homogenization. Oversampling . . . . . . . . . . . . . . . . 2.2.5 Increasing representative volume size . . . . . . . . . . . 2.2.6 Improving numerical homogenization . . . . . . . . . . . 2.2.7 Numerical homogenization for space-time heterogeneous problems . . . . . . . . . . . . . . . . . . . . . 2.3 Homogenization in perforated regions . . . . . . . . . . . . . . . . . 2.3.1 Homogenization of Stokes equations . . . . . . . . . . . . 2.4 Numerical homogenization in perforated domains . . . . . . . . . Local model reduction. Introduction to Multiscale Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Multiscale finite element methods . . . . . . . . . . . . . . . . . . . . 3.1.1 Finite element with multiscale basis functions . . . . . 3.1.2 Basic idea of MsFEM . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Using smaller regions in computing multiscale basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Reducing boundary effects . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Oversampling technique . . . . . . . . . . . . . . . . . . . . . 3.3 Comparison to other multiscale methods . . . . . . . . . . . . . . .

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3.3.1 Comparison to numerical homogenization . . . . . 3.3.2 Comparison to variational multiscale . . . . . . . . . 3.3.3 Comparison to heterogeneous multiscale method Performance and implementation issues . . . . . . . . . . . . . 3.4.1 Cost and performance . . . . . . . . . . . . . . . . . . . . Convergence of multiscale finite element methods . . . . . 3.5.1 The analysis of conforming multiscale finite element method . . . . . . . . . . . . . . . . . . . . . . . . Mixed MsFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MsFEM for parabolic equations . . . . . . . . . . . . . . . . . . . MsFEM using limited global information . . . . . . . . . . . .

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Generalized multiscale finite element methods. Main concepts and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Parameter-independent case . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Examples of snapshot spaces. Oversampling and non-oversampling . . . . . . . . . . . . . . . . . . . . . 4.3.2 Offline spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 A numerical example . . . . . . . . . . . . . . . . . . . . . 4.4 Online space for parameter-dependent case . . . . . . . . . . . . 4.5 An example of enrichment. The importance of local spectral problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Reduced-dimensional coarse spaces . . . . . . . . . . . 4.6 Iterative solvers - online correction of fine-grid solution . . 4.7 Some numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Case with no parameter . . . . . . . . . . . . . . . . . . . 4.7.2 Elliptic equation with the parameter . . . . . . . . . . 4.8 Randomized snapshots . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Randomized oversampling . . . . . . . . . . . . . . . . . 4.8.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . Adaptive strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 A-posteriori error estimates and adaptive enrichment . 5.4 Numerical results for offline adaptivity . . . . . . . . . . . 5.4.1 Comparison with uniform enrichment . . . . . 5.4.2 Performance study . . . . . . . . . . . . . . . . . . . 5.5 Residual-based online adaptivity . . . . . . . . . . . . . . .

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Numerical results for online adaptivity . . . . . . . . . . . . . . . . . . . 148 5.6.1 Comparison of using different numbers of initial basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.6.2 Adaptive online enrichment . . . . . . . . . . . . . . . . . . . . 153

Selected global formulations for GMsFEM and energy stable oversampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Global formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Mixed GMsFEM . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 GMsDGM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Nonconforming GMsFEM . . . . . . . . . . . . . . . . . 6.2.5 GMsHDG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6 General concept of energy stable (minimizing) oversampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Basis construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Multiscale basis functions in mixed GMsFEM . . . 6.3.2 Multiscale basis functions in GMsDGM . . . . . . . 6.3.3 Multiscale basis functions in nonconforming GMsFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Multiscale basis functions in GMsHDG . . . . . . . . 6.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Mixed GMsFEM . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 GMsDGM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Constraint energy minimizing concepts . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Construction of multiscale basis functions . . . 7.4 Numerical results . . . . . . . . . . . . . . . . . . . . . 7.5 Relaxed CEM-GMsFEM . . . . . . . . . . . . . . . . 7.6 Construction of online basis functions . . . . . . 7.7 Numerical results using online basis functions

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Non-local multicontinua upscaling . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The non-local multicontinua upscaling . . . . . . . . . . 8.3.1 Multicontinua functions . . . . . . . . . . . . . . 8.3.2 Transmissibility computations . . . . . . . . . . 8.3.3 Approximation using local multiscale basis 8.4 Time-dependent problem . . . . . . . . . . . . . . . . . . . .

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10 Multiscale methods for perforated domains . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Problem setting . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Coarse- and fine-grid notations . . . . . . . . . . . 10.2.3 Outline of GMsFEM . . . . . . . . . . . . . . . . . . 10.3 The construction of offline and online basis functions . 10.3.1 Elasticity problem . . . . . . . . . . . . . . . . . . . . 10.3.2 Stokes problem . . . . . . . . . . . . . . . . . . . . . . 10.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Elasticity equations in perforated domain . . . . 10.4.2 Stokes equations in perforated domain . . . . . . 10.5 Convergence results . . . . . . . . . . . . . . . . . . . . . . . . .

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Numerical results . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Steady state case . . . . . . . . . . . . . . . . . . 8.5.2 Time-dependent case . . . . . . . . . . . . . . . Coupled GMsFEM-NLMC at different resolutions

Space-time GMsFEM . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Space-time GMsFEM . . . . . . . . . . . . . . . . . . . 9.2.1 Preliminaries and motivation . . . . . . . . 9.2.2 Construction of offline basis functions . 9.2.3 Error estimates . . . . . . . . . . . . . . . . . . 9.3 Numerical results for offline GMsFEM . . . . . . . 9.4 Residual-based online adaptive procedure . . . . . 9.5 Numerical results for online GMsFEM . . . . . . .

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11 Multiscale stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Generalized multiscale finite element method for Petrov-Galerkin approximations . . . . . . . . . . . . . . . . . 11.3.1 Construction of the multiscale trial space . . . . . . . 11.3.2 Construction of the multiscale test space . . . . . . . 11.3.3 Global coupling . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 Summary of the procedures for the offline method 11.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.6 Online test basis construction (residual-driven correction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 GMsFEM for selected applications . . . . . . . . . . . . . . . . . . . . . . . 12.1 Multiscale methods for elasticity equations . . . . . . . . . . . . . . 12.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Construction of multiscale basis functions . . . . . . . . 12.1.3 Numerical result . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Multiscale methods for multi-phase flow and transport . . . . . 12.3 Multiscale methods for acoustic wave propagation: Mixed formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Multiscale basis functions . . . . . . . . . . . . . . . . . . . . 12.3.3 The mixed GMsFEM . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport . . . . . . . . . . . . . . . . . . . . 12.4.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Fine-scale discretization . . . . . . . . . . . . . . . . . . . . . 12.4.3 Coarse-grid discretization using GMsFEM: Offline spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4 Randomized oversampling GMsFEM . . . . . . . . . . . 12.4.5 Residual-based adaptive online GMsFEM . . . . . . . . 12.5 Non-local multicontinua upscaling for poroelasticity in fractured media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Embedded fracture model for poroelastic medium . . 12.5.2 Fine-grid approximation of the coupled system . . . . 12.5.3 Coarse-grid upscaled model for coupled problem . . . 12.5.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Multiscale methods for elastic wave propagation in fractured media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . 12.6.2 Fine-scale discretization . . . . . . . . . . . . . . . . . . . . . 12.6.3 Coarse-scale discretization . . . . . . . . . . . . . . . . . . . 12.6.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 GMsFEM for stochastic problems using clustering . . . . . . . . 12.7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.2 Outline of the method . . . . . . . . . . . . . . . . . . . . . . . 12.7.3 The construction of offline space . . . . . . . . . . . . . . . 12.7.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 GMsFEM for uncertainty quantification in inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8.2 GMsFEM for parameter-dependent problem . . . . . . . 12.8.3 Multilevel Monte Carlo methods . . . . . . . . . . . . . . .

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265 265 266 268 272 278

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12.8.4 Multilevel Markov chain Monte Carlo . . . . . . . . . . . . . 374 12.8.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 12.9 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 13 Homogenization and numerical homogenization of nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Monotone and pseudomonotone operators . . . . . . . . . . . . . 13.2 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Numerical homogenization (computation of effective parameters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Pre-computing the effective coefficients . . . . . . . . . 13.3.2 Parabolic equation . . . . . . . . . . . . . . . . . . . . . . . . 13.4 MsFEM for nonlinear problems . . . . . . . . . . . . . . . . . . . . . 13.4.1 Multiscale finite volume element method (MsFVEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Examples of VH . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.3 MsFEM for parabolic equations . . . . . . . . . . . . . . 13.5 Remark on the analysis of MsFEM for nonlinear problems . 14 GMsFEM for nonlinear problems . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . 14.2 Preliminaries and motivation . . . . . . . 14.2.1 Preliminaries and notations . . 14.2.2 Motivation . . . . . . . . . . . . . . 14.3 The GMsFEM . . . . . . . . . . . . . . . . . 14.3.1 Partition of unity functions . . 14.3.2 Multiscale basis . . . . . . . . . . 14.4 Convergence of the method . . . . . . . . 14.5 Numerical implementation and results 14.5.1 Numerical results . . . . . . . . .

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15 Nonlinear non-local multicontinua upscaling . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Preliminaries. A brief overview of NLMC for linear problems . . . . . . . . . . . . . . . . . . . . 15.3 Nonlinear non-local multicontinua model (NLNLMC) . 15.3.1 General concept . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Nonlinear non-local multicontinuum approach 15.4 Linear approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Linear transport . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Single-phase flow . . . . . . . . . . . . . . . . . . . . . 15.4.3 Two-phase flow . . . . . . . . . . . . . . . . . . . . . . 15.5 Nonlinear approach . . . . . . . . . . . . . . . . . . . . . . . . . .

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15.6 RVE-based non-local multicontinua 15.6.1 NLNLMC on RVE-scale . 15.6.2 RVE-based NLNLMC . . . 15.6.3 Examples . . . . . . . . . . . . .

approaches . .......... .......... ..........

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434 436 437 438

16 Global-local multiscale model reduction using GMsFEM . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 Discrete empirical interpolation method (DEIM) 16.2.3 Generalized multiscale finite element method (GMsFEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Global-local nonlinear model reduction . . . . . . . . . . . . . 16.3.1 Local multiscale model reduction . . . . . . . . . . . 16.3.2 Global-local nonlinear model reduction approach 16.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Single offline parameter . . . . . . . . . . . . . . . . . . 16.4.2 Multiple offline parameters . . . . . . . . . . . . . . . .

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17 Multiscale methods in temporal splitting. Efficient implicit-explicit methods for multiscale problems . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Partially explicit temporal splitting scheme . . . 17.3 Spaces construction . . . . . . . . . . . . . . . . . . . . 17.3.1 Construction of VH;1 . . . . . . . . . . . . . 17.3.2 Construction of VH;2 . . . . . . . . . . . . . 17.4 Numerical results . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

Notations

Throughout the book, we use the notations listed below. In some places, our notations may deviate if there are no ambiguities. • • • • • • • • • • • • • • • • • • • • • • • • •

› domain; x space; t time T h is a partition of › into fine finite elements h is the fine mesh size T H is a coarse partition of › such that every element in T H is a union of connected fine-mesh grid blocks H is the coarse mesh size xi is the coarse grid vertex wi is a subdomain with a common vertex at xi, or with an edge Ei wiþ is an oversampled subdomain for wi wE is a subdomain with a common edge E K is a coarse-grid block K + is an oversampled coarse-grid domain that contains K. u is the primal variable uh is the fine-grid solution uH is the coarse-grid multiscale solution ðnÞ uH is the coarse-grid space-time multiscale solution defined on ðTn1 ; Tn Þ  is small scale V—continuous function space Vh—fine-scale space spanned by polynomials Nc is the number of coarse-grid nodes Ne is the number of coarse-grid edges v—partition of unity function (v0 —standard, vms —multiscale) Ni —numerical homogenization cell solution ovs Ni —numerical homogenization cell solution in oversampled cell wj ` —standard multiscale basis function in wj (continuous), 'wj —standard multiscale basis function in wj (discrete) `i —local solutions used in numerical homogenization

xiii

xiv

Notations w

w

• ˆi j —local snapshot solutions in wj (continuous), “ i j —discrete local snapshot solution in wj. þ ;w • ˆi j —local snapshot solutions in the oversampled region wjþ (continuous), þ ;w

• • • • •

“ i j —discrete local snapshot solution in wjþ . w w `i j —local offline basis in wj (continuous), 'i j —discrete local offline basis in wj . þ ;w þ ;w `i j —local offline basis in the oversampled region wjþ (continuous), 'i j — discrete local offline basis in wjþ . lj is the number of offline basis in wj þ ;wi wi VH,off is the offline space in ›; VH;off is the offline space in wi; VH;off is the þ offline space in wi ; þ ;wi wi is the snapshot space in wi; VH;snap is VH,snap is the snapshot space in ›; VH;snap þ the snapshot space in wi ; ðT

;T n Þ

n1 • VH;snap

w ;ðT

i n1 is the snapshot space in ›  ðTn1 ; Tn Þ; VH;snap

þ ;x ;ðT ;T Þ VH;snapi n1 n

;Tn Þ

is the snap-

shot space in xi  ðTn1 ; Tn Þ; is the snapshot space in xiþ  ðTn1 ; Tn Þ ðTn1 ;Tn Þ wi ;ðTn1 ;Tn Þ • VH;off is the offline space in ›  ðTn1 ; Tn Þ; VH;off is the offline space in xi  ðTn1 ; Tn Þ • K is the smallest eigenvalue among all coarse blocks that the corresponding eigenvector is not included in the offine space.

Chapter 1

Introduction

1.1 Challenges and motivation 1.1.1 Multiscale problems A broad range of scientific and engineering problems involve multiple scales. Traditional approaches have been known to be valid for limited spatial and temporal scales. Multiple scales dominate simulation efforts wherever large disparities in spatial and temporal scales are encountered. Such disparities appear in virtually all areas of modern science and engineering, for example, composite materials, porous media, turbulent transport in high Reynolds number flows, etc. A complete analysis of these problems is extremely difficult. In this book, our main goal is to discuss systematic model reduction techniques for problems with multiple scales, complex heterogeneities, and high contrast. We will discuss some examples, which contain multiple scales, complex heterogeneities, and high contrast. First, we will briefly illustrate examples with scale separation, where one can use homogenization techniques, in general. Problems with the scale separation We briefly illustrate some spatial fields that have scale separation. Our first example is a function with two scales 1 and  with   1 x  . 2 + x cos  We depict this function in Figure 1.1 (left plot) with  = 0.003, where one can clearly observe two scales. More precisely, we see that there is a slowly changing amplitude and a high oscillatory component. Generally, a function F(x) having the form F(x) := f (x, x ) is said to have two separable scales 1 and . An example is given above. A function F(x) having the © Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_1

1

2

1 Introduction 3

12

2.8

10

2.6

8

2.4

6

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4 2

2 1.8

0

1.6

−2

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−4

1.2 1 0

0.25

0.5 x

Fig. 1.1 Left: 2 + x cos

x 

0.75

−6 0

1

. Right: 2 + x +

10

i=1 cos

0.25

 xri  

0.5 x

0.75

1

.

form F(x) = f (x, x1 , x2 , · · · ) is said to have multiple separable scales 1, 1 , 2 , · · · . Our next example is a function with multiple separable scales that are of the same order. We consider 10  xr   i , 2+x + cos  i=1 where ri ’s are random numbers between 1 and 2 and  = 0.003. (see Figure 1.1, right plot). In a more general setting, we can take F(x) = 2 + x +

10  i=1

cos

 xr  i



+

20  i=11

cos

 xr  i

2

,

where ri ’s are random numbers between 1 and 2 and  = 0.003. In this case, the function F(x) is said to have three scales. There are various other ways to introduce scales, e.g., via Fourier series and we limit ourselves to the above discussions. In our discussions below, homogenization or numerical homogenization techniques can be used for finding effective parameters and equations for problems with the scale separation. The examples described above are intended to demonstrate simple functions with scale separation. One can similarly generate multidimensional examples both periodic and non-periodic with scale separation. For example, a case with multiple highconductivity inclusions dispersed in the media (Figure 1.2) can be regarded as a case with the scale separation, where one can use numerical homogenization techniques.

Physical and computational scales. No scale separation and high contrast In the previous section, we discussed examples with scale separation. These examples can, for example, be illustrated using Fourier series or some other well-known basis

1.1 Challenges and motivation

3

Fig. 1.2 A heterogeneous media. The dark color region is the high-conductivity region, the light color region is the low conductivity region.

functions, e.g., wavelets. In this section, we would like to discuss the media properties that do not have scale separation and contain high contrast. We note that high contrast is a relative concept. Because our intention is to solve the problems numerically, the computational grid enters as an additional scale and high contrast is with respect to the grid size. We note that both grid and contrast are dimensionless quantities. In Figure 1.10, we show a computational grid (depicted by red color). For simplicity, we choose a Cartesian grid with the size H , where we would like to perform simulations. In this figure, the black regions are high-conductivity regions. We can observe that they have different length scales, moreover, the contrast in the conductivity can be much higher compared to the physical scales and the mesh size. We call the latter high contrast, i.e., the contrast in the media properties is comparable or larger the quantities related to scales (physical or computational). Next, we present some simplistic examples, to demonstrate no scale separation and high contrast with respect to the computational grid. In general, no scale separation can be understood as infinite scales, also; however, here, we restrict ourselves to multiple scales that are much smaller compared to the computational grid. In our book, no scale separation and high contrast are understood with respect to the computational mesh size. To present a simple spatial field, we first consider two dimensional example. Assume that X l (s) = (x1l (s), x2l (s)) is a curve, s ∈ [s0l : s1l ] is l-th curve that is randomly selected in the region of interest with a fixed computational grid size H (see

4

1 Introduction

Figure 1.10). Next, we consider regions RRl = {x; x ∈ [X l (s) + δl (x)n : X l (s) + γl (x)n]}, where δl (s) and γl (s) are two “small” valued functions along the curve and n denotes the unit normal vector of the curve. As a result, RRl is a region around the curve. In this region, we have two scales, one is associated with the length of the curve and the other with the width. We take the length associated with the width to be small. As a result, we have two distinct scales. By adding many such regions, l = 1, ..., we introduce many scales, which are associated with the lengths and widths of these regions (see Figure 1.3). Thus, we have introduced the scales: (1) computational grid H ; (2) physical scales that include length of each inclusion and width of each inclusion (length and width can be thought as sizes in longer and shorter directions). Next, we assign the conductivity κ l to each region RRl and assume the background conductivity is 1. We define the contrasts contrast l = κ l /1, where contrast l is the contrast of l-th region. We note that the contrast can be large compared to the reciprocals of computational grid H and the physical scales that include length and width. This is a simplified but a challenging situation, where one deals with many scales and the contrast at the same time, and, where the coarse-mesh size does not resolve the scales and contrast. In general, one can consider higher dimensional cases that are similar to the two dimensional example. In this case, we consider the surfaces described by X l (s) in Rd , s ∈ [s0l : s1l ], s ∈ Rd−1 , where X l (s) is l-th surface that is randomly selected in the region of interest with a fixed computational grid size H . Next, we consider regions RRl = {x; x ∈ [X l (s) + δl (x)n : X l (s) + γl (x)n]}, where δl (s) and γl (s) are two “small” valued functions along the surface. As a result, RRl is a region around the surface, where we can define contrast and scales as before. Consequently, we have introduced the scales: (1) computational grid H ; (2) physical scales which include sizes along dimensions for each inclusion. In this case, the width can have a rich hierarchy of scales, for example.

Examples with complex heterogeneities, multiple scales, and high contrast Next, we list a few examples with complex heterogeneities, multiple scales, and high contrast with illustrations and discussions. We define high contrast being the case where the ratio of media properties, such as the permeability can be very high (possibly comparable to the small spatial scales). Our first example is a spatial field that is generated using a fractal field (see Figure 1.4). Here, we assume that the

1.1 Challenges and motivation

5

Fig. 1.3 Figure represents scales and contrast. The conductivity in each shaded region is assumed to be large compared to the computational grid (shown) and scales of the inclusions (which include length and width scales).

light color represents low conductivity regions, while dark color represents highconductivity regions. As we can observe from this figure that the media contain many high and low conductivity regions that have different sizes, lengths, and widths, as well as different geometries. One can simplify these complex multiscale geometries and use a high-conductivity inclusions with a small width and different lengths (see Figure 1.5). Different lengths will introduce multiple scales, while the smallest scale is due to the width. High contrast is due to the ratio between the background conductivity and the conductivity of the inclusions. This is discussed earlier. The above examples were computationally generated to illustrate multiple scales. To show a more realistic example, which contain some of the features mentioned above, we depict a rock at three different resolutions in Figure 1.6. As we can observe that at every resolution, the heterogeneities are complex with multiple length scales. High-contrast features can be observed at the pore-scale distribution. For example, at the very detailed resolution, we can observe irregular geometries for valleys that can have multiscale scales. These types of complex heterogeneities are also present in porous media formations such as rocks and groundwater. We would like to mention that there are many examples with both space and time scales. In these cases, the spatial heterogeneities change in time. We will study these examples in the book.

6

1 Introduction

Fig. 1.4 A heterogeneous fractal media. The dark color region is the high-conductivity region, the light color region is the low conductivity region.

Fig. 1.5 A heterogeneous media with high-conductivity inclusions. The different lengths of highconductivity regions introduce a hierarchy of length scales.

1.1 Challenges and motivation

7

Fig. 1.6 Schematic description of hierarchy of heterogeneities in subsurface formations.

1.1.2 Numerical challenges of solving multiscale problems The direct numerical solution of multiple scale problems is difficult even with the advent of supercomputers. The major difficulty of direct solutions is the size of computation. The direct numerical simulation techniques need to resolve smallest scale (e.g.,  for the example with the scale separation), which can require a tremendous amount

8

1 Introduction

of computer memory and CPU time. On the other hand, many of these problems can be approximated using some type of model reduction. For example, homogenization and numerical homogenization techniques can approximate the solution by homogenized equations, where the coefficients of the homogenized equations are computed using the solutions in local representative volumes. From an application perspective, it is often sufficient to predict the macroscopic properties of the multiscale systems. Therefore, it is desirable to develop a method that captures the small-scale effect on the large scales, but does not require resolving all the small-scale features. Approaches discussed in this book provide novel models for complex heterogeneous systems.

1.2 Multiscale model reduction concepts Our objective is to solve linear and nonlinear problems in the media with multiple scales. We will assume that the fine-scale dynamics is governed by L(u) = f in Ω, where L(·) is a partial differential equation with multiscale coefficients and Ω is a spatial domain, which can also contain multiple scales. The equation is equipped with appropriate boundary and initial conditions. Below, we will denote the multiscale field by κ (x) and the heterogeneous domain by Ω  . If domain is not heterogeneous, then it will be denoted by Ω.

1.2.1 Exemplary problems Diffusive spatial heterogeneities In these examples, the media properties are heterogeneous. Examples are L(u) = div(κ (x)∇u); ∂u − div(κ (x)∇u); L(u) = ∂t ∂ 2u L(u) = 2 − div(κ (x)∇u). ∂t These examples include elliptic, parabolic, and wave equations.

(1.1)

Advective spatial heterogeneities In these examples, we consider the heterogeneous convective processes, where the substance is convected with a heterogeneous convection field. Examples are L(u) = κ (x) · ∇u + Δu; ∂u − κ (x) · ∇u − Δu; L(u) = (1.2) ∂t ∂ 2u L(u) = 2 − κ (x) · ∇u − Δu. ∂t In general, one can also include heterogeneous diffusion and reaction terms.

1.2 Multiscale model reduction concepts

9

Diffusive space-time heterogeneities In these examples, the media properties contain heterogeneities both in space and time. Examples are L(u) = div(κ (x, t)∇u); ∂u − div(κ (x, t)∇u); L(u) = (1.3) ∂t 2 ∂ u L(u) = 2 − div(κ (x, t)∇u). ∂t One can also have advection and reaction terms, which are heterogeneous in space and time. Nonlinear diffusive heterogeneities Nonlinear problems constitute a large class of problems. It is difficult to discuss all of these examples. We consider two representative examples for the problems that are studied in the book. The first case is a nonlinear diffusion, where the diffusion depends on the solution, but not the gradient. Examples are L(u) = div(κ (x, u)∇u); ∂u − div(κ (x, u)∇u); L(u) = ∂t ∂ 2u L(u) = 2 − div(κ (x, u)∇u). ∂t

(1.4)

One can consider space and time heterogeneities as well as convection and reaction terms. The next set of examples include the cases, when the coefficients depend on the gradient of the solution. Examples are L(u) = div(κ (x, ∇u)∇u); ∂u − div(κ (x, ∇u)∇u); L(u) = ∂t ∂ 2u L(u) = 2 − div(κ (x, ∇u)∇u). ∂t

(1.5)

These cases are challenging for multiscale problems because the gradient of the solution introduces small-scale features. Parameter-dependent problems In these examples, the media properties contain a parameter dependence. Examples can include

10

1 Introduction

Fig. 1.7 An example of a perforated domain.

L(u) = div(κ (x; μ)∇u); ∂u − div(κ (x; μ)∇u); L(u) = ∂t ∂ 2u L(u) = 2 − div(κ (x; μ)∇u). ∂t

(1.6)

Here, μ is a parameter representing different heterogeneities as a result of randomness or a parametrization. One can include a parameter in many different examples.

Perforated domains In these examples, one deals with domains that contain heterogeneities at smallest scales and the computations attempt to solve the problem on a computational grid, which is much larger than the small-scale heterogeneities. In Figure 1.7, we depict a domain with many inclusions of different sizes. In these problems, the partial differential equations in these domains may or may not have any small-scale parameters. For simplicity, we consider cases when the underlying equations do not contain small scales. Examples are −∇ · u = 0 in Ω  (1.7) −∇ p + μΔv = f in Ω  .

1.2 Multiscale model reduction concepts

11

1.2.2 Fine and coarse grids To discretize the problem, we will use the notions of fine and coarse grids. We assume that the problem is resolved using the fine grid (e.g., see Figure 1.7, where the domain is resolved with the fine grid) and the approximation is satisfactory. One can also consider multiscale methods with adaptive fine grid; however, we will not study this in the book. We denote by T h a fine-grid partition with h being the fine-mesh size. In general, h resolves the small-scale heterogeneities (the perforations of the domain in the case of problems in perforated domains, see Figure 1.7). We do not assume that the fine-grid is uniform. As for the coarse grid, we will use the notation T H . This is a usual conforming partition of the computational domain Ω into finite elements (triangles, quadrilaterals, tetrahedrals, etc.). We assume that each coarse subregion is partitioned into a connected union of fine-grid blocks. In general, we will deal with various coarsegrid neighborhoods depending on the discretization. For the continuous Galerkin discretization, we will use coarse-grid blocks that share a common vertex, while for the mixed discretization, we will use coarse-grid blocks that share common faces. We depict the coarse and fine grids in Figure 1.8. In presenting the main ideas of the method, we will restrict ourselves to conforming Galerkin method, though we would like to emphasize the proposed concept (1) can be easily used with various discretizations (2) is observed to work very well with Nv (where Nv the some discontinuous Galerkin methods (see Chapter 6). We use {xi }i=1 number of coarse nodes) to denote the vertices of the coarse mesh T H , and define the neighborhood of the node xi by  ωi = {K j ∈ T H ; xi ∈ K j }. (1.8) See Figure 1.8 for an illustration of neighborhoods and elements subordinated to the coarse discretization. Furthermore, we introduce a notation for an oversampled region. We denote by ωi+ an oversampled region of ωi ⊂ ωi+ . In general, we will consider oversampled regions ωi+ defined by adding several fine-grid or coarse-grid layers around ωi .

1.2.3 Scale separation approaches Many current approaches for handling complex multiscale problems typically limit themselves to two or more distinct idealized scales (e.g., pore and Darcy scales). These include homogenization and numerical homogenization methods [176, 352, 419]. To demonstrate some of main concepts from the point of view of numerical homogenization, we consider L(u) = f,

(1.9)

12

1 Introduction

ωE K+

+ i

K

ω K

1

K

ωi

K3

2

K4

Fig. 1.8 Illustration of fine grid, coarse grid, coarse neighborhood, and oversampled domain.

where L(u) is a differential operator representing the fine-scale processes. One can use many different examples for L(u). To convey our main idea, we consider a simple and well-studied heterogeneous diffusion L(u) = −div(κ(x)∇u), where κ(x) is assumed to be a multiscale field representing the media properties. Our objective is to define a macroscopic (a.k.a., upscaled or homogenized) property for each coarse block, in general without assuming periodicity. Based on homogenization techniques [352], the local problems for each coarse block is solved using an auxiliary equation

∂ ∂ κi j (x) (1.10) Nl = 0 in K , ∂ xi ∂x j where K is a coarse block (see Figure 1.8 for illustration). The choice of boundary conditions is important and various boundary conditions can be used (see e.g., [419]). For example, Dirichlet boundary conditions N = xl on ∂ K are often used. are defined by averaging the fluxes The upscaled coefficients, κi∗,nh j  K

κi∗,nh j

∂ N∗ = ∂x j l

 κi j (x) K

∂ Nl . ∂x j

The motivation behind this upscaling is to state that the average flux response for the fine-scale local problem with prescribed boundary conditions is the same as that for the upscaled solution. Because Nl∗ = xl in K ,

1.2 Multiscale model reduction concepts

13

we have an explicit expression for macroscopic coefficients  1 ∂ κi j (x) Nl . κil∗,nh = |K | K ∂x j Once the homogenized coefficients are computed, the macroscopic solution is found by solving

∂ ∂ ∗ ∗,nh κi j (x) u (x) = f. (1.11) ∂ xi ∂x j The proximity of u and u ∗ can be shown in the case of scale separation [419]. The above numerical homogenization concepts (for linear and nonlinear problems) are often used to derive macroscopic equations. These techniques assume that the solution space has a reduced dimensional structure. For example, as we can observe from the above example and derivations, the solution in each coarse block is approximated by three local fields (in three dimension). In fact, this “effective” dimension is related to the number of elements (constants) in the effective properties. Homogenization and numerical homogenization techniques derive or postulate macroscopic equations and formulate local problems for computing the macroscopic parameters. For example, in the heterogeneous diffusion example, the effective properties κ ∗ (x) are computed on a coarse grid (see Figure 1.8 for illustration of coarse and fine grids) via solving local problems. Using κ ∗ (x), the global problem (1.9) is solved. The number of macroscopic parameters represents the effective dimension of the local solution space. Consequently, these (numerical homogenization) approaches cannot necessarily represent many important local features of the solution space unless they are identified apriori in a modeling step. Extensions that include multicontinuum concepts have been considered in the literature, which we will discuss later. These approaches introduce multiple macroscopic variables in each coarse block and write “conservation” laws to derive macroscopic equations. Our approaches use “first’ principles” to derive macroscale equations by appropriately introducing macroscopic variables and coupling them via fine-grid equations.

1.2.4 Multiscale finite element methods and some related methods The methods discussed in the book take their origin in Multiscale Finite Element Methods [255] and Generalized Finite Element Methods [329]. Similar methods are introduced earlier in [35, 207]. The main idea of MsFEM is to construct local multiscale basis functions and replace macroscopic equations by using a limited number of basis functions. More precisely, for each coarse node i (see Figure 1.8), we construct multiscale basis function φ ωi and seek an approximate solution of (1.9)

14

1 Introduction

uH =



ci φ ωi .

i

The approximate solution u H is substituted into the original equation and using test functions, for example, φ ωi , the coarse-grid system is formulated. The resulting coarse system can be thought as an upscaled model. These approaches together with the ideas of global model reduction motivate our new techniques as MsFEMs are the first methods that replace macroscopic equation conceptswithcarefullydesignedmultiscalebasisfunctionswithinfiniteelementmethods. Multiscale finite element approaches are shown to be powerful and have many advantages over numerical homogenization methods as MsFEMs can recover finescale information and be flexible in terms of gridding. However, these methods do not contain a systematic way to add degrees of freedom locally and adaptively, which are main contributions of the proposed methods. Some other approaches, such as variational multiscale methods [266], can also be considered as a general multiscale framework. These approaches typically correct a coarse-grid solution (or stabilize it as it is originallyproposed)byformulatingmodelequationsforthesubgridcorrections.These approaches do not contain a systematic way of finding reduced solution representation. Other class of approaches that are built on numerical homogenization methods (e.g., [160, 212, 247, 324]) are limited to problems where macroscopic equations can be formulated and lack a framework to add degrees of freedom.

1.2.5 The need for a systematic multiscale model reduction approach Above approaches do not have a systematic way of constructing multiscale basis functions. Such systematic approaches are needed for problems with complex heterogeneities, high contrast, and multiple scales. The development of these systematic approaches and associated ingredients is a main goal of this book. Our objective is to solve complex problems and to choose the coarse grid independent of small scales and contrast. We do not assume limiting cases for small scales and high contrast and would like our approaches to be robust with respect to these parameters. We will consider Generalized Multiscale Finite Element and present main ideas and ingredients below.

1.3 The basic concepts of Generalized Multiscale Finite Element Method (GMsFEM) 1.3.1 General idea of GMsFEM. In this book, we will discuss novel solution strategies centered around adaptive multiscale model reduction. The GMsFEM was first presented in [190] and later

1.3 The basic concepts of Generalized Multiscale Finite Element Method (GMsFEM)

15

investigated in several other papers (e.g., [72, 101, 111, 124, 125, 172, 191, 219, 220]). The need for such systematic methods will be discussed in the examples of various applications. Our proposed approaches add local degrees of freedom as needed and provide numerical macroscopic equations for problems without scale separation and high contrast. Because of local nature of proposed multiscale model reduction, the degrees of freedom can be added adaptively based on error estimators and the errors can be rigorously estimated.

1.3.2 Multiscale basis functions and snapshot spaces The main idea of our multiscale approach is to systematically select important degrees of freedom for the solution in each coarse block. More precisely, for each coarse block ωi (or K ), we identify local multiscale basis functions φ ωj i ( j = 1, ..., Nωi ) and seek the solution in the span of these basis functions. For problems with scale separation, one needsalimitednumberofdegreesoffreedom.However,astheheterogeneitiesgetmore complicated,oneneedsasystematicapproachtofindtheadditionaldegreesoffreedom. The procedure starts with a coarse-grid layout (see Figure 1.8). In each coarse grid, we first build the snapshot space, VH,snap = ψ ωj i . The choice of the snapshot space depends on the global discretization and the particular application. The appropriate snapshot space (1) yields faster convergence, (2) imposes problem relevant restrictions on the coarse spaces (e.g., divergence free solutions), and (3) reduces the computational cost of building the offline spaces. Each snapshot can be constructed, for example, using random boundary conditions or source terms [71].

1.3.3 Reducing the degrees of freedom. Once we construct the snapshot space VH,snap , we identify the offline space which is a principal component subspace of the snapshot space and derived based on analysis. To obtain the offline space, we perform a dimension reduction of the snapshot space. This reduction identifies dominant modes which are used to construct a multiscale space. If the snapshot space and the local spectral problems are appropriately chosen, we obtain reduced dimensional spaces corresponding to numerical homogenization in the case of scale separation. When there is no scale separation, we have a constructive method to add the necessary extra degrees of freedom that capture the relevant interactions between scales.

1.3.4 Constraint minimization concepts and mesh dependent convergence The GMsFEM approach provides an efficient way to reduce the degrees of freedom in each coarse region. The GMsFEM has spectral convergence as 1/Λ, where Λ is the

16

1 Introduction

smallest eigenvalue that the corresponding eigenfunction is not included in the coarse space. To obtain mesh-dependent convergence (e.g., H/Λ), in [113], the authors propose Constraint Energy Minimizing GMsFEM. In this approach, oversampling regions are used to compute the multiscale basis functions. Multiscale basis functions are supported in the oversampled regions, i.e., φ ωj i are supported in the regions ωi+ . This construction starts with auxiliary space containing spectral basis functions of the GMsFEM. This auxiliary space contains the information related to channels or features that cannot be localized with respect to the coarse-mesh size is used to take care of the non-decaying component of the oversampled local solutions. The construction of multiscale basis functions is done by seeking a minimization of a functional subject to a constraint such that the minimizer is orthogonal (in a certain sense) to the auxiliary space. This allows handling non-decaying component of the oversampled local solutions. This construction allows obtaining the convergence rate H/Λ1/2 , where Λ is the minimal eigenvalue that the corresponding eigenvector is not included in the space. Our analysis also shows that the size of the oversampling domain depends on the contrast weakly (logarithmically).

1.3.5 The relation to upscaling and novel upscaled concepts The degrees of freedom in multiscale methods do not have physical meanings, in general, since they are coordinates in the multiscale space. To be more precise, we seek the solution u H as  u ij φ ωj i . uH = i, j

Consequently, u ij are coordinates. In order to obtain an upscaled model, we discussed computing multiscale basis functions such that u ij ’s are related to the averages of the ω solution. For example, we can take K (l) φm j = δim δ jl , where δ being the Kronecker i

symbol and K i(l) are subregions characterizing high and low conductivity regions. Then j Ui = K (l) U . As a result, the coarse-grid model is an equation for the solutions averi

agedovereachcontinua K i(l) .Thus,constructingmultiscalebasisfunctionswithcertain properties results to the degrees of freedom, which represents the physical quantities. Using these basis function concepts, we design upscaled models. In particular, we have focused on the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) [106, 113, 115] concept. We construct multiscale basis functions, which are supported in oversampled regions such that ωj (l) φm = δim δ jl . Because the local calculations are done in an oversampled domain, Ki the upscaled quantities (e.g., transmissibilities in flows) are non-local and extend to the oversampled region. The resulting upscaled equation for single-phase porous media flow is written in a discrete form as

1.3 The basic concepts of Generalized Multiscale Finite Element Method (GMsFEM)



i, j Tmn (u nj − u im ) = qmi ,

17

(1.12)

j,n i, j

where Tmn are non-local transmissibilities for different continua (associated with basis functions) m and n, and i, j correspond to different coarse blocks. In this regard, our approaches share some similarities with multicontinuum methods (e.g., [28, 41, 348, 409, 420]), where multiple continua are related to basis functions. Next, we would like to discuss the upscaled equation (1.12). First, we note that this equation differs from a standard homogenized equation in two ways. First, each coarse block contains several macroscale quantities related to averages of the solution in our general approach. These macroscopic quantities arise because one cannot localize some features related to high contrast. Secondly, our equation has non-locality. In i, j standard homogenization discretization, Tmn in (1.12) has only next neighbor connection and has a form T i, j , while our novel upscaled model, that provides an accurate approximation for any heterogeneities, has beyond next neighbor connection.

1.3.6 Adaptivity The algorithmic framework proposed is general and can be used for different multiscale, high contrast and perforated problems [72, 110, 125, 219]. The multiscale basis functions are constructed locally and using an adaptive criteria. Thus, in different regions, we expect a different number of basis functions depending on the local features of the problem, such as heterogeneities and high contrast. For example, in the regions with scale separation, we expect only a limited number of degrees of freedom. The adaptivity will be achieved using error indicators.

1.3.7 Nonlinearities In nonlinear problems, one can construct and use multiscale basis functions in obtaining coarse-grid solution. This will involve performing local nonlinear interpolation to approximate the Jacobian or other nonlinear terms (e.g., [72]). The above multiscale procedure can be complemented with online basis functions which helps to converge to the fine-scale solution by constructing multiscale basis functions in the simulations [111, 124] (also, see Figure 1.9 for the ingredients of proposed multiscale method). In the book, we will discuss various approaches for nonlinear upscaling and multiscale methods. Nonlinear upscaling methods can be traced back to nonlinear homogenization [184, 350]. The main idea of nonlinear homogenization is to formulate coarse-grid equations based on nonlinear local problems formulated in each coarse block. In these approaches, the local problems are solved with periodic boundary conditions and some constraints on averages of the solutions of gradients. In non-periodic

18

1 Introduction

cases, these methods are extended by solving local problems in coarse block subject to some boundary conditions. Because of nonlinearity, these local problems need to be solved for all possible average values, which can make the computations expensive. One can compute the upscaled fluxes on-the-fly using the values at previous iteration or previous time. These approaches are limited to problems without high contrast and scale separation. In the book, we will discuss multiscale approaches to problems with high contrast and non-separable scales by introducing multiple macroscopic variables for each coarse-grid block, formulating appropriate local constrained problems that determine the downscaling map; formulating macroscopic equations. Before discussing some ingredients of our approach, we give a brief overview. We assume the nonlinear problem is given by F(u h ) = 0,

(1.13)

where u h is a fine-grid discrete solution and F is a solution operator. In our approaches, we first construct coarse to fine-grid nonlinear operators wh = G(w H ),

(1.14)

where wh is a fine-grid vector and w H is a coarse grid vector. The map G presents a downscaling from coarse-grid discretization to the fine grid. We note that one main step is to identify coarse-grid quantities. Once this map is defined, the coarse-grid solution u H is sought such that functions RF(G(u H )) = 0,

(1.15)

where R consists of test functions. The equation (1.15) is a coarse-grid model. The main ingredients for designing efficient coarse-grid models are finding macroscopic variables, downscaled maps, and test functions that will be discussed here. Next, we present a more detailed discussion on nonlinear multiscale methods. To extend the non-local upscaling methods to nonlinear equations, we first identify macroscopic quantities for each coarse-grid block. These variables are typically found via local spectral decomposition and represent the features that cannot be localized. Next, we consider local problems formulated in the oversampled regions with constraints. These local problems allow identifying the downscaling map from average macroscopic quantities to the fine-grid variables. By imposing the constraints for each continuum variable via the source term, we define effective fluxes and the homogenized equation. Using the local solutions in the oversampled regions with constraints allows localizing the global downscaled map, which provides an accurate representation of the solution; however, it is expensive as it involves solving the global problem. By using the constraints in the oversampled regions, we can guarantee the proximity between the global and local downscaled maps for a given set of oversampled constraints. The resulting homogenized equation significantly differs from standard homogenization. First, there are several variables per coarse block, which represent each continuum. Secondly, the local problems are formulated in

1.3 The basic concepts of Generalized Multiscale Finite Element Method (GMsFEM)

19

Fig. 1.9 Multiscale model reduction. Ingredients and Applications.

oversampled regions with constraints. Finally, the nonlinear homogenized fluxes depend on all averages in oversampled regions, which bring non-local behavior for the equation. These ingredients are needed to perform upscaling in the absence of scale separation and high contrast.

1.3.8 Methodological ingredients of GMsFEM Our computational framework will rely on several important ingredients (see Figure 1.9) that are general for various discretizations (such as mixed methods and discontinuous Galerkin) and applications. These ingredients include (1) procedure for identifying local snapshot spaces and multiscale basis (2) developing global coupling mechanisms for multiscale basis functions (3) adaptivity strategies (4) local nonlinear interpolation tools (5) non-local approaches (6) upscaling methods and (7) online basis functions. These ingredients are building blocks that are needed for constructing an accurate and robust multiscale model reduction. Next, we discuss some of the ingredients of the proposed adaptive multiscale framework in the example of elliptic type problems. To construct multiscale basis functions, the local multiscale structure of the solution space needs to be accounted for. As for the snapshot space, we can use local κ-harmonic functions subject to random boundary conditions [71]. Randomized boundary conditions require a small oversampling region. Once the snapshot space is identified, an appropriate local spectral decomposition in the snapshot spaces is designed to extract the local modes.

20

1 Introduction

The local spectral decomposition is motivated by the analysis and is derived such that one can control the error to yield fast convergence. In continuous Galerkin approach, the multiscale basis functions are multiplied by a partition of unity to preserve conformity. To solve nonlinear equations with these basis functions, the adaptive multiscale procedure is used to represent the solution on a coarse grid with the multiscale modes computed offline. Computing the multiscale solution involves calculating the system residuals and Jacobians. We use empirical interpolation [81] to evaluate these residuals and Jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse-scale problem rather than the fully-resolved fine scale one [72]. The empirical interpolation method uses basis functions, which are built by sampling the nonlinear function we approximate a limited number of times. The coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system. Our methods will use adaptive strategies to identify the regions that require more degrees of freedom and the regions where only a few degrees of freedom suffice. For this purpose, we will develop error indicators and develop an adaptive enrichment algorithms. The error indicators are computed based on some norms of the residual, which involve the spectral decomposition [101]. The error indicators identify the regions where more resolution is needed. The adaptive algorithm automatically enriches the approximation space in regions where the solution requires more degrees of freedom. The GMsFEM has spectral convergence and to obtain mesh-dependent convergence, we propose approaches where basis functions are supported in the oversampled regions. This construction uses auxiliary space containing spectral basis functions of the GMsFEM and constraint optimization problems in computing basis functions. The construction of multiscale basis functions is done by seeking a minimization of a functional subject to a constraint such that the minimizer is orthogonal (in a certain sense) to the auxiliary space. The resulting approach provides a better approximation; however, it is computationally more expensive. Another important ingredient is the development of special multiscale basis functions which can connect multiscale and upscaling methods. These multiscale basis functions are constructed using some constraints such that the resulting discrete system is formulated for physically relevant quantities. Offline computation is an essential component in most multiscale model reduction techniques. This offline process can give a sufficient accuracy. However, there are cases in which the offline procedure is insufficient to give accurate representations of the solution, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. These phenomena

1.3 The basic concepts of Generalized Multiscale Finite Element Method (GMsFEM)

21

Fig. 1.10 A schematic illustration of a multiscale media and a coarse grid. This illustration is intended to show that it is difficult for a coarse grid to capture scales and contrast.

occur locally and in some of these regions that are identified using the proposed error indicators, we will need to develop online basis functions [111]. We will motivate the use of multiscale model reduction by showing how it can take us beyond conventional macroscopic modeling and discuss some relevant ingredients. The proposed framework provides a promising tool and capability for solving a large class of multiscale problems without scale separation and high contrast. This framework is tested and applied in various applications where we have followed the general framework to construct local multiscale spaces. We identify main ingredients of the framework, which can be further investigated for speed-up and accuracy.

1.3.9 High contrast and scale issues In the book, we assume that the coarse mesh does not resolve the scales and the contrast. Resolving the contrast means that the mesh size is much smaller compared to the reciprocal of the contrast. This is very important for many applications. For example, in Figure 1.10, we present a coarse grid and a multiscale media. We can, for example, assume that the conductivity of dark regions are many orders higher compared to the conductivity of white regions, e.g., 106 order of magnitude. In this case, resolving the contrast means that the coarse mesh size is less than 10−6 . it is very difficult to construct a coarse grid, which will resolve the contrast and scales, in this case. The main idea of our proposed methods is to construct multiscale basis functions on a fixed coarse grid that can provide an accurate representation of the solution.

22

1 Introduction

In general, the coarse grids can be tuned based on a physical problem; however, for complex multiphysics problems, it is difficult to adjust the coarse grid as it is problem dependent. For example, for the coupled flow and transport, “optimal” coarse grid for flow equation can substantially differ from “optimal” coarse grid for the transport. In our approaches, coarse grids do not resolve spatial/temporal scales and contrast. The latter implies that when overlaying the coarse grid, we do not take into account the relation between the coarse-grid size and the local contrast (in the media properties). The latter is regarded as high contrast (relative to the coarse-mesh size). Consequently, in the error analysis, the contrast needs to be taken into account. Note that high-contrast regions can be thin and long channels and cross many coarse blocks. This is one of the main reasons that one needs to deal with multiple basis functions.

1.3.10 Modeling with multiscale methods and applications General modeling concepts In the book, we will demonstrate that using our multiscale concepts one can derive novel models. These models are derived by using multiple macroscopic parameters to represent the solution over each coarse-grid block. These degrees of freedom are called continua, which are important for achieving a high order accuracy. As we mentioned earlier, an important step that connects multiscale methods and upscaling techniques includes using basis functions such that the resulting degrees of freedom have physical meanings, typically averages of the solution. For nonlinear problems, using linear basis functions is not very suitable. In general, once the models are known, one can use inverse problem techniques or machine learning techniques to learn these models [87, 400, 405, 406]. In Figure 1.11, we illustrate the main steps of our modeling approach. Below, we briefly describe them. In the first step, we identify continua in each coarse block. This is done with the help of test functions, which can separate the features that cannot be localized within the region of influence (oversampling region designated with green color in Figure 1.11). For nonlinear problems, each continua is defined by a corresponding test function. Continua play the role of macroscale variables. In Step 2, once we identify the continua, we use oversampling regions to define downscaling maps. The oversampling region represents the region of influence and thus, the macroscopic parameter interactions are defined within oversampling regions. The local nonlinear problems are formulated in the oversampled regions using constraints. However, these computations are expensive and require appropriate local problems. Instead, we propose to use local space-time models of the original PDEs and perform many tests with various boundary conditions and sources. These local solutions are used to train macroscopic parameters as a function of multiple

1.3 The basic concepts of Generalized Multiscale Finite Element Method (GMsFEM)

23

Fig. 1.11 Schematic description of the method.

macroscale continua variables. For machine learning, we use deep learning algorithms, which allow approximating complex multicontinua dependent functions. In Step 3, we seek a coarse-grid solution (the values in each continua) such that the downscaled global fine-scale solution satisfies the variational formulation that uses the test functions defined in Step 1. An example of test functions that we use is piecewise constant functions in each subregions (defined as channels). Then, the macroscale variables are average solutions defined in these subregions. The corresponding downscaled maps represent the local fine-grid solutions given these constraints. The global coarse-grid formulation can be thought as a mass balance equation formulated for each continua. We note that our modeling techniques can also be used in modeling with representative volume elements (RVE) [109]. In some applications, representative volumes can be used to extract the macroscale information and macroscopic variables. Though representative volumes typically assume scale separation, one can extract multicontinua information based on representative volumes. For example, using RVE, we can identify several continua and attempt to connect them to derive macroscopic equations. In this case, representative volume allows extracting macroscale information and RVEs are used in deriving macroscale equations. This modeling is discussed in [109].

Applications We will apply the proposed adaptive multiscale framework to several challenging and distinct applications where one deals with a rich hierarchy of scales. These include (1) flows in heterogeneous porous media (2) diffusion in fractured media (3) multiscale processes in perforated regions (4) wave propagation in heterogeneous media (5) multiphysics problems in heterogeneous media. We will show how the proposed framework can be used in designing multiscale methods and common aspects in designing multiscale methods.

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1 Introduction

Limitations. The proposed approaches can be used in many applications, where the solution space has a low-dimensional structure locally. Mathematical and numerical analysis presented in the book allow understanding problems, where our proposed approaches are efficient. In particular, the proposed methods can go beyond, previously known periodic problems or problems with scale separation, as our results show. For example, standard GMsFEM approaches (see Chapter 4) will work well for problems, where there is a strong decay of eigenvalues. If the latter does not hold, we can apply constraint energy minimizing concepts and oversampling ideas to obtain first-order convergence independent of the contrast and scales. These methods allow deriving novel upscaled models for a large class of problems, which can also be learned using available data. In general, if there are many (comparable to the inverse of the fine-mesh size) local features, which can be localized, our approaches will use many degrees of freedom and the computational gain may not be significant in those regions. Many multiscale application problems we have encountered can benefit from local adaptive multiscale model reduction.

1.3.11 Efficient temporal discretizations with multiscale methods Multiscale models play an important role in many directions (see Figure 1.12 for illustration) For example: (1) multiscale methods provide macroscopic laws; (2) multiscale models can be used in inverse problems; (3) multiscale spaces are used in solvers. One other application is the use of multiscale methods in time discretizations. Because of high contrast and multiple scales, one needs to choose a small time step when performing explicit time discretizations. High contrast naturally introduces small time scales in dynamics, which require small time steps to resolve them. For this reason, in many high-contrast multiscale discretizations, implicit methods are used. Explicit methods have many advantages and allow following the dynamics and use much less communications among the degrees of freedom. Choosing larger time steps for explicit methods is remaining challenging. Using our multiscale approaches, we identify the degrees of freedom that require implicit time discretizations. The rest of degrees of freedom can be handled explicitly. Separating the degrees of freedom to implicit and explicit is done using multiscale approaches discussed in the book. In [116–118, 196], we design novel temporal splitting algorithms that allow treating only few degrees of freedom implicitly and the rest are treated explicitly. In these methods, one can take the time step to be independent of the contrast and the time step scales as the coarse-mesh size. The stability of these approaches is shown with appropriate choices of multiscale spaces. With our splitting algorithms, one can use a pre-defined time step and identify the degrees of freedom that need to be treated implicitly.

1.3 The basic concepts of Generalized Multiscale Finite Element Method (GMsFEM)

25

Fig. 1.12 Selected applications of multiscale modeling. Illustration.

1.3.12 Learning and multiscale methods In some cases, one can take an advantage of multiscale models that are derived based on our concepts and learn these models without spending computational efforts to evaluate macroscopic quantities. We discuss two applications of machine learning and deep learning techniques in our applications and refer to [87, 400, 405, 406] for more details. We assume, as before (cf. (1.13)), the fine-grid equation is given by Fh (u h ) = 0,

(1.16)

and denote the coarse-grid equation FH (u H ) = 0,

(1.17)

where u H is the macroscopic solution, which consists of multiple continua. As discussed before, to derive (1.17), we compute the downscaled maps and compute effective nonlinear fluxes, which are functions of multiple macroscale quantities. For example, in the case of nonlinear diffusion equation (when permeability depends on i, j the solution), the macroscale equations has a form (1.12), with the Tmn that depends on solutions in neighboring coarse blocks. For this reason, one can use local learning in computing effective properties. Local learning is introduced and applied in [299, 400]. The main idea is to learn macroscale quantities based on local simulations. Our techniques require complex local computations, which involve solving local problems in oversampled regions subject to constraints. The solutions of these local problems can be replaced by solving original problem on a coarse (oversampled) region for many input parameters

26

1 Introduction

(boundary and source terms) and computing effective properties derived by nonlinear non-local multicontinua approaches. The effective properties depend on many variables (oversampled region and the number of continua), thus their calculations require some type of machine learning techniques. One can use machine learning to effectively compute macroscopic parameters. The proposed method can be regarded as local machine learning and complements our earlier approaches on global machine learning. In [400], we present results for two model problems in heterogeneous and fractured porous media and show that the presented method is highly accurate and provides fast coarse grid calculations. Global learning can be used to learn macroscale equation (1.17) without computing effective properties, while exploiting special structure of the equation. This is done in [406], where multi-layer neural network architectures for multiscale simulations of flows are constructed that take into account the observed data and physical modeling concepts. We have used deep learning algorithms to train the elements of the reduced model discrete system. This shows how our proposed approaches can be combined with deep learning methods to achieve more efficiency.

1.4 Relevant literature review The proposed methods are motivated by numerical homogenization, MsFEM, robust domain decomposition preconditioners, and global model reduction techniques. In particular, the use of multiscale basis functions is due to the MsFEM [176, 255], the idea of snapshots is due to global model reduction methods [250], and the choice of local spectral problems is motivated by two-level domain decomposition approaches for multiscale problems [168]. Many multiscale numerical methods have been developed and studied in the literature. These include generalized finite element methods [32, 33, 35], waveletbased numerical homogenization methods [64, 148, 151, 287], methods based on the homogenization theory (cf. [59, 133, 158]), equation-free computations (e.g., [281, 300, 353, 368, 370, 371]), variational multiscale methods [65, 265, 266, 277, 340], heterogeneous multiscale methods [7, 9, 160, 199, 331, 412], coarse graining [163, 164], matrix-dependent multigrid based homogenization [148, 287], multicontinuum approaches [28, 41, 348, 409, 420], generalized p-FEM in homogenization [322, 323], mortar multiscale methods [27, 361, 362], upscaling methods (cf. [154, 326]), network methods [52–54, 57], localized orthogonal decomposition methods [66, 246, 318], gamblets [345, 346], and other methods [74, 76, 77, 134, 215, 260, 308, 309, 317, 342, 354]. Some of these methods based on the homogenization theory have been successfully applied to determine the effective properties of heterogeneous materials. However, their range of applications is usually limited by restrictive assumptions on the media, such as scale separation and periodicity [49, 275]. Among these approaches are generalized continua concepts [203], which include generalized continuum theories (e.g., [203]), computational continua framework (e.g., [210, 213]), and other approaches. Some other approaches, e.g., [318,

1.4 Relevant literature review

27

345], solve problems without scale separation, but they are limited to the cases with low contrast. Before we present a brief discussion about various multiscale methods, we would like to mention that multiscale finite element methods (MsFEMs) share similarities with upscaling methods. Upscaling procedures have been commonly applied and are effective in many cases. There are many upscaling methods have been developed depending on the nature of heterogeneities and the underlying problem. This is a very rich subject and many methods have been developed and discussed. We cite some of the papers [10, 84, 92, 92, 138, 154, 156, 166, 188, 197, 206, 249, 285, 286, 339, 349, 366, 413, 415, 419]. The main idea of upscaling techniques is to form coarse-scale equations with a prescribed analytical form that may differ from the underlying fine-scale equations. The upscaled equations can be derived based on physical intuition or derived via homogenization or perturbation techniques using some closure arguments. In multiscale methods, the fine-scale information is carried throughout the simulation and the coarse-scale equations are generally not expressed analytically but rather formed and solved numerically. For problems with scale separation, one can establish the equivalence between upscaling and multiscale methods. For example, if only one multiscale basis function is used, one can show that the coarse-grid stiffness matrix obtained using the MsFEM is the same as the stiffness matrix computed using piecewise linear basis functions for the upscaled equation [176]. The MsFEMs discussed in this book, take their origin from a pioneering work of Babuška and Osborn [35]. In this paper, the authors propose the use of multiscale basis functions for elliptic equations with a special multiscale coefficient which is the product of one-dimensional fields. This approach is extended in the work of Hou and Wu [255] to general heterogeneities. Hou and Wu [255] showed that boundary conditions for constructing basis functions are important for the accuracy of the method. They further proposed an oversampling technique to improve the subgrid capturing errors. Later on, the MsFEM of Hou and Wu were generalized to nonlinear problems in [177, 184]. In these papers, various global coupling approaches and subgrid capturing mechanisms are discussed. There are a number of approaches with the purpose of forming a general framework for multiscale simulations. Among them are Equation-Free [281] and Heterogeneous Multiscale Method (HMM) [160]. These approaches are intended for solving macroscopic equations based on the information in RVE (in space and time) and cover a wide range of applications. When applied to partial differential equations, MsFEMs are similar to these approaches. For such problems, multiscale basis functions presented in the book are approximated using the solutions in RVE [176]. We note that for MsFEMs the local problems can be described by the set of equations different from the global equations. An important step in multiscale simulations is often to determine the form of the macroscopic equations and the variables which the basis functions depend upon. In many linear problems and problems with scale separation, these issues are well understood. Many general numerical approaches for multiscale simulations do not address the issues related to determining the vari-

28

1 Introduction

ables that the macroscopic quantities (e.g., multiscale basis functions) depend on (see [300] where some of these issues are discussed). MsFEMs also share similarities with variational multiscale methods [65, 265, 266, 318]. In this approach, the solution of the multiscale problem is divided into resolved (coarse) and unresolved parts. The objective is to compute the resolved part via the unresolved part of the solution and then approximate the unresolved part of the solution. This is shown in the framework of linear equations. Typically, the approximation of the unresolved part of the solution requires some type of localization (e.g., [24, 265]). The localization leads to methods similar to the MsFEM [176]. Some other approaches, e.g., [318, 345], construct multiscale basis functions with the aim of solving problems without scale separation. Our approaches share some similarities with these approaches, in particular, CEM-GMsFEM, which will be discussed in relevant sections. The multiscale methods considered in this book pre-compute the effective parameters which are repeatedly used for different sources and boundary conditions. In this regard, these methods can be classified as upscaling methods where upscaled parameters are pre-computed. An illustration for the concept of upscaling is presented in Figure 1.13. In multiscale approaches discussed in this book, one can re-use pre-computed quantities to form coarse-scale equations for different source terms, boundary conditions and etc. Moreover, adaptive and parallel computations can be carried out with these methods where one can downscale the computed coarsescale solution in the regions of interest. The multiscale methods considered in this book differ from domain decomposition methods and solvers (e.g., [394]), which are designed to find an exact solution via iterative methods.

pre−computation of multiscale (or coarse−scale) quantities

External parameters (forcing, boundary conditions, mobilities,....)

Coupling of multiscale parameters (coarse−scale problem)

Fig. 1.13 A schematic illustration of upscaling concept.

Simulation results

1.4 Relevant literature review

29

Many recent research has been devoted to improving multiscale basis functions. For example, one of the recent directions in multiscale simulations has been the use of some type of limited global information. The use of limited global information is not new in upscaling methods. The main idea of these approaches is to use some simplified surrogate models to extract important information about non-local multiscale behavior of physical processes. The surrogate models are typically solved offline in a pre-computation step and their computations can be expensive. Similar to upscaling methods using global information, multiscale finite element methods using limited global information are introduced in [1, 175, 344]. The work of [344] provides a theoretical foundation for upscaling using limited global information. These methods use limited global information to construct multiscale basis functions. Similarly, novel approaches, which use oversampling techniques to construct coarse-grid approximation are proposed recently in [66]. The methods, we study in the book, develop a systematic approach for the construction of multiscale basis functions. By designing local snapshot spaces and local spectral decomposition, along with several fundamental ingredients, we propose a general adaptive strategy to approximate the solution of multiscale equations at any prescribed accuracy at a reduced cost. The use of snapshots is motivated by global model reduction techniques [43, 82, 250], which construct global basis functions. The global model reduction approaches construct the snapshots by solving the global problem with some selected input data. Furthermore, using Proper Orthogonal Decomposition (POD), the global basis functions are computed. These approaches allow achieving a small dimensional approximating spaces. Our approach uses local dimension reduction techniques. However, there are many important differences as we will discuss. First, the computation of local snapshots requires some judicious choices for local problems that can avoid expensive global solves and many local solves. Secondly, the local spectral problems need to take into account the global coupling discretization and to selected to achieve a fast convergence. We also note that global model reduction approaches, though are powerful in reducing the degrees of freedom, they lack of local adaptivity and numerical discretization properties (e.g., conservations of local mass and energy,...) that local approaches enjoy. Many successful macroscopic laws (e.g., Darcy’s law, and so on) are possible because the solution space admits a large compression locally. We will present several distinct examples to demonstrate this. For this reason, it is important to construct local multiscale model reduction techniques that can identify local degrees of freedom and be consistent with homogenization when there is scale separation. The proposed method is a systematic step in developing such approaches. There are many other multiscale methods in the literature which discuss bridging scales in various applications. In this book, we mostly focus on methods which are most relevant to local multiscale model reduction. We note that a main feature of these methods is (1) the use of variational formulation at the coarse scale which allows to couple multiscale basis functions and (2) the construction of relevant multiscale basis functions. Fine-scale formulation of the problem, which allows computing

30

1 Introduction

multiscale basis functions, is not necessarily based on partial differential equations and can have a discrete formulation.

1.5 Overview of the content of the book In Chapter 2, our goal is to present some basics of homogenization and numerical homogenization techniques. This chapter is intended to show homogenization and numerical homogenization of linear equations using formal asymptotic expansion tools. We consider elliptic and parabolic equations. Numerical homogenization gives basic background on applying homogenization tools for problems without scale separation. As we pointed out these approaches are limited due to the fact that they use only a limited degrees of freedom to represent the fine-scale features. Chapter 3 is dedicated to MsFEM. In this chapter, we discuss the basic concepts of MsFEM and multiscale basis functions. We discuss some convergence issues related to resonance errors. The oversampling technique is motivated and introduced in this chapter. More discussions on MsFEM can be found in our earlier book [176]. In Chapter 4, we present basic outline of Generalized Multiscale Finite Element Method. We discuss offline and online procedure and a general concept of constructing a systematic framework for constructing multiscale basis functions. The main idea of the method is presented in the exemplary problem. We consider both parameter-dependent and parameter-independent problems. In next several chapters and sections, we discuss some basic ingredients of GMsFEM. In Chapter 5, we present adaptivity for GMsFEM. Because multiscale features of the solution can be complex, some adaptivity is needed to select appropriate number of basis functions in different spatial regions. In this chapter, we present a basic adaptive strategy on the example of elliptic equations. Based on the a-posteriori error estimator, we develop an adaptive enrichment algorithm. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. This chapter shows that one can achieve a better convergence using an adaptivity. We use this idea in different discretizations and different applications later in the book. Previous approaches construct multiscale basis functions locally without using any global information. In the same chapter, Chapter 5, we present residual-based online multiscale basis functions. The main idea of the online multiscale basis functions is to use the residual information and construct multiscale basis functions in the online stage. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. We show that if one chooses a sufficient number of offline basis functions,

1.5 Overview of the content of the book

31

one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast. We note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects. Using the residual information, we can incorporate the distant information provided the offline approximation satisfies certain properties. We show that using online multiscale basis functions, one can significantly accelerate the convergence. In this chapter, this idea is presented for elliptic equations and later it is used for time-dependent problems, where we construct online multiscale basis functions in space and time. The GMsFEM can be used in various global discretizations. In Chapter 6, we present three different global discretizations which include mixed, Interior Penalty Discontinuous Galerkin, and Hybridized Discontinuous Galerkin methods. These discretizations are important for many applications. In all these cases, we show that one can use a general framework and construct multiscale basis functions that are suited to the discretization. We note that these discretizations are motivated by practical applications and are used to conserve the properties or the structure of the solution. We observe that HDG coupling gives a better accuracy compared to IPDG coupling due to the fact that they use multiscale weight functions in the penalization. We also develop a concept for multiscale basis construction, which involves energy minimization concepts. In previous approaches, the support of multiscale basis functions is limited to the coarse-grid block. In Chapter 7, we discuss Constraint Energy Minimizing concepts, where multiscale basis functions are constructed in oversampled regions using special constraints related to local spectral problems. These methods are shown to have both spectral and mesh convergence and more accurate compared to standard GMsFEM. The basis functions in these approaches are supported in oversampled regions. We present both offline and online techniques in Chapter 7. In Chapter 8, we discuss the relation between upscaling and multiscale methods. In particular, we propose a general concept for multiscale basis function construction, which results to discrete systems formulated for physically relevant quantities. In these approaches, multiscale basis functions are computed with certain constraints that make the degrees of freedom have physical meanings. In this chapter, we develop new upscaled models based on Constraint Energy Minimizing concepts. In these upscaled models, there are several macroscopic variables in each coarse block and the upscaled models are non-local (in oversampled regions). We discuss these models for some applications later on. In Chapter 9, we consider local multiscale model reduction for problems with multiple scales in space and time. Previous research in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. Our main objective is to develop a multiscale model reduction framework within GMsFEM that uses space-time coarse cells. We construct space-time snapshot and offline spaces. We compute these snapshot solutions by solving local problems. A complete snapshot space will use all possible boundary conditions; however, this can be very expensive. We propose using randomized boundary con-

32

1 Introduction

ditions and oversampling. We construct the local spectral decomposition based on our analysis and present numerical results for space-time heterogeneous problems. In Chapter 10, we develop and analyze generalized multiscale finite element methods for problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, sizes, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. We present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. Our approaches follow GMsFEM framework and start with the offline procedure, which constructs multiscale basis functions in each coarse region and formulates coarse-grid equations. We present an online procedure, which allows adaptively incorporating global information. To illustrate the performance of our method, we present numerical results with both small and large perforations. In Chapter 11, we discuss a Petrov-Galerkin stabilization method for multiscale convection-diffusion transport systems. Existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion, which may not be sufficient to stabilize multiscale systems. We seek a local reduced-order model for this kind of multiscale transport problems. PetrovGalerkin framework using optimal weighting functions is discussed. We introduce an auxiliary variable to a mixed formulation of the problem, where the auxiliary variable stands for the optimal weighting function. The problem reduces to finding a test space (a dimensionally reduced space for this auxiliary variable), which guarantees that the error in the primal variable (representing the solution) is close to the projection error of the full solution on the dimensionally reduced space that approximates the solution. We introduce appropriate snapshots and local spectral problems that appropriately define local weight and trial spaces. We also discuss online basis functions. We present a numerical example, which shows that one needs a few test functions to achieve an error similar to the projection error in the primal variable irrespective of the Peclet number. The applications of GMsFEM to selected problems are presented in Chapter 12. We choose a few applications to demonstrate the performance of GMsFEM. First, we consider the applications to elasticity equations in Section 12.1. In Section 12.2, we use GMsFEM to solve multi-phase flow and transport. In particular, the mixed formulation of GMsFEM is used to solve the flow equation. This is coupled to solving the transport equation on the fine grid without updating multiscale basis functions. Our numerical results show that one can obtain an accurate solution using only a few multiscale basis functions. In Section 12.3, we present the application of GMsFEM to acoustic wave propagation. The mixed formulation is used and several challenging cases are studied. The applications of GMsFEM to the flow and transport in fractured media are studied in Section 12.4. We consider the transport in shale gas that is described by nonlinear equations, which are coupled to the flow in fractures. The fractures are modeled using Discrete Fracture Models. In Section 12.5, we present

1.5 Overview of the content of the book

33

an application of non-local multicontinua upscaling to poroelastic equations in fractured media. In Section 12.6, we present the application of GMsFEM to seismic wave propagation in fractured media. The linear slip model is used to model the fracture in the media. We present the application of GMsFEM for quantifying the uncertainty in Section 12.8. We study the use of GMsFEM within multilevel uncertainty quantification methods, such as multilevel Monte Carlo (MLMC) and multilevel Markov Chain Monte Carlo (MLMCMC). GMsFEM provides a hierarchical approximation of the solution, which can be used to speed up MLMC and MLMCMC. In Chapter 13, we present basics of homogenization and numerical homogenization for nonlinear equations. We consider quasilinear PDES. Our main objective is to show how the solution behavior can affect the homogenization and how the solution enters into homogenization problems. We also present MsFEM framework. More discussions about MsFEM can be found in our earlier book [176]. In Chapter 14, we present a multiscale model reduction framework within Generalized Multiscale Finite Element Method (GMsFEM) for nonlinear elliptic problems. We consider an exemplary problem, which consists of nonlinear p-Laplacian with heterogeneous coefficients. The main challenging feature of this problem is that local subgrid models are nonlinear involving the gradient of the solution (e.g., in the case of scale separation, when using homogenization). Our main objective is to develop snapshots and local spectral problems. We re-cast the multiscale model reduction problem onto the boundaries of coarse cells. This is important and allows capturing separable scales as discussed. We also introduce nonlinear eigenvalue problems in the snapshot space for these nonlinear “harmonic” functions. The proposed methods can, in general, be used for more general nonlinear problems, where one needs nonlinear local spectral decomposition. In Chapter 15, we discuss multiscale methods for nonlinear problems. The main concept consists of (1) identifying macroscopic quantities; (2) constructing appropriate oversampled local problems with coarse-grid constraints; (3) formulating macroscopic equations. We consider two types of approaches. In the first approach, the solutions of local problems are used as basis functions (in a linear fashion) to solve nonlinear problems. This approach is simple to implement; however, it lacks the nonlinear interpolation, which we present in our second approach. In this approach, the local solutions are used as a nonlinear forward map from local averages (constraints) of the solution in oversampling region. This local fine-grid solution is further used to formulate the coarse-grid problem. Both approaches are discussed on several examples and applied to single-phase and two-phase flow problems, which are challenging because of convection-dominated nature of the concentration equation. The numerical results show that we can achieve good accuracy using our new concepts for these complex problems. In Chapter 16, we present a brief introduction to global-local approaches. In these approaches, one uses local multiscale approaches, such as GMsFEM, in approximating the snapshots that are used in global model reduction techniques. In Chapter 17, we present multiscale methods for temporal splitting. In particular, using our multiscale concepts, we design efficient implicit-explicit schemes for multiscale problems.

34

1 Introduction

1.6 Objectives In this book, we will present novel multiscale model reduction techniques for solving many challenging problems with multiple scales and high contrast. Our approach systematically and adaptively adds degrees of freedom in local regions as needed and go beyond scale separation cases. We also present mathematical analysis of our approaches and show that they converge independent of scales and contrast, which is important first results in this direction. The objectives of the book are the following. • To demonstrate the main concepts of our unified approach for local multiscale model reduction • To discuss main ingredients of our proposed methods and future developments • To demonstrate its applications to a variety of challenging multiscale problems

Chapter 2

Homogenization and numerical homogenization of linear equations

2.1 Homogenization for linear problems with oscillatory coefficients. Main concepts 2.1.1 Elliptic equations with heterogeneous coefficients First, we consider the second-order elliptic equation   ∂ ∂ L(u  ) ≡ − κi j (x/) u  = f, u  |∂Ω = 0, ∂ xi ∂x j

(2.1)

where κi j (y) are 1-periodic in both variables of y, and satisfy κi j (y)ξi ξ j ≥ αξi ξi , with α > α0 > 0. Here, we have used the Einstein summation notation, i.e., repeated index means summation with respect to that index. Homogenization theory studies the limiting behavior u  → u as  → 0. The main task is to find the homogenized coefficients, κi∗j , and the homogenized equation for the limiting solution u   ∂ ∂ κi∗j u = f, u|∂Ω = 0. − (2.2) ∂ xi ∂x j We will derive the limiting equations using formal expansion with the purpose of introducing some of the main concepts. We refer, e.g., to [49, 275] for in-depth homogenization theory. Special case: One-dimensional problem Let Ω = (x0 , x1 ). We have

  d du  κ(x/) = f, in Ω , (2.3) dx dx where u  (x0 ) = u  (x1 ) = 0, and κ(y) > α0 > 0 is y-periodic with period 1. −

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_2

35

36

2 Homogenization and numerical homogenization of linear equations

A heuristic homogenization −

d dx

  du  κ(x/) = 0, dx

(2.4)

where u  (x0 ) = 0, u  (x1 ) = 1. Assume that we have a homogenized equation with a homogenized coefficient κ ∗ that is constant. Moreover, we assume that the homogenized equation has the same form  ∗ d ∗ du κ = 0, (2.5) − dx dx where u ∗ (x0 ) = 0, u ∗ (x1 ) = 1. Even though κ ∗ is unknown, this equation can be easily solved and the solution u ∗ is a linear function. We assume that κ ∗ is such that the average flux of homogenized equation and oscillatory equation are the same, i.e.,  x1  x1 du ∗ du  = . κ(x/) κ∗ dx dx x0 x0 From here, we can compute the homogenized coefficient: −1   x1 1 1 ∗ κ = dx . x1 − x0 x0 κ(x/) However, this does not guarantee the closeness of the homogenized solution and the fine-scale solution. A more systematic asymptotic expansion is needed, which is discussed next. One-dimensional case We shall look for u  (x) in the form of asymptotic expansion u  (x) = u 0 (x, x/) + u 1 (x, x/) +  2 u 2 (x, x/) + · · · , where the functions u j (x, y) are periodic in y with period 1. Denote by A the second-order operator du  (x) d ) = f. A = − (κ(x/) dx dx When differentiating a function φ(x, x/) with respect to x, we have d d 1 d = + , dx dx  dy where y is evaluated at y = x/.

(2.6)

(2.7)

2.1 Homogenization for linear problems with oscillatory coefficients. Main concepts

37

With this notation, we can expand A as follows: A =  −2 A1 +  −1 A2 +  0 A3 , where

  d d κ(y) , dy dy     d d d d κ(y) − κ(y) , A2 = − dy dx dx dy   d d κ(y) . A3 = − dx dx A1 = −

(2.8)

(2.9) (2.10) (2.11)

Substituting the expansions for u  and A into A u  = f , and equating the terms of the same power, we get A1 u 0 = 0,

(2.12)

A1 u 1 + A2 u 0 = 0, A1 u 2 + A2 u 1 + A3 u 0 = f.

(2.13) (2.14)

Equation (2.12) can be written as   d d κ(y) u 0 (x, y) = 0, − dy dy

(2.15)

where u 0 is periodic in y. The theory of second-order elliptic PDEs implies that u 0 (x, y) is independent of y, i.e., u 0 (x, y) = u 0 (x). This simplifies equation (2.13)     for u 1 , d du 0 d d κ(y) u1 = κ(y) (x). − dy dy dy dx Define N = N (y) as the solution to the following cell problem:   d d d κ(y) N = − κ(y) , dy dy dy

(2.16)

where N is periodic in y. Note the average of the right-hand side must be zero in order to have a solution. The general solution of equation (2.13) for u 1 is then given du 0 by (x) + u(x) ˜ . (2.17) u 1 (x, y) = N (y) dx Finally, we note that the equation for u 2 is given by   d d κ(y) u 2 = A2 u 1 + A3 u 0 − f . dy dy

(2.18)

The solvability condition implies that the right-hand side of (2.18) must have mean zero in y over one periodic cell Y = [0, 1], i.e.,  (A2 u 1 + A3 u 0 − f ) dy = 0. Y

38

2 Homogenization and numerical homogenization of linear equations

Remark 2.1. Note that for any function F(x, y) that is periodic with respect to y, we have  d d F(x, y) = F(x, y = 1) − F(x, y = 0) = 0  F(x, y) = dy Y =[0,1] dy because of periodicity. We take the average of (2.18) and get     d du 1 d du 0 − κ(y) − κ(y)  = f. dx dy dx dx From here

d − dx

    dN du 0 d du 0 κ(y)  − κ(y) = f. dy d x dx dx

Thus, we have the homogenized equation   d ∗ d κ u= f , − dx dx   where 1 dN ∗ κ = (κ(y) + κ(y) ) dy . |Y | Y dy

(2.19)

(2.20)

(2.21) (2.22)

Note that the expansion can be written as u  (x) = u 0 (x) + N (x/) One can estimate R (x) and show that

 Ω

d u 0 + R (x). dx

|d R /d x|2 ≤ C.

Homogenization as a numerical tool to approximate the solution   d d κ(x/) u  = f dx dx with u  (0) = u  (1) = 0. For fine-scale simulations, we can take, say, M, which is of order 1, points per  (the wavelength is π ). The fine grid will be h = /M. The total computational cost is Consider

Nfine = O(h −1 ) = O( −1 ) if finite element or finite difference methods are used. For the cost of solving the homogenized problem with an accuracy similar to the fine-scale problem, we need to have, M points per wavelength. The computational cost of solving the coarse problem is Nhom = O(H −1 ) = O(1). The cost of performing offline simulations, i.e., computing the homogenized equation is also O(1).

2.1 Homogenization for linear problems with oscillatory coefficients. Main concepts

39

Multidimensional case We consider

−

∂ ∂ xi

  ∂ κi j (x/) u  (x) = f (x) in Ω, ∂x j

u  (x) = 0 on ∂Ω. We shall look for u  (x) in the form of asymptotic expansion u  (x) = u 0 (x, x/) + u 1 (x, x/) +  2 u 2 (x, x/) + · · · , where the functions u j (x, y) are double periodic in y with period 1. Denote by A the second-order elliptic operator   ∂ ∂  κi j (x/) . A =− ∂ xi ∂x j

(2.23)

(2.24)

When differentiating a function φ(x, x/) with respect to x, we have ∂φ ∂φ 1 ∂φ = + , ∂x j ∂x j  ∂yj where y is evaluated at y = x/. With this notation, we can expand A as follows: A =  −2 A1 +  −1 A2 +  0 A3 , where

  ∂ ∂ κi j (y) , A1 = − ∂ yi ∂yj     ∂ ∂ ∂ ∂ κi j (y) − κi j (y) , A2 = − ∂ yi ∂x j ∂ xi ∂yj   ∂ ∂ κi j (y) . A3 = − ∂ xi ∂x j

(2.25)

(2.26) (2.27) (2.28)

Substituting the expansions for u  and A into A u  = f , and equating the terms of the same power, we get A1 u 0 = 0,

(2.29)

A1 u 1 + A2 u 0 = 0, A1 u 2 + A2 u 1 + A3 u 0 = f.

(2.30) (2.31)

Equation (2.29) can be written as   ∂ ∂ κi j (y) u 0 (x, y) = 0, − ∂ yi ∂yj

(2.32)

where u 0 is periodic in y. The theory of second-order elliptic PDEs implies that u 0 (x, y) is independent of y, i.e., u 0 (x, y) = u 0 (x). This simplifies Equation (2.30)

40

2 Homogenization and numerical homogenization of linear equations

for u 1 , −

∂ ∂ yi

    ∂ ∂ ∂u 0 κi j (y) u1 = κi j (y) (x). ∂yj ∂ yi ∂x j

Define N j = N j (y) as the solution to the following cell problem: ∂ ∂ yi

  ∂ ∂ κi j (y) Nj = − κi j (y) , ∂yj ∂ yi

(2.33)

where N j is double periodic in y. The general solution of equation (2.30) for u 1 is then given by ∂u 0 (x) + u˜ 1 (x) . (2.34) u 1 (x, y) = N j (y) ∂x j Finally, we note that the equation for u 2 is given by ∂ ∂ yi



∂ κi j (y) ∂yj

 u 2 = A2 u 1 + A3 u 0 − f .

(2.35)

The solvability condition implies that the right-hand side must have mean zero in y over one periodic cell Y = [0, 1] × [0, 1], i.e.,  (A2 u 1 + A3 u 0 − f ) dy = 0. Y

Integrating (2.35) over the period, we get homogenized equation ∂ − ∂ xi κi∗j =

1 |Y |

  ∗ ∂ κi j u= f , ∂x j

 (κi j + κik Y

 ∂N j ) dy . ∂ yk

(2.36)

(2.37)

Exact computations There are many examples, where one can derive exact formulas for homogenized coefficients. One case is layered media κ(x1 , x2 ) = κ(x2 /)δi j . In this case,

∗ ∗ ∗ ∗ = κ, κ12 = κ21 = 0, κ22 = κ −1 −1 . κ11

2.1 Homogenization for linear problems with oscillatory coefficients. Main concepts

41

2.1.2 Homogenization of parabolic equations In this section, we consider a parabolic equation with the coefficients, which depend on x and x/:   ∂ ∂u  ∂ κi j (x, x/) u  = f, u  |∂Ω = 0. − (2.38) ∂t ∂ xi ∂x j We shall look for u  (x) in the form of asymptotic expansion u  (x, t) = u 0 (x, x/, t) + u 1 (x, x/, t) +  2 u 2 (x, x/, t) + · · · , where the functions u j (x, y, t) are double periodic in y with period 1. Denote by A the second-order elliptic operator   ∂ ∂ κi j (x, x/) . A = − ∂ xi ∂x j

(2.39)

(2.40)

With this notation, we can expand A as follows: A =  −2 A1 +  −1 A2 +  0 A3 , where

  ∂ ∂ κi j (x, y) , A1 = − ∂ yi ∂yj     ∂ ∂ ∂ ∂ κi j (x, y) − κi j (y) , A2 = − ∂ yi ∂x j ∂ xi ∂yj   ∂ ∂ κi j (x, y) . A3 = − ∂ xi ∂x j

(2.41)

(2.42) (2.43) (2.44)

Substituting the expansions for u  and A into A u  = f , and equating the terms of the same power, we get A1 u 0 = 0,

(2.45)

A1 u 1 + A2 u 0 = 0,

(2.46)

∂u 0 . A1 u 2 + A2 u 1 + A3 u 0 − f = − ∂t Equation (2.45) can be written as   ∂ ∂ − κi j (x, y) u 0 (x, y, t) = 0, ∂ yi ∂yj

(2.47)

(2.48)

where u 0 is periodic in y. The theory of second-order elliptic PDEs implies that u 0 (x, y, t) is independent of y, i.e., u 0 (x, y, t) = u 0 (x, t). This simplifies equation (2.46) for u 1 ,

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2 Homogenization and numerical homogenization of linear equations

−

∂ ∂ yi

    ∂ ∂ ∂u 0 κi j (x, y) u1 = κi j (x, y) (x, t). ∂yj ∂ yi ∂x j

Define N j = N j (x, y) as the solution to the following cell problem: ∂ ∂ yi

  ∂ ∂ κi j (x, y) Nj = κi j (x, y) , ∂yj ∂ yi

(2.49)

where N j is double periodic in y. The general solution of equation (2.46) for u 1 is then given by ∂u 0 (x, t) + u˜ 1 (x, t) . (2.50) u 1 (x, y, t) = −N j (x, y) ∂x j Finally, we note that the equation for u 2 is given by   ∂ ∂u 0 ∂ κi j (x, y) u 2 = A2 u 1 + A3 u 0 − f + . ∂ yi ∂yj ∂t

(2.51)

The solvability condition implies that the right-hand side of (2.51) must have mean zero in y over one periodic cell Y = [0, 1] × [0, 1], i.e.,  (A2 u 1 + A3 u 0 − f + Y

∂u 0 ) dy = 0. ∂t

Next, we use a different trick. In particular, the equation at  0 is averaged over the period. We note that  ∂ F(x, y, t)dy = 0 ∂ Y yi for any F(x, y) which is periodic with respect to y. This can be easily verified using the divergence theorem. Then, the averages over the terms which start ∂∂yi will disappear and we get ∂ ∂u 0 − ∂t ∂ xi

    ∂ ∂ ∂ κi j (x, y) κi j (y) u0 − u 1  = f. ∂x j ∂ xi ∂yj

Substituting the expression for u 1 into this equation, we obtain ∂ ∂u 0 − ∂t ∂ xi where κi∗j

1 = |Y |





κi∗j

∂ ∂x j

 u0 = f ,

 ∂N j (κi j − κik ) dy . ∂ yk Y

(2.52)

(2.53)

2.1 Homogenization for linear problems with oscillatory coefficients. Main concepts

2.1.3

43

Homogenization of convection-diffusion equation

In this section, we present an example of a heterogeneous convection and consider ∂u  1 x  ∂ u  − DΔu  = f. (2.54) + vi ∂t   ∂ xi We assume that

vi (y) = 0,

∂vi = 0. ∂ yi

As before, we start with a formal expansion u  (x, t) = u 0 (x, y, t) + u 1 (x, y, t) +  2 u 2 (x, y, t) + .....

(2.55)

Substituting this equation into (2.54), we get ∂ u 0 − DΔ yy u 0 = 0. O( −2 ) : vi (y) ∂ yi From here, we conclude that u 0 (x, y, t) = u 0 (x, t). O( −1 ) : vi (y)

∂ ∂ u 1 − DΔ yy u 1 = −vi (y) u 0 (x, t). ∂ yi ∂ xi

From this equation, we express u 1 in terms of u 1 (x, y, t) = Nl (y)

∂ ∂ xi

u 0 (x, t)

∂ u 0 (x, t) +  u 1 (x, t), ∂ xl

(2.56)

where Nl (y) satisfies vi (y)

∂ Nl − DΔ yy Nl = −vl (y). ∂ yi

(2.57)

Note that this equation is solvable because vl  = 0. Finally, at O( 0 ), we have O( 0 ) :

∂u 0 ∂ ∂ + vi (y) u 2 + vi (y) u 1 − DΔx x u 0 − DΔx y u 1 − DΔ yy u 2 = f (x). ∂t ∂ yi ∂ xi

Taking the average of this equation over the period and noticing that ∂ Fi (x, y)  = 0,  ∂ yi we have

∂u 0 ∂ + vi (y)u 1  − DΔx x u 0 = f (x). ∂t ∂ xi

Using the expression (2.56), we have ∂ ∂ ∂u 0 + (Nl (y)vi (y) u 0 ) − DΔx x u 0 = f (x). ∂t ∂ xi ∂ xl

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2 Homogenization and numerical homogenization of linear equations

∂ ∂ ∂u 0 + (κi∗j u 0 ) = f (x), ∂t ∂ xi ∂x j

Or

where

κi∗j = −D + N j (y)vi (y).

The diffusion is enhanced due to the convection. Remark 2.2. Note that Ni (y)vi (y) ≤ 0. To show that we multiply the cell equation (2.57) by Nl and integrate · −vl (y)Nl (y) = D|∇Nl |2  ≥ 0. Remark 2.3. We can consider a special case, laminar flow v(y) = (v(y2 ), 0). Note that because the flow is divergence-free, v1 (y) will only depend on y2 . In this case, one can solve the cell problem exactly and show that 1 1 ∗ ∗ ∗ , κ ∗ = −D, κ12 = −D − cos(y2 )2  = −D − = κ21 = 0. κ11 D 2D 22 If D is small, the enhanced diffusion is large. Remark 2.4. Note that if we consider x  ∂ ∂u  + vi u  − DΔu  = f, ∂t  ∂ xi then the homogenized equation becomes ∂u 0 ∂ + vi (y) u 0 − DΔu 0 = f. ∂t ∂ xi

(2.58)

(2.59)

2.1.4 Homogenization of convection-diffusion reaction equations We consider

We assume that

1 x  ∂ 1 ∂u  + vi u  − DΔu  + r (x/)u  = f. ∂t   ∂ xi  vi (y) = 0,

(2.60)

∂vi = 0, r (y) = 0. ∂ yi

Remark 2.5. For solvability, we need r ≥ 0. As before, we start with a formal expansion u  (x, t) = u 0 (x, y, t) + u 1 (x, y, t) +  2 u 2 (x, y, t) + .....

(2.61)

2.1 Homogenization for linear problems with oscillatory coefficients. Main concepts

45

Substituting this equation into (2.60), we get O( −2 ) : vi (y)

∂ u 0 − DΔ yy u 0 = 0. ∂ yi

From here, we conclude that u 0 (x, y, t) = u 0 (x, t): O( −1 ) : vi (y)

∂ ∂ u 1 − DΔ yy u 1 = −vi (y) u 0 (x) − r (y)u 0 . ∂ yi ∂ xi

From this equation, we express u 1 in terms of ∂∂xi u 0 (x, t) and u 0 (x, t): ∂ u 0 (x, t) + N (y)u 0 (x, t) +  u 1 (x, t), u 1 (x, y, t) = Nl (y) ∂ xl

(2.62)

where Nl (y) satisfies vi (y) and N (y) satisfies

∂ Nl − DΔ yy Nl = −vl (y) ∂ yi

vi (y)

(2.63)

∂ N − DΔ yy N = r (y). ∂ yi

Note that these equations are solvable because vl  = 0 and r  = 0. Finally, at O( 0 ), we have O( 0 ) :

∂u 0 ∂ ∂ + vi (y) u 2 + vi (y) u 1 − DΔx x u 0 − DΔx y u 1 ∂t ∂ yi ∂ xi −DΔ yy u 2 + r (y)u 1 = f (x).

Taking the average of this equation over the period and noticing that  we have

∂ Fi (x, y)  = 0, ∂ yi

∂u 0 ∂ + vi (y)u 1  − DΔx x u 0 + r (y)u 1  = f (x). ∂t ∂ xi

Using the expression (2.62), we have ∂ ∂u 0 ∂ (Nl (y)vi (y) u 0 ) − DΔx x u 0 + r (y)N (y)u 0 = f (x). + ∂t ∂ xi ∂ xl Or

where

∂ ∂ ∂u 0 + (κ ∗ u 0 ) + r ∗ u 0 = f (x), ∂t ∂ xi i j ∂ x j κi∗j = −D + N j (y)vi (y), r ∗ = r (y)N (y).

(2.64)

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2 Homogenization and numerical homogenization of linear equations

2.2 Numerical homogenization for linear problems with oscillatory coefficients: Main concepts 2.2.1 A motivation The discussions in the previous chapter assume periodic structures. In many applications, we do not have periodicity. In these cases, one can still use numerical techniques provided “scale-separation” holds. For simplicity, we consider du d (κ(x/) ) = f, dx dx u(0) = u(1) = 0 with  −1 κ(x/) = 3 + cos(x (2)/) + cos(x/) , or, one can take

κ(x/) = a0 +

n 

−1 ai cos(xβi /)

.

i=1

If we compute the “homogenized” coefficients in [0, H ], H , using a standard homogenization formula, we get κ ∗,H =



1 H



H 0

dx κ(x/)

−1 .

From here, −1  √   sin(H/) κ ∗,H = 3 + √ , sin(H 2/) + H 2H

κ ∗,H

n   ai = a0 + sin(Hβi /) H i=1 βi

−1 .

As we see from here, κ ∗,H converges to a constant as H increases. This value can, in general, be regarded as a “homogenized coefficient” if the region of integration contains sufficient scale information. In many scenarios, one can still use homogenization-based techniques and localize the computations. More importantly, in these cases, one can still use homogenized coefficients and homogenization theory. However, it is not important to use periodic boundary conditions, as will be discussed next.

2.2 Numerical homogenization for linear problems with oscillatory …

47

2.2.2 Local problems and macroscopic equations The main idea of numerical homogenization is to identify the homogenized coefficients in each coarse-grid block. The basic underlying principle is to compute these upscaled quantities such that they preserve some averages for a given set of local boundary conditions. We discuss it on the example of elliptic equations. We consider   ∂ ∂ κi j (x) u = f. (2.65) ∂ xi ∂x j Our objective is to define an upscaled (or homogenized) conductivity for each coarse block, in general without assuming periodicity. We will follow the homogenization technique and will solve the local problem for each coarse block subject to some boundary condition (see Figure 2.1 for illustration): ∂ ∂ xi

  ∂ κi j (x) φl = 0 in K , ∂x j

(2.66)

where K is a coarse block. Here, we denote φl to be the solution to local problems (similar to multiscale basis functions, as they are related). The choice of boundary conditions can be important for the accuracy in non-periodic cases. Below, we list some commonly used boundary conditions. Choice 1. Dirichlet boundary conditions. In this case, we impose Dirichlet boundary conditions φl = xl on ∂ K . This is one of most commonly used boundary conditions, and it can be used with an arbitrary shape coarse region. Choice 2. Mixed Dirichlet-Neumann boundary conditions. In this case, we impose mixed Dirichlet-Neumann boundary conditions where Neumann boundary conditions are imposed on lateral boundaries. φ1 = x1 , on ∂ K that is parallel to x2 axis and κi j (x)

∂ φ1 · n 2 = 0, on ∂ K that is parallel to x1 axis. ∂x j

Similarly, for φ2 . These boundary conditions are more suited to rectangular coarse regions. Choice 3. Periodic boundary conditions. This follows the homogenization theory and uses φl = xl + periodic function on ∂ K . These boundary conditions can give large errors if periodicity is not present.

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2 Homogenization and numerical homogenization of linear equations

Fig. 2.1 Illustration of numerical homogenization.

In all these examples, if the local problem is homogenized and the homogenized coefficients are constant, then φl∗ = xl in K . The upscaled coefficients are defined by averaging the fluxes  K

κi∗,nh j



∂ ∗ φ = ∂x j l

κi j (x) K

∂ φl . ∂x j

(2.67)

The motivation behind this upscaling is that the average flux response for the finescale local problem with prescribed boundary conditions is the same as that for the local upscaled solution. If we+ use φl∗ = xl in K , we have κil∗,nh =

1 |K |

 κi j (x) K

∂ φl . ∂x j

(2.68)

Remark 2.6. One can show that ∗,nh κlm

1 = |K |

 κi j (x) K

∂ ∂ φl φm . ∂ x j ∂ xi

This can be used to prove that if κi j is symmetric, then κi∗j is also symmetric. Remark 2.7. (Simplified Computations). In the layered case, κi j (x) = a(x2 )δi j , one can easily compute ∗,nh κ11

1 = |K |

 a(x2 )d x, K

∗,nh κ12

=

∗,nh κ21

= 0,

∗,nh κ22

 =

1 |K |

 K

1 dx a(x2 )

−1

.

2.2 Numerical homogenization for linear problems with oscillatory …

49

Remark 2.8. (A possible generalization). A possible generalization for numerical homogenization will be to calibrate homogenized coefficients for a set of boundary conditions. Assume that the local problems are solved subject to some boundary conditions Bl , i.e.,   ∂ ∂ κi j (x) (2.69) φl = 0 in K , ∂ xi ∂x j and φl = Bl (x) on ∂ K , where l = 1, .., N . Further, we select an average quantity, E, which we would like to match. For example.  ∂ ∂ κi j (x) φl φm , Elm = ∂ x j ∂ xi K l, m = 1, .., N , is used for numerical upscaling, before. Then, we can compute the corresponding homogenized averaged response E ∗ . For the example Elm , E ∗ is  ∂ ∗ ∂ ∗ ∗ Elm = κi∗,nh φ φ , j ∂ x j l ∂ xi m K where

∂ ∂ xi

  ∗,nh ∂ ∗ κi j φ = 0 in K , ∂x j l

and φl∗ = bl on ∂ K . Finding κi∗,nh can be set as a minimization problem for j  (E i j − E i∗j )q , min ij

for some q. This minimizer is zero for the example Elm by choosing the numerical homogenization coefficients presented above. However, for more complex boundary conditions and average output, one can use this minimization to honor the selected integrated responses for selected boundary conditions. An application example for numerical homogenization We present an example of the application of numerical homogenization to shale gas transport (see [388] for full details). We consider the following equation: ∂c = ∇ · (b(x, c)∇c) , in Ω, t > 0, (2.70) a(x, c) ∂t where a(x, c) and b(x, c) are the fine-grid functions. For the numerical solution, we will use the numerical homogenization technique. We present the coarse T H and fine grid T h descriptions in Figure 2.2. To define an upscaled coefficient for each coarse block K , we solve a fine-scale problem for some fixed c¯ in local block K , as before for a fixed c (cf., [15]):

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2 Homogenization and numerical homogenization of linear equations

Fig. 2.2 Unstructured computational coarse grid T

H

with coarse cell K .

− ∇ · (b(x, c)∇φ ¯ l ) = 0, in K ,

(2.71)

with Dirichlet boundary conditions φl = xl on ∂ K . The upscaled coefficient b is defined by  1

= b(x, c)∇φ ¯ b·,l l d x. |K | K

(2.72)

(2.73)

For calculation of the upscaled coefficient a , we use volume averaging 1 a = |K |

 a(x, c) ¯ d x.

(2.74)

K

To handle the nonlinearities, we assume that the average c¯ over the coarse block is bounded above and below, i.e., c ∈ [cmin , cmax ], where cmin and cmax are pre-defined constants [177]. The interval [cmin , cmax ] is divided into N equal regions: cmin = c0 < c1 < ... < c N −1 < c N = cmax , N . After the calculation of the and we calculate upscaled coefficients for c¯ ∈ {ci }i=0

upscaled coefficients a (x, c¯i ) and b (x, c¯i ) for i = 1, N , we solve the following equation on the coarse grid

 ∂c = ∇ · b (x, c)∇c , in Ω, t > 0, a (x, c) (2.75) ∂t

where b is the anisotropic upscaled coefficient  

bx,x bx,y

. b = b y,x b y,y

2.2 Numerical homogenization for linear problems with oscillatory …

51

In Eq. (2.75), for the nonlinear coefficients, we use a linear interpolation to approximate the coefficient value. Next, we present numerical results to test the accuracy of the proposed method. We consider the solution of the problem with nonlinear coefficients in Ω = Ωi ∪ Ωk , where Ωi refers to the inorganic region (blue in Figure 2.3) and Ωk refers to the organic region (red in Figure 2.3). The coefficients are taken to be  a (c) in Ωk , a(x, c) = k ai in Ωi ,

 b (c) in Ωk , b(x, c) = k bi (c) in Ωi ,

where for kerogen subdomain ak (c) = (φk + (1 − φk )Fck ), bk (c) = (φk Dk + (1 − φk )Ds Fck ), and for inorganic matrix   km . ai = φi , bi (c) = φi D + (1 − φi )c RT μ The corresponding values are Dk = 10−7 [m 2 /s], Ds = Di = 10−8 [m 2 /s], φi = 0.01 φk = 0.5, T = 413[K ], μ = 2 · 10−5 [kg/(m · s)], and km = 10−20 [m 2 ]. We use s , F(c) = cμs (1 + sc)2 where s = 0.26 · 10−3 [m 3 /mol] and cμs = 0.25 · 10−5 [mol/m 3 ]. Kerogen is dispersed randomly in an inorganic frame and shown in Figure 2.3.

Fig. 2.3 Random distribution of the kerogen in inorganic frame with different kerogen distribution. Red—kerogen, and blue—shale matrix. Left: small kerogen inclusions, Case 1. Right: larger kerogen inclusions compared to the domain, Case 2.

For the numerical solution, we use Tmax = 300 hours with 50 time steps. Computational domain Ω = [0, 1] m × [0, 1] m. Initial condition is c(x, 0) = c0 with c0 = 10000 [mol/m 3 ]. On the left and the right boundaries of Ω, we set

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2 Homogenization and numerical homogenization of linear equations

Dirichlet boundary conditions (c0 = 10000 [mol/m 3 ] on right boundary and c1 = 5000 [mol/m 3 ] on left boundary). For nonlinear coefficients, we use cmin = c1 , cmax = c0 , N = 21. For the test cases, we consider two porosity fields (Figure 2.3) and use 3 different structured and 3 unstructured coarse grids (Figures 2.4–2.5), which we designate by “5s”, “5u”, “10s”, “10u”, “20s”, and “20u”, where “s” stands for structured and “u” stands for unstructured and the numbers represent the complexity of the grid: • grid 5s and 5u: structured grid “5s” contains 50 cells and 36 vertices; unstructured grid “5u” contains 68 cells and 45 vertices. • grid 10s and 10u: structured grid “10s” contains 200 cells and 121 vertices; unstructured grid “10u” contains 242 cells and 142 vertices. • grid 20s and 20u: structured grid “20s” contains 800 cells and 441 vertices; unstructured grid “20s” contains 1054 cells and 568 vertices. Fine grid contains approximately 80000 cells and 40000 vertices.

Fig. 2.4 Structured coarse grids. Left: grid “5s”. Middle: grid “10s”. Right: grid “20s”.

Fig. 2.5 Unstructured coarse grids. Left: grid “5u”. Middle: grid “10u”. Right: grid “20u”.

2.2 Numerical homogenization for linear problems with oscillatory …

53

For the numerical comparison of the accuracy of the presented method, we calculate the fine-scale reference solution and compare it with the coarse-grid solution. Fine- and coarse-scale solutions are shown in Figures 2.6 and 2.7 for kerogen distribution from Case 1 and Case 2, respectively. In Figure 2.8, we present the solution on a fine grid along line y = 0.5 for both cases for different time levels at t = 12, 120, and 300 hours. We solve the macroscopic model on the coarse grid with upscaled coefficients that are calculated for each coarse cell. As for the coarse grid in both cases, we use grid 5s. We observe that we obtain similar solutions for fine- and coarsegrid calculations. We note that the size of the discrete problem for the approximation on the fine grid is approximately D O F = 40000 and for coarse grid it is DOF = 36, which equals the number of vertices for grid 5s.

Fig. 2.6 Fine- and coarse-grid solutions for Case 1 for different time steps t = 12, 120, and 300 hours (left to right). Top: fine-grid solution. Bottom: coarse-grid solution.

To verify the accuracy of the calculations, we plot the solutions for coarse and fine grids along line y = 0.5 for three time steps t = 12, 120, and 300 for Case 1 in Figure 2.9 and for Case 2 in Figure 2.10. We observe that the solutions are similar for the coarse-grid solution and fine-grid solution. The difference between solutions is slightly bigger for small times for grid 5s, which is the coarsest grid. In Figure 2.11, we show the solutions for coarse and fine grids along line y = 0.5 for Cases 1 and 2 for the final time, t = 300 hours. We obtain very

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2 Homogenization and numerical homogenization of linear equations

Fig. 2.7 Fine- and coarse-grid solutions for Case 2 for different time steps t = 12, 120, and 300 hours (left to right). Top: fine-grid solution. Bottom: coarse-grid solution.

Fig. 2.8 Fine-grid solution along line y = 0.5 for Case 1 (Left) and Case 2 (Right) for different time steps t = 12, 60, 120, 180, 240, and 300 hours.

Fig. 2.9 Fine- and structured coarse-grid solutions along line y = 0.5 for Case 1 for different time steps t = 12 (Left), 120 (Middle), and 300 hours (Right). Black—fine grid. Green—grid 5s. Red—grid 10s. Blue—grid 20s.

2.2 Numerical homogenization for linear problems with oscillatory …

55

Fig. 2.10 Fine- and structured coarse-grid solutions along line y = 0.5 for Case 2 for different time steps t = 12 (Left), 120 (Middle), and 300 hours (Right). Black: fine grid. Green: grid 5s. Red: grid 10s. Blue: grid 20s.

good predictions with the coarse-grid models for structured and unstructured coarse grids.

Fig. 2.11 Fine- and coarse (structured and unstructured)-grid solutions along line y = 0.5 for Case 1 (Left) and Case 2 (Right) for time step t = 300 hours. Black: fine grid. Green: grid 5s. Red: grid 5u.

 An average value in Ω (cav = Ω c d x/|Ω|) of the fine and coarse solutions as a function of time is presented in Figures 2.12 and 2.13 for Cases 1 and 2, respectively. We can conclude from this figure that the the proposed upscaling can give a good solution using a very coarse grid.

2.2.3 Convergence results for numerical homogenization In this section, we present a convergence rate for numerical homogenization if the medium is periodic and the Dirichlet boundary condition is used. This convergence shows that there is a resonance error and the error has components that are generic and occur when coefficients are no longer periodic. We assume that κi j (x) = κi j (x/). More general cases with multiple small scales can also be studied. In this case, the local solution has the expansion φl (x) = φl∗ (x) + Nm

∂ ∗ φ + Rl , ∂ xm l

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2 Homogenization and numerical homogenization of linear equations

Fig. 2.12 Fine- and coarse structured (Left) and unstructured (Right) grid average solutions by time for Case 1.

Fig. 2.13 Fine- and coarse structured (Left) and unstructured (Right) grid average solutions by time for Case 2.

where φl∗ (x) = xl is a linear function because κi∗,nh is constant. In this case, j φl (x) = xl + Nl + Rl . The equation for Rl can be easily written ∂ ∂ xi



∂ κi j (x) Rl ∂x j

 = 0, in K .

On the boundary ∂ K , we have Rl = −Nl on ∂ K . Furthermore,

2.2 Numerical homogenization for linear problems with oscillatory …

57

 ∂ 1 κi j φl = |K | K ∂x j   ∂ ∂ 1 1 κi j (xl + Nl ) + κi j R= |K | K ∂x j |K | K ∂x j    ∂ ∂ ∂ 1  1 1 κi j (xl + Nl ) + κi j (xl + Nl ) + κi j R, |K | ∂x j |K | K /  ∂x j |K | K ∂x j  κil∗,nh =



(2.76)

where  are  periods in K . It can be easily shown that  ∂ κi j (xl + Nl ) = κil∗ , ∂ xj  where κil∗ is the true homogenized coefficient. Then,     ∂ ∂ |K | 1 1 1 floor ||κil∗ + (xl + Nl ) + κi j R=  κi j |K | || |K | K /  ∂x j |K | K ∂x j       ∂ ∂ |K | 1 1 1 floor || − 1 κil∗ + κi j (xl + Nl ) + κi j R. κil∗ + |K | || |K | K /  ∂x j |K | K ∂x j κil∗,nh =

(2.77)

Thus, the error is given by κil∗,nh − κil∗ =



1 floor |K |



|K | ||



   ∂ ∂ 1 1 || − 1 κil∗ + κ (x +  N ) + κi j Rl . i j l l |K | K /  ∂x j |K | K ∂x j

(2.78)

We will estimate the terms on the right-hand sides of (2.78). For the first term, we have     |K | 1 floor || − 1 κil∗ | ≤ C(/H )d , | |K | || where d is the dimension of the space. For the second term on the right-hand side of (2.78), we have  1 ∂ | κi j (xl + Nl )| ≤ C(/H )d . |K | K /  ∂ x j Next, we estimate the third term on the right-hand side of (2.78): |

1 |K |

 κi j K

∂ 1 R| ≤ C ∂x j H

1/2

 |∇ Rl |2

.

K

From the equation for Rl , we can introduce the usual cut-off functions τ and show that

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2 Homogenization and numerical homogenization of linear equations

 |∇ Rl |2 ≤ C H.

(2.79)

K

Thus, |κil∗,nh − κil∗ | ≤ C



 . H

We note that the largest source of the error is due to the mismatch in the boundary. Next, we discuss the use of the oversampling technique to reduce the effects of artificial boundary conditions in numerical homogenization. The oversampling for numerical homogenization has been extensively discussed [84, 231, 419].

2.2.4 The choice of boundary conditions in numerical homogenization. Oversampling The above analysis suggests that the boundary layers due to artificial linear boundary conditions cause resonance errors. The resonance errors are first studied in the MsFEM; see [179, 255]. To reduce these resonance errors, the oversampling technique has been proposed. The main idea of this method is to solve local problems in a larger domain. In the oversampling method, the local problem that is analogous to (2.66) is solved in a larger domain (see Figure 2.14). In particular, if we denote the large domain by S while the target coarse block by K , then   ∂ ovs ∂ κi j (x) = 0 in S. (2.80) φl ∂ xi ∂x j For simplicity, we use φlovs = xl on ∂ S. The upscaled conductivity is computed by equating the fluxes on the target coarse   block: ∂ ovs ∗,ovs ∂ ∗,ovs κi j φl = κi j (x) φ . (2.81) ∂x j ∂x j l K K Noting that the domain S is only slightly larger, we can use that φl∗,ovs = xl , and thus, κil∗,ovs

1 = |K |

 κi j (x) K

∂ ovs φ . ∂x j l

(2.82)

The advantage of this approach is that one reduces the effects of the oscillatory boundary conditions. √ One can show that the error for the residual is small and scales as /H instead of /H if no oversampling is used (see [232, 419]).

2.2 Numerical homogenization for linear problems with oscillatory …

59

Fig. 2.14 Coarse and fine grids.

Fig. 2.15 Increasing representative volume.

2.2.5 Increasing representative volume size In the above discussions, we use the entire target coarse-grid information to compute effective properties. In a number of applications, one cannot afford to solve the local problem on a coarse grid and instead use a representative volume (see Figure 2.15 for the illustration). In this case, large representative volumes can be chosen to compute effective properties. If we consider the problem with a small scale but with no periodicity (e.g., almost periodic case or random homogeneous case), it can be shown that as the representative volume size increases, the effective conductivity converges. More

60

2 Homogenization and numerical homogenization of linear equations

precisely, we consider

∂ ∂ xi

 κi j (x)

∂ η φ ∂x j l

 = 0 in K η ,

(2.83)

η

and φl = xl on ∂ K η . Here, K η is increasing size representative volumes as η → ∞. Then, we define  1 ∂ η ∗,η κil = κi j (x) φ . (2.84) |K | K ∂x j l It is expected that

∗,η

κi j → κi∗j , as η → ∞. ∗,η

To understand this numerically, we can compute κi j for increasing values of η and ∗,η we can observe that the value κi j stops changing after some large η. These ideas are used in stochastic homogenization with multilevel Monte Carlo methods (see [194]).

2.2.6 Improving numerical homogenization One approach for improving the accuracy of numerical homogenization methods further is to use global information. We refer, e.g., to [61, 84, 156, 253, 413, 414] for more discussions. There are various ways to perform global upscaling. In global approaches, the solutions of a limited number of global problems are used to construct the upscaled conductivities. One approach is to take the domain S in the oversampling to be a large region or the entire domain. Note that the boundary conditions are chosen independent of the boundary conditions of the flow equations. Global approaches are found not always to be robust if not implemented carefully. The choice of a coarse grid can improve the accuracy of upscaling methods. In particular, the coarse grid needs to be chosen to minimize the error in the numerical upscaling. A number of approaches are proposed in this direction and many of them rely on the use of limited global information (e.g., [3, 92, 154, 156, 157]). In particular, a coarse grid is chosen such that to minimize the variations of certain fields within. These fields can include conductivity or some limited global spatial fields.

2.2.7 Numerical homogenization for space-time heterogeneous problems We consider a diffusion in the media with space-time heterogeneities   ∂ ∂u ∂ κi j (x, t) − u = f. ∂t ∂ xi ∂x j In the homogenization of these equations, various homogenization limits are obtained depending on the time scales. This is due to the fact that the cell problem needs

2.2 Numerical homogenization for linear problems with oscillatory …

61

to be independent of  and the effects of weak temporal or spatial processes are eliminated. However, in numerical homogenization, one can deal with weak temporal and spatial processes. As a result, the precise scalings of these processes are not very important for numerical homogenization processes at the expense of performing extra computations at the cell problem level. In numerical homogenization, we consider a cell problem in K × [tn , tn+1 ], (see Figure 2.16)   ∂ ∂φl ∂ φl = 0, κi j (x, t) − ∂t ∂ xi ∂x j with φl = xl on ∂ K and φl (x, t = tn ) = xl . Then, denoting (x∗ , t∗ ) a representative point for the coarse grid, we have  tn+1 1 1 ∂ ∗ κil (x∗ , t∗ ) = κi j (x, t) φl . (2.85) |K | tn+1 − tn tn ∂x j The coarse-grid equation is ∂ ∂u ∗ − ∂t ∂ xi

  ∂ ∗ ∗ κi j (x, t) u = f, ∂x j

(2.86)

where κi∗j (x, t) are defined for each coarse block (in space and time) by (2.85) at quadrature points. For example, if we assume that (2.86) is discretized with implicit Euler using piecewise linear elements, then, if u ∗ (x, t) = i u i (t)φi0 (x), we have 

tn+1 tn

 Ω

du l (t) 0 0 φl φm + dt



tn+1

 Ω

tn

u l (t)κi∗j (x, t)

This can be written as

∂ 0 ∂ 0 φ φ = ∂ x j l ∂ xi m

 u l (tn+1 ) 

u l (tn+1 ) tn

tn+1

 K

κi∗j (x, t)

 Ω

f φm0 . (2.87)

 Ω



φl0 φm0

− u l (tn )

Ω

∂ 0 ∂ 0 φ φ = ∂ x j l ∂ xi m

φl0 φm0 −

 Ω

(2.88) f φm0 .

In the last integral, we noted that ∂∂xi φm0 is constant (assuming piecewise linear basis t  functions on a triangulation). The integral tnn+1 K κi∗j (x, t) is computed using numerical homogenization as above. One can simply use one quadrature point, for example or use multiple quadrature points. In this case, we can use a simple approximation  tn

tn+1

 K

u l (t)κi∗j (x, t) = u l (tn+1 )κi∗j (x∗ , t∗ )(tn+1 − tn )

and the resulting system can be written as

62

2 Homogenization and numerical homogenization of linear equations

Fig. 2.16 Time and space coarse grids.

Mu(tn+1 ) − Mu(tn ) − Au(tn+1 ) = b. Here, 

 Mlm =

Ω

φl0 φm0 , Aml =

tn+1

tn

 K

u l (t)κi∗j (x, t)



∂ 0 ∂ 0 φ φ ∂ x j l ∂ xi m



and bm =

Ω

f (x)φm0 .

We note that the homogenization of parabolic equations with space and time heterogeneities involves several cases. In this case, one considers ∂ ∂u − ∂t ∂ xi

  x t ∂ κi j ( β , α ) u = f.   ∂x j

The cell problems differ for the cases: (1) α = 2β; (2) α < 2β; (3) α > 2β; (4) α = 0; (5) β = 0.

2.3 Homogenization in perforated regions In homogenization in perforated domains, the resulting averaged equations are formulated in the domains without perforations. In this section, we present an example of fluid flow in porous media (see Figure 2.17). In this case, the domain depends on  and we are interested in the limiting equations when  is small.

2.3 Homogenization in perforated regions

63

Fig. 2.17 The illustration of porous media.

2.3.1 Homogenization of Stokes equations We consider

− ∇ p + μ 2 Δv = f in Ω  , div(v ) = 0,

(2.89)

v = 0 on ∂Ω  . Here v = (vi ) is the velocity of the fluid and p is the pressure (see Figure 2.17). The viscosity is rescaled as  2 in order to avoid rescaling the velocity field with  2 . As before, our objective is to show the homogenized equations without giving an in-depth review of this subject area. We write a formal expansion p (x) = p0 (x, y) + p1 (x, y) + · · · , v (x) = v0 (x, y) + v1 (x, y) + · · · , v,i (x) = v0,i (x, y) + v1,i (x, y) + · · · . Substituting this into the equation (2.89) and collecting the same powers, we get O( −1 ) : ∇ y p0 (x, y) = 0, div y (v0 (x, y)) =

∂ v0,i (y) = 0. ∂ yi

From here, we have p0 (x, y) = p0 (x). Furthermore, O( 0 ) : −

∂ ∂ ∂ ∂ p1 (x, y) − p0 (x) + μΔ yy v0,i = f i , v0,i (x, y) + v1,i (x, y) = 0. ∂ yi ∂ xi ∂ xi ∂ yi

From the first equation, we can write

64

2 Homogenization and numerical homogenization of linear equations

p1 (x, y) = Q i (y)( f i +

∂ 1 ∂ p0 (x)), v0,i = Ni j (y)( f i + p0 (x)), ∂ xi μ ∂ xi

where Q and N solve −

∂ Q j (y) + Δ yy Ni j (y) = δi j , ∂ yi

where N = 0 on ∂Ys . This equation is solved for every j (i.e., for every j we have a vector solution (Ni j )). Because of ∂ v0,i (y) = 0, ∂ yi we have ∂ Ni j (y) = 0. ∂ yi Thus, the cell problem is ∂ ∂ − Q j (y) + Δ yy Ni j (y) = δi j , Ni j (y) = 0, ∂ yi ∂ yi with Ni j = 0 on ∂Ys . Next, we take an average of the equation for the divergence ∂ v (x, y) = 0 and arrive at ∂ yi 1,i ∂ v0,i (x, y) = 0, ∂ xi κi j ∂ where v0,i (x, y) = ( fi + p0 (x)). μ ∂ xi Here

κi j = Ni j (y).

∂ ∂ xi

v0,i (x, y) +

(2.90) (2.91)

The equation (2.90) is called Darcy’s law where the permeability is given by (2.91). This equation is supplemented with incompressibility condition ∂ v0,i (x, y) = 0, ∂ xi with v0,i (x, y)n i = 0 on ∂Ω. Remark 2.9. (About scaling of viscosity.) If we consider − ∇ p + μΔv = f in Ω  , div(v ) = 0,

(2.92)

then, we will find out that v (x) = v0 (x) + v1 (x) +  2 v2 (x, y) + .... Because of zero boundary conditions on the inclusions, we conclude v0 (x) = v1 (x) = 0.

2.4 Numerical homogenization in perforated domains

65

Simple permeability computations Permeability can be computed for simple regions such as cylindrical pipes or other shapes. The permeability of the planar fracture can be computed. We consider Figure 2.18. In 2D, there are two cell problems −∇ Q j + ΔN j = e j div(N j ) = 0, e1 = (1, 0) and e2 = (0, 1). If j = 2, then N2 = 0 and the permeability is zero in x2 direction. If j = 1, we consider −∇ Q + ΔN = (1, 0), div(N ) = 0, Q 1 = 0, N1 = N1 (y2 ), N2 = 0. We have

d2 N1 = 1, dy22

N1 (0) = 0, and N1 (δ) = 0. The solution is given by N1 (y2 ) = 

The permeability is

1

κ11 = −

y2 (y2 − δ) . 2

N1 (y2 )dy2 =

0

1 3 δ . 12

2.4 Numerical homogenization in perforated domains The above procedure can be implemented numerically and the permeability can be computed numerically. In this case, one can solve the cell problem −∇ Q j (x) + μΔN j (x) = e j in K , with N j = 0 on ∂ K . Here, K is a coarse-grid block. This is equivalent to −

∂ Q j (x) + μΔNi j (x) = δi j in RVE, ∂ xi

Ni j = 0 on ∂RVE. The permeability is defined as κi j =

1 |K |

 Ni j (x). K

(2.93)

66

2 Homogenization and numerical homogenization of linear equations

The Darcy law is v i = −κi j

∂ . ∂x j

This equation is supplemented by the incompressibility condition ∂ vi = 0. ∂ xi

Fig. 2.18 A period.

Chapter 3

Local model reduction. Introduction to Multiscale Finite Element Methods

3.1 Multiscale finite element methods 3.1.1 Finite element with multiscale basis functions In this section, we will motivate MsFEM on the example of one-dimensional elliptic equation. We will start with a standard finite element method and consider −

du d (κ(x) ) = f dx dx

and u(0) = u(1) = 0. We assume that the interval [0, 1] is divided into N segments 0 = x0 < x1 < x2 < ... < xi < xi+1 < ... < x N = 1. We assume that for each node xi , a basis function, φ0ωi , is assigned that is supported in ωi = [xi−1 , xi+1 ]. 1 0.9 0.8 0.7 0.6

0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3.1 An illustration of one-dimensional basis functions.

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_3

67

68

3 Local model reduction. Introduction to Multiscale Finite Element Methods

The solution is represented as u(x) =

N 

u i φ0ωi ,

i=1

where u i ’s are unknowns to be determined. To determine u i , we substitute u(x) into ω the original equation and multiply by a test function φ0 j and integrate over [0, 1]: N   i=1

d dφ ωi ω (κ(x)u i 0 )φ0 j d x = dx dx

1 0



1

ω

f (x)φ0 j d x,

0

j = 1, ..., N . Integrating by parts and noting that φ0ωi (0) = φ0ωi (1) = 0 (no boundary terms), we get  1  1 ω N  dφ0ωi dφ0 j ω ui κ(x) f (x)φ0 j d x. dx = d x d x 0 0 i=1 This is a matrix equation in the form Au = b, where A = (ai j ) and b = (bi ), 

1

ai j =

ω

κ(x)

0

dφ0ωi dφ0 j d x, b j = dx dx



1

0

ω

f (x)φ0 j d x.

We compute the elements of the matrix for piecewise linear basis functions as in Figure 3.1. For each fixed j, we will have 3 basis functions that will have overlapping ω support with φ0 j . That is,  1 ω dφ ωi dφ0 j (3.1) ai j = d x = 0 if i > j + 1, or i < j − 1. κ(x) 0 dx dx 0 We compute for each i, i = j, i = j − 1, i = j + 1. Let H j = x j+1 − x j . If i = j, 

1

ajj =

κ(x)

0

If i = j − 1, 

1

a j−1, j =

ω

κ(x)

0

 x j+1 ω ω ω ω dφ0 j dφ0 j dφ j dφ0 j κ(x) 0 dx = dx = dx dx dx dx x j−1  xj  x j+1 1 1 κ(x) + κ(x). 2 H j−1 H j2 x j x j−1

ω

dφ0 j−1 dφ0 j dx = dx dx



xj

ω

κ(x)

x j−1

ω

dφ0 j−1 dφ0 j 1 dx = − 2 dx dx H j−1

(3.2)



xj

κ(x)d x.

x j−1

(3.3) If i = j + 1,  a j+1, j = 0

1

ω

ω

1 dφ j+1 dφ0 j dx = − 2 κ(x) 0 dx dx Hj



x j+1 xj

κ(x)d x.

(3.4)

3.1 Multiscale finite element methods

69

Multiscale basis functions The multiscale basis function for the node i is given by d dφ ωi (κ(x) )=0 dx dx

(3.5)

with the support in [xi−1 , xi+1 ]. In the interval [xi−1 , xi ], the boundary conditions for the basis function φ ωi are defined as φ ωi (xi−1 ) = 0, φ ωi (xi ) = 1. In the interval [xi , xi+1 ], the boundary conditions for the basis function φ ωi are defined as φ ωi (xi ) = 1, φ ωi (xi+1 ) = 0. Note that for the computation of the elements of the stiffness matrix, we do not need explicit expression of φ ωi and instead, we simply need to compute κ(x) ddx φ ωi . From (3.5), it is easy to see that κ(x) ddx φ ωi = const, where the constants are different in [xi−1 , xi ] and [xi , xi+1 ]. The constants can be easily computed by writing ddx φ ωi = const/κ(x) and integrating it over [xi−1 , xi ] and [xi , xi+1 ], respectively. This yields 1 dφ ωi =  xi d x κ(x) dx x κ(x) i−1

on [xi−1 , xi ] and κ(x)

1 dφ ωi = −  xi+1 d x dx x κ(x) i

on [xi , xi+1 ]. Then, the elements of the stiffness matrix A = (ai j ) are given by  xi+1  xi ω ω dφ ωi dφ0 j dφ ωi dφ0 j dx + dx = κ(x) κ(x) ai j = dx dx dx dx xi−1 xi (3.6)  xi  xi ω ω 1 1 dφ0 j dφ0 j  xi d x d x −  xi+1 d x d x. dx dx x x κ(x) xi−1 κ(x) xi−1 i−1

i

Thus, we have ai,i−1 = −  xi

1

xi−1

dx κ(x)

, aii =  xi

1

xi−1

dx κ(x)

1 +  xi+1 xi

dx κ(x)

1 , ai,i+1 = −  xi+1 xi

dx κ(x)

.

Consequently, the stiffness matrix has tri-diagonal form and the right-hand side 1 b = (bi ), where bi = 0 f φ0ωi d x. In Figures 3.2 and 3.3, we illustrate the solution and a few multiscale basis functions. We refer to [258] for the analysis in one-dimensional case.

3.1.2 Basic idea of MsFEM In this section, we give a brief introduction to MsFEM. We refer to [176] (and references therein) for more details. We consider linear elliptic equations Lu := −∇ · (κ(x)∇u) = f in Ω, u = 0 on ∂Ω,

(3.7)

70

3 Local model reduction. Introduction to Multiscale Finite Element Methods 1 0.9 0.8 0.7 0.6

0.5 0.4 0.3 0.2 0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3.2 An illustration of one-dimensional multiscale basis functions. Basis

Exact solution 0.3

1.2

0.25

1

0.2

0.8

0.15

0.6

φ

φ

φ

3

2

p

1

0.4

0.1 −(k (x) p’ )’=1 0.05

0.2

0

0

−0.05 0

0.2

0.4

0.6 x

0.8

1

−0.2 0

0.2

0.4

0.6

0.8

1

x

Fig. 3.3 An illustration of one-dimensional basis functions and the solution.

where Ω is a domain in R d (d = 2, 3) and κ(x) is a heterogeneous field varying over multiple scales. Again, we note that κ = (κi j (x)). MsFEM basically consists of two parts: • basis function construction; • a choice of the global formulation that couples these basis functions (Figures 3.4 and 3.5). First, we discuss the basis function construction. We remind that T H denotes a coarse-grid partition of Ω and T h denotes a fine-grid partition of Ω. Let xi be the interior nodes of the mesh T H and φ0ωi be the nodal basis of the standard finite element space VH0 . For simplicity, one can assume V0H the space of piecewise linear functions for a rectangular partition. Denote by ωi = supp(φ0ωi ) and define φ ωi as L(φ ωi ) = 0 in K , φ ωi = φ0ωi on ∂ K , ∀ K ∈ T H , K ⊂ ωi .

(3.8)

3.1 Multiscale finite element methods

71

Note that even though the choice of φi0 can be quite arbitrary, our main assumption is that the basis functions satisfy the leading- order homogeneous equations when the right-hand side f is a smooth function. We would like to remark that MsFEM formulation allows one to take an advantage of scale separation which is discussed later in the book. In particular, K can be chosen to be a volume smaller than the coarse grid. To illustrate the basis functions, we depict them in Figure 3.6. On the left, the basis function is constructed by taking K to be an element smaller than the coarse-grid block size and on the right the basis function is constructed when K is a coarse partition element. Indeed, in the presence of scale separation, one can use the solution in RVE to represent the solution in the entire region as it is done in classical homogenization. Once the basis functions are constructed, we let VH be the finite element space spanned by φi .

Fig. 3.4 An illustration of two-dimensional multiscale basis functions.

Next, we discuss the global formulation of MsFEM. In general, the global formulation of MsFEM is derived from standard finite element methods. In the case of Galerkin finite element methods and assuming the basis functions are conforming, the multiscale finite element method is to find u H ∈ VH such that   κ(x)∇u H · ∇v H d x = f v H d x ∀ v H ∈ VH . (3.9) Ω

Ω

72

3 Local model reduction. Introduction to Multiscale Finite Element Methods

Fig. 3.5 Multiscale basis functions: Illustration. 1 0.9

K

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

bilinear

0

Fig. 3.6 Example of basis functions. Left: basis function with K being RVE. Right: basis function with K being a coarse element.

One can choose the test functions from VH0 and arrive at the Petrov-Galerkin version of multiscale finite element method as introduced in [259]: find u H ∈ VH such that   κ(x)∇u H · ∇v H d x = f v H d x ∀ v H ∈ VH0 . (3.10) Ω

Ω

In the above discussion, we presented the simplest basis function construction and a global formulation. In general, the global formulation can be easily modified and various global formulations based on finite volume, mixed finite element, discontinuous Galerkin finite element, and other methods can be derived. Many of them are studied in the literature and some of them will be discussed here. As for basis functions, the choice of boundary conditions in defining the multiscale basis functions plays a crucial role in approximating the multiscale solution.

3.1 Multiscale finite element methods

73

Intuitively, the boundary condition for the multiscale basis function should reflect the multiscale oscillation of the solution u across the boundary of the coarse-grid element. By choosing a linear boundary condition for the basis function, we will create a mismatch between the exact solution u and the finite element approximation u H across the element boundary. In the next sections, we will discuss this issue further and introduce an oversampling technique to alleviate this difficulty. This technique enables us to remove the artificial numerical boundary layer across the coarse-grid boundary element. We would like to note that in the one-dimensional case, this issue is not present since the boundaries of the coarse element consist of isolated points. Comments on the assembly of stiffness matrix. One can use the representation of multiscale basis functions via fine-scale basis functions to assemble the stiffness matrix. This is particularly useful in code development. Assume that multiscale basis function (in discrete form) φi can be written as φ ωi = di j φ j , 0, f

0, f

where D = (di j ) is a matrix and φ j are fine-scale finite element basis functions (e.g., piecewise linear functions). The ith row of this matrix contains the fine-scale representation of the ith multiscale basis function. Substituting this expression into the formula for the stiffness matrix ai j , we have  ai j =

Ω

a∇φ ωi ∇φ ω j d x = dil

 0, f

Ω

a∇φl

∇φm0, f d x d jm . f

f

Denoting the stiffness matrix for the fine-scale problem by A f = (alm ), alm =  0, f 0, f Ω a∇φl ∇φm d x we have A = D A f DT .  Similarly, for the right-hand side, we have b = Ω φ ωi f d x = Db f , where b f =  f f 0, f (bi ), bi = Ω f φi d x. This simplification can be used in the assembly of the stiffness matrix. A similar procedure can be done for the Petrov-Galerkin MsFEM.

3.1.3 Using smaller regions in computing multiscale basis functions If the local computational domain is chosen to be smaller than the coarse-grid block, then one can use the approximation of u H in RVE to represent the left- hand side of (3.10). In this case, there is often no need to compute the integral over the entire coarse-grid block K , and one can approximate this  integral via the integral over   RVE. That is, Ω κ(x)∇u H · ∇v H d x ≈ K |K|Kloc| | Kloc κ(x)∇u H · ∇v H d x, the solution of the local leading-order partial differential equations is approximated. Similar approximation can be done for the right-hand side. With this formulation, one has

74

3 Local model reduction. Introduction to Multiscale Finite Element Methods

 |K |   |K |  κ(x)∇u H · ∇v H d x = f v H d x ∀ v H ∈ VH0 . (3.11) |K | |K | loc loc K K loc loc K K

3.2 Reducing boundary effects 3.2.1 Motivation The boundary conditions for the basis functions play a crucial role in capturing small-scale information. If the local boundary conditions for the basis functions do not reflect the nature of the underlying heterogeneities, MsFEM can have large errors. These errors result from the resonance between the coarse-grid size and the characteristic length scale of the problem. When the coefficient κ(x) is a periodic function varying over  scale (κ(x) = κ(x/)), the convergence rate of MsFEM contains a term /H (see [258]), which is large when H ≈ . We remind that H is the coarse mesh size. As illustrated by the error analysis of [258], the error due to the resonance manifests as a ratio between the wavelength of the small-scale oscillation and the grid size; the error becomes large when the two scales are close. A deeper analysis based on the homogenization theory shows the main source of the resonance effect. By a judicious choice of boundary conditions for basis functions, we can reduce the resonance errors significantly. Some approaches include the use of reduced problems based on the solutions of one-dimensional problems along the boundaries (e.g., [255, 273, 274]), and oversampling methods (e.g., [85, 179, 255]) are studied in the literature with the goal of reducing resonance errors. Here, we will focus on oversampling methods. Next, we present an outline of the analysis which motivates the oversampling method. We consider a simple case with two distinct scales, i.e., κ(x) = κ(x, x/) and assume that κ is a periodic function with respect to x/. In this case, the solution has a well-known multiscale expansion (see, for example, [49, 275]) ∂ u 0 + θp , u = u 0 + N j ∂x j where u 0 satisfies the homogenized equation −div(κ ∗ (x)∇u 0 ) = f . The homogenized coefficients are defined via an auxiliary (cell) problem over a period of size . To illustrate this, we denote the fast variable by y = x/ and, thus, the coefficients have the form κ(x, y). Then, κ ∗ (x) = |Y1 | Y κ(x, y)(I + ∇ y N (x, y))dy, where N = (N1 , ..., Nd ) is a solution of div y (κ(x, y)(I + ∇ y N (x, y))) = 0

(3.12)

in the period Y for a fixed x (see [49, 275] for more details). For simplicity, one can assume that x represents a coarse-grid block. If there is no slow dependence with respect to x in the coefficients, κ = κ(x/) = κ(y), then there is only one

3.2 Reducing boundary effects

75

1

0.1

0.5

0.05

0

0

−0.5

−0.05

−1 1

−0.1 1 0.8

1 0.6

0.8

0.8

1 0.6

0.6

0.4

0.4

0.2

0.4

0.2

0.2 0

0.8 0.6

0.4 0.2 0

0

0

Fig. 3.7 An illustration of boundary layer function θ . Left: θ in the coarse element with oscillatory boundary conditions. Right: θ in  distance away from the boundaries.

cell problem (3.12) for the entire domain Ω. It can be shown that u 0 + N j ∂∂x j u 0 approximates the solution u in H 1 norm for small  (see [49, 275]). Following multiple scale expansion, as discussed above, we can write a similar expansion for the basis function φ ωi = φi1 + θ ,

(3.13)

where φi1 = φ0ωi + N j ∂∂x j φ0ωi is the part of the basis function which has the same nature of oscillations near boundaries as the approximation of the fine-scale solution u 0 + N j ∂∂x j u 0 . Assuming φ0ωi is a linear function, it can be easily shown that θ satisfies div(κ∇θ ) = 0 in K and θ = −N j ∂∂x j φ0ωi on ∂ K . If one can ignore θ in (3.13), then MsFEM will converge independent of the resonance error. The term θ is due to the mismatch between the fine-scale solution and multiscale finite element solution along the boundaries of the coarse-grid block where the multiscale finite element solution is linear. This mismatch error propagates into the interior of the coarse-grid block. The analysis shows that the MsFEM error is dominated by θ . In Figure 3.7, we depict θ (x) and the same θ (x) which is  distance away from the boundaries. It is clear from this figure that the oscillations decay fast as we move away from the boundaries. To avoid these oscillations, one needs to sample a larger domain and use only interior information to construct the basis functions. The decay of these oscillations basically dictates how large the sampling region should be chosen (Figure 3.8).

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3 Local model reduction. Introduction to Multiscale Finite Element Methods

Fig. 3.8 No oversampling solution.

3.2.2 Oversampling technique Motivated by the above discussion and the convergence analysis of [258], Hou and Wu proposed an oversampling method in [255] to overcome the difficulty due to scale resonance. Since the boundary layer in the first-order corrector is thin, we can sample in a domain with a size larger than h and use only the interior sampled information to construct the basis functions. By doing this, we can reduce the influence of the boundary layer in the larger sample domain on the basis functions significantly. It is intuitively clear from Figure 3.7 that the effects of artificial boundary conditions are significantly reduced for this special two-scale example. Specifically, let φ +,ω j be the basis functions satisfying the homogeneous elliptic equation in the larger domain K + ⊃ K (see Figure 3.9). We then form the actual basis φ ωi by linear combination of φ +,ω j , φ ωi =



ci j φ +,ω j .

j

The coefficients ci j are determined by condition φ +,ωi (x j ) = δi j , where x j are nodal points. Extensive numerical experiments have demonstrated that the oversampling technique does improve the numerical error substantially in many applications. On the other hand, the oversampling technique results in a nonconforming MsFEM method, where the basis functions are discontinuous along the edges of coarse-grid blocks. In [179], we perform a careful estimate of the nonconforming errors. The analysis shows that the nonconforming error is indeed small, and consistent with our numerical results [255, 257]. Our analysis also reveals another source of resonance, which is the mismatch between the mesh size and the “perfect” sample size. In the case of a periodic structure, the “perfect” sample size is the length of an integer multiple of the period. We call the new resonance the “cell resonance”. In the error expansion, this resonance effect appears as a higher order correction.

3.3 Comparison to other multiscale methods

77

K KE

Oversampled domain

Coarse−grid

Fine−grid

Fig. 3.9 Schematic description of oversampled region.

3.3 Comparison to other multiscale methods 3.3.1 Comparison to numerical homogenization MsFEMs share similarities with many other multiscale methods. One of the early approaches is the upscaling technique (e.g., [60, 154, 419]) which is based on the homogenization method. The main idea of upscaling techniques is to form a coarsescale equation and pre-compute the effective coefficients, as discussed earlier. In the case of linear elliptic equations, the coarse-scale equation has the same form as the fine-scale equation except that the coefficients are replaced by effective, homogenized coefficients. The effective coefficients in upscaling methods are computed using the solution of the local problem in a representative volume. Various boundary conditions can be used for solving local problems and, for simplicity, we consider div(κ∇φe ) = 0 in K

(3.14)

with φe (x) = x · e on ∂ K , where e is a unit vector. It is sufficient  to solve (3.14)  for d linearly independent vectors e1 , ..., ed in R d because φe = i βi φei if e = i βi ei . Here K denotes a coarse-grid block, though one can use a smaller region. The effective coefficients are computed in each K as  1 κ∇φe d x. (3.15) κ∗ e = |K | K We note that k∗ (which is not the same as the homogenized coefficients) is a symmetric matrix provided κ is symmetric and (3.15) can be computed for any point in the domain by placing the point at the center of K , i.e.,  1 κ∇φe d x, κ∗ (x0 )e = |K x0 | K x0

78

3 Local model reduction. Introduction to Multiscale Finite Element Methods

where K x0 is RVE with the center at x0 and φe is the local solution defined by (3.14) in K x0 . One can use various boundary conditions, including periodic boundary conditions as well as oversampling methods. We refer to [154, 419] for the discussion on the use of various boundary conditions. Once the effective coefficients are calculated, the coarse-scale equation − div(κ∗ ∇u ∗ ) = f

(3.16)

is solved over the entire region. To show the similarity to MsFEM, we write down the discretization of (3.16) using the Galerkin finite element method. Find u ∗h ∈ VH0 , such that   f v H d x, ∀v H ∈ VH0 . (3.17) κ∗ ∇u ∗H · ∇v H d x = K

Ω

K

Next, we write down the Petrov-Galerkin MsFEM discretization (see (3.10)) ai j pi = b j ,

 

(3.18) ω

where ai j = K K κ∇φ ωi d x · ∇φ0ωi (recall φ0 j is piecewise linear) and b j =  ωj Ω f φ0 d x. One can show that  ω ai j = κ∗ ∇φ0ωi · ∇φ0 j d x 

K

K

κ∗ ∇φ0ωi . We assumed that φ0ωi

because |K1 | K κ∇φ ωi d x = are piecewise linear functions. Thus, (3.18) and (3.17) are equivalent. This shows that MsFEM can be derived from traditional upscaling methods. However, the concept of MsFEM differs from traditional upscaling methods, since the local information is directly coupled via a variational formulation, and we do not assume a specific form for coarse-scale equations. Moreover, MsFEMs allow to recover the local information adaptively which makes it a powerful tool, e.g., for porous media flow simulations. More advantages of MsFEM will be discussed in later chapters.

3.3.2 Comparison to variational multiscale Next, we briefly discuss the relation between variational multiscale approaches and MsFEM. These similarities are also shown in [26] in the context of mixed finite element methods. Here, we will discuss Galerkin finite element methods. We assume that the fine-scale solution space, X F , is partitioned into the coarse-dimensional space X C (e.g., VH0 ), and the space containing the unresolved scales, X U , X F = XC ⊕ XU . We assume also that these spaces are the subspaces of H01 (Ω), for simplicity. The fine-scale solution can be written accordingly as

3.3 Comparison to other multiscale methods

79

u = uC + uU . Substituting this into the original equation and multiplying by the test functions from X C , we obtain the equation for the coarse-scale solution 

 Ω

κ∇(u C + u U ) · ∇v H d x =

Ω

f v H d x, ∀v H ∈ X C .

(3.19)

Similarly, multiplying the original equation by the test functions from X U , we obtain the equation for unresolved part of the solution 

 Ω

κ∇u U · ∇v H d x =

 Ω

f vH d x −

Ω

κ∇u C · ∇v H d x, ∀v H ∈ X U .

(3.20)

To find the coarse-scale solution, u C , one first solves u U from (3.20) and substitutes it into (3.19) to compute u C . We note that (3.19) is exact and the solution of (3.20) is non-local. In general, the solution of (3.20) can be localized by imposing local boundary conditions. One can use various choices for the boundary conditions. Noting the solution to the local problem can be written via generic basis functions, one can derive a formulation similar to MsFEM. To show the similarity between MsFEM and variational multiscale methods, as an example, we consider the localization based on u U = 0 on the boundaries of the coarse-grid block K and X C = VH0 . In this case, it is evident that the solution u C + u U satisfies the local problem div(κ∇(u C + u U )) = f in K and u C + u U is piecewise linear on ∂ K . This solution can be approximated by multiscale finite element basis  functions defined by (3.8). Thus, we can seek u C + u U = i u i φ ωi . Substituting this expression into (3.19), we obtain a Petrov-Galerkin formulation of MsFEM if φ ωi are chosen with zero right-hand side. We note that one of the differences between variational multiscale method and MsFEM is that the former uses source terms in the formulation of the local problems. The representation of source terms with MsFEMs has been extensively studied in the literature (e.g., see [2, 295]) within the context of subsurface flows. Typically, only singular source terms require special treatment with multiscale basis functions.

3.3.3 Comparison to heterogeneous multiscale method As we mentioned earlier, one can take advantage of scale separation in MsFEM. There are various ways to do so, and these approaches will share similarities, e.g., with the application of heterogeneous multiscale methods (HMM) ([160]) and multiscale enrichment methods ([209]). HMM has been extensively studied in the literature (e.g., see [4, 5, 161, 330] for the applications to elliptic equations). The main idea of this approach is to use small regions at quadrature points for the computation of effective coefficients. This is performed on-the-fly when the stiffness matrix corresponding

80

3 Local model reduction. Introduction to Multiscale Finite Element Methods

to the coarse-scale problem is assembled. As we mentioned earlier, multiscale basis functions can be approximated when there is scale separation. The basic idea behind this localization is that elements of the stiffness matrix, e.g., 1 |K | can be approximated by 1 |K loc |



κ∇φ ωi d x, K



ωi d x, κ∇ φ K loc

ωi is the where φ ωi is the solution of div(κ∇φ ωi ) = 0 in K , φ ωi = φ0ωi on ∂ K , and φ ωi ω ω   i i solution of div(κ∇ φ ) = 0 in K loc , φ = φ0 on ∂ K loc . This can be shown using the general G-convergence theory (e.g., [275]) when κ ∗ (x) is a smooth function. For periodic problems, κ = κ(x/), it is not difficult to show that

1 | |K |

 K

κ∇φ ωi d x − κ ∗ ∇φ0ωi | ≤ C(

 + H ), H

where κ ∗ is computed for the coarse-grid block according to (3.15). Similarly, |

1 |K loc |

 K loc

ωi d x − κ ∗ ∇φ ωi | ≤ C( κ∇ φ 0

 + Hloc ). Hloc

Based on these results, one can show the convergence of MsFEMs using the local information in K loc . We refer to [181] for the details where more general problems are studied. This approximation of the basis functions and the corresponding approximation of the stiffness matrix elements can save CPU time if there is a strong scale separation. The method obtained in this way is very similar to the application of HMM to elliptic equations, though it differs in some details (e.g., the computations are not performed at quadrature points). We would like to note that one can also use first-order corrector approximation for the basis functions as discussed earlier. In this case, the local solution in RVE can be used as a cell problem solution χ . We would like to mention that there are other approaches (e.g., [208, 209, 262]) which use the solution of the cell problem to construct multiscale basis functions based on partition of unity method. As we mentioned in Section 1.4, multiscale methods considered in this book differ from domain decomposition methods (e.g., [394]) where the local problems are solved many times iteratively to obtain an accurate approximation of the fine-scale solution. Multiscale methods studied in this book share similarities with upscaling/homogenization methods, where the basis functions are computed based on coarse-grid information. Figure 3.10 illustrates the main concept of MsFEM and its advantages (see also Section 3.4). The multiscale methods attempt to find the coarsescale solution and can also compute an approximation of the fine-scale solution via

3.4 Performance and implementation issues

81

downscaling. One can use iterations (e.g., [155]) similar to domain decomposition methods or some type of global information to improve the accuracy of multiscale methods when there is no scale separation. Finally, we remark that we restricted ourselves only to a few multiscale methods. One can find similarities between multiscale finite element methods and other multiscale methods known in the literature. Some of these similarities may not be so apparent. Some of these algorithms are designed for periodic problems and have advantages when the underlying heterogeneities are periodic. For example, the approach proposed in [323] is based on the two-scale convergence concept ([18]). This approach is generalized to problems with multiple separable scales ([251]) using hierarchical basis functions. In this book, we do not want to discuss the similarities between different multiscale methods to a great extent and instead focus on our work on extensions and applications of various MsFEMs. We again stress that the main idea of MsFEM stems from earlier work of Babuška and Osborn [35]. One can show that MsFEM can take advantage of global information and can be naturally extended to nonlinear problems.

3.4 Performance and implementation issues We outline the implementation of Galerkin MsFEM for a simple test problem (following [255]) and define some notations that are used in the discussion below. We consider solving problems in a unit square domain. Let N be the number of elements in each coordinate direction. The mesh size is thus H = 1/N . To compute the basis functions, each element is discretized into M × M subcell elements with mesh size h = H/M. To implement the oversampling method, we partition the domain into sampling domains where each of them contains many elements. Analysis and numerical tests indicate that the size of the sampling domains can be chosen freely as long as the boundary layer is avoided. In practice, though, one wants to maximize the efficiency of oversampling by choosing the largest possible sample size which reduces the redundant computation of overlapping domains to a minimum. In general, the multiscale basis functions are constructed numerically, except for certain special cases. They are solved in each K or K + using standard FEM. The global linear system on Ω is solved using the same method. Numerical tests show that the accuracy of the final solution is weakly insensitive to the accuracy of basis functions. Since the basis functions are independent of each other, their construction can be carried out in parallel perfectly. In a parallel implementation of oversampling, the sample domains are chosen such that they can be handled within each processor without communication. More implementation details can be found in [255].

82

3 Local model reduction. Introduction to Multiscale Finite Element Methods Precomputation of multiscale basis functions (RVE or coarse grid block is used)

External parameters (forcing, boundary conditions, mobilities...)

Coupling of basis functions (simulation on the coarse grid)

Advantages

Parallel construction of basis functions

Adaptive simulations

Adaptive downscaling

Re−use of basis functions

Fig. 3.10 A schematic illustration of multiscale simulations and advantages.

3.4.1 Cost and performance In computations, a large amount of overhead time comes from constructing the basis functions. This is also true for classical upscaling methods discussed in Section 3.3. On a sequential machine, the operation count of the MsFEM is about twice that of a conventional FEM for a two-dimensional problem. However, due to good parallel efficiency, this difference is reduced significantly on a massively parallel computer (see [255] for a detailed study of MsFEM’s parallel efficiency). This overhead can be reduced if there is scale separation. In practice, multiple solutions are often required for different source terms, boundary conditions, mobilities, etc. MsFEMs have advantages in such situations and the overhead of basis construction can be negligible since the basis functions can be reused. This is illustrated in Figure 3.10, where pre-computed multiscale basis functions can be re-used for different external parameters, such as source terms and boundary conditions. Moreover, multiscale basis functions can be used to reconstruct the finescale features of the solution in the regions of interest. This adaptivity is often used in subsurface applications where the fine-scale features of the velocity (−k∇ p) are reconstructed in some regions where detailed velocity information is needed, e.g., for updating sharp interfaces. In summary, MsFEMs provide the following advantages in simulations: (1) parallel multiscale basis function construction (which can be very cheap if there is scale separation); (2) re-use of basis functions for different external parameters and inexpensive coarse-scale solution; (3) adaptive downscaling of the fine-scale features of the solution in the regions of interest. Significant computational savings are obtained for time-dependent problems such as those that occur in subsurface applications. In these problems, the heterogeneities

3.5 Convergence of multiscale finite element methods

83

representing porous media properties do not change, and the basis functions are pre-computed at the initial time. These basis functions are used throughout the simulations, and the elliptic equations are solved on the coarse-grid each time. In this sense, our approaches are similar to classical upscaling methods where the upscaled quantities are pre-computed before solving the equations on the coarse grid. In some situations, a local basis function update is required, for example, if there is a sharp interface dividing two propagating fluids. The interface modifies the permeability, and this requires local updates of the basis functions.

3.5 Convergence of multiscale finite element methods For the analysis here, we restrict ourselves to a periodic case κ(x) = (κi j (x/ε)). We assume κi j (y), y = x/, are smooth periodic functions in y in a unit cube Y . We assume that f ∈ L 2 (Ω). The assumptions on ai j can be relaxed, and one can extend the analysis to the cases κ = κ(x, x ), where κ( x ) is a homogeneous random field, and other cases. For simplicity, we consider the analysis in two dimensions. Let u 0 be the solution of the homogenized equation L 0 u 0 := −∇ · (κ ∗ ∇u 0 ) = f in Ω, u 0 = 0 on ∂Ω, where κi∗j

1 = |Y |

 κil (y)(δl j − Y

(3.21)

∂N j ) dy, ∂ yl

and N j (y) is the periodic solution of the cell problem ∇ y · (κ(y)∇ y N j ) =

∂ κi j (y) in Y, ∂ yi

 N j (y) dy = 0. Y

We note that u 0 ∈ H 2 (Ω) since Ω is a convex polygon. Denote by u 1 (x, y) = and let θε be the solution of the problem −N j (y) ∂u∂0x(x) j Lθε = 0 in Ω, θε (x) = u 1 (x, xε ) on ∂Ω.

(3.22)

Our analysis of the multiscale finite element method relies on the following homogenization result obtained by Moskow and Vogelius [332]. Lemma 3.1. Let u be the solution of (3.7) and u 0 ∈ H 2 (Ω) be the solution of (3.21), θε ∈ H 1 (Ω) be the solution to (3.22), and u 1 (x) = −N j (x/ε)∂u 0 (x)/∂ x j . Then there exists a constant C independent of u 0 , ε and Ω such that  u − u 0 − ε(u 1 − θε ) 1,Ω ≤ Cε(| u 0 |2,Ω +  f 0,Ω ).

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3 Local model reduction. Introduction to Multiscale Finite Element Methods

3.5.1 The analysis of conforming multiscale finite element method The analysis of conforming multiscale finite element methods uses Cea’s lemma [62]. Lemma 3.2. Let u be the solution of (3.7) and u H be the solution of (3.9). Then we have  u − u H 1,Ω ≤ C inf  u − v H 1,Ω . v H ∈VH

Proof. We have a(u H , v H ) = ( f, v H ) and a(u, v H ) = ( f, v H ). When subtracting a(u − u H , v H ) = 0, ∀v H . Then, Cu − u H 2H 1 ≤ a(u − u H , u − u H ) = a(u − u H , u − v H ) ≤ c1 u − u H  H 1 u − v H  H 1 .

From here, we get our main assertion. One-dimensional analysis We consider an interpolant u H (x) =



u(xi )φ ωi (x).

i

Then, in each K = [xi , xi+1 ], we have d d (κ(x) u H ) = 0, dx dx u H (xi ) = u(xi ), and u(xi+1 ) = u H (xi+1 ). For the error, we have d d (κ(x) (u − u H )) = f, dx dx (u − u H )(xi ) = (u − u H )(xi+1 ) = 0. Multiplying both sides by (u − u H ) and integrating over K = [xi , xi+1 ], we get   d κ(x)| (u − u H )|2 = | f (u − u H )| ≤  f  L 2 (K ) u − u H  L 2 (K ) ≤ dx K K (3.23) d C H  f  L 2 (K )  (u − u H ) L 2 (K ) . dx

3.5 Convergence of multiscale finite element methods

85

From here d (u − u H ) L 2 (K ) ≤ C H  f  L 2 (K ) . dx Summing this over all K , we have d  (u − u H ) L 2 (D) ≤ C H  f  L 2 (D) . dx 

Error estimates (H > ε) In this section, we will show that the multiscale finite element method gives a convergence result uniform in  as  tends to zero. This is the main feature of this multiscale finite element method over the traditional finite element method. The main result in this subsection is the following theorem. Theorem 3.3. Let u ∈ H 2 (Ω) be the solution of (3.7) and u H ∈ VH be the solution of (3.9). Then we have  ε 1/2  u − u H 1,Ω ≤ C(H + ε) f 0,Ω + C  u 0 1,∞,Ω , (3.24) H where u 0 ∈ H 2 (Ω) ∩ W 1,∞ (Ω) is the solution of the homogenized equation (3.21). ¯ → VH0 ⊂ H01 (Ω) as the usual To prove the theorem, we first denote H : C(Ω) Lagrange interpolation operator:

H u(x) =

J 

¯ u(x j )φ0ωi (x) ∀ u ∈ C(Ω)

j=1

¯ → VH as the corresponding interpolation operator defined through and I H : C(Ω) the multiscale basis functions φ ω j I H u(x) =

J 

¯ u(x j )φ ω j (x) ∀ u ∈ C(Ω).

j=1

From the definition of the basis functions φ ω j in (3.8), we have L(I H u) = 0 in K , for any K ∈ T H . Let u I (x) = I H u 0 (x) =

I H u = H u on ∂ K ,

 j

u 0 (x j )φ ω j (x) ∈ VH .

(3.25)

86

3 Local model reduction. Introduction to Multiscale Finite Element Methods

From (3.25), we know that Lu I = 0 in K and u I = h u 0 on ∂ K for any K ∈ Th . The homogenization theory (see ([332])) implies that  u I − u I0 − ε(u I1 − θIε ) 1,K ≤ Cε( f 0,K + | u I0 |2,K ),

(3.26)

where u I0 is the solution of the homogenized equation on K : L 0 u I0 = 0 in K , u I0 = H u 0 on ∂ K ,

(3.27)

u I1 is given by the relation u I1 (x, y) = −N j (y)

∂u I0 in K , ∂x j

(3.28)

and θIε ∈ H 1 (K ) is the solution of the problem: LθIε = 0 in K , θIε (x) = u I1 (x, xε ) on ∂ K .

(3.29)

It is obvious from (3.27) that u I0 = H u 0 in K ,

(3.30)

since H u 0 is linear on K . From (3.26), we obtain that  u − u I 1,Ω ≤  u 0 − u I0 1,Ω +  ε(u 1 − u I1 ) 1,Ω + ε(θε − θIε ) 1,Ω + Cε f 0,Ω ,

(3.31)

where we have used the regularity estimate  u 0 2,Ω ≤ C f 0,Ω . Now it remains to estimate the terms at the right-hand side of (3.31). Lemma 3.4. We have  u 0 − u I0 1,Ω ≤ C H  f 0,Ω ,  ε(u 1 − u I1 ) 1,Ω ≤ C(H + ε) f 0,Ω .

(3.32) (3.33)

Proof. The estimate (3.32) is a direct consequence of the standard finite element interpolation theory since u I0 = H u 0 by (3.30). Next, we note that N j (x/ε) satisfies  N j 0,∞,Ω + ε ∇N j 0,∞,Ω ≤ C

(3.34)

for some constant C independent of H and ε. Thus we have, for any K ∈ T H ,

3.5 Convergence of multiscale finite element methods

87

∂ (u 0 − H u 0 ) 0,K ≤ C H ε| u 0 |2,K , ∂x j ∂(u 0 − H u 0 ) = ε ∇(N j ) 0,K ∂x j ≤ C ∇(u 0 − H u 0 ) 0,K + Cε| u 0 |2,K

 ε(u 1 − u I1 ) 0,K ≤ Cε N j  ε∇(u 1 − u I1 ) 0,K

≤ C(H + ε)| u 0 |2,K . This completes the proof.



Lemma 3.5. We have √  εθε 1,Ω ≤ C ε u 0 1,∞,Ω + Cε| u 0 |2,Ω .

(3.35)

Proof. Let ζ ∈ C0∞ (R2 ) be the cut-off function which satisfies ζ ≡ 1 in Ω\Ωδ/2 , ζ ≡ 0 in Ωδ , 0 ≤ ζ ≤ 1 in R2 , and |∇ζ | ≤ C/δ in Ω, where for any δ > 0 sufficiently small, we define Ωδ as Ωδ = {x ∈ Ω : dist(x, ∂Ω) ≥ δ}. With this definition, it is clear that θε − ζ u 1 = θε + ζ (N j ∂u 0 /∂ x j ) ∈ H01 (Ω). Multiplying the equation in (3.22) by θε − ζ u 1 , we get 

∂u 0 x (a( )∇θε , ∇(θε + ζ N j ))d x = 0, ε ∂x j Ω

which yields, by using (3.34),  ∇θε 0,Ω ≤ C ∇(ζ N j ∂u 0 /∂ x j ) 0,Ω ≤ C ∇ζ · N j ∂u 0 /∂ x j 0,Ω + C ζ ∇N j ∂u 0 /∂ x j 0,Ω +C ζ N j ∂ 2 u 0 /∂ 2 x j 0,Ω



D D ≤ C |∂Ω| · δ + C |∂Ω| · δ + C| u 0 |2,Ω , δ ε

(3.36)

where D =  u 0 1,∞,Ω and the constant C is independent of the domain Ω. From (3.36), we have √ ε  εθε 0,Ω ≤ C( √ + δ) u 0 1,∞,Ω + Cε| u 0 |2,Ω δ √ ≤ C ε u 0 1,∞,Ω + Cε| u 0 |2,Ω . (3.37) Moreover, by applying the maximum principle to (3.22), we get  θε 0,∞,Ω ≤  N j ∂u 0 /∂ x j 0,∞,∂Ω ≤ C u 0 1,∞,Ω . Combining (3.37) and (3.38) completes the proof.

(3.38) 

88

3 Local model reduction. Introduction to Multiscale Finite Element Methods

Lemma 3.6. We have  εθIε 1,Ω ≤ C

 ε 1/2  u 0 1,∞,Ω . H

(3.39)

Proof. We recall that for any K ∈ T H , θIε ∈ H 1 (K ) satisfies x ∂( H u 0 ) LθIε = 0 in K , θIε = −N j ( ) on ∂ K . ε ∂x j

(3.40)

By applying maximum principle and (3.34), we get  θIε 0,∞,K ≤  N j

∂( H u 0 ) 0,∞,∂ K ≤ C u 0 1,∞,K . ∂x j

Thus we have  εθIε 0,Ω ≤ Cε u 0 1,∞,Ω .

(3.41)

On the other hand, since the constant C in (3.36) is independent of Ω, we can apply the same argument leading to (3.36) to obtain



√  ε∇θIε 0,K ≤ Cε H u 0 1,∞,K ( |∂ K |/ δ + |∂ K |δ/ε) + Cε| H u 0 |2,K √ √ ε ≤ C H  u 0 1,∞,K ( √ + δ) δ √ ≤ C H ε u 0 1,∞,K , which implies that  ε∇θIε 0,Ω ≤ C This completes the proof.

 ε 1/2  u 0 1,∞,Ω . H 

Proof (of Theorem 3.2.) The theorem is now a direct consequence of (3.31), Lemmas 3.2, 3.4–3.6, and the regularity estimate  u 0 2,Ω ≤ C f 0,Ω .  Remark 3.7. As we pointed out earlier, the multiscale FEM indeed gives correct homogenized result as  tends to zero. This is in contrast with the traditional FEM which does not give the correct homogenized result as  → 0. The L 2 error would grow like O(H 2 / 2 ). On the other hand, we also observe that when H ∼ , the multiscale method attains large error in both H 1 and L 2 norms. This is called the resonance effect between the grid scale (H ) and the small scale () of the problem. This estimate reflects the intrinsic scale interaction between the two scales in the discrete problem. Our extensive numerical experiments confirm that this estimate is indeed generic and sharp. From the viewpoint of practical applications, it is important to reduce or completely remove the resonance error for problems with many scales since the chance of hitting a resonance sampling is high.

3.6 Mixed MsFEM

89

Error estimates (H < ε) Lemma 3.8. Let u ∈ H 2 (Ω) be the solution of (3.7). Then there exists a constant C independent of H, ε such that  u − I H u 0,Ω + H  u − I H u 1,Ω ≤ C H 2 (| u |2,Ω +  f 0,Ω ).

(3.42)

The proof can be found in [176, 258]. In conclusion, we have the following estimate by using Lemmas 3.2 and 3.8. Theorem 3.9. Let u ∈ H 2 (Ω) be the solution of (3.7) and u H ∈ VH be the solution of (3.9). Then we have  u − u H 1,Ω ≤ C H (| u |2,Ω +  f 0,Ω ).

(3.43)

Note that the estimate (3.43) is insufficient for practical applications because | u |2,Ω blows up like O(1/) as ε → 0. In contrast, (3.24) gives an error estimate which is uniform as ε → 0.

3.6 Mixed MsFEM In porous media applications, mass conservative methods are often employed. In particular, it is important to construct multiscale basis functions for the velocity field defined as v = −κ∇u on a coarse grid. For this purpose, a mixed finite element framework can be used with multiscale basis functions for the velocity. Next, we describe this framework. Our presentation of the mixed multiscale finite element method follows [86] (see also [1, 25] and [26]). First, we rewrite the elliptic equation in the form κ −1 v + ∇u = 0 in Ω div(v) = f in Ω

(3.44)

with non-homogeneous Neumann boundary conditions v · n = g on ∂Ω. In mixed multiscale finite element methods, the basis functions for the velocity field, v = −κ∇u, are needed. As in the case of MsFEM, one can use known mixed finite element spaces to construct these basis functions. For simplicity, we consider multiscale basis functions corresponding to lowest order Raviart-Thomas elements (following [25, 86]). The basis functions for velocity in each coarse block K are given by div(κ(x)∇wiK ) = κ(x)∇wiK n K =

1 in K |K | K gi on eiK 0 else,

(3.45)

90

3 Local model reduction. Introduction to Multiscale Finite Element Methods

Fig. 3.11 Example of basis functions. Left: basis function with K being RVE. Right: basis function with K being a coarse element.

where giK = |e1K | and eiK are the edges of K in the absence of singular sources. i Furthermore, we can define the finite-dimensional space for velocity by VH =

{iK }, K

where iK = κ(x)∇wiK . The basis functions for the pressure are piecewise constant functions over each K (Figure 3.11). To formulate a mixed multiscale finite element method, we use the numerical approximation associated with the lowest order Raviart-Thomas mixed multiscale finite element. That is, we find (v H , u H )∈ VH × Q H such that  v H · n = g H on ∂Ω, where g H = g0,H n on ∂Ω and g0,H = e∈{∂ K  ∂Ω,K ∈T H } ( e gds)Ne , Ne , is corresponding basis function to edge e,  Ω

κ −1 v H · w H d x −

 div(w H )u H d x = 0, ∀w H ∈ VH0  Ω (3.46) div(v H )q H d x = f q H d x, ∀q H ∈ Q H , Ω

Ω

where Q H is the space of piecewise constant functions and VH0 is a subspace of VH with elements that are homogeneous on the boundary. The above formulation is a mixed multiscale finite element method as introduced in [86]. This method is modified later by Aarnes in [1] with the purpose of applications to porous media flow simulations. Detailed numerical studies can be found in the literature (e.g., [1, 2, 86]).

3.7 MsFEM for parabolic equations

91

3.7 MsFEM for parabolic equations In this section, we briefly describe the extension of MsFEM to parabolic equation ∂ u − div(κ(x, t)∇u) = f ∂t

(3.47)

with appropriate boundary conditions on the finite time interval [0, T ]. In general, when there are space and time heterogeneities, basis functions are the solutions of the leading-order homogeneous parabolic equations. In the absence of time heterogeneities, one can use spatial basis functions developed for elliptic equations. To introduce MsFEM, we assume that the interval [0, T ] is divided into M equal parts 0 = t1 < t2 < ... < t M = T . These intervals are coarse-scale intervals, i.e., Δt = ti+1 − ti is larger than the characteristic time interval. The basis functions are constructed in [tn , tn+1 ] as the solution of ∂ ωi φ (x, t) − div(κ(x, t)∇φ ωi (x, t)) = 0 ∂t

(3.48)

in each K such that φ ωi = φ0ωi on ∂ K and φ ωi (x, t = tn ) = φ0ωi . We seek the finitedimensional approximation of the solution in [tn , tn+1 ] as u n+1 H (x, t) =



cin+1 φ ωi (x, t),

(3.49)

i

where cin+1 will be determined. Then, substituting (3.49) into the original equation, multiplying it by φ0ωi (as in Petrov-Galerkin formulation), and integrating over the space and [tn , tn+1 ], we have   ωj ω n+1 ωi n ci φ (x, tn+1 )φ0 (x)d x − ci φ ωi (x, tn )φ0 j (x)d x+ Ω  tn+1  Ω (3.50) tn+1  ωj ω κ(x, t)∇u H (x, t) · ∇φ0 (x)d x = f φ0 j (x)d x. Ω

tn

Ω

tn

The third term on the right-hand side can be treated implicitly or explicitly. In particular, the implicit method is given by   ωj ω n+1 ωi n ci φ (x, tn+1 )φ0 (x)d x − ci φ ωi (x, tn )φ0 j (x)d x+ Ω Ω  tn+1   tn+1  (3.51) ω ωj j n+1 ωi κ(x, t)∇φ (x, t) · ∇φ0 (x)d x = f φ0 (x)d x. ci tn

Ω

tn

Ω

The third term is evaluated explicitly in the case of the explicit method. We note that if there are no temporal heterogeneities, the basis functions can be the solution of elliptic equations as in the case of elliptic equations. The equations

92

3 Local model reduction. Introduction to Multiscale Finite Element Methods

for the basis functions can be simplified depending on the relation between spatial and temporal heterogeneities (see Chapter 4). (3.48) defines the basis functions independent of the relation between spatial and temporal heterogeneities. Furthermore, in the case of scale separation, (3.48) can be solved in a smaller volume, RVE, and this solution can be used in (3.50) similar to the elliptic case. Finally, we would like to note that one can couple the basis functions using different methods, such as finite volume elements.

3.8 MsFEM using limited global information The previous multiscale finite element methods can be extended to take into account additional global information. Next, we present an extension of the Galerkin multiscale finite element method that uses the partition of unity method [34] (also see, for example, [209, 262, 386]). We assume |u − G(v1 , v2 , ..., v N )|1,Ω is sufficiently small for a priori selected global fields v1 , v2 ,..., v N . Later, we will discuss the use of multiple global information for stochastic porous media equations. Here, v1 , ..., v N can also be possible pressure snapshots for different mobility λ(S) or pressure fields corresponding to different source terms and/or boundary conditions. Let ωi be a patch (see Figure 3.12), and define φ0ωi to be piecewise linear basis function in patch ωi , such that φ0ωi (x j ) = δi j . For simplicity of notation, denote v1 = 1. Then, the multiscale finite element method for each patch ωi is constructed by (3.52) φ ωj i = φi0 v j , where j =  1, .., N and i is the index of nodes (see Figure 3.12). First, we note that n φ ωj i = v j is the desired single-phase flow solution. in each K , i=1

K

ωi xi

Fig. 3.12 Schematic description of patch.

3.8 MsFEM using limited global information

93

For the analysis of the multiscale finite element method, we will use the following assumption. There exists a sufficiently smooth scalar-valued function G(η), η ∈ R N 2s (G ∈ W 3, s−2 , s > 2), such that |u − G(v1 , ..., v N )|1,Ω ≤ Cδ,

(3.53)

where δ is sufficiently small. The form of the function G is not important for the computations, however, it is crucial that the basis functions span v1 ,..., v N in each coarse block. It can be shown that MsFEM converges for problems without scale separation. Theorem 3.10. Assume (3.53) and vi ∈ W 1,s (Ω), s > 2, i = 1, ..., N . Then |u − u H |1,Ω ≤ Cδ + C H 1−2/s .

Chapter 4

Generalized multiscale finite element methods. Main concepts and overview

4.1 Introduction 4.1.1 Overview Some of the previous multiscale finite element concepts focus on constructing one basis function per node. As we discussed that these approaches are similar to numerical homogenization (upscaling), i.e., one can use one basis function to localize the effects of local heterogeneities. However, for many complex heterogeneities, multiple basis functions are needed to represent the local solution space. For example, if the coarse region contains several high-conductivity regions, one needs multiple multiscale basis functions to represent the local solution space. The construction of multiple basis functions via global information is discussed in the previous section; however, we did not provide a systematic way of constructing these basis functions. In this chapter, we give an overview of a generalization of MsFEM, which presents a general framework for constructing multiscale basis functions (Figures 4.1 and 4.2). Generalized Multiscale Finite Element Method (GMsFEM) incorporates complex input space information and the input-output relation. It systematically enriches the coarse space through our local construction. Our approach, as in many multiscale and model reduction techniques, divides the computation into two stages: offline and online. In the offline stage, we construct a small-dimensional space that can be efficiently used in the online stage to construct multiscale basis functions. These multiscale basis functions can be re-used for any input parameter to solve the problem on a coarse grid. Thus, this provides a substantial computational saving at the online stage. Below, we present an outline of the algorithm and a chart that depicts our algorithm in Figure 4.2. 1. Offline computation: – 1.0. Coarse grid generation;

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_4

95

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4 Generalized multiscale finite element methods. Main concepts and overview

source terms boundary conditions f

Forward Model

Output

Lm (u)=f external parameters m Fig. 4.1 Flow chart.

OFFLINE Generate a coarse grid

For each coarse region Compute local snapshots

Reduce the dimension of local snapshot space using a spectral decomposition

ONLINE Given a parameter; a right hand side; and boundary condition For each coarse region

Compute local basis using a spectral decomposition Solve a coarse−scale problem

Iterative process (if needed)

Output: Reduced dimensional offline space and downscaling operators

Fig. 4.2 Flow chart.

– 1.1. Construction of snapshot space that will be used to compute an offline space. – 1.2. Construction of a small-dimensional offline space by performing dimension reduction in the space of local snapshots. 2. Online computations: – 2.1. For each input parameter, compute multiscale basis functions; – 2.2. Solution of a coarse-grid problem for any force term and boundary condition; – 2.3. Iterative solvers, if needed.

4.2 Setup

97

In the offline computation, we first set up a coarse grid consisting of a connected union of fine-grid blocks. A starting point for constructing the offline space is the snapshot space. The snapshot space consists of local functions that can represent the solution space. This task is very different from constructing the snapshot space for global approaches. In particular, we need to identify the local features of the solution space without computing the solution vectors. The construction of snapshot space in Step 1.1 involves solving local problems for various choices of input parameters or it could use the local fine-grid functions. The snapshot spaces are important for some of the reasons listed below. • Allows robust and fast convergent multiscale spaces. • Allows imposing physical properties of the problem, e.g., mass conservation. • Allows imposing geometrical information, e.g., perforations. The snapshot space is used to construct the offline space in Step 1.2 via a spectral decomposition of the snapshot space. The snapshot space in a coarse region can be replaced by the fine-grid space associated with this coarse space; however, in many applications, one can judiciously choose the space of snapshots to avoid expensive offline space construction. The offline space in Step 1.2. is constructed by spectrally decomposing the space of snapshots. This spectral decomposition is typically based on the offline eigenvalue problem. The spectral decomposition enables us to select the low-energy elements from the snapshot space by choosing those eigenvectors corresponding to the smallest eigenvalues. More precisely, we seek a subspace of the snapshot space such that it can approximate any element of the snapshot space in the appropriate sense defined via auxiliary bilinear forms. • Appropriate local spectral decomposition in the snapshot space is needed for accurate approximation. • The local spectral decomposition should take into account the smoothness of the solution and is motivated by the analysis. • The local spectral decomposition depends on global discretization. In the online step 2.1, for a given input parameter, we compute the required online coarse space. In general, we want this to be a small-dimensional subspace of the offline space. This space is computed by performing a spectral decomposition in the offline space via an eigenvalue problem. Furthermore, the eigenvectors corresponding to the smallest eigenvalues are identified and used to form the online coarse space. The online coarse space is used within the finite element framework to solve the original global problem.

4.2 Setup We will demonstrate some of the main concepts of GMsFEM on an example problem   − div κ(x; μ) ∇u = f,

in Ω,

(4.1)

98

4 Generalized multiscale finite element methods. Main concepts and overview

where u is  prescribed on ∂Ω and μ is a parameter. For simplicity, we assume that Q Θq (μ)κq (x) and that the coefficient κ(x; ·) has multiple scale and κ(x; μ) = q=1 high variations (e.g., see Figure 4.3 for κ1 (x) and κ2 (x) used in simulations). 4

x 10

1

18 16

0.8 14 12

0.6

10 8

0.4

6 4

0.2

2 0

(a) κ1 (x)

0

0.2

0.4

0.6

0.8

1

(b) κ2 (x)

Fig. 4.3 Decomposition of permeability field.

The GMsFEM has a structure similar to that of MsFEM. The main difference between the two approaches is that we systematically enrich coarse spaces in GMsFEM and generalize it by considering an input space consisting of parameters and source terms. In the first step of GMsFEM (offline stage), we construct the space of ωi , a large-dimensional space of local solutions. In the next step of ‘snapshots’, VH,snap ωi the offline computation, we reduce the space VH,snap via some spectral procedures ωi to VH,off . In the second stage (online stage), for each input parameter, we construct ωi that is used to solve the problem at the online a corresponding local space, VH,on stage for the given input parameter. Our systematic approach allows us to increase the dimension of the coarse space and achieve a convergence. Next, we briefly outline the global coupling and the role of coarse basis functions for the respective formulations that we consider. Throughout, we use the continuous Galerkin formulation, and use ωi as the support of basis functions even though ωi+ will be used in constructing multiscale basis functions. For the purpose of this ωi description, we formally denote the basis functions  i ofωi the online space VH,on by ψk . The solution will be sought as u H (x; μ) = i,k ck φk (x; μ). Once the basis functions are identified, the global coupling is given through the variational form (4.2) a(u H , v; μ) = ( f, v), for all v ∈ VH,on , and

4.3 Parameter-independent case

99

 a(u, v; μ) =

Ω

κ(x; μ)∇u · ∇v.

We note that in the case when the coefficient is independent of the parameter, then VH,on = VH,off .

4.3 Parameter-independent case First, we describe the basis construction for parameter-independent problems L(u) = f,   where L(u) = −div κ(x) ∇u . We refer to Figure 1.8 for the description of coarse and fine grids.

4.3.1 Examples of snapshot spaces. Oversampling and non-oversampling All fine-grid functions We can use local fine-scale spaces consisting of fine-grid basis functions within a coarse region. More precisely, the snapshot vectors consist of unit vectors defined on a fine grid within a coarse region. In this case, the offline spaces will be computed on a fine grid directly. The local fine-grid space has an advantage if the dimension of ωi computed by solving the local fine spaces is comparable to the dimension of VH,snap local problems as in the subsequent choices. “Harmonic extensions” This choice of snapshot space consists of a harmonic extension of fine-grid functions defined on the boundary of ωi . More precisely, for each xk ∈ Jh (ωi ), where Jh (ωi ) denotes the set of fine-grid boundary nodes on ∂ωi , let δlh be the piecewise linear function on ∂ωi defined by δlh (xk ) = δl,k , ∀xk ∈ Jh (ωi ). The snapshot basis, ψlωi , corresponding to δlh is the function satisfying L(ψlωi ) = 0, in ωi , subject to boundary condition, ψlωi = δlh on ∂ωi . Oversampling approaches To describe the oversampling approach, we consider the harmonic extension as described above in the oversampled region (see, e.g., Figure 1.8). This choice of snapshot space consists of harmonic extension of fine-grid functions defined on the boundary of ωi+ . More precisely, for each xk ∈ Jh (ωi+ ), where Jh (ωi+ ) denotes the set

100

4 Generalized multiscale finite element methods. Main concepts and overview

of fine-grid boundary nodes on ∂ωi+ , let δl+,h be the piecewise linear function on ∂ωi+ defined by δl+,h (xk ) = δl,k , ∀xk ∈ Jh (ωi+ ). The snapshot basis, ψl+,ωi , corresponding to δl+,h is the function satisfying L(ψl+,ωi ) = 0, in ωi+ , subject to boundary condition, ψl+,ωi = δl+,h on ∂ωi+ . Randomized boundary conditions In the above construction of snapshot vectors, many local problems are solved. This is not necessary and one can solve only a necessary number of snapshot vectors. The number of snapshot vectors is defined by the number of offline multiscale basis functions. In this case, we will use random boundary conditions. More precisely, let rl+,h be defined by rl+,h (xk ) = rl,k , ∀xk ∈ Jh (ωi+ ), where rl,k are random numbers. The snapshot basis, ψl+,ωi , corresponding to rl+,h is the function satisfying L(ψl+,ωi ) = 0 in ωi+ subject to boundary condition, ψl+,ωi = rl+,h on ∂ωi+ . We describe this procedure in more detail in Section 4.8.

4.3.2 Offline spaces ωi The offline space, denoted by VH,off for a generic domain ωi , is defined for each coarse region ωi (with elements of the space denoted φlωi ) and used to approximate the solution. The offline space is constructed by performing a spectral decomposition in the snapshot space and selecting the dominant eigenvectors (corresponding to the smallest eigenvalues). The choice of the spectral problem is important for the convergence and is derived from the analysis as it is described below. The convergence rate of the method is proportional to 1/Λ∗ , where Λ∗ is the smallest eigenvalue among all coarse blocks whose corresponding eigenvector is not included in the offline space. Our goal is to select the local spectral problem to remove as many small eigenvalues as possible so that we can obtain smaller- dimensional coarse spaces to achieve higher accuracy.

General concept and example The construction of the offline space requires solving an appropriate local spectral eigenvalue problem. The local spectral problem is derived from the analysis. The first step in the analysis is to decompose the energy functional corresponding to

4.3 Parameter-independent case

101

the error into coarse subdomains. For simplicity, we denote the energy functional corresponding to the region D by a D (u, u), e.g., a D (u, u) = D κ∇u · ∇u. Then aΩ (u − u H , u − u H ) 



aω (u ω − u ωH , u ω − u ωH ),

ω

(4.3)

where ω are coarse regions (ωi ), u ω is the localization of the solution using the snapshot vectors defined for ω and u ωH is the component of the solution u H spanned by the basis supported in ω. The local spectral decomposition is chosen to bound aω (u ω − u ωH , u ω − u ωH ). We ω ω ω (since u ω ∈ VH,snap ) such that for any ψ ∈ VH,snap , there seek the subspace VH,off ω exists ψ0 ∈ VH,off with aω (ψ − ψ0 , ψ − ψ0 )  δsω (ψ − ψ0 , ψ − ψ0 ),

(4.4)

where sω (·, ·) is an auxiliary bilinear form, which needs to be chosen and δ is a threshold related to eigenvalues. First, we would like the fine-scale solution to be bounded in sω (·, ·). We note that, in computations, (4.4) involves solving a generalized eigenvalue problem with a mass matrix defined using sω (·, ·) and the basis functions are selected based on dominant eigenvalues as described above. The threshold value δ is chosen based on the eigenvalue distribution. Secondly, we would like the eigenvalue problem to have a fast decay in the spectrum (this typically requires using oversampling techniques). Thirdly, we would like to bound 

sω (u ω − u ωH , u ω − u ωH )   aΩ (u, u),

(4.5)

ω

where  aΩ (u, u) can be bounded independent of physical parameters and the mesh sizes. Remark 4.1 (Energy minimizing snapshots). The step presented in (4.5) often requires a special construction for the snapshot space. In particular, the snapshots with the minimum energy are needed, see Chapter 6 and [95] for details. In these approaches, the snapshot vectors are computed by solving locally constrained minimization problems. More precisely, first, we use an oversampled region to construct the snapshot space. We denote them ψ1+ , ..., ψ + N for simplicity and consider one coarse block. Next, we consider the restriction of these snapshot vectors in a target coarse block and identify linearly independent components, ψ1 , ..., ψ M , M ≤ N . The next important ingredient is to construct minimum energy snapshot vectors that represent the snapshot functions in the target coarse region. These snapshot vectors are conM . 1 , ..., ψ structed by solving a local minimization problem. We denote them by ψ In the final step, we perform a local spectral decomposition to compute multiscale basis functions ([95]).

102

4 Generalized multiscale finite element methods. Main concepts and overview

Next, we discuss a choice for aω (·, ·) and sω (·, ·), which can be used in continuous Galerkin formulation in the case of elliptic equations. Later on, we will discuss appropriate choices for various other discretization and problems. Recall that for (4.4), we need a local spectral problem, which is to find a real number λ and v ∈ ωi such that VH,snap ωi ∀w ∈ VH,snap ,

aωi (v, w) = λsωi (v, w),

(4.6)

where aωi is a symmetric non-negative definite bilinear operator and sωi is a symωi ωi × VH,snap . Based on our metric positive definite bilinear operators defined on VH,snap analysis, we can choose   aωi (v, w) = κ∇v · ∇w, sωi (v, w) =  κvw, ωi

ωi

 Nc

where  κ = k=1 κ∇φωk · ∇φωk and φωk are multiscale basis functions (see (3.8)). We let λωj i be the eigenvalues of (4.6) arranged in ascending order. We will use the ωi first li eigenfunctions to construct our offline space VH,off . The choice of the eigenvalue problem is motivated by the convergence analysis. The convergence rate is proportional to 1/Λ∗ (with the proportionality constant depending on H ), where Λ∗ is the smallest eigenvalue that the corresponding eigenvector is not included in the offline space. We would like to remove as many small eigenvalues as possible. The eigenvectors of the corresponding small eigenvalues represent important features of the solution space. Remark 4.2. In general, aω and sω contain partition of unity functions, penalty terms, and other discretization factors that appear in coarse-grid finite element formulations as it will be discussed in later sections. Remark 4.3 (On S norm). In this remark, we show eigenvalue decay for different choices of aω and sω for a domain ω. We consider a target domain ω = [0.4, 0.6] × [0.4, 0.6] within Ω = [0, 1] × [0, 1]. The snapshot space is generated by solving problems with force terms located outside ω. Our first choice for aω and sω is  Nc   κ(x)|∇(φω0 i u)|2 , sω1 = κ(x)|∇u|2 , aω1 = ω

i=1

ω

φω0 i

are bilinear basis functions on the coarse grid. We also consider the folwhere lowing two choices   2 2 2 aω = κ(x)|u| , sω = κ(x)|∇u|2 , ω

and

ω

 aω3

=

 κ(x)|∇u| , 2

ω

where ω + = [0.3, 0.7] × [0.3, 0.7].

sω3 +

=

ω+

κ(x)|∇u|2 ,

4.3 Parameter-independent case

103 0.4

1 0.9

0.35

0.8 0.3 0.7 0.25

0.6

0.2

0.5 0.4

0.15

0.3 0.1 0.2 0.05

0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0

1

2

4

6

8

10

12

14

16

18

−3

4

x 10

0.7

3.5

0.6

3 0.5 2.5 0.4 2 0.3 1.5 0.2 1 0.1

0.5

0 0

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4

6

8

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18

0 0

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4

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18

Fig. 4.4 Top left: Force locations. Top right: Eigenvalues when using a 1 and s 1 . Bottom left: Eigenvalues when using a 2 and s 2 . Bottom right: Eigenvalues when using a 3 and s 3 .

We take κ(x) to be 102 in [0.45, 0.55] × [0.45, 0.55] and 1 elsewhere. In Figure 4.4, we plot the eigenvalues corresponding to the above choices of aωn and sωn for n = 1, 2, 3. The space of snapshots is generated using unit force terms in the locations shown in Figure 4.4, top left. As we see in all the cases, the eigenvalues decay and one can choose a threshold for the cut-off. The decay of eigenvalues depends on the choices of aωn and sωn , and also on the choice of φω0 i . By choosing the multiscale basis functions, φωi , in the construction of local spectral problem, some localizable important features are taken into consideration through φωi and, as a result, we have fewer small eigenvalues (see [168] for more discussions). ωi . Once The global offline space VH,off is the union of all VH,off  the offline space is constructed, we solve the fine-scale equation and find u H = ci, j φωj i ∈ VH,off such that (4.7) a(u H , v H ) = ( f, v H ), ∀v H ∈ VH,off .

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4 Generalized multiscale finite element methods. Main concepts and overview

An implementation view Next, we present an implementation view of the local spectral decomposition. For each snapshot basis function ψlωi , let Ψlωi be the corresponding coefficient vector in terms of the basis for fine space Vh . The space formed by the coefficient vectors is ωi = span{Ψlωi : VH,snap

+,ωi 1 ≤ l ≤ L i }, VH,snap = span{Ψl+,ωi :

1 ≤ l ≤ L i+ }

for each coarse neighborhood ωi and for each oversampled coarse neighborhood ωi ωi+ , respectively, where L i and L i+ are the dimensions of the snapshot spaces VH,snap +,ωi and VH,snap . We note that in the case when ωi is adjacent to the global boundary, no oversampled domain is used. We can put all snapshot functions using a matrix representation

ωi = Ψ1ωi , . . . , Ψ Lωii , Rsnap

 +,ωi i . Rsnap = Ψ1+,ωi , . . . , Ψ L+,ω + i

The local spectral problems (4.6) can be written in a matrix form as Aωi Φkωi = λωk i S ωi Φkωi ,

A+,ωi Φkωi = λωk i S +,ωi Φkωi

(4.8)

for the k-th eigenpair. We present two choices: one with no oversampling and one with oversampling, though one can consider various options [191]. In (4.8), with no ωi ωi oversampling, we can choose Aωi = [amn ] and S ωi = [smn ], where ωi amn

 =

ωi ωi κ∇ψmωi · ∇ψnωi = (Rsnap )T A Rsnap

ωi



and ωi smn

=

ωi

ωi ωi  κ ψmωi ψnωi = (Rsnap )T S Rsnap .

+,ωi +,ωi With oversampling, we can choose A+,ωi = [amn ] and S +,ωi = [smn ], where +,ωi amn

 =

ωi+

and +,ωi = smn

+,ωi T +,ωi κ∇ψm+,ωi · ∇ψn+,ωi = (Rsnap ) A Rsnap

 ωi+

+,ωi T +,ωi  κ ψm+,ωi ψn+,ωi = (Rsnap ) S Rsnap .

To generate the offline space, we then choose the smallest li eigenvalues and form the corresponding eigenvectors in the respective space of snapshots by setting φωk i =  L i+ +,ωi +,ωi L i ωi ωi +,ωi = j=1 Φk j ψ j (for k = 1, . . . , li or k = 1, · · · , li+ ), j=1 Φk j ψ j or φk ωi +,ωi where Φk j and Φk j are the coordinates of the vector Φkωi and Φk+,ωi , respectively. Collecting all offline basis functions and using a single index notation, we then create the offline matrices

4.4 Online space for parameter-dependent case

105

 +,off + Roff = Φ1+,off , . . . , Φ M off and

off , Roff = Φ1off , . . . , Φ M off

where Moff is the total number of offline basis functions. The discrete system corresponding to (4.7) is T T T T Rsnap A Rsnap Roff u discrete = Roff Rsnap f Roff H

where u discrete is the discrete version of u H . H

4.3.3 A numerical example We present a numerical result that demonstrates the convergence of the GMsFEM. More detailed numerical studies can be found in the literature. We consider the permeability field, κ(x) that is shown in Figure 4.5, the source term f = 0, and the boundary condition to be x1 . The fine grid is 100 × 100 and coarse grid is 10 × 10. We consider the snapshot space spanned by harmonic functions in the oversampled domain (with randomized boundary conditions) and vary the number of basis functions per node. The numerical results are shown in Table 4.1, in which ea and e2 denote relative errors in energy and L 2 norms, repsectively. As we observe from these numerical results, the GMsFEM converges as we increase the number of basis functions. We also show convergence when using polynomial basis functions. It is clear that the FEM method with polynomial basis functions does not converge and one can only observe this convergence only after a very large number of basis functions are chosen so that we cross the fine-scale threshold. We note that the convergence with one basis function per node does not perform well for the GMsFEM because of the high contrast. The first, second, and third smallest eigenvalues, Λ∗ , among all coarse blocks are 0.0024, 24.0071, and 35.6941, respectively. We emphasize that if the eigenvalue problem (and the snapshot space) is not chosen appropriately, one can get very small eigenvalues ([168]). As we see that the first smallest eigenvalue is very small, and as a result, the error is large when using one basis function. The eigenvalue distribution is important for online basis construction, and will be discussed later.

4.4 Online space for parameter-dependent case First, we remark that in Chapter 5, we will construct online spaces using the residual information. This will be called residual-based online spaces. In this section, we discuss online multiscale spaces that are constructed for parameter-dependent problems. In general, we refer to online spaces, when the multiscale basis functions depend on the online information, such as right-hand side or the parameter. We only describe

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4 Generalized multiscale finite element methods. Main concepts and overview 4

x 10 5.5

10

5

20

4.5

30

4

40

3.5

50

3 2.5

60

2

70

1.5

80

1

90 100

0.5 40

20

60

80

100

Fig. 4.5 The permeability field κ(x). #basis (DOF) 1 (81) 2 (162) 3 (243) 4 (324) 5 (405)

ea 69.05% 22.55% 19.86% 16.31% 14.20%

e2 12.19% DOF ea e2 1.19% 81 103.2% 23% 0.99% 361 100% 23% 0.70% 841 80.33% 15% 0.65%

Table 4.1 Left table: The convergence for the GMsFEM using multiscale basis functions. Right table: The convergence for FEM using polynomials basis functions.

the online space using a single spectral problem. One can analogously construct the online space using multiple spectral problems. For the parameter-dependent case, we next construct the associated online coarse space Von (μ) for each fixed μ value on each coarse subdomain. In principle, we want this to be a small-dimensional subspace of the offline space for computational efficiency. The online coarse space will be used within the finite element framework to solve the original global problem, where a continuous Galerkin coupling of the multiscale basis functions is used to compute the global solution. In particular, we seek a subspace of the respective offline space such that it can approximate any element of the offline space in an appropriate sense. We note that at the online stage, the bilinear forms are chosen to be parameterdependent. Similar analysis motivates the following eigenvalue problems posed in the offline space: on on Aon (μ)Φkon = λon k S (μ)Φk or on on A+,on (μ)Φkon = λon k A (μ)Φk

Aon (μ)Φkon where

or +,on = λon (μ)Φkon , k S

(4.9) (4.10) (4.11)

4.5 An example of enrichment. The importance of local spectral problem

107



T κ(x; μ)∇φωmi · ∇φωn i = Roff A(μ)Roff ,  T S on (μ) = [s on (μ)mn ] =  κ(x; μ)φωmi φωn i = Roff S(μ)Roff ,

Aon (μ) = [a on (μ)mn ] =

ωi

ωi



 + T + + i i κ(x, μ)∇φ+,ω · ∇φ+,ω = Roff A (μ)Roff , m n + ωi  + T + + +,on i +,ωi S +,on (μ) = [smn (μ)] =  κ(x, μ)φ+,ω = Roff S (μ)Roff , m φn

+,on A+,on (μ) = [amn (μ)] =

ωi+

and κ(x; μ) and κ(x; μ) are now parameter dependent. Again, we will take κ(x, μ) = κ(x, μ) in our simulations though one can use multiscale partition of unity functions to compute κ(x, μ) (cf. [173]). To generate the online space, we then choose the smallform the corresponding est Mon eigenvalues from one of Equations (4.9)–(4.11)  and on ωi eigenvectors in the offline space by setting φon j Φk j φ j (for k = 1, . . . , Mon ), k = where Φkonj are the coordinates of the vector Φkon . At the online stage, for each parameter value, multiscale basis functions are computed based on each local coarse region. In particular, for each ωi and for each input ωi (μ) where parameter, we will formulate a quotient for finding a subspace of Von the space will be constructed for each μ (independent of source terms). We seek a ωi ωi ωi ωi (μ) of Voff such that for each φ ∈ Voff , there exists φ0 ∈ Von (μ), such subspace Von that (4.12) aωoni (φ − φ0 , φ − φ0 ; μ)  δsωoni (φ − φ0 , φ − φ0 ; μ) for some prescribed error tolerance δ (different from the one in the offline stage), and the choices of aωoni and sωoni . The corresponding eigenvalue problem is formed in the space of offline basis functions. We note that an assumption as in Remark 4.2 (see also Remark 4.3) is needed for obtaining a convergence result, and in general, aωoni and sωoni contain partition of unity functions, penalty terms, and other discretization factors that appear in finite element formulations. In particular, we assume that a K (χi φ, χi φ; μ)  aωoni (φ, φ; μ). Remark 4.4. The bilinear form aωoni can be chosen based on finite element variational formulation in ωi . In particular, it can be chosen to be the same as the coarse-grid finite element formulation. As before, one can also take sωoffi to be weaker, provided the error in s-norm between the solution and its approximation can be estimated.

4.5 An example of enrichment. The importance of local spectral problem In this section, we will present an offline space that uses fine-grid snapshot space. We will consider complementing the coarse spaces described above by finding appropri-

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4 Generalized multiscale finite element methods. Main concepts and overview

ate local fields in ωi and by multiplying them with our multiscale functions. We start with the coarse space generated by one basis function per node, e.g., φω0 i (standard linear or bilinear finite element basis functions) or φωi (multiscale finite element i basis functions) or φωemf (energy minimizing basis functions) or so on, and further, we complement this space by adding basis functions in each ωi . For simplicity from now on, unless otherwise stated, we denote the initial basis function for node i by χi where χi can be computed with any of the above discussed methods. We emphasize that this initial choice of basis function is crucial for determining the dimension of the coarse space needed to obtain an accurate coarse-scale approximation and robust preconditioners. We consider the eigenvalue problem κφlωi , − div(κ∇φlωi ) = λlωi 

(4.13)

where λlωi (or simply λli ) and φlωi (or simply φli ) are eigenvalues and eigenvectors in κ is defined by ωi and  Nc 1   κ = 2κ |∇χ j |2 . (4.14) H j=1 We recall that χi are initial multiscale basis functions, e.g., χi = φω0 i or χi = φωi or i , and Nc is the number of the coarse nodes. One can choose other multiscale χi = φωemf basis functions, for example, multiscale basis functions that employ limited global information. The eigenvalue problem considered above is solved with zero Neumann boundary condition and it is understood in a discrete setting. Note that the eigenvalues of (4.13) depend on the initial basis functions χi . Assume eigenvalues are given by 0 = λω1 i ≤ λω2 i ≤ .... Basis functions are computed by selecting a number of eigenvalues (starting with small ones) and multiplying corresponding eigenvectors by χi (see Figure 4.6 for the illustration). Thus, multiscale space is defined for each i as the span of χi φlωi , l = 1, ..., li , where li is the number of selected eigenvectors. We note that the dimension of the coarse space depends on the choice of κ, and thus it is important to have a good choice of  κ when solving the local eigenvalue problem. We design a good choice for  κ via initial multiscale basis functions. The essential ingredient in designing  κ is to guarantee that there are fewer small, asymptotically vanishing (when the contrast increases), eigenvalues of (4.13). With an initial choice of multiscale basis functions that contain many small-scale localizable features of the solution, one can reduce the dimension of the coarse space. In particular, we note that the eigenvectors corresponding to small, asymptotically vanishing when contrast increases, eigenvalues represent the local features of the solution that is not captured by the initial multiscale basis functions. This gives a natural way to complement

4.5 An example of enrichment. The importance of local spectral problem

109

Fig. 4.6 Schematic description of basis function construction. Left: subdomain ωi . Right-Top: Selected eigenvector φli with small eigenvalue. Right-Bottom: product χi φli where χi is the initial basis function of node i.

initial coarse spaces and emphasizes the importance of the initial multiscale spaces. If initial basis functions are not chosen carefully, it can result in large- dimensional coarse spaces. Thus, the use of advanced multiscale techniques in constructing initial basis helps reduce the dimension of the coarse space needed to achieve contrastindependent two-level domain decomposition preconditioners and more accurate coarse-grid solutions. In [174], we show that under some conditions, the convergence rate is inversely related to the smallest eigenvalue whose eigenvector is not included in the coarse space. This is also observed in our numerical results. More precisely, we show that  the convergence rate in the energy norm Ω κ(x)|∇(u − u 0 )|2 is proportional to max i

H 1+β , i λlωi +1

(4.15)

where H is the coarse mesh size and β ≥ 0 is related to the smoothness of the coefficients. For example, for smooth coefficients, one can have β = 1 and thus recover classical error estimates. We note that these results assume that the right hand side f is a smooth function. We see from here that one needs to reach larger, O(1), eigenvalues as fast as possible. More precisely, with a choice of initial basis functions χi , we need to get a rapid increase in eigenvalues. We note that (4.15) also is a main factor in the condition number of the preconditioned matrix that is computed using two-level additive Schwarz (see discussions later and [168, 169, 217]). Thus, our goal is to choose initial basis functions, such that eigenvalues increase rapidly and more precisely we have fewer asymptotically small eigenvalues. Next, we discuss how various choices of initial multiscale basis functions can affect the eigenvalue behavior. We start with a piecewise linear basis functions, κ and κ have similar high-contrast i.e., χi are linear basis functions. In this case,  regions because ∇χi are piecewise constant functions over coarse-grid regions. In particular, if there are m separated inclusions and channels, then one can observe

110

4 Generalized multiscale finite element methods. Main concepts and overview −4

7

−4

x 10

x 10

x 10

4

10 8

5

2

6 0

0

4 −2

2

−4 1

0 1

−5 1 1

1 0.8

0.5

0.6

0.8

0.5

0.6

0

0.2 0

1 0.8

0.5

0.6

0.4

0.4 0

0.2 0

0.4 0

0.2 0

Fig. 4.7 Illustration of eigenvectors in a unit square coarse block. Left: Permeability field κ. Middle: Second eigenvector. Right: Third eigenvector. The first four eigenvalues using  κ = κ are [0 0.0745e–5 0.0918e–5 0.2416e–5 ]. The fifth eigenvalue is 9.88.

m small (asymptotically vanishing as contrast increases) eigenvalues. These small eigenvalues represent high-conductivity regions of the weight function (this can be seen from the corresponding Rayleigh-Quotient). We recall that the inclusions are isolated high-conductivity regions within a coarse block, while channels are isolated high-conductivity regions that connect the boundaries of the coarse blocks (ωi in this case). In fact, these small eigenvalues are inversely related to the high-conductivity values of the coefficients. We assume throughout that the coefficient takes values 1 and η, where η is large. The corresponding eigenvectors have constant values (in the asymptotic sense when the contrast increase to infinity) within each high-contrast region. In Figure 4.7, we depict a region with three high-conductivity inclusions and one high-conductivity channel. The first four eigenvalues are small and are given by [0, 0.0745e–5, 0.0918e–5, and 0.2416e–5] for η = 1e + 8. Here we used piecewise κ in (4.14). We depict eigenvectors corresponding linear functions χ j to compute  to the second and third smallest eigenvalues. The eigenvectors corresponding to the small, asymptotically vanishing with increasing contrast, eigenvalues represent all possible constants within high-conductivity regions. The fifth eigenvalue is 9.88, and thus there is a large gap in the spectrum. The choice of initial multiscale basis functions, and consequently the choice of  κ can result in large coarse-dimensional spaces that are needed to eliminate small eigenvalues. Note that without including eigenvectors that correspond to small eigenvalues, one expects very large errors that are proportional to the contrast. Thus, it is essential that we reduce the number of small, asymptotically vanishing as the contrast increases, eigenvalues. Note that in many cases, one can have a very large number of small isolated high-conductivity regions within the domain. To illustrate this, in Figure 4.8, we schematically depict a coarse region with many small isolated high-conductivity inclusions. In this case, the dimension of the coarse space is very large. For example, if the size of inclusions is of order , then we may need a coarse space with the

4.5 An example of enrichment. The importance of local spectral problem

111

ω i Fig. 4.8 Schematic description of a coarse region with many isolated inclusions. This will lead to a large-dimensional coarse space unless the initial multiscale space is chosen properly.

dimension that can scale as 1γ (for some γ > 0 depending density of the inclusions) when there are many inclusions. However, it turns out that one can encapsulate the effects of isolated inclusions on a single basis function per coarse node that will ultimately reduce the dimension of the coarse space. This is done via the initial choice of partition of unity functions resulting to a new weight matrix  κ. This is discussed in the next section.

4.5.1 Reduced-dimensional coarse spaces In this section, we will discuss how one can reduce the dimension of the coarse space that is spanned by eigenvectors corresponding to asymptotically small eigenvalues. In particular, we we will show that one can reach O(1) eigenvalues with fewer basis functions if initial multiscale basis functions are chosen appropriately. As we mentioned, if the partition of unity functions χ j are piecewise linear polynomials, then  κ and κ have the same high-contrast structure. We are interested in the partition of unity functions that can “eliminate” isolated high-conductivity inclusions. This can be achieved by minimizing the number of high-conductivity components in  κ. We note that the high-conductivity regions of the weight  κ determine the number of small eigenvalues. By choosing multiscale finite element basis functions or energy minimizing basis functions as defined above, we can eliminate all isolated high-conductivity inclusions, while preserving the channels. The elimination of isolated high-conductivity inclusions is understood in a sense that their fine-scale features are incorporated into a single basis function. Indeed, the energy of local solutions remains bounded in the high-conducting regions that are isolated. This can be observed in our numerical experiments. In Figure 4.9, we depict the coarse grid

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4 Generalized multiscale finite element methods. Main concepts and overview

Fig. 4.9 Left: Coarse mesh and original coefficient. Here η = 108 . Right: Coarse mesh and coefficient  κ computed as in (4.14) using (linear) multiscale basis functions.

and κ (left plot) and  κ (right picture) using multiscale basis functions on the coarse grid. One can observe that isolated inclusions are removed in κ, and consequently, the coarse space contains only long channels that connect the boundaries of the coarse grid. We are interested in the partition of unity functions that can “eliminate” isolated high-conductivity inclusions and thus reduce the dimension of the coarse space. In particular, the effects of isolated inclusions are represented by a coarse-scale basis function in each ωi . This can be achieved by minimizing high-conductivity components in  κ. In particular, by choosing multiscale finite element basis functions or energy minimizing basis functions, we can eliminate all isolated (i.e., not touching the boundaries of the coarse grids) high-conductivity inclusions, while preserving the channels. To remove the degrees of freedom associated with isolated inclusions that intersect only a single boundary segment of the coarse-grid block, we can use oversampling, energy minimizing basis functions, multiscale basis functions with limited global information or basis functions proposed in [93]. In our numerical results, we will use energy minimizing basis functions. We note that one cannot remove the channels with the partition of unity functions. Indeed, because the partition of unity functions have “unit” gradient flow within coarse-grid blocks, there will be a non-zero gradient within channels, and thus  κ will remain high. Thus, an optimal initial partition of unity functions will be those that can eliminate all isolated inclusions. As we show numerically, multiscale finite element basis functions can achieve that. For example, for the permeability field depicted in Figure 4.7, we can remove all three inclusions and κ will contain only one channel. In fact, more sophisticated multiscale basis functions can further reduce the dimension by eliminating the inclusions that touch boundaries.

4.6 Iterative solvers - online correction of fine-grid solution

113

4.6 Iterative solvers - online correction of fine-grid solution In the previous approach, the coarse spaces are designed to achieve the desired accuracy. One can also iterate (on residual and/or the basis functions) for a given source term and converge to the true solution without increasing the dimension of the coarse space. Both approaches have their application fields and allows computing the fine-grid solution with an increasing accuracy. Next, we briefly describe the solution procedure based on the concept of two-level iterative methods. M the overlapping decomposition obtained from the original We denote by {Ωi }i=1 M of Ω by enlarging each subdomain Ωi to non-overlapping decomposition {Ωi }i=1 Ωi = Ωi ∪ {x ∈ Ω, dist(x, Ωi ) < δi }, i = 1, . . . , M,

(4.16)

where dist is some distance function and let V0h (Ωi ) be the set of finite element functions with support in Ωi and zero trace on the boundary ∂Ωi . We also denote by RiT : V0h (Ωi ) → V h the extension by zero operator, where V h is an appropriate finite element space defined on the fine grid. Let u 0 be the coarse grid component of the solution u. With the help of (u − u 0 ), we can correct the coarse-grid solution. A number of approaches can be used. For instance, we write the solution in the form 

χi vi , u = u0 + i

where vi are defined in ωi (though it can be taken to be supported in a different domain) and χi is a partition of unity. Suppose that vi has zero trace on ∂Ωi . We can solve the local problems L μ (vi ) = f − L μ (u 0 ), with zero boundary condition. Here, L μ (u) = −div(κ(x; μ)∇u). Other correction schemes can be implemented to correct the coarse-grid solution. For example, we can use the traces of u 0 to correct the solution in each subdomain in a consecutive fashion. When the bilinear form is symmetric and positive definite, we can consider twolevel domain decomposition additive methods to find the solution, u, of the fine-grid finite element problem a(u, v; μ) = ( f, v), for all v ∈ V h ,

(4.17)

114

4 Generalized multiscale finite element methods. Main concepts and overview

where V h is the fine-grid finite element space of piecewise linear polynomials. The matrix of this linear system is written as A(μ)u(μ) = b. Here, A is the stiffness matrix associated to the bilinear form a and b is defined so that v T b = ( f, v) for all v ∈ V h . We can solve the fine-scale linear system iteratively with the preconditioned conjugate gradient (PCG) method (or any other Krylov type method for a non-positive problem). Any other suitable iterative scheme can be used as well. We introduce the two-level additive preconditioner of the form T

−1 (μ) A B −1 (μ) = R0,on 0 (μ)R0,on (μ) +

M 

RiT Ai−1 (μ)Ri ,

(4.18)

i=1

where the local matrices are defined by v T Ai (μ)w = a(v, w; μ)

for all v, w ∈ V0h (Ωi ).

(4.19)

T T T is defined by R0,on = R0,on (μ) = [Φ1 , . . . , Φ Nv ] The coarse projection matrix R0,on T

0 (μ) = R0,on (μ)A(μ)R0,on and the online coarse matrix A (μ). The columns Φi ’s are fine-grid coordinate vectors corresponding to the basis functions, e.g., in the Galerkin Ω ,on formulations they correspond to the basis functions {χi (x)φ j i (x; μ)}. See [394] and references therein for more details on various domain decomposition methods. The application of the preconditioner involves solving local problems in each iteration. In domain decomposition methods, our main goal is to reduce the number of iterations in the iterative procedure. It is well known that a coarse solve needs to be added to the one-level preconditioner in order to construct robust methods. The appropriate construction of the coarse space V0 plays a key role in obtaining robust iterative domain decomposition methods. Our methods provide inexpensive coarse solves and efficient iterative solvers for general parameter-dependent problems. This will be discussed in the next sections.

4.7 Some numerical studies 4.7.1 Case with no parameter In this section, we consider the elliptic equation (4.1) with the heterogeneous coefficients κ independent of the parameter μ. For the spectral problem in the construction of the offline space (4.6), we can choose   aωi (ψ, ψ) = κ∇(χ ψ) · ∇(χ ψ), (4.20) ,ω



ωi =0

ω

4.7 Some numerical studies

115

where χ is the partition of unity corresponding to ω , and  sωi (ψ, ψ) =

ωi

κ∇ψ · ∇ψ.

The corresponding eigenvalue problem can be explicitly written as before. In  our numerical implementation, we choose aωi (ψ, ψ) = ,ω  ωi =0 ω κ|∇χ |2 |ψ|2 (see [174] that shows that this is not smaller than (4.20) in the space of local κharmonic functions). The basis functions are constructed by multiplying the eigenvectors corresponding to the dominant eigenvalues by partition of unity functions. Note that the stiffness matrix is pre-computed in the offline stage and there is no need for any stiffness matrix computation in the online stage. We point out that the choice of initial partition of unity basis functions χi are important in reducing the number of very large eigenvalues. We note that the dimension of the coarse space depends on the choice of χi , and thus it is important to have a good choice of χi . The essential ingredient in designing them is to guarantee that there are fewer large eigenvalues, and thus the coarse space dimension is small. With an initial choice of multiscale basis functions χi that contain many localizable small-scale features of the solution, one can reduce the dimension of the resulting coarse space. Next, we briefly discuss a few numerical examples (see [168] for more discussions). We present a numerical result for the coarse-scale approximation and for the two-level additive preconditioner (4.18) with the local spectral multiscale coarse spaces as discussed above. The equation −div(κ∇u) = 1 is solved with boundary conditions u = x + y on ∂Ω. For the coarse-scale approximation, we vary the dimension of the coarse spaces by adding additional basis functions corresponding to the largest eigenvalues. We investigate the convergence rate, while for preconditioning results, we will investigate the behavior of the condition number as we increase the contrast for various choices of coarse spaces. The domain Ω = [0, 1] × [0, 1] is divided into 10 × 10 equal square subdomains. Inside each subdomain, we use a finescale triangulation, where triangular elements constructed from 10 × 10 squares are used. We consider the scalar coefficient κ(x) depicted in Figure 4.10 that corresponds to a background one and high conductivity channels and inclusions. We test the accuracy of GMsFEMs when coarse spaces include eigenvectors corresponding to the small eigenvalues. We implement GMsFEM by choosing the initial partition of unity functions which consist of multiscale functions with linear boundary conditions (MS), see (3.8). We use the following notation. GMsFEM+0 refers to the GMsFEM where the coarse space includes all eigenvectors that correspond to eigenvalues which are asymptotically unbounded as the contrast increases, i.e., these eigenvalues increase as we increase the contrast. One of these eigenvectors corresponds to a constant function in the coarse block. GMsFEM+n refers to the GMsFEM where in addition to eigenvectors that correspond to asymptotically unbounded eigenvalues, we also add n eigenvectors corresponding to the next n eigenvalues.

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4 Generalized multiscale finite element methods. Main concepts and overview

In previous studies [169, 174, 217], we discussed how the number of these asymptotically unbounded eigenvalues depends on the number of inclusions and channels. In particular, we showed that if there are n inclusions (isolated regions with high conductivity) and m channels (isolated high-conductivity regions connecting boundaries of a coarse grid), then the number of asymptotically unbounded eigenvalues is n + m when standard bilinear partition of unity function, φω0 i , is used. However, if the partition of unity χi is chosen as multiscale finite element basis functions ([255]), φωi , defined in (3.8), then the number of asymptotically unbounded eigenvalues is m. We can also use energy minimizing basis functions that are defined (see [424]) as min

 i

ωi

i κ|∇φωemf |2

(4.21)

 i i subject to i φωemf = 1 with supp(φωemf ) ⊂ ωi , i = 1, . . . , Nc , to achieve even smallerdimensional coarse spaces. In all numerical results, the errors are measured in the energy norm (Hκ1 (Ω)), and 2 L -weighted norm (L 2κ (Ω)), respectively. We present the convergence as we increase the number of additional eigenvectors. In Table 4.2, we present the numerical results when the initial partition of unity consists of multiscale basis functions with linear boundary conditions for the contrast η = 106 . We note that the convergence is robust with respect to the contrast and the error is reduced. The error is proportional to the largest eigenvalue (Λ∗ ) whose eigenvector is not included in the coarse space as one can observe from the table (correlation coefficient between Λ∗ and the energy error is 0.99). We observe that the errors are smaller compared to those obtained using MsFEM with piecewise linear initial conditions. H = 1/10 Hκ1 (Ω) L 2κ (Ω) Λ∗ GMsFEM+0 (81) 37.8 0.24 0.0704 GMsFEM+1 (162) 24.0 0.1 0.0117 GMsFEM+2 (243) 23.0 0.09 0.0071 GMsFEM+3 (324) 21.0 0.08 0.0043 GMsFEM+4 (405) 20.0 0.069 0.0032 Table 4.2 Convergence results (in %) for GMsFEM with MS with increasing dimension of the coarse space. Here, the high conductivity value is η = 106 and the background permeability is 1. The initial coarse space is spanned by multiscale basis functions with piecewise linear boundary conditions (φωi ). In the parentheses, the dimensions of the coarse spaces are shown. The coefficient is depicted in Figure 4.10.

Next, we present the results for a two-level preconditioner. We implement a twolevel additive preconditioner with the following coarse spaces: multiscale functions with linear boundary conditions (MS); energy minimizing functions (EMF); spectral

4.7 Some numerical studies

117

coarse spaces using piecewise linear partition of unity functions as an initial space (GMsFEM with Lin); spectral coarse spaces where multiscale finite element basis functions with linear boundary conditions (φωi ) are used as an initial partition of unity (GMsFEM with MS); spectral coarse spaces with  κ where energy minimizing basis i ) are used as an initial partition of unity (GMsFEM with EMF). In functions (φωemf Table 4.3, we show the number of PCG iterations and estimated condition numbers. We also show the dimensions of the coarse spaces. Note that the standard coarse space with one basis per coarse node has the dimension 81 × 81. The smallest dimension can be achieved by using energy minimizing basis functions as an initial partition of unity. We observe that the number of iterations does not change as the contrast increases when spectral coarse spaces are used. This indicates that the preconditioner is optimal. On the other hand, when using multiscale basis functions (one basis per coarse node), the condition number of the preconditioned matrix increases as the contrast increases.

Fig. 4.10 Coefficient κ(x). The dark region has high conductivity η and the white background has conductivity 1. Green lines show the coarse grid.

η MS EMF GMsFEM with Lin GMsFEM with MS GMsFEM with EMF 103 83(2.71e+002) 69(1.43e+002) 31(8.60e+000) 31(9.34e+000) 32(9.78e+000) 105 130(2.65e+004) 74(1.29e+004) 33(8.85e+000) 33(9.72e+000) 34(1.02e+001) 107 189(2.65e+006) 109(1.29e+006) 34(8.85e+000) 35(9.60e+000) 37(1.02e+001) Dim

81

81

165

113

113

Table 4.3 Number of iterations until convergence and estimated condition number for the PCG and different values of the contrast η with the coefficient depicted in Figure 4.10. We set the tolerance to 1e-10. Here H = 1/10 with h = 1/100.

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4 Generalized multiscale finite element methods. Main concepts and overview

4.7.2 Elliptic equation with the parameter We consider a parameter-dependent elliptic equation (4.1). As before, we start the computation with the space of snapshots consisting of local fine-grid functions and compute offline basis functions. In this example, we will construct offline multiscale spaces for some selected values of μ, μi (i = 1, ..., Nr b ), where Nr b is the number of selected values of μ used in constructing multiscale basis functions. Note that these values of μ are selected via an inexpensive RB procedure [173]. We briefly describe the offline space construction. For each selected μ j , we choose the offline space as in the previous example,   κ(x; μ j )∇(χ ψ) · ∇(χ ψ), aωi , j (ψ, ψ) = ,ω



ω

ωi =0

where χ is the partition of unity corresponding to ω . As for sωi , we select  κ(x; μ j )∇ψ · ∇ψ. sωi , j (ψ, ψ) = ωi

Then, the selected eigenvectors are orthogonalized with respect toH 1 -like inner  product. In our numerical implementation, we choose aωi , j (ψ, ψ) = ,ω  ωi =0 ω κ(x, μ j )|∇χ |2 |ψ|2 . and sωi (ψ, ψ) can also be defined following way. The resulting aωi (ψ, ψ)  in the ωi ωi ωi , ψ = α ψ , we define ψ = α ψ . For any ψ ∈ V j H,snap l, j l, j l, j l l, j l, j Note that ψ =  ψ . Then, s and a are defined in the following way: j ω ω j sωi (ψ, ψ) =

 j

 tj

ωi

κ(x, μ j )|∇ψ j |2 , aωi =

 j

 tj

ωi

κ(x, μ j )|∇(χi ψ ωj i )|2 ,

where t j are non-negative weights. In this case, the weights do not play any role as the resulting algebraic eigenvalue problem is block diagonal. At the online stage, for each parameter value, multiscale basis functions are computed based on the solution of a local problem. In particular, for each ωi and for each ωi (μ), where the space input parameter, we formulate a quotient to find a subspace Von will be constructed for each μ. For the construction of the online space, we choose aωoni (ψ, ψ; μ)

=



 ,ω



ω

ωi =0

κ(x; μ)∇(χ ψ) · ∇(χ ψ).

For the bilinear form for sωoni , we choose  sωoni (ψ, ψ; μ)

=

ωi

κ(x; μ)∇ψ · ∇ψ.

In this case, the online space is a subspace of the offline space and computed by solving an eigenvalue problem for a given value of the parameter μ. The online

4.7 Some numerical studies

119

ωi space is computed by solving an eigenvalue problem in ωi using Voff . Using dominant eigenvectors, we form a coarse space and solve the global coupled system. For the numerical example, we will consider the coefficient that has an affine representation. The online computational cost of assembling the stiffness matrix involves summing Q pre-computed matrices corresponding to coarse-grid systems. We point out that the choice of initial partition of unity functions, χi , are important in reducing the number of very large eigenvalues (we refer to [173] for further discussions). We present a numerical example for −div(κ(x; μ)∇u) = 1 which is solved with boundary conditions u = x + y on ∂Ω. We take Ω = [0, 1] × [0, 1] that is divided into 10 × 10 equal square subdomains. As in Section 4.7.1, in each subdomain, we use a fine-scale triangulation, where triangular elements constructed from 10 × 10 squares are used.

One-dimensional input parameter We consider a permeability field which is the sum of two permeability fields with each containing inclusions such that their sum gives a channelized permeability field. The permeability field is described by κ(x; μ) := (1 − μ)κ0 (x) + μκ1 (x).

(4.22)

We can represent 3 distinct different features in κ(x; μ): inclusions (left), channels (middle), and shifted inclusions (right), see Figure 4.11. There exists no single value of μ that has all the features. Furthermore, we will use a trial set for the reduced basis algorithm that does not include μ = 0.5.

Fig. 4.11 From left to right: μ = 0, μ = 1 and μ = 1/2.

As there are three distinct spatial fields in the space of conductivities, we will choose several functions in our reduced basis. In the case of insufficient number of samples in the offline space, we will observe that when the online permeability field does not contain appropriate features, then we can not obtain the convergence.

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4 Generalized multiscale finite element methods. Main concepts and overview

We observe in Table 4.4 that we indeed need Nr b ≥ 3 to capture all the details of the solution. In these tables, we compare the errors obtained with GMsFEM when the online problem is solved with a corresponding number of basis functions. We observe a convergence with respect to the number of local eigenvectors when Nr b is chosen such that it contains spatial features included in the conductivity space. The error is proportional to the largest eigenvalue whose eigenvector is not included in the coarse space as one can observe from the table (correlation coefficient between Λ∗ and the energy error is 0.99). We have also computed weighted L 2 error which shows a similar trend and these errors are much lower. We use this conductivity field example to test the two-level additive Schwarz preconditioners. In this example, we only use coarse spaces based on reduced models. The numerical results are presented in Table 4.5. In these numerical results, we observe that when the dimension of the reduced space is 3 (and more), the condition number of the preconditioned system is independent of the contrast. In this example, we only choose the eigenvectors that correspond to asymptotically unbounded eigenvalues. Note that, in this example, we only choose basis functions corresponding to the interior nodes, while in the coarse-grid approximation, we choose basis functions that also represent boundary nodes. We observe that the number of iterations does not change as the contrast increases when spectral coarse spaces are used. On the contrary, when using multiscale basis functions (one basis per coarse node), the condition number of the preconditioned matrix increases as the contrast increases. The latter is due to the fact that the coarse space does not contain enough degrees of freedom (Figure 4.12). H = 1/10 Nr b = 2 Λ∗ Nr b = 3 Λ∗ GMsFEM+0 65.20(121) 0.4469 25.92(235) 0.0311 GMsFEM+1 18.87(462) 0.0874 7.22(356) 0.0040 GMsFEM+2 15.37(566) 0.0035 6.07(477) 0.0028 Table 4.4 Convergence results (energy norm in % and space dimension) for GMsFEM with the increasing dimension of the coarse space. Here, h = 0.01, η = 106 , and μ = 1/2 (error with MsFEM 39.29%).

η MS Nr b = 2 Nr b = 3 104 80(7.26e + 002) 38(1.76e + 001) 33(9.64e + 000) 106 120(7.17e + 004) 44(2.64e + 001) 35(9.85e + 000) Dim

81

310

338

Table 4.5 Number of iterations until convergence and estimated condition number for the PCG and different values of the contrast η and μ = 1/2. We set the tolerance to 1e-10. Here H = 1/10 with h = 1/100.

4.7 Some numerical studies

121

Four-dimensional input parameter We consider a permeability field which is the sum of four permeability fields each of which contains inclusions such that their sum gives several channelized permeability field scenarios. The permeability field is described by κ(x; μ) := μ1 κ1 (x) + μ2 κ2 (x) + μ3 κ3 (x) + μ4 κ4 (x).

(4.23)

There are several distinct features in this family of conductivity fields which include inclusions and the channels that are obtained by choosing μ1 = μ2 = 1/2 or μ3 = μ4 = 1/2. There exists no single value of μ that has all the features. Furthermore, we will use a trial set for the reduced basis algorithm that does not include μi = 1/2 (i = 1, 2, 3, 4).

Fig. 4.12 From left to right and top to bottom: κ1 , κ2 , κ3 , κ4 .

As there are several distinct spatial fields in the space of conductivities, we will choose multiple functions in our reduced basis. In the case of insufficient number of samples in the offline space, we observe that the online permeability field does not contain appropriate features and this affects the convergence rate. We observe in Table 4.6 that we indeed need Nr b ≥ 4 to capture all the details of the solution. In this table, we compare the errors obtained by GMsFEM with a different number of online basis functions. We observe convergence with respect to the number of local eigenvectors when Nr b increases. We note that, in these computations, we also have an error associated with the fact that the offline space is not sufficiently large and thus the error decay is slow as we increase the number of basis functions. We have

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4 Generalized multiscale finite element methods. Main concepts and overview

also computed weighted L 2 error which shows a similar trend and the L 2 errors are generally much smaller. H = 1/10 GMsFEM+0 GMsFEM+1 GMsFEM+2 GMsFEM+3

Nr b = 2 45.5(478) 39.4(599) 38.4(720) 36.2(841)

Nr b = 3 40.3(565) 27.5(686) 26.8(807) 26.2(928)

Nr b = 4 9.2(588) 5.3(709) 5.1(830) 4.9(951)

Table 4.6 Convergence results (energy norm in % and space dimension) for GMsFEM with the increasing dimension of the coarse space. Here, h = 0.01, η = 106 , and μ1 = μ2 = μ3 = μ4 = 1/2 (error with MsFEM 96.77%).

4.8 Randomized snapshots 4.8.1 Overview In this chapter, we discuss the use of random boundary conditions in constructing snapshot vectors to greatly reduce the computational cost associated with the construction of multiscale basis functions. We show that by using only a few of these randomly generated snapshots, we can adequately approximate the dominant modes of the solution space. To avoid oscillations near the boundary, the oversampling technique is needed. Typically, the oversampling domains are larger by several layers of fine-grid blocks around the target coarse block. Furthermore, we perform a local spectral decomposition using the restriction of the randomly generated snapshots to the target coarse-grid domain. The use of random boundary conditions (to generate the snapshot spaces) is motivated by the randomized SVD methodology [240, 321]. In general, randomized SVD algorithms allow computing dominant eigenvectors by considering a random linear combination of the columns (or rows) of a given matrix. The random linear combinations typically have a component in the dominant modes, and thus by performing a spectral decomposition in the span of these random combinations, we can achieve an accurate approximation of dominant eigenvectors. We take advantage of the idea of randomized linear combinations to considerably reduce the computational cost associated with the computation of snapshot vectors. We propose solving local problems with random boundary conditions and perform the local spectral decomposition in the space of these snapshots. The cost reduction is due to the fact that the snapshot spaces were constructed by solving local problems for every possible boundary condition in each coarse region. Using this new methodology, the number of snapshots to be generated is only slightly larger than the number

4.8 Randomized snapshots

123

of desired eigenvectors. Our experience suggests that for GMsFEM modeling, in general, it suffices to include four additional random boundary conditions to the number of eigenvectors sought. For instance, in the numerical experiments, when three basis functions per coarse grid are needed, we compute only seven snapshot vectors (i.e., only seven random boundary conditions are generated). This new methodology can provide substantial computational savings in the offline stage as we compute much fewer snapshots. We show that one needs to use randomized boundary conditions on the oversampled region to avoid oscillations near the boundaries. Indeed, if random boundary conditions are imposed on the target coarse grid (and no oversampling is used), the computed solution has oscillations near the boundaries which can cause large errors. Moreover, oversampling snapshots have several additional advantages as they allow faster convergence for GMsFEM discretizations. We compare the results obtained by using randomized snapshots to those obtained when all snapshot vectors are used. In the latter, we employ all possible boundary conditions on the oversampled region to construct the snapshot vectors. The numerical results show that one can achieve similar accuracy when using fewer random snapshots instead of using all possible snapshot vectors. We discuss approaches that can improve the results obtained by using randomized snapshots; however, at an additional computational cost. The proposed method is analyzed in [71], where we estimate the approximation error between the full snapshots and randomized snapshots in each coarse neighborhood in a certain norm. One can also use adaptive strategies with randomized snapshots, which is discussed in [71]. In the adaptive methods, additional multiscale basis functions are adaptively computed by considering only a few extra random snapshots. The main idea of randomized snapshots is demonstrated in continuous Galerkin coupling. This idea can be used for any other discretization. We remind some preliminaries. We consider the example of linear elliptic equations   − div κ(x) ∇u = f in Ω, (4.24) where u is prescribed on ∂Ω. As discussed earlier, we use T H as a conforming coarse-grid partition of the computational domain Ω into finite elements denoted by {K j } (see Figure 1.8). Assume that each coarse subregion is partitioned into a connected union of fine-grid blocks, denoted as T h . Throughout this chapter, we use the continuous Galerkin formulation, and use ωi as the support of basis functions. Randomized snapshots can be employed in other discretizations, such as mixed, discontinuous Galerkin, and so on. The regions ωi+ are used to construct the multiscale basis functions. As before, we formally denote the basis functions of the offline space VH,off by φωk i . The solution is sought as u H (x) =



cki φωk i (x),

i,k

where k denotes the basis function index in the domain ωi . Once the basis functions are identified, as before, we solve

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4 Generalized multiscale finite element methods. Main concepts and overview

a(u H , v) = ( f, v), for all v ∈ VH,off ,



(4.25)

and a(u, v) = Ω κ(x)∇u · ∇v. For the sake of the completeness, we briefly repeat the description of the GMsFEM that is more relevant to the randomized snapshots. We consider oversampling for GMsFEM that uses harmonic snapshots. That is, snapshots vector are obtained as harmonic extensions of some subset of all possible boundary conditions on the +,ωi by solving local proboversampled domain. We construct a snapshot space VH,snap lems as described in this chapter. The snapshot space consists of harmonic extensions of fine-grid functions defined on the boundary of ωi+ . For each fine-scale function with support on the boundary of the oversampled coarse domain, δlh (x), we solve a local problem. Let δlh (xk ) = δlk be one of these functions where for all l, k ∈ Jh (ωi+ ), where Jh (ωi+ ) is the fine-grid boundary nodes on ∂ωi+ and δlk is Kronecker’s delta with value 1 for k = l and value 0 otherwise. Thus, the local problem to solve is − div(κ(x)∇ψl+,ωi ) = 0 in ωi+

(4.26)

subject to boundary condition, ψl+,ωi = δlh (x) on ∂ωi+ . We form the snapshot matrices by placing the solutions of these local problems as the rows of this matrix (throughout, for notational convenience, we do not distinguish between the fine-grid vectors and their continuous representations) Ψ +,ωi = [Ψ1+,ωi ; ...; Ψl+,ωi ; ....]. We define the vectors ψlωi as the restrictions of the snapshot vectors Ψl+,ωi to degrees of freedom in ωi by taking their values at the fine-grid nodes of ωi . Considering these vectors, we form the snapshot matrix in ωi Ψωi = [Ψ1ωi ; ...; Ψlωi ; ....].

(4.27)

Next, we mention the construction of a smaller offline space using an eigenvalue problem [190]. In order to construct an offline space Voff , we reduce the dimension of the snapshot space using an auxiliary spectral decomposition. For each ωi , we define off ωi Aoff Θkωi = λoff k S Θk ,



where A

off

=

off [amn ]

=

ωi

κ(x)∇ψm+,ωi · ∇ψn+,ωi = Ψωi A(Ψωi )T



and S

off

=

off [smn ]

=

ωi

 κ(x)ψm+,ωi ψn+,ωi = Ψωi S(Ψωi )T .

(4.28)

4.8 Randomized snapshots

125

The coefficient κ(x) uses a multiscale partition of unity functions described in (4.30). Here, A and S are fine-grid stiffness and mass matrices in the coarse region, as described in this chapter. As before, to generate the offline space, we choose the smallest Moff eigenvalues of Equation (4.28) for each ωi+ and form the corresponding  k+,ωi = j Θkωji Ψ j+,ωi eigenvectors in the respective space of snapshots by setting Φ (for k = 1, . . . , Moff ), where Θkωji are the components of the vector Φkωi . We then create the offline matrices   +,ωi ωi and Φ ωi = Φ˜ 1ωi , . . . , Φ˜ M , Φ +,ωi = Φ˜ 1+,ωi , . . . , Φ˜ M off off where Φ˜ kωi is the restriction of Φ˜ k+,ωi to ωi . To construct multiscale basis functions, we multiply the dominant eigenvectors by a partition of unity functions χi that are supported in ωi , such that i χi = 1. More precisely, the offline space is composed of the following basis functions: φωk i = χi φ˜ ωk i .

(4.29)

We can choose the partition of unity functions to be multiscale finite element basis functions, denoted by χims as described in Chapter 3, or we can choose the standard κ is chosen as finite element basis function χi0 . As before,   κ=



κ|∇χi+ |2 .

(4.30)

i

4.8.2 Randomized oversampling As described above, a usual choice for the snapshot space consists of the harmonic extension of fine-grid functions defined on the boundary of ωi+ . This type of snapshot is complete in the sense that it captures all the boundary information of the solution. However, the computational cost is expensive since, in each local coarse neighbor+ + hood, O(n ωi ) number of local problems is required to solve. Here, n ωi denotes the + number of fine grids on the boundary of ωi . A smaller yet accurate snapshot space is needed to build a more efficient multiscale method. In the following, we generate inexpensive snapshots using random boundary conditions. That is, instead of solving Equations (4.26) for each fine boundary node, we solve a small number of local problems imposed with random boundary conditions ψl+,ωi ,r = rl on ∂ωi+ ,

(4.31)

where rl are independent identically distributed (i.i.d.) standard Gaussian random vectors on the fine-grid nodes of the boundary. Then, we can obtain the local random snapshot on the target domain ωi by restricting the solution of this local problem,

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4 Generalized multiscale finite element methods. Main concepts and overview

ψl+,ωi ,r to ωi (which is denoted by ψlωi ). The space generated by ψl+,ωi ,r is a subspace of the space generated by all local snapshots Ψωi . Therefore, there exists a randomized matrix R with rows composed by the random boundary vectors rl , such that Ψ ωi ,r = RΨ ωi .

(4.32)

Using these snapshots, we follow the procedure in the previous section to generate multiscale basis functions. Below, we summarize the algorithm. We denote the buffer ωi ωi for each ωi and the number of local basis functions by knb for each ωi . number pbf Later on, we use the same buffer number for all ωi and simply use the notation pbf (Table 4.7). Table 4.7 Randomized GMsFEM Algorithm. Input:

ωi Fine-grid size h, coarse-grid size H , oversampling size t, buffer number pbf for each ωi ,

ωi the number of local basis functions knb for each ωi ;

Output: Coarse-scale solution u H . 1. 2.

Generate oversampling region for each coarse block: T

H,

T h , and ωi+ ;

ωi ωi Generate knb + pbf random vectors rl and obtain randomized snapshots in ωi+ (Equation (4.31));

Add a snapshot that represents the constant function on ωi+ ; 3.

ωi Obtain knb offline basis by a spectral decomposition (Equation (4.28) restricted to random snapshots);

4.

Construct multiscale basis functions (Equation (4.29)) and solve (Equation (4.25)).

4.8.3 Numerical results In this section, we present representative numerical experiments that demonstrate the performance of the randomized snapshots algorithm. These numerical results and some additional tests can be found in [71]. We take the domain Ω as a square, set the forcing term f = 0 and use a linear boundary condition for the problem (4.24), that is, u = x1 + x2 on ∂Ω where xi are the Cartesian components of each point. In the numerical simulations, we use a coarse grid of 10 × 10 blocks, and each coarse grid block is divided into 10 × 10 fine-grid blocks. Thus, the whole computational domain is partitioned by a 100 × 100 fine grid. We use a few multiscale basis functions per coarse block. These coarse basis set defines the problem size. We assume that the finescale solution is obtained by discretizing problem (4.24) by the classical conforming piecewise bilinear elements on the fine grid. To test the performance of the algorithm, we consider the permeability fields κ as depicted in Figure 4.13. In Table 4.8, a comparison between using all snapshots and the randomized snapshots is shown. The first column shows the dimension of the offline space for each test. We choose 5, 10, 15, 20, and 25 basis functions per each interior node (in addition to the constant eigenvectors) and use an oversampling layer that consists of three fine-grid blocks (t = 3). The offline space Voff is defined via a local spec-

4.8 Randomized snapshots

127 1

5

0.8

4

0.6

3

0.4

2

0.2

1

0

0

0.2

0.4

0.6

0.8

0

1

Fig. 4.13 Permeability fields in log10 -scale.

tral decomposition as specified in Section 4.8.2. The snapshot ratio is calculated as the number of randomized snapshots divided by the number of the full snapshots. This ratio is displayed in the second column. Here, the total number of snapshots refers to the number of boundary nodes of the oversampled region. In the numerical results, an oversampled region has 26 × 26 fine-grid dimension and there are total 104 snapshots if all boundary nodes are used. For example, when the dimension of the offline space is 931, we only compute 14 snapshots instead of 104. This ratio gives the information on the computational savings of the algorithm compared to the previous algorithm using all snapshots. The next two columns shows the relative weighted L 2 error and relative energy error using the full snapshots. The weighted L 2 norm and energy norm are defined as  u L 2κ =

Ω

κu

2

 21

 and

u Hκ1 =

Ω

κ|∇u|

2

 21

,

respectively. Further, the relative weighted L 2 error and relative energy error using the randomized snapshots are shown in the last two columns. From this table, we observe that the randomized algorithm converges in the sense that the relative error decreases as we increase the dimension of the coarse space. Comparing the fourth column with the last column, we conclude that the accuracy when using the randomized snapshots is similar to using all snapshot vectors. The latter has much larger dimension as shown in the second column that shows the percentage of the snapshots computed. Therefore, the proposed method is an order of magnitude faster while having comparable accuracy. For example, when the dimension of the offline space is 931, the accuracy of the methods is comparable while randomized snapshot approach uses only 13.46% of the snapshots. Similar results are obtained when the fine mesh is refined to 200 × 200. In particular, with the offline space with the dimension 931 and the snapshot ratio of 10%, we obtain similar L 2κ (Ω) and Hκ1 (Ω) errors which

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4 Generalized multiscale finite element methods. Main concepts and overview

are 1.28% and 24.02%. The results are displayed in Table 4.9. Here, pbf refers to the buffer that is used to compute the eigenvectors. For example, pbf = 4 means that we use n + 4 snapshots to compute n basis functions for each coarse block. Table 4.8 Numerical results comparing the performance between using all harmonic snapshots and the snapshots generated by random boundary conditions with pbf = 4, κ as shown in Figure 4.13. In the parenthesis, we show a higher value of the snapshot ratio. dim(VH,off ) Snapshot ratio All snapshots Few randomized snapshots % L 2κ (Ω) Hκ1 (Ω) L 2κ (Ω) Hκ1 (Ω) 526 931 1336 1741 2146

8.65(15.38) 13.46 18.27 23.08 27.88

0.87 0.64 0.55 0.50 –

18.15 14.85 13.59 12.69 –

2.81(1.38) 1.04 0.70 0.64 0.54

44.95(26.04) 23.61 18.08 15.91 14.16

In Figure 4.14, the fine-scale solution, coarse-scale solution using all snapshots, and coarse-scale solution using randomized snapshots are shown. They are obtained using the second test (when the dimension of the offline space is 931) in Table 4.8. These two coarse-scale solutions are a good approximation of the fine-scale solution. 2

1

0.8

1.5

2

1

0.8

1.5

0

0.4

0.4 0.5

0.2

0

0.2

0.4

0.6

0.8

1

0

(a) Fine-scale solution.

1

1

1 0.4

1.5

0.6

0.6

0.6

2

1

0.8

0.5

0.2

0

0

0.2

0.4

0.6

0.8

1

0

0.5

0.2

0

0

0.2

0.4

0.6

0.8

1

0

(b) coarse-scale solution (c) coarse-scale solution using the randomized using the full snapshots snapshots

Fig. 4.14 The fine-scale solution and coarse-scale solutions correspond to Figure 4.13.

Next, we investigate the effect of the buffer number pbf on the accuracy of the coarse solution. We test a series of simulations with different pbf while keeping the coefficients and meshes fixed. The results are presented in Table 4.10, which shows that a larger buffer coefficient decreases the relative energy error. However, there is no need for very large values. If we take pbf = 4, we can get a coarse solution with

4.8 Randomized snapshots

129

Table 4.9 Numerical results comparing the results between using all harmonic snapshots and the snapshots generated by random boundary conditions with pbf = 4. dim(VH,off ) Snapshot All snapshots (%) Using the randomized snapshots (%) ratio (%) L 2κ (Ω) Hκ1 (Ω) L 2κ (Ω) Hκ1 (Ω) 526 931 1336 1741 2146

8.65(15.38) 13.46 18.27 23.08 23.88

0.71 0.51 0.45 0.40 –

20.98 17.33 15.83 14.66 –

1.33(0.80) 0.66 0.53 0.48 0.43

33.76(24.14) 21.67 18.26 17.13 15.39

an error of 15.51%, while obtaining a 14.49% error if using pbf = 20 at the cost of solving 16 extra local problems for each inner coarse node. Table 4.10 Numerical results for different pbf and using 20 local basis in each coarse neighborhood, κ as shown in Figure 4.13. pbf u − u H  (%) L 2κ (Ω) Hκ1 (Ω) 4 10 15 20

0.62 0.62 0.57 0.57

15.51 15.08 14.70 14.49

Table 4.11 Numerical results for different oversampling domain ωi+ = ωi + t and using 20 local basis in each coarse neighborhood, pbf = 4, κ as shown in Figure 4.13. t u − u H  (%) L 2κ (Ω) Hκ1 (Ω) 0 2 4 7

1.52 0.61 0.62 0.59

23.26 15.63 15.56 15.24

Lastly, numerical tests are conducted to study the influence of oversampling effects on the accuracy of the randomized snapshots. The simulation results are shown in Table 4.11. From this table, we observe that oversampling technique is needed to obtain an accurate solution. However, a larger oversampling domain is not necessary since it increases the computational cost of the solution, while no significant improvement in the solution accuracy is observed.

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4 Generalized multiscale finite element methods. Main concepts and overview

T H (Coarse Grid) i K3

Li Boundary Layer

K4

i

K2

K1

Fig. 4.15 Illustration of a skin layer Li that is used for computing boundary conditions for the snapshots in ω.

Comparison of results of different spectral problems One can use solution-based boundary conditions to achieve higher accuracy compared to the random boundary conditions. The main idea behind this algorithm is to select boundary modes using a small spectral decomposition over the boundary layer Li instead of the oversampling region ωi+ that surrounds the boundary in the spectral problem Equation (4.28). More precisely, we consider a local spectral problem in the layer of a few fine-grid blocks in the region that contains the boundary of ωi (see Figure 4.15). We choose a layer that has a thickness of five fine-grid elements (two interior to ωi and three on the immediate neighborhood of ωi ). Furthermore, we select dominant eigenvectors (corresponding to smallest eigenvalues) by solving local eigenvalue problem in the strip. The local eigenvalue problem uses local stiffness and mass matrices (as in [174, 217]). This approach provides correct fine-scale features and we expect higher accuracy compared to the randomized snapshots. The numerical results are shown in Table 4.12. Comparing the fourth column with the last column of Table 4.12, we observe that this new algorithm is more accurate compared to the previous one. Taking the fifth row as an example, for the same dimension of the offline space, the new algorithm gives 14.97% error, while the previous algorithm ends with 17.13%. In general, one can apply randomized snapshot algorithms to reduce the computational cost associated with our new algorithm. That is, one can use randomized snapshots for the strip Li to reduce the computational cost further. Remark 4.5. We note that one can easily use other ingredients of the GMsFEM, such as adaptivity, online basis functions, and so on, in conjunction with the randomized

4.8 Randomized snapshots

131

Table 4.12 Numerical results comparing the results between the snapshots obtained from skin layer spectral problems and the snapshots generated by random boundaries with pbf = 4. dim(VH,off ) Snapshot ratio Snapshots from skin layer (%) Randomized snapshots (%) (%) L 2κ (Ω) Hκ1 (Ω) L 2κ (Ω) Hκ1 (Ω) 526 931 1336 1741 2146

8.65 13.46 18.27 23.08 27.88

1.03 0.63 0.48 0.42 0.39

26.51 18.64 16.29 14.97 14.40

1.33 0.66 0.53 0.48 0.43

33.76 21.67 18.26 17.13 15.39

snapshots. The convergence analysis of the method can be found in [71].

Chapter 5

Adaptive strategies

5.1 Introduction The success of local multiscale model reduction techniques strongly depends on the local adaptivity, which can guide an appropriate number of multiscale basis functions in each coarse block. To determine the number of basis functions, computable a-posteriori error indicators are needed. In this chapter, we present an a-posteriori error indicator for the Generalized Multiscale Finite Element Method (GMsFEM) framework. We show it for continuous Galerkin formulation; however, this concept can be generalized to other discretizations. This error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high-contrast coefficients. In previous findings [174, 191], a-priori error bounds for the GMsFEM are derived for linear elliptic equations. It was shown that the convergence rate is proportional to the inverse of the eigenvalue that corresponds to the first eigenvector which is not included in the coarse space. Thus, adding more basis functions will improve the accuracy and it is important to include the eigenvectors that correspond to very small eigenvalues [174]. Rigorous a-posteriori error indicators are needed to perform an adaptive enrichment. We would like to point out that there are many related activities in designing a-posteriori error estimates [7, 150, 153, 268, 335, 393] for global reduced models. The main difference is that the error estimators presented in this chapter are based on special local eigenvalue problem and use the eigenstructure of the offline space. We will discuss two error indicators that are based on the L 2 -norm of the local residual and a weighted H −1 -norm (we will also call it H −1 -norm-based) of the local residual where the weight is related to the coefficient of the elliptic equation. The convergence analysis of the method is briefly discussed (we refer to [101] for more details). The proposed error indicators allow adding multiscale basis functions in the regions detected by the error indicator. The multiscale basis functions are selected by choosing next important eigenvectors (based on the increase of the eigenvalues)

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_5

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5 Adaptive strategies

from the offline space. The convergence proof of our adaptive enrichment algorithm is based on the techniques used for proving the convergence of adaptive refinement method for classical conforming finite element methods for second-order elliptic problems [63, 328]. If we consider the use of snapshot space in GMsFEM, the residual contains an irreducible error due to the difference between the snapshot solution and the fine-grid solution. We present numerical tests by considering high-contrast multiscale permeability fields. We have studied both error indicators based on the L 2 -norm of the local residual and the weighted H −1 -norm of the local residual. While L 2 -norm residual is easier to compute, our numerical results show that the use of weighted H −1 -norm residual gives a more robust error indicator which works well for cases with highcontrast media. In our numerical results, we also compare the results obtained by the proposed indicators and the exact error indicator which is computed by considering the energy norm of the difference between the fine-scale solution and the offline solution. Our numerical results show that the use of the exact error indicator gives nearly similar results to the case of using weighted H −1 error indicator. In our studies, we also consider the errors between the fine-grid solution and the offline solution as well as the snapshot solution and the offline solution. Adaptivity is important for local multiscale methods as it identifies regions with large errors. However, after adding some initial basis functions, one needs to take into account some global information as the distant effects can be important. In this chapter, we will also discuss the development of online basis functions that substantially accelerate the convergence of GMsFEM. More detailed results can be found in [111]. The online basis functions are constructed based on a residual and motivated by the analysis. We discuss it for the continuous Galerkin framework as an example for steady state equation; however, this concept can be used in various discretizations and in space and time also. We show, both theoretically and numerically, that one needs to have a sufficient number of initial basis functions in the offline space to guarantee an error decay independent of the contrast. We define such spaces as having online error reduction property (ONERP) and show that the eigenvalue, that the corresponding eigenvector is not included in the offline space, controls the error decay of the multiscale method. The Larger such eigenvalue, the larger the decrease in the error. Consequently, one needs to guarantee that eigenvectors associated with small (asymptotically small) eigenvalues are included in the initial coarse space. As we have discussed (see also [168, 218]), many multiscale problems with high contrast can have very small eigenvalues, and thus we need to include the eigenvectors associated with small eigenvalues in the initial coarse space. Numerical results are presented to demonstrate that one needs to have a sufficient number of initial basis functions in the offline space before constructing online multiscale basis functions. Moreover, we study how different dimensional offline spaces can affect the error decay when online multiscale basis functions are added. We consider several examples where we vary the dimension of the offline space and add multiscale basis functions based on the residual. Our numerical results show that without a sufficient number of offline basis functions, the error decay is not

5.2 Preliminaries

135

substantial. We study the proposed online basis construction in conjunction with adaptivity [111], where online basis functions are added in some selected regions. Indeed, adaptivity is an important step to obtain an overall efficient local multiscale model reduction as it is essential to reduce the cost of online multiscale basis computations. Our numerical results show that the adaptive addition of online basis functions substantially improves GMsFEM. To reduce the computational cost associated with online multiscale basis computations, we propose computing the online basis functions in a reduced dimensional space consisting of several consequent offline basis functions. Our results show that one can still achieve a substantial error reduction this way. Because the online multiscale basis functions are not sparse in the offline space, approaches based on sparsity are not very helpful in our methods as our numerical results show.

5.2 Preliminaries We will first review some main steps of GMsFEM before we introduce a-posteriori error indicators in the next section. As before, we will focus on the elliptic equation   − div κ(x)∇u = f, in Ω, (5.1) subjected to the homogeneous Dirichlet boundary condition u = 0 on ∂Ω where κ(x) is a heterogeneous coefficient with high contrast. We assume a fine grid T h and a coarse grid T H as defined earlier (see Figure 1.8 for the illustration). The corresponding weak form of (5.1) is to find u ∈ V = H01 (Ω), such that a(u, v) = ( f, v), ∀v ∈ V.

(5.2)

We denote the k-th basis functions by φkωi , which is supported in the coarse neighborhood ωi . In particular, we note that the GMsFEM will employ the use of multiple basis functions per coarse neighborhood, and the index k represents the numbering of these basis functions. The GMsFEM solution in the CG formulation  will be sought as u H (x) = i,k cki φkωi (x). Once the basis functions are identified, we will find u H by solving a(u H , v) = ( f, v), for all v ∈ VH,off ,

(5.3)

where VH,off is used to denote the space spanned by those offline basis functions and a(·, ·) is the standard bilinear form corresponding to (5.1). ω for the coarse neighborIn the offline computation, we use a snapshot space VH,snap hood ω, and the construction and motivation of such space are discussed in Chapter 4. Recall that the snapshot space can be the space of all fine-scale basis functions or

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5 Adaptive strategies

the solutions to some local problems with various choices of boundary conditions, as mentioned in Chapter 4. Let li be the number of functions in the snapshot space in the region ωi , and ωi = span{ψ ωj i : VH,snap

1 ≤ j ≤ li },

for each coarse neighborhood ωi . As before, in a discrete setup, we denote   Rsnap = Ψ1ωi , . . . , Ψlωi i , where the vector Ψ jωi contains the coefficients of the function ψ ωj i expanded in an appropriate fine-scale basis functions. ωi , we perform a dimension reduction of the space To obtain the offline space VH,off of snapshots using an auxiliary spectral decomposition. off ωi Aoff Θkωi = λoff k S Θk ,

where

 off ]=[ Aoff = [amn

ωi

(5.4)

T κ(x)∇ψmωi · ∇ψnωi ] = Rsnap A Rsnap

and  off S off = [smn ]=[

ωi

T  κ (x)ψmωi ψnωi ] = Rsnap S Rsnap ,

where A and S denote fine-scale stiffness and mass matrices, respectively. We remind ωi that we choose the smallest Moff eigenvalues from the eigensystem (5.4) and form the i kωi = lj=1 Θkωji Ψ jωi corresponding eigenvectors in the space of snapshots by setting Φ ωi ωi ωi (for k = 1, . . . , Moff ), where Θk j are the coordinates of the vector Θk . We denote by χims , the standard multiscale partition of unity functions defined by (3.8). The function  κ is defined as N H 2 |∇χims |2 ,  κ=κ i=1 ωi where H denotes the coarse mesh size. The offline space VH,off is constructed as the linear span of the following local basis functions: ωi kωi for 1 ≤ i ≤ N and 1 ≤ k ≤ Moff φkωi = χims φ , (5.5) ωi denotes the number of offline eigenvectors that are chosen for each coarse where Moff node i. We define the continuous Galerkin spectral multiscale space as ωi }. VH,off = span{φkωi : 1 ≤ i ≤ N and 1 ≤ k ≤ Moff

(5.6)

Nc Using a single index notation, we may write VH,off = span{φi }i=1 , where Nc = N ωi M denotes the total number of basis functions in the space V H,off . We seek off i=1 u H (x) ∈ VH,off such that (5.3) holds.

5.3 A-posteriori error estimates and adaptive enrichment

137

5.3 A-posteriori error estimates and adaptive enrichment In this section, we present a-posteriori error indicators for the error u − u H in the energy norm. We will then use the error indicators to develop an adaptive enrichment algorithm. The offline a-posteriori error indicator gives an estimate of the local error on the coarse-grid regions ωi , and we can then add more basis accordingly to improve the solution. We will give two kinds of error indicators, one is based on the L 2 -norm of the local residual and the other is based on the weighted H −1 -norm of the local residual (for simplicity, we will also call it H −1 -norm based indicator). The L 2 -norm residual is also used in the classical adaptive finite element method. In our case, this type of error indicator works well when the coefficient does not contain high-contrast region. We will provide a quantitative explanation for this in the next section. On the other hand, the H −1 -norm-based residual gives a more robust error indicator which works well for cases with high-contrast media. This section is devoted to the presentation of the a-posteriori error indicators and the corresponding adaptive enrichment algorithm. We will also state some theoretical results. Assume that a GMsFEM solution u H is obtained. Then we give the following definitions of the residuals. L 2 -based residual: Let ωi be a coarse neighborhood. We define a linear functional Q i (v) on L 2 (ωi ) by   Q i (v) =

ωi

f vχims −

ωi

κ∇u H · ∇(vχims ).

(5.7)

The norm of Q i is defined as |Q i (v)| . v L 2 (ωi ) v∈L 2 (ωi )

Q i  = sup

(5.8)

H −1 -based residual: Let ωi be a coarse neighborhood and let Vi = H01 (ωi ). We define a linear functional Ri (v) on Vi by   fv− κ∇u H · ∇v. (5.9) Ri (v) = ωi

ωi

The norm of Ri is defined as Ri Vi∗ = sup v∈Vi

|Ri (v)| , vVi

(5.10)

1 where vVi = ( ωi κ(x)|∇v|2 d x) 2 . ωi To simplify notations, we denote li = Moff . We recall that, for each ωi , the ωi eigenvalues λ j are ordered increasingly, and the eigenfunctions corresponding to λω1 i , · · · , λlωi i are used in the construction of VH,off . We also define κi = min x∈ωi  κ (x).

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5 Adaptive strategies

The following theorem (see [101]) gives the a-posteriori error bounds for the two types of residuals. Theorem 5.1. Let u be the weak solution of (5.1) and u H ∈ VH,off be the solution of the multiscale problem (5.3). Assume that the space VH,off is defined as in (5.6). Then we have N i u − u H 2V ≤ Cerr Q i 2 ( κi λlωi +1 )−1 , (5.11) i=1

u − u H 2V ≤ Cerr

N

i Ri 2Vi∗ (λlωi +1 )−1 ,

(5.12)

i=1

where Cerr is a uniform constant, Q i  and Ri Vi∗ are, respectively, the L 2 -based i denotes the (li + 1)-th eigenvalue over and H −1 -based residuals. Moreover, λlωi +1 coarse neighborhood ωi , and corresponds to the first eigenvector that is not included in the construction of VH,off . From (5.11) and (5.12), we see that the norms Q i  and Ri Vi∗ give indications on the size of the energy norm error u − u H V , where u2V = a(u, u). Even though (5.11) and (5.12) have the same form, we emphasize that they give different convergence behaviors in the high-contrast case. We will next present the adaptive enrichment algorithm. We use m ≥ 1 to represent m be the solution space at level m. For each coarse region, the enrichment level and VH,off m we use li to denote the number of eigenfunctions used at the enrichment level m for the coarse region ωi . Adaptive enrichment algorithm: Choose a fixed number θ with, 0 < θ < 1. 1 by specifying a fixed number of basis Choose also an initial offline space VH,off functions for each coarse neighborhood, and this number is denoted by li1 . Then, m and a sequence of multiscale solutions we will generate a sequence of spaces VH,off m u H obtained by solving (5.3). Specifically, for each m = 1, 2, · · · , we perform the following calculations: m , Step 1: Find the multiscale solution in the current space. That is, find u mH ∈ VH,off such that m . (5.13) a(u mH , v) = ( f, v) for all v ∈ VH,off

Step 2: Compute the local residual. For each coarse region ωi , we compute  ηi2

=

Q i 2 ( κi λlωmi +1 )−1 , i

2V ∗ (λlωmi +1 )−1 , i

where

for L 2 -based residual

for H −1 -based residual

i

  ms f vχi − κ∇u mH · ∇(vχims ) Q i (v)= ωi ωi   Ri (v)= fv− κ∇u mH · ∇v ωi

ωi

5.3 A-posteriori error estimates and adaptive enrichment

139

and their norms are defined in (5.8) and (5.10), respectively. Next, we reenumerate the coarse neighborhoods so that the above local residuals ηi2 are arranged in decreasing order η12 ≥ η22 ≥ · · · ≥ η2N . That is, in the new enumeration, the coarse neighborhood ω1 has the largest residual η12 and the coarse neighborhood ω N has the least residual η2N . Step 3: Find the coarse regions where enrichment is needed. We choose the smallest integer k, such that N k ηi2 ≤ ηi2 . (5.14) θ i=1

i=1

The above inequality says that the total residual in the coarse neighborhoods ω1 , ω2 , · · · , ωk is just larger than a percentage of the total residual, and the percentage θ is a user-defined quantity chosen at the beginning of the simulation. These coarse neighborhoods ω1 , ω2 , · · · , ωk are the regions where the solution contains the largest error. Step 4: Enrich the space. For each i = 1, 2, · · · , k, we add basis functions for the region ωi according to the following rule. Let s be the smallest positive integer, such that λlim +s+1 is large enough (see the proof of Theorem 5.3 in [101]) compared with λlim +1 . Then we include the eigenfunctions Φlωmi+1 , · · · , Φlωmi+s i i in the construction of the basis functions. The resulting space is denoted as m+1 m+1 VH,off . Mathematically, the space VH,off is defined as l m +s

ωi m+1 m k i VH,off = VH,off + span ∪i=1 ∪ j=l m +1 {φ j }, i

where ψi, j = χims ψ˜ ωj i , and φ˜ ωj i =

li

Φ ωjri ψrωi ,

r =1

with j = lim + 1, · · · , lim + s, denote the new basis functions obtained by the eigenfunctions Φlωmi+1 , · · · , Φlωmi+s of (5.4). In addition, we set lim+1 = i i lim + s. Remark 5.2. The algorithm above can be described as follows. We start with an initial space with a small number of basis functions for each coarse neighborhood. Then we solve the problem and compute the error estimator. We locate the coarse-grid blocks with large errors and add more basis functions for these coarse neighborhoods. This procedure is repeated until the error goes below a certain tolerance. We remark that the adaptive strategy belongs to the online process because it is the actual simulation. On the other hand, the generation of basis functions belongs to the offline process. About stopping criteria for this algorithm, one can stop the algorithm when the total number of basis functions reach a certain level. On the other hand, one can stop the algorithm when the value of the error indicator goes below a certain tolerance. We remark that the choice of s above will ensure the convergence of the enrichment algorithm, and in practice, the value of s is easy to obtain. We also remark that the

140

5 Adaptive strategies

choice of k defined in (5.14) is called the Dorlfer’s bulk marking strategy [150]. Moreover, contrary to classical adaptive refinement methods, the total number of basis functions that we can add is bounded by the dimension of the snapshot space. Thus, the condition (5.14) can be modified as follows. We choose the smallest integer k, such that N ηi2 ≤ ηi2 , θ i=1

i∈I

where the index set I is a subset of {1, 2, · · · , k} and contains indices j such that l mj is less than the maximum number of eigenfunctions for the region ω j . We now describe how the norms Q i  and Ri Vi∗ are computed. Let Wi be the diagonal matrix containing the nodal values of the fine-grid cut-off function χims in the diagonal. Then the norm Q i  can be computed as Q i  = Wi (R0T F0 − A R0T U0 ).

(5.15)

According to the Riez representation theorem, the norm Ri Vi∗ can be computed as follows. Let z i be the solution of  κ∇z i · ∇v = Ri (v), for all v ∈ Vi . (5.16) ωi

Then we have Ri Vi∗ = z i Vi . Thus, to find the norm Ri Vi∗ , we need to solve a local problem on each coarse region ωi . Finally, we state the convergence theorem of the adaptive enrichment algorithm (see [101] for the proof). Theorem 5.3. Let u be the weak solution of (5.1) and let u mH , m = 1, 2, · · · , be the sequence of solutions obtained by the adaptive enrichment algorithm. Then, there is a positive constant C, such that 2 u − u m+1 H V + C

N i=1

N

Sm+1 (ωi )2 ≤ ε u − u mH 2V + C Sm (ωi )2 , i=1

where 0 < ε < 1. We remark that the precise definitions of the constants ε and C are given in [101]. We also remark that the convergence rate ε is robust with respect to the contrast. The m quantity Sm (ωi ) is related to the approximation property of the space VH,off . Remark 5.4. (Goal Oriented Adaptivity). For some practical problems, one is interested in approximating some functions of the solution, known as the quantity of interest, rather than the solution itself. Examples include an average or weighted average of the solution over a particular subdomain, or some localized solution response. In these cases, goal-oriented adaptive methods yield a more efficient approximation

5.4 Numerical results for offline adaptivity

141

than standard adaptivity, as the enrichment of degrees of freedom is focused on the local improvement of the quantity of interest rather than across the entire solution. In [162], we study goal-oriented adaptivity for multiscale methods, and in particular, the design of error indicators to drive the adaptive enrichment based on the goal function. In this methodology, one seeks to determine the number of multiscale basis functions adaptively for each coarse region to efficiently reduce the error in the goal functional. Two estimators are studied. One is a residual-based strategy and the other uses a dual weighted residual method for multiscale problems. The method is demonstrated on high-contrast problems with heterogeneous multiscale coefficients and is seen to outperform the standard residual based strategy with respect to efficient reduction of error in the goal function.

5.4 Numerical results for offline adaptivity In this section, we will present some numerical results. We will first compare our adaptive enrichment algorithm with uniform enrichment (Section 5.4.1). Then, in Section 5.4.2, we will study the performance of the adaptive algorithm. We will only consider the use of H −1 -based residual in this section, and refer the reader to [101] for the other type of residual. Below, we summarize the indicators used in our simulations. For comparison purpose, we also use an indicator computed by the exact error in energy norm. • The indicator constructed using the weighted H −1 -based residual is ηωEni = Ri 2Vi∗ (λlωmi +1 )−1 i

(5.17)

and we name it the H −1 indicator. • The indicator constructed using the exact energy error is ηωExi = u − u H 2Vi

(5.18)

and name it the exact indicator. For the comparison of the results, we will compute relative errors using the fol1 κ 2 w L 2 (Ω) and the energy norm wV := lowing weighted L 2 norm w L 2κ (Ω) =  1 a(w, w) 2 . We denote the relative error in weighted L 2 norm by e2 and the relative error in the energy norm by ea . We remark that the reference solution is computed on the fine mesh.

142

5 Adaptive strategies

5.4.1 Comparison with uniform enrichment In this example, we will compare the use of adaptive and uniform enrichments using the H −1 -based residual. We consider the permeability field shown in Figure 4.5 for (5.1). This is the same example as in Section 4.3.3. The forcing term is shown in Figure 5.1. The fine grid is 100 × 100 and the coarse grid is 10 × 10. We consider the snapshot space spanned by harmonic functions in the oversampled domain (with randomized boundary conditions) and vary the number of basis functions per node. In Table 5.1, we present the relative errors with respect to #DOF and in Figure 5.1 (right plot), we present a log-log plot of relative error versus #DOF to compare the convergence of the adaptive GMsFEM and the GMsFEM, which uses the same number of basis functions for all coarse neighborhoods. We observe that the adaptive GMsFEM converges faster. We remark that #DOF denotes the total number of multiscale basis functions over the whole domain. 1

−0.4

0.1

0.9

−0.6

0.2

0.8

0.3

0.7

0.4

0.6

0.5

0.5

0.6

0.4

0.7

0.3

0.8

0.2

0.9

0.1

1 0

0.2

0.4

0.6

0.8

1

0

Adaptive enrichment Uniform enrichment

−0.8

log(Energy Error)

0

−1 −1.2 −1.4 −1.6 −1.8 −2 −2.2

4

4.5

5

5.5

6

6.5

7

7.5

log(DOF)

Fig. 5.1 Left: Source function f . Right: The convergence of adaptive versus uniform enrichment.

#DOF 81 151 245 334 395

ea 63.62% 32.61% 24.18% 21.15% 18.86%

e2 31.69% 7.20% 4.82% 3.85% 3.21%

#DOF 81 162 243 324 405

ea 63.62% 33.94% 31.57% 30.35% 24.40%

e2 31.69% 7.28% 6.44% 5.80% 3.99%

Table 5.1 Errors for the adaptive GMsFEM. Left: Adaptive enrichment. Right: Uniform enrichment.

5.4 Numerical results for offline adaptivity

143

5.4.2 Performance study In this section, we present some detailed numerical experiments (see also [101]) to show the performance of the error indicators and the adaptive enrichment algorithm. We take the domain Ω as a square, set the forcing term f = 1 and use a linear boundary condition for the problem (5.1). In our numerical simulations, we use a 20 × 20 coarse grid, and each coarse-grid block is divided into 5 × 5 fine-grid blocks. Thus, the whole computational domain is partitioned by a 100 × 100 fine grid. To test the performance of our algorithm, we consider two permeability fields κ as depicted in Figure 5.2. We obtain similar numerical results for these cases, and therefore, we will only demonstrate the numerical results for the first permeability field (Figure 5.2a). 5

4

x 10

1

x 10 10

1

9 8

0.8

9 0.8

8

0.6

6

0.4

4

0.2

2

7 6

0.6

7

5 4

0.4

5

3 0.2

2

3

1 0

0

0.2

0.4

0.6

0.8

(a) Permeability field 1

1

1 0

0

0.2

0.4

0.6

0.8

1

(b) Permeability field 2

Fig. 5.2 Permeability fields.

Numerical results with harmonic basis In this section, we present numerical examples to test the performance our adaptive enrichment algorithm with θ = 0.7. We will also compare our results with the use of the exact indicator. In the simulations, we take a snapshot space consisting of harmonic functions of dimension 7300 giving errors of 0.05% and 3.02% in weighted L 2 and weighted H 1 norms, respectively. Thus, the snapshot solution u snap , which is obtained by solving the problem in the snapshot space, is as accurate as the fine-scale reference solution u h . For the adaptive enrichment algorithm, the initial offline space has 4 basis functions for each coarse-grid node. In Table 5.2, we present the convergence history of the adaptive enrichment algorithm for θ = 0.7. At the 18th iteration, the dimension of the corresponding offline space is 3378. Moreover, the error u h − u off in relative weighted L 2 and energy norms are 0.54% and 7.83%, respectively, while the error u snap − u off in relative weighted

144 dim(VH,off ) 1524 1711 2434 2637 3378

5 Adaptive strategies u h − u H  (%) L 2κ (Ω)

Hκ1 (Ω)

u snap − u H  (%) L 2κ (Ω) Hκ1 (Ω)

4.50 4.24 2.34 1.64 0.54

31.29 27.37 20.13 15.43 7.83

4.49 4.23 2.36 1.61 0.51

34.14 27.19 20.31 15.13 7.22

Table 5.2 Convergence history for harmonic basis with θ = 0.7 and 18 iterations. The snapshot space has dimension 7300 giving 0.05% and 3.02% weighted L 2 and weighted energy errors. When using 12 basis per coarse inner node, the weighted L 2 and the weighted H 1 errors will be 2.34% and 19.77%, respectively, and the dimension of offline space is 3378.

L 2 and energy norms are 0.51% and 7.22%, respectively. And we see the similarity of the errors u h − u H and u snap − u H . To test the reliability and efficiency of the proposed indicator, we apply the adaptive enrichment algorithm with the exact energy error as an indicator and θ = 0.7. The results are shown in Table 5.3. dim(VH,off ) 1524 1762 2333 2522 3466

u h − u H  (%) L 2κ (Ω)

Hκ1 (Ω)

u snap − u H  (%) L 2κ (Ω) Hκ1 (Ω)

4.50 3.96 2.07 1.38 0.46

31.29 27.09 19.00 15.12 7.52

4.49 3.95 2.04 1.36 0.44

34.14 26.91 18.75 14.81 6.89

Table 5.3 Convergence history for harmonic basis with θ = 0.7 and the exact indicator. The number of iterations is 23. The snapshot space has dimension 7300 giving 0.05% and 3.02% weighted L 2 and weighted energy errors.

Comparing the results in Tables 5.2 and 5.3 for the use of the proposed and the exact indicator, respectively, we see that both indicators give similar convergence behavior and offline space dimensions. However, we also see that the exact indicator is a little inefficient. This is due to the fact that we only enrich the coarse region ωi if ηωExi is large enough according to (5.14), but the error ηωExi = u − u H Vi contains contributions of basis functions from neighboring coarse regions. In [101], we also compare the distribution for the number of basis functions and observe that the dimension distribution follows a similar pattern. Remark 5.5. (Other numerical results). In [101], we repeat the above tests using the spectral snapshot space instead of the harmonic snapshot space, and we observe similar behavior. Furthermore, we present in [101] numerical tests to show that our adaptive method is equally good when the H −1 indicator ηωEni is computed in the snapshot space.

5.5 Residual-based online adaptivity

145

Remark 5.6. (Numerical results with the L 2 indicator). The numerical results with the L 2 indicator are performed in [101]. We note that this is the most natural error indicator, as it is more efficient to compute and is widely used for classical adaptive finite element methods. However, this indicator does not work well for high-contrast coefficients. The L 2 indicators work well for mixed methods (see [79]).

5.5 Residual-based online adaptivity As we mentioned earlier in this chapter, some online basis functions are necessary to obtain a coarse representation of the true solution and give a rapid convergence of the corresponding adaptive enrichment algorithm. In this section, we will give the precise meaning of online basis functions and the corresponding adaptive enrichment algorithm following our work [111]. We will first derive a framework for the construction of online multiscale basis functions. Based on our derivations, we will argue that one also needs offline basis functions to satisfy some properties in order to guarantee that adding online basis functions will decrease the error. We will use the same problem formulation and offline basis construction as in previous sections. We use the index m ≥ 1 to represent the enrichment level. At the m to denote the corresponding GMsFEM space and enrichment level m, we use VH,on m u H the corresponding solution obtained in (5.13). The sequence of functions {u mH }m≥1 0 will converge to the true solution. The initial space VH,on consists of some offline basis functions selected in a suitable way which will be presented in the following paragraphs. Our online adaptive procedure will construct a strategy for getting the m+1 m from VH,n . space VH,on Next, we present a framework for the construction of online basis functions. By online basis functions, we mean basis functions that are computed during the iterative process, contrary to offline basis functions that are computed in the offline stage. The online basis functions are computed based on some local residuals for the current multiscale solution, that is, the function u mH . Thus, we see that some offline basis functions are necessary for the computations of online basis functions. We will also see how many of these offline basis functions are needed in order to obtain a rapidly converging sequence of solutions. Consider a given coarse neighborhood ωi . Suppose that we need to add a basis m+1 m = VH,on + span{φ} function φ ∈ Vi on the i-th coarse neighborhood ωi . Let VH,on m+1 m+1 be the new approximation space, and u H ∈ VH,on be the corresponding GMsFEM solution. By using the standard finite element convergence theory, it is easy to see satisfies from (5.13) that u m+1 H 2 2 u − u m+1 H V = inf u − vV . m+1 v∈VH,on

Taking v = u mH + αφ, where α is a scalar to be determined, we have

146

5 Adaptive strategies

2 m 2 m 2 m 2 u − u m+1 H V ≤ u − u H − αφV = u − u H V − 2αa(u − u H , φ) + α a(φ, φ).

The last two terms in the above inequality measure the amount of reduction in m . To determine error when the new basis function φ is added to the space VH,on φ, we first assume that the basis function φ is normalized so that a(φ, φ) = 1. In order to maximize the reduction in error, we will find α in order to maximize the quantity 2αa(u − u mH , φ) − α 2 . Clearly, one needs to take α = a(u − u mH , φ). Using this choice of α, we have 2 m 2 m 2 u − u m+1 H V ≤ u − u H V − |a(u − u H , φ)| .

Since φ ∈ Vi ⊂ V , by using (5.2), we have 2 m 2 m 2 u − u m+1 H V ≤ u − u H V − |( f, φ) − a(u H , φ)| .

We will then find φ ∈ Vi to maximize the local residual |( f, φ) − a(u mH , φ)|2 . Clearly, the maximum of the quantity |( f, φ) − a(u mH , φ)| equals to the functional norm of the residual Ri . Moreover, the required φ ∈ Vi is the solution of a(φ, v) = ( f, v) − a(u mH , v), ∀v ∈ Vi

(5.19)

and φVi = Ri Vi∗ . Hence, the new online basis function φ ∈ Vi can be obtained by solving (5.19). In addition, the residual norm Ri Vi∗ provides a measure of the amount of reduction in energy error. We remark that we call this algorithm the online adaptive GMsFEM since only online basis functions are used. Now, we study the convergence of the above online adaptive procedure. To simplify notations, we write ri = Ri Vi∗ . From the above constructions, we have 2 m 2 2 u − u m+1 H V ≤ u − u H V − ri .

(5.20)

m , m ≥ 1, contains n j offline basis functions We assume that each of the spaces VH,on for the coarse neighborhood ω j . Then, similar to (5.12), we have

u − u mH 2V ≤ Cerr

N

ω

r 2j (λn jj+1 )−1 .

(5.21)

j=1

Combining (5.20) and (5.21), we obtain λωnii+1 ri2 (λωnii+1 )−1 2 u − u m+1 u − u mH 2V .  ≤ 1 −  V H N 2 ωj −1 Cerr r (λ ) n +1 j=1 j j

The above inequality gives the convergence of the online adaptive GMsFEM with a precise convergence rate for the case when one online basis function is added per iteration. To enhance the convergence and efficiency of the online adaptive GMsFEM, we consider enrichment on non-overlapping coarse neighborhoods. Let I ⊂ {1, 2, · · · , N } be the index set of some non-overlapping coarse neighborhoods.

5.5 Residual-based online adaptivity

147

For each i ∈ I , we can obtain a basis function φi ∈ Vi using (5.19). We define m+1 m = VH,on + span{φi , i ∈ I }. Following the same argument as above and using VH,on the fact that the coarse neighborhoods ωi , i ∈ I , are non-overlapping, we obtain 2 m 2 u − u m+1 H V ≤ u − u H V −



ri2 .

(5.22)

i∈I

Consequently, we have u −

2 u m+1 H V

 ) 2 ωi −1 Λ(I i∈I ri (λn i +1 ) min m 2 ≤ 1−  N 2 ω j −1 u − u H V , Cerr r (λ ) n +1 j=1 j

(5.23)

j

where ) ωi Λ(I min = min λn i +1 . i∈I

(5.24)

Inequality (5.23) shows that we are able to obtain a better convergence of our online adaptive GMsFEM by adding more online basis functions per iteration. The con) vergence rate depends on the factors Cerr and Λ(I min . We will, therefore, need to take (I ) enough offline basis functions so that Λmin is large enough and  ) 2 ωi −1 Λ(I i∈I ri (λn i +1 ) min  N 2 ω j −1 ≥ θ0 Cerr j=1 r j (λn +1 ) j

for some 0 < θ0 < 1 which is independent of the contrast in κ(x). Hence, we obtain the following convergence for the online adaptive GMsFEM: 2 m 2 u − u m+1 H V ≤ (1 − θ0 )u − u H V . ) We note that Λ(I min can be very small when there are channels in the domain. This is extensively discussed in [168]. For this reason, we introduce a definition. 0 Definition 5.7. We say the initial space VH,on satisfies Online Error Reduction Property (ONERP) if  ) 2 ωi −1 Λ(I i∈I ri (λn i +1 ) min  N 2 ωi −1 ≥ θ0 , Cerr i=1 ri (λn +1 ) i

for some θ0 > δ > 0, where δ is independent of physical parameters such as contrast. 0 We remark that if the initial space VH,on is ONERP, then the error of the corresponding sequence of solutions {u mH } using online basis functions will decrease independent of physical parameters, such as contrast and scales. We will show in 0 with ONERP, the online basis our numerical results that if we do not choose VH,on 0 functions will not decrease the error. One of easiest way to determine VH,on being

148

5 Adaptive strategies

) ONERP is to guarantee that Λ(I min is sufficiently large. In general, one can use the  (I ) ω sizes of Λmin and i∈I ri2 (λnii+1 )−1 to determine the switching between offline and online. 2 We remark that one can derive a-priori error estimate for u − u m+1 H V . To do so, we can use an error estimate for the GMsFEM that uses initial multiscale basis functions [101, 174, 191] and obtain estimate for u − u 0H 2V , where u 0H denotes the multiscale solution that uses the initial basis functions. This convergence rate depends on the use of oversampling and is proportional to 1/Λ∗ , where Λ∗ is the largest eigenvalue that the corresponding eigenvector is not used in constructing the initial multiscale space.

5.6 Numerical results for online adaptivity In this section, we will present numerical examples to analyze the performance of the online basis functions. These numerical results can also be found in [111]. The domain Ω is taken as the unit square [0, 1]2 and is divided into 16 × 16 coarse blocks consisting of uniform squares. Each coarse block is then divided into 16 × 16 fine blocks consisting also of uniform squares. That is, the whole domain is partitioned by 256 × 256 fine-grid blocks. The medium parameter κ and the source function f are shown in Figure 5.3. We will use the following error quantities to measure the accuracy of our algorithm 10000

0

1

0

0.1

9000

0.1

0.9

0.2

8000

0.2

0.8

0.3

7000

0.3

0.7

0.4

6000

0.4

0.6

0.5

5000

0.5

0.5

0.6

4000

0.6

0.4

0.7

3000

0.7

0.3

0.8

2000

0.8

0.2

0.9

1000

0.9

0.1

1

0

0.2

0.4

0.6

0.8

1

1

0

0.2

0.4

0.6

0.8

1

0

Fig. 5.3 Left: Permeability field κ. Right: Source function f .

e2 =

u − u H  L 2 (Ω) u − u H V , ea = , u L 2 (Ω) uV

which are the relative error measured in the L 2 norm defined as v2L 2 (Ω) = Ω v 2 d x and the relative error measured in the energy norm defined by v2V = a(v, v). The online adaptive GMsFEM is implemented as follows. We will first choose a fixed number of offline basis functions for every coarse neighborhood, and denote the 0 = VH,off . In addition, the basis functions resulting offline space as VH,off . We set VH,on

5.6 Numerical results for online adaptivity

149

0 in VH,on are called initial basis. We will enumerate the coarse neighborhoods by a two-index notation. More precisely, the coarse neighborhoods are denoted by ωi, j , where i = 1, 2, · · · , N x and j = 1, 2, · · · , N y and N x and N y are the number of coarse nodes in the x and y directions, respectively. We let Ix,odd and Ix,even be the set of odd and even indices from {1, 2, · · · , N x }. We use similar definitions for I y,odd and I y,even . Each iteration of our online adaptive GMsFEM contains 4 sub-iterations. In particular, these 4 sub-iterations are defined by adding online basis functions in the non-overlapping coarse neighborhoods ωi, j with (i, j) ∈ Ix,odd × I y,odd , (i, j) ∈ Ix,odd × I y,even , (i, j) ∈ Ix,even × I y,odd and (i, j) ∈ Ix,even × I y,even , respectively.

5.6.1 Comparison of using different numbers of initial basis In Table 5.4, we present the convergence history of our algorithm for using one initial basis per coarse neighborhood. Notice that we have shown the number of basis functions (#basis) used for each coarse neighborhood and the total degrees of freedom (DOF), which are the numbers in parentheses. We use the standard multiscale basis functions as the initial basis. We consider two different contrasts for the coefficient κ in Figure 5.3 (left figure) in our numerical simulations. In the right table, we present the results when we increase the contrast by 100 times. More precisely, the conductivity of inclusions and channels in Figure 5.3 (left figure) is multiplied by 100. In this case, the first few eigenvalues that are in the regions with channels become 100 times smaller [168]. This decrease in the eigenvalues will make the error decay slower. This can be observed by comparing the left and the right tables of Table 5.4, where we can see that the errors in case of the higher contrast decrease much slower. This is also observed when we use 2 initial basis functions (see Table 5.5). When using two initial basis functions, there are contrast-dependent small eigenvalues (these eigenvalues decrease as we increase the contrast), and thus, by increasing the contrast, the decay becomes slower. This can be observed in Table 5.5. However, if we choose 3 initial basis functions, then Λmin is independent of the contrast and, thus, for larger contrasts, we observe a similar fast error decay behavior. We observed a similar convergence when using 4 initial basis functions, see Figure 5.4, where we plot the relative errors in the energy norm (two graphs on the top) and the logarithm of relative errors in the energy norm (two graphs at the bottom) against the m for various choices on the initial basis and for two types of condimensions of VH,on trasts, 1e4 (left) and 1e6 (right). From this figure, we also see clearly the exponential m (Table 5.6). convergence of the method with respect to the dimension of VH,on From Figure 5.4, we can also observe the following facts. First, we observe if we choose the number of initial basis functions to be 2 (which satisfies ONERP), it will give the smallest error for a fixed coarse space dimension. Indeed, if we start with the smallest number of initial basis functions (that satisfy ONERP), then at every iteration of the online stage, the error will reduce more compared to the offline-stage basis addition (i.e., adding the basis functions computed in the offline stage). On the other hand, if we would like to reduce the online cost associated with computing

150

5 Adaptive strategies

basis functions, then it is more advantageous to choose more offline basis functions. Indeed, choosing a larger number of initial offline basis functions will give us a better result, as we observe from Figure 5.4. For example, if we compare the errors for one online basis addition, we will find that the case with 4 initial offline basis functions gives the smallest error. This is because the initial error for the four offline basis functions is the smallest among those shown in Figure 5.4. In general, one needs to be careful as more offline basis functions will increase the dimension of the online system and, consequently, the cost of solving the coarse-grid system. #basis (DOF) 1 (225) 2 (450) 3 (675) 4 (900) 5 (1125) 6 (1350) 7 (1575) 8 (1800) 9 (2025)

ea 60.71% 33.10% 14.38% 4.28% 1.33% 0.065% 0.00083% 1.59e-05% 2.35e-07%

e2 33.87% 13.38% 3.25% 1.02% 0.24% 0.0028% 2.96e-05% 4.87e-07% 2.10e-08%

#basis (DOF) 1 (225) 2 (450) 3 (675) 4 (900) 5 (1125) 6 (1350) 7 (1575) 8 (1800) 9 (2025)

ea 60.90% 35.90% 35.00% 25.77% 14.17% 7.79% 6.83% 4.15% 2.60%

e2 34.15% 15.87% 15.29% 8.77% 4.39% 2.78% 2.06% 1.20% 0.64%

Table 5.4 Convergence history for the permeability field in Figure 5.3 and for the case with one initial basis. Left: Lower contrast(1e4). Right: Higher contrast(1e6).

#basis (DOF) 2 (450) 3 (675) 4 (900) 5 (1125) 6 (1350)

ea 26.60% 1.46% 0.017% 0.000021% 3.56e-06%

e2 6.92% 0.060% 0.000079% 1.06e-05% 1.65e-07%

#basis (DOF) 2 (450) 3 (675) 4 (900) 5 (1125) 6 (1350)

ea 27.17% 4.99% 0.20% 0.0017% 2.71e-05%

e2 7.53% 0.79% 0.0073% 8.16e-05 1.09e-06%

Table 5.5 Convergence history for the permeability field in Figure 5.3 and for the case with two initial basis. Left: Lower contrast(1e4). Right: Higher contrast(1e6).

Next, we will present an example with a different medium parameter κ shown in Figure 5.5 to show the importance of ONERP. The source function f is taken as the constant 1. The domain Ω is divided into 8 × 8 coarse blocks consisting of uniform squares. Each coarse block is then divided into 32 × 32 fine blocks consisting also of uniform squares. The convergence history for the use of one initial basis and the corresponding total number of degrees of freedom (DOF) are shown in Table 5.7. In this case, Λmin = 0.0033, which is considered to be very small, and we observe a very slow convergence of the online adaptive procedure. In Table 5.8, we present the convergence history for the use of two to five initial basis, where we only show the results for the last 4 iterations. We see that the values of Λmin increase as we increase

5.6 Numerical results for online adaptivity

#basis (DOF) 3 (675) 4 (900) 5 (1125) 6 (1350) 7 (1575)

ea 16.95% 0.54% 0.011% 9.07e-05% 1.38e-06%

e2 2.53% 0.023% 0.00040% 3.79e-06% 6.05e-08%

151

#basis (DOF) 3 (675) 4 (900) 5 (1125) 6 (1350) 7 (1575)

ea 16.96% 0.54% 0.011% 9.07e-05% 1.58e-06%

e2 2.54% 0.023% 0.00041% 3.79e-06% 5.49e-07%

Table 5.6 Convergence history for the permeability field in Figure 5.3 and for the case with three initial basis. Left: Lower contrast(1e4). Right: Higher contrast(1e6). 0.7

0.7 1 basis 2 basis 3 basis 4 basis

0.5

0.4

0.3

0.2

0.1

400

600

800 1000 Dimensions of Vms

Log of relative errors in energy norm

1200

0.4

0.3

0.2

0

−5

−10

−15

−20

400

600

800 1000 Dimensions of V

ms

1200

1400

0 200

1400

1 basis 2 basis 3 basis 4 basis slope = −0.018

5

−25 200

0.5

0.1

1600

400

600

800 1000 Dimensions of Vms

1200

1400

1 basis 2 basis 3 basis 4 basis slope = −0.018

5

Log of relative errors in energy norm

0 200

1 basis 2 basis 3 basis 4 basis

0.6 Relative errors in energy norm

Relative errors in energy norm

0.6

0

−5

−10

−15

−20 200

400

600

800 1000 Dimensions of V

1200

1400

1600

ms

Fig. 5.4 Convergence comparison for the permeability field in Figure 5.3 and for different choices of the number of initial basis. Left: The contrast is 1e4. Right: The contrast is 1e6.

the number of initial basis. We also observe that the convergence rate increase when we raise the number of initial basis from 2 to 4. For the use of 5 initial basis, we again see rapid convergence and a faster convergence compared when using 4 initial basis functions. In particular, we observe (based on 3 iterations following the initial one) that the error decays at 130-fold when 5 initial basis functions are selected, while the error decay is about 90-fold when 4 initial basis functions are selected. A comparison of error decay for the use of 1 to 5 initial basis functions is shown in Figure 5.6. We have also tested harmonic basis functions and the results are similar, i.e., the convergence rate is very slow unless a sufficient number of offline basis functions is selected. In conclusion, we observe

152

5 Adaptive strategies 5

x 10 10

0 0.1

9

0.2

8

0.3

7

0.4

6

0.5

5

0.6

4

0.7

3

0.8

2

0.9

1

1

0

0.2

0.4

0.6

0.8

1

Fig. 5.5 Permeability field κ. DOF 81

ea 17.24%

e2 4.35%

162

2.80%

1.02%

243

2.65%

0.88%

323

2.64%

0.87%

401

1.09%

0.081%

478

0.74%

0.094%

555

0.73%

0.090%

632

0.48%

0.039%

709

0.37%

0.026%

Table 5.7 Convergence history for the permeability field in Figure 5.5 and for the case with one initial basis. (The value of Λmin = 0.0033). 0.18 1 basis 2 basis 3 basis 4 basis 5 basis

Relative errors in energy norm

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

100

200

500 400 300 Dimensions of Vms

600

700

800

Fig. 5.6 Convergence comparison for the permeability field in Figure 5.5 and for different numbers of initial basis functions.

5.6 Numerical results for online adaptivity

153

DOF 162 243 324 405 486 567 648 725 765

ea 13.29% 2.00% 1.79% 1.60% 0.33% 0.30% 0.012% 0.00012% 3.62e-06%

e2 2.90% 0.32% 0.23% 0.17% 0.025% 0.022% 0.00057% 4.99e-06% 2.45e-06%

DOF 243 324 405 486 567 638 644

ea 10.26% 1.78% 1.75% 0.25% 0.0016% 8.34e-06% 3.69e-06%

e2 1.51% 0.23% 0.23% 0.0088% 0.000089% 2.50e-06% 2.49e-06%

DOF 324 405 486 563 568

ea 7.95% 0.074% 0.0010% 1.10e-05% 3.71e-06%

e2 1.06% 0.0035% 3.24e-05% 2.55e-06% 2.52e-06%

DOF 405 486 567 635

ea 7.24% 0.0684% 0.00049% 3.80e-06%

e2 0.92% 0.0028% 1.51e-05% 2.49e-06%

Table 5.8 Convergence history for the permeability field in Figure 5.5. Top-left: Two initial basis (Λmin = 0.026). Top-right: Three initial basis (Λmin = 0.18), Bottom-left: Four initial basis (Λmin = 199.12). Bottom-right: Five initial basis (Λmin = 319.32).

0 • If the initial space VH,on does not satisfy ONERP, then the error decay is slower as the contrast becomes larger. 0 does not satisfy ONERP, in some cases, we have observed • If the initial space VH,on the error does not decrease as we add online basis functions (see Tables 5.7, 5.8). 0 satisfies ONERP, then we observe a fast convergence, • If the initial space VH,on which is independent of contrast.

5.6.2 Adaptive online enrichment In this section, the online enrichment is performed only for regions with the residual that is larger than a certain threshold. In the first case, the online enrichment is performed for the coarse regions with a residual error bigger than a certain threshold which will be taken 10−3 , 10−4 , and 10−5 . In the second case, the online enrichment is performed for coarse regions that have cumulative residual (see Remark 5.8) that is θ fraction of the total residual. One of our objectives is to show that one can drive the error down to a number below a threshold, adaptively. In our numerical results, we will consider three tolerances (tol) 10−3 , 10−4 and −5 10 . We will enrich coarse regions, if the H −1 -norm of the residual is bigger than the tolerance. In Table 5.9, we show the errors when using 1 initial basis function for tolerances 10−3 , 10−4 and 10−5 . We first observe a very slow reduction in errors

154

5 Adaptive strategies

similar to the results presented in the previous section. Another observation is that the energy error of the multiscale solution is in the same order of the tolerance, and the error cannot be further reduced if we perform more iterations. This allows us to compute a multiscale solution with a prescribed error level by choosing a suitable tolerance in the adaptive algorithm. In Tables 5.10 and 5.11, we show the errors for the last three iterations when using 2 and 3 initial basis functions, respectively, for tolerances 10−3 , 10−4 , and 10−5 . We observe that the convergences are much faster. In addition, the energy errors again have the same magnitude as the tolerances. From these results, we obtain the following conclusions. • Using smaller tolerances, we can reduce the final error below desired threshold errors. • We have observed that the number of initial basis functions is important to achieve better results. For example, we observe a slow decay of the error when 1 initial basis function is selected. Moreover, if the contrast is higher, the decay becomes slower.

DOF ea e2 225 60.71% 33.87% 447 33.10% 13.39% 652 14.43% 3.28% 776 4.37% 1.06% 824 1.83% 0.37% 847 1.10% 0.25% 863 0.50% 0.029%

DOF 225 449 674 883 1031 1125 1136

ea e2 60.71% 33.87% 33.10% 13.38% 14.38% 3.25% 4.28% 1.02% 1.33% 0.24% 0.082% 0.0036% 0.052% 0.0023%

DOF ea e2 225 60.701% 33.87% 450 33.10% 13.38% 675 14.38% 3.25% 899 4.28% 1.02% 1114 1.33% 0.24% 1275 0.065% 0.0028% 1338 0.0048% 0.00017%

Table 5.9 Convergence history with a fixed tolerance (tol) and one initial basis for the permeability field in Figure 5.3. Left: tol = 10−3 . Middle: tol = 10−4 . Right: tol = 10−5 .

DOF ea e2 450 26.60% 6.92% 649 1.49% 0.063% 666 0.53% 0.028%

DOF ea e2 450 26.60% 6.92% 674 1.46% 0.059% 802 0.048% 0.0022%

DOF ea e2 675 1.46% 0.060% 885 0.017% 0.00079% 925 0.0043% 0.00019%

Table 5.10 Convergence history with a fixed tolerance (tol) and two initial basis for the permeability field in Figure 5.3. Left: tol = 10−3 . Middle: tol = 10−4 . Right: tol = 10−5 .

Remark 5.8. (Cumulative residual). In [111], we have presented the numerical results, when the online enrichment is performed for coarse regions that have a cumulative residual that is θ fraction of the total residual. More precisely, we assume that the local residuals are arranged so that r1 ≥ r2 ≥ r3 ≥ · · · .

5.6 Numerical results for online adaptivity DOF ea e2 675 16.96% 2.54% 863 0.63% 0.027% 867 0.44% 0.018%

155

DOF ea e2 675 16.96% 2.54% 898 0.054% 0.023% 993 0.046% 0.0015%

DOF ea e2 900 0.54% 0.023% 1087 0.011% 0.00043% 1106 0.0050% 0.00019%

Table 5.11 Convergence history with a fixed tolerance (tol) and three initial basis for the permeability field in Figure 5.3. Left: tol = 10−3 . Middle: tol = 10−4 . Right: tol = 10−5 .

Then, we only add the basis φ1 , · · · , φk for the coarse neighborhoods ω1 , · · · , ωk such that k is the smallest integer with θ

N i=1

ri2 ≤

k

ri2 .

i=1

In Table 5.12, we present numerical results for the last 4 iterations when using 1, 2, and 3 initial basis functions with the tolerance 10−4 and θ = 0.7. We observe that one can reduce the total number of basis functions compared to the previous case to achieve a similar error. Our conclusions regarding the importance of ONERP 0 is the same as before. condition for the initial space VH,on

DOF ea e2 620 1.10% 0.23% 709 0.49% 0.082% 787 0.050% 0.0024% 789 0.046% 0.0022%

DOF ea e2 450 26.60% 6.92% 576 1.94% 0.12% 690 0.20% 0.0099% 744 0.051% 0.0023%

DOF ea e2 675 16.96% 2.54% 827 1.02% 0.045% 957 0.091% 0.0033% 987 0.048% 0.0017%

Table 5.12 Convergence results using cumulative errors with θ = 0.7, tol = 10−4 and the permeability field in Figure 5.3 Left: One initial basis. Middle: Two initial basis. Right: Three initial basis.

Remark 5.9. (Inexpensive online basis construction). In [111], we discuss approaches that can use inexpensive online basis construction using a smaller dimensional sets. In particular, the online basis functions are computed in the next N0 snapshot vectors that are eigenvectors of the local spectral problem defined in the offline stage. We test various choices for N0 and observe that the convergence is fast and its behavior is comparable to the use of the whole local snapshot space for the computation of online basis functions. Remark 5.10. In all the above examples, we consider a snapshot space that consists of local eigenvectors. It can be that in different regions, one needs to use different eigenvalue problems. For example, if the heterogeneities are very localized, one can use polynomial basis functions away from heterogeneous regions. In this case, we

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5 Adaptive strategies

consider an adaptive strategy that can identify which class of basis functions to use in a given region. We have considered several basis sets and used the residual to decide which set to use in a given ωi for the permeability field shown in Figure 5.5 with 16 × 16 coarse mesh and each coarse block is subdivided into 16 × 16 fine blocks. The set is identified using a few initial basis functions (e.g., using multiscale basis functions). Then, for each set, we consider the residual using only a few basis and estimate the error. We choose the set that gives the largest reduction in the residual. Our numerical results show that by selecting an appropriate class of basis functions in each region, we can improve the accuracy of GMsFEM.

Chapter 6

Selected global formulations for GMsFEM and energy stable oversampling

6.1 Introduction In previous chapters, we use continuous Galerkin coupling for multiscale basis functions. In many applications, one needs to use various discretizations. For example, for flows in porous media, mass conservation is very important. Standard continuous Galerkin approaches do not provide mass conservation unless some post-processing is performed. In this case, mixed finite element methods can be used in obtaining mass conservative velocity field. In another application involving the time-explicit discretization of wave equations, one needs block-diagonal mass matrices, which can be obtained using discontinuous Galerkin approaches. In this chapter, we present several global couplings, mixed finite element (mixed GMsFEM), Interior Penalty Discontinuous Galerkin (GMsDGM), nonconforming Galerkin, and Hybridized Discontinuous Galerkin couplings (GMsHDG). For each discretization, we define snapshot spaces and local spectral decompositions in the snapshot spaces. As we emphasized earlier that the construction of the snapshot spaces and local multiscale spaces depends on global discretization. For example, for mixed GMsFEM, the multiscale spaces are constructed for the velocity field using two neighboring ccoarse elements. In this chapter, we give an overview of the mixed GMsFEM, the GMsDGM, the nonconforming GMsFEM, and the GMsHDG. We present some ingredients of GMsFEM introduced earlier in the construction of multiscale basis functions. In particular, we discuss oversampling techniques and online multiscale basis functions. In this chapter, we also present a new ingredient for multiscale basis construction— energy stable (minimizing) snapshots following [95]. The energy stable (minimizing) snapshots are motivated by the analysis of multiscale methods. In the analysis of GMsFEM, one first decomposes and localizes the error in coarse grids. Assuming the error functional is a(·, ·), the first step in the error analysis is

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_6

157

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6 Selected global formulations for GMsFEM and energy stable oversampling

aΩ (u − u H , u − u H ) 



aω (u ω − u ωH , u ω − u ωH ),

ω

where u is the solution, u H is the multiscale solution, ω’s are coarse regions, u ω is localized components of the solution, and u ωH is the local components of the multiscale solution. To achieve fast convergence, one often uses oversampling concepts and constructs snapshot spaces in larger regions. The local errors are estimated +,ω − u +,ω aω (u ω − u ωH , u ω − u ωH )  δa+,ω (u +,ω − u +,ω H ,u H ),

where +, ω is an oversampled region and δ represents the error. Once the local errors are estimated, one needs to combine the contributions of local errors together and estimate  +,ω a+,ω (u +,ω − u +,ω − u +,ω H ,u H ) ω+

by aΩ (u, u). This requires energy stable (minimizing) properties in the construction of multiscale basis functions. These snapshot vectors are constructed by solving a local minimization problem. We discuss this concept that oversampling can be used and, at the same time, maintaining stable decomposition property. The proposed multiscale basis functions can be used in constructing online basis functions (see Section 12.7.3). Numerical results are presented, which study both offline and online performance of GMsFEM.

6.2 Global formulations 6.2.1 Preliminaries In a mixed formulation, the flow problem can be formulated as κ−1 v + ∇u = 0

in Ω,

div(v) = f

in Ω,

(6.1)

with Neumann boundary condition v · n = vΩ on ∂Ω, where κ is a high-contrast permeability field, Ω is the computational domain in Rd and n is the unit outward normal vector of the boundary of Ω. In the primal formulation, the above single-phase problem can be formulated as − div(κ∇u) = f, in Ω, with Dirichlet boundary condition u = u Ω on ∂Ω.

(6.2)

6.2 Global formulations

159

We will present various coarse-grid formulations for the problems (6.1) and (6.2). We denote T H be a partition of Ω into standard finite elements (triangles, quadrilaterals, tetrahedra, etc.), where H is called the coarse mesh size, and we call T H the coarse grid (see also Figure 1.8). Our coarse-grid discretizations are defined with respect to this coarse grid. We will construct basis functions with supports on coarse elements or unions of coarse elements. These basis functions are obtained by solving some local problems, which require a finer mesh in order to represent these basis functions accurately. To define the fine grid, we further divide every coarse element K ∈ T H into a union of finer elements such that the resulting partition T h of Ω with size h is conforming across coarse-grid edges. We call T h the fine grid, and h the fine mesh size. Depending on the discretization technique, the coarse-grid configuration will vary. For example, for a mixed formulation, coarse-grid blocks that share common faces will be used in constructing multiscale basis functions. In a discontinuous Galerkin formulation, the support of multiscale basis functions is limited to coarse blocks. Correspondingly, the choice of oversampled regions will be different depending on a discretization. In the following sections, we will introduce these discretizations.

6.2.2 Mixed GMsFEM For the mixed GMsFEM, we will construct our basis functions whose supports are ωi , which are the two coarse elements that share a common face E i . In particular, we let E H be the set of all faces of the coarse grid and let Ne be the total number of faces of the coarse grid. We define the coarse-grid neighborhood ωi of a face E i ∈ E H as ωi =

 {K ∈ T

H

: E i ∈ ∂ K }, i = 1, 2, · · · , Ne ,

which is a union of two coarse-grid blocks if E i is an interior edge. Furthermore, we denote by ωi+ an oversampling region for ωi (see Figure 6.1). One can also take the oversampled region ωi+ with an arbitrary geometry. Next, we define the notations for the solution spaces for the pressure u and the velocity v. Let Q H,off be the space of functions that are constant on each coarsegrid block. We will use this space to approximate u. For the multiscale approximation of the velocity, we will follow the general framework of mixed GMsFEM ωi supported [79]. In particular, we will first construct a set of basis functions βsnap functions in the coarse-grid neighborhood ωi . We call the span of all these basis    ωi ωi ωi VH,snap = Ei ∈E H VH,snap the snapshot space, where VH,snap = span βsnap is the local snapshot space for the coarse neighborhood ωi . The snapshot space is an extensive set of functions that can be used to approximate the solution v. However, this space is large and we will reduce it to a smaller one before we solve the equation. ωi ωi , we select a subset of basis functions βoff by using an appropriate From each VH,snap spectral problem to form a smaller-dimensional offline space. We denote the local

160

6 Selected global formulations for GMsFEM and energy stable oversampling

 ωi  ωi offline space corresponding to ωi by VH,off = span βoff and the global offline space ωi . The size of VH,off is generally much smaller than VH,snap , as VH,off = Ei ∈E H VH,off but still contains the most important features about the heterogeneous coefficient κ. We will use the space VH,off to approximate the velocity v. With the pressure space Q H,off and the velocity space VH,off , we solve for u H ∈ Q H,off and v H ∈ VH,off , such that Ω

κ−1 v H · w −

div(w)u H = 0 div(v H )q =

∀w ∈ V˙ H,off ,

Ω

Ω

Ω

(6.3) fq

∀q ∈ Q H,off ,

with boundary condition v H · n = vΩ,H on ∂Ω, where V˙ H,off = {v ∈ VH,off : v · n = 0 on ∂Ω}, and vΩ,H is the projection of vΩ in the sense that (vΩ,H − vΩ )φ · n = 0 Ei

ωi ∀φ ∈ βsnap and E i ⊆ ∂Ω,

and vΩ,H is constant on each fine-grid face. We remark that we will define the snapshot and offline spaces so that the functions are globally H (div)-conforming, so that the normal components of the basis are continuous across coarse-grid edges. We remark that the mixed GMsFEM can be seen as a generalization of the classical Raviart-Thomas mixed finite element method. The novelty is that we will construct the snapshot space VH,snap by using an oversampling approach using ωi+ and energy minimizing snapshot functions. The resulting scheme gives a smaller, yet rich enough, dimensional snapshot space and a more accurate representation of the solution by multiscale basis.

ω+i ω+i Ei

Fig. 6.1 A coarse neighborhood (red) and its corresponding oversampling region (green) for mixed GMsFEM and GMsHDG. An example of fine-grid partition (blue) for a coarse element is also shown.

6.2 Global formulations

161

6.2.3 GMsDGM For the GMsDGM, the general methodology for the construction of multiscale basis functions is similar to the above mixed GMsFEM. We will construct multiscale basis functions for the approximation of the pressure u in (6.2). The main difference is that the functions in the snapshot space VH,snap and the offline space VH,off are supported in coarse element K , instead of the coarse neighborhood ωi . Moreover, in the oversampling approach, the oversampled regions K + are defined by enlarging coarse elements K by one coarse-grid block (see Figure 6.2). The detailed construction of basis functions will be given in the next section. When the offline space VH,off is available, we can find the solution u H ∈ VH,off , such that (see [112, 124]) aDG (u H , q) = ( f, q), ∀q ∈ VH,off ,

(6.4)

where the bilinear form aDG is defined as    { κ∇u · n E } [[q]] + { κ∇q · n E } [[u]] + aDG (u, q) = a H (u, q) − E∈E H

E

 γ κ[[u]][[q]] h E H

(6.5)

E∈E

with a H (u, q) =

 K ∈T

a HK (u, q),

a HK (u, q)

=

H

κ∇u · ∇q,

(6.6)

K

where γ > 0 is a penalty parameter, n E is a fixed unit normal vector defined on the coarse edge E ∈ E H . Note that, in (6.5), the average and the jump operators are defined in a classical way. Specifically, consider an interior coarse edge E ∈ E H and let K L and K R be the two coarse-grid blocks sharing the edge E. For a piecewise smooth function G, we define { G}} =

1 (G R + G L ), 2

[[G]] = G R − G L ,

on E,

where G R = G| K R and G L = G| K L and we assume that the normal vector n E is pointing from K R to K L . Moreover, on the edge E, we define κ = (κ K R + κ K L )/2 where κ K L is the maximum value of κ over K L and κ K R is defined similarly. For a coarse edge E lying on the boundary ∂Ω, we define { G}} = [[G]] = G,

and

κ = κK

on E,

where we always assume that n E is pointing outside of Ω. We note that the DG coupling (6.4) is the classical interior penalty discontinuous Galerkin (IPDG) method with the multiscale basis functions as the approximation space.

162

6 Selected global formulations for GMsFEM and energy stable oversampling

K+ K

Fig. 6.2 A coarse element (red) and its corresponding oversampling region (green) for GMsDGM. An example of fine-grid partition (blue) for a coarse element is also shown.

6.2.4 Nonconforming GMsFEM In nonconforming GMsFEM, we use the same bilinear form as the continuous Galerkin methods, but with nonconforming approximation spaces. Two spaces are needed in the characterization of the nonconforming GMsFEM space, namely, the local approximation space VH,off and the moment space M H . The construction of M H will be discussed in the next section. On the other hand, the space VH,off can be obtained in a similar fashion as in the DG coupling. Note that the functions in VH,off are discontinuous across coarse elements. In the DG coupling, they are coupled by the DG bilinear form through jumps and averages on the interfaces. In contrast, the basis functions in nonconforming GMsFEM are coupled through certain weak continuity requirements related to the moment space M H . More precisely, the nonconforming GMsFEM space VHN C is defined as VHN C = { q ∈ VH,off :

[[q]]s = 0, ∀s ∈ M H (E), ∀E ∈ E H }. E

The discrete problem of nonconforming GMsFEM is to find the solution u H ∈ VHN C such that (see [294]) a H (u H , q) = ( f, q), ∀q ∈ VHN C , where a H (u, q) =

 K ∈T

H

(6.7)

κ∇u · ∇q. K

Notice that, unlike DG, even though the approximation space is nonconforming, there is no penalty parameter in the bilinear form of nonconforming GMsFEM. In fact, the nonconforming GMsFEM has a simple global coupling just like the continuous

6.2 Global formulations

163

Galerkin formulation, while the construction of its approximation space is as local as the GMsDGM.

6.2.5 GMsHDG For the GMsHDG, the general methodology for the construction of multiscale basis functions combines the ideas from mixed GMsFEM and GMsDGM. We will consider the construction of multiscale basis functions of the pressure u on the coarse edges. Once the multiscale solution u H is obtained on the coarse edges, the solution in the interior of coarse elements K can then be found by solving κ−1 v + ∇u = 0 in K , (6.8) div(v) = f in K , u = uH

on ∂ K .

We remark that the problem (6.8) can be solved by any fine-scale solver or any multiscale solver. We also remark that our GMsDGM will give a multiscale approximation of the pressure variable u on the coarse edges. Using notations for standard HDG methods [336], we will now use the notation

u to denote quantities defined on the coarse-grid ωi

H,snap for the edge E i is edges. Consider a coarse edge E i ∈ E H . The snapshot space V + defined by solving a problem on the oversampled region ωi (see Figure 6.1). A local offline spectral problem defined on ωi will then be solved to obtain a lower-dimensional  ωi ωi

H,off . The global offline space is defined as VH,off = Ei ∈E H VH,off . To write space V

H,off , we can write down the for our GMsHDG, we see that for any

uH ∈ V  Nformulation ωi e

H,off

u ωHi where

u ωHi ∈ V . Moreover, using

u H , we can define (v K , u K ) by

u H = i=1 κ−1 v K + ∇u K = 0 div(v K ) = f uK =

uH

in K , in K ,

(6.9)

on ∂ K ,

which can be solved by any numerical method defined on the fine mesh, or any multi H,off is obtained by the following equations: scale method. Finally, the solution

uH ∈ V   ωi

H,off vK L · n K L + vK R · n K R

q = 0, ∀

q∈V , ∀E ∈ E H , (6.10) E

where K L and K R are the two coarse elements having the edge E. cWe remark that for a boundary edge, one of the two terms v K L · n K L and v K R · n K R is replaced by the given Neumann boundary data vΩ .

6.2.6 General concept of energy stable (minimizing) oversampling In this section, we discuss the motivations for using the energy stable (minimizing) oversampling technique for the proposed basis construction procedure [95]. Let ω

164

6 Selected global formulations for GMsFEM and energy stable oversampling

be a generic coarse-grid domain and ω + be an oversampling region (see Figure 6.3). Our objective is to use oversampling and minimum energy snapshot functions to guarantee the stability in the local decomposition and fast convergence. As we mentioned earlier, these properties are essential. To start, we discuss the main ideas in the error analysis. The first step in the analysis is to decompose the energy functional corresponding to the error in local energies on coarse subdomains. For simplicity, we denote the energy functional corresponding to the domain Ω by aΩ (u, u). For example, aΩ (u, u) = Ω κ−1 u · u for the mixed GMsFEM case and aΩ (u, u) = aDG (u, u) for the GMsDGM case. Then aΩ (u − u H , u − u H ) 



aω (u ω − u ωH , u ω − u ωH ),

ω

(6.11)

where ω are coarse regions (with ω = ωi in the mixed GMsFEM and ω = K in the GMsDGM), u ω is the localization of the solution on ω and u ωH is a representation of the solution in the local offline space for ω.

Fig. 6.3 A generic coarse region and its oversampled region.

Next, we define oversampling snapshots and corresponding snapshots in the target in the oversampled domain ω. As a first step, we consider all possible snapshots ψ +,ω j region ω + , such that + L(ψ +,ω j ) = 0, in ω , with boundary condition ψ +,ω = δ j , j = 1, ..., N , where δ j are discrete delta funcj tions and L is the underlying PDE. These snapshots are slightly different in a mixed formulation and use unit fluxes and constant source terms. These snapshot vectors span all harmonic functions and can represent the solution with an accuracy of coarse mesh size. on E (see Figure 6.3) for Next, we take the traces of these snapshot functions ψ +,ω j a mixed formulation and ∂ω for a discontinuous Galerkin formulation. For simplicity,

6.2 Global formulations

165

we denote them ψ +,ω j | E , j = 1, .., M, M ≤ N , where we perform a suitable Proper Orthogonal Decomposition and remove linearly dependent functions. Once these traces functions are defined, we identify harmonic extensions, with respect to the operator L, to ω (if needed) and denote them by ψ ωj , which are snapshot functions. In the next step, energy stable (minimizing) snapshots are calculated, which have +,ω the traces ψ +,ω j | E (or ψ j |∂ω in a discontinuous Galerkin framework). These functions have the smallest energy in the norm aω+ (·, ·) and are sought in the snapshot +,ω

+,ω space span{ψ +,ω j } with the trace to be ψ j | E . We denote these snapshots by ψ j , j = 1, .., M. As a final step, we identify the offline multiscale space by taking the dominant eigenmodes (corresponding to the smallest eigenvalues) of the following Rayleigh Quotient:

+,ω , ψ

+,ω ) a ω + (ψ . ω aω (ψ , ψ ω ) This local spectral problem allows obtaining an estimate aω (u ω − u ωH , u ω − u ωH ) 

1 + + + + aω+ (u ω − u ωH , u ω − u ωH ). Λ

(6.12)

Further estimates use energy minimizing property of snapshot vectors and allows estimating the local error contributions by the global error as  + + + + aω+ (u ω − u ωH , u ω − u ωH )  aΩ (u, u). (6.13) ω

This is an important step that allows combining the local error contributions. This construction of the offline space is important to guarantee that if we additionally compute multiscale basis functions in the online step using the residuals (cf. [111, 124])), then the resulting convergence is independent of contrast and small scales. As we discussed earlier, online basis functions can improve the convergence significantly as they bring distant effects. Online basis functions are constructed by solving local problems with the local residual as a right-hand side. This property guarantees obtaining an estimate  on aΩ (u − u on Rω 2V ∗ , H , u − u H )  aΩ (u − u H , u − u H ) − (6.14) ω

 where Rω is defined by f − u H . One of the main tasks is to bound below ω Rω V ∗ by aΩ (·, ·). This bound follows similar steps as above, where we divide aΩ (·, ·) into subregions and use local spectral problems and obtain    −1 2 Rω V ∗ . aΩ (u − u H , u − u H )  Λ (6.15) ω

Here, Λ is the smallest eigenvalue that the corresponding eigenvector is not included in the coarse space. This error allows us to write

166

6 Selected global formulations for GMsFEM and energy stable oversampling on aΩ (u − u on H , u − u H )  (1 − CΛ)aΩ (u − u H , u − u H ),

(6.16)

where C depends on the number of subdomains, where online basis functions are added and the maximum of Λ such that 1 − CΛ > 0. From here, we can see that one needs to include a sufficient number of offline basis functions (such that Λ is not close to 0) constructed in an appropriate way.

6.3 Basis construction 6.3.1 Multiscale basis functions in mixed GMsFEM We will consider the system (6.1). Our focus is the construction of basis functions. We will consider various strategies including oversampling and energy minimizing oversampling. Construction of basis functions. Non-oversampling We start with the construction of the snapshot space for the approximation of the velocity v. The space is a large function space containing basis functions whose normal traces on coarse-grid edges are resolved up to the fine-grid level. Let  E i ∈ E H be a coarse edge. We can write the boundary of the coarse region i e j , where the e j ’s are the fine-grid faces contained in ∂ωi and Ji is the ∂ωi = Jj=1 ωi for the local total number of those fine-grid faces. To construct the basis set βsnap snapshot space, for each j = 1, 2, · · · , Ji , we will solve the following local problem: κ−1 ψ ωj i + ∇η ωj i = 0 div(ψ ωj i )

= αi, j

in ωi , in ωi ,

(6.17)

subject to the Neumann boundary condition ψ ωj i · n i = δ ωj i on ∂ωi , where δ ωj i is a fine-scale discrete delta function defined by  1 on e j , ωi δj = (6.18) 0 on ∂ωi \e j , αi, j is constant on and n i is the outward unit-normal vector on ∂ωi . The function each coarse-grid block and it should satisfy the condition ωi αi, j = ∂ωi ψ ωj i · n i . We denote the number of snapshots from (6.17) by Ni and the space spanned by these ωi . functions by VH,snap ωi Now, we give the local spectral problem for VH,snap for the construction of the ωi offline space VH,off . The local spectral problem is to find real number λ and v ∈ ωi such that VH,snap ωi ai (v, w) = λsi (v, w), ∀w ∈ VH,snap , (6.19)

6.3 Basis construction

167

ωi where ai and si are symmetric positive definite bilinear operators defined on VH,snap × ωi VH,snap . Motivated by the analysis, we choose



κ−1 (v · n i )(w · n i ),

ai (v, w) = si (v, w) =

Ei

ωi

κ−1 v · w + div(v)div(w).

We let λωj i be the eigenvalues of (6.19) arranged in ascending order, and ωj i be the corresponding eigenfunctions. We will use the first li eigenfunctions to construct our ωi . We note that it is important to keep the eigenfunctions with small offline space VH,off ωi . eigenvalues in the offline space. The global offline space VH,off is the union of all VH,off Construction of basis functions. Oversampling Let E i ∈ E H be a coarse edge. We can write the boundary of the oversampled region  Ji+ e j , where the e j ’s are the fine-grid faces contained in ∂ωi+ and Ji+ ∂ωi+ = j=1 ωi is the total number of those fine grid faces. To construct the basis set βsnap for the + local snapshot space, for each j = 1, 2, · · · , Ji , we will solve the following local problem: i i + ∇η +,ω = 0, in ωi+ , κ−1 ψ +,ω j j (6.20) i div(ψ +,ω ) = αi,+j , in ωi+ , j ω+

ω+

i · n i = δ j i on ∂ωi+ , where δ j i is subject to the Neumann boundary condition ψ +,ω j a fine-scale discrete delta function defined by

ω+ δj i

 =

1 on e j , 0 on ∂ωi+ \e j ,

(6.21)

and n i is the outward unit-normal vector on ∂ωi+ . The function αi,+j is constant on i · ni . each coarse-grid block and it should satisfy the condition ω+ αi,+j = ∂ω+ ψ +,ω j i i + We denote the number of snapshots from (6.20) by Ni and the space spanned by +,ωi . these functions by VH,snap +,ωi The functions ψ j , j = 1, 2, · · · , Ji+ , are supported in ωi+ . We will use the i · m i on the edge E i (see Figure 6.1) and perform POD to normal traces of ψ +,ω j remove linearly dependent vectors, where m i is a unit-normal vector on E i . We denote these traces by ψ ωj i · m i | Ei , and define their “harmonic extension” H(ψ ωj i ) in ωi by in ωi , κ−1 H(ψ ωj i ) + ∇η ωj i = 0, (6.22) ωi div(H(ψ j )) = αi, j , in ωi ,

168

6 Selected global formulations for GMsFEM and energy stable oversampling

with appropriately chosen piecewise constant αi, j and H(ψ ωj i ) · m i = ψ ωj i · m i | Ei and H(ψ ωj i ) · n = 0 on the other coarse edges lying on the boundary of ∂ωi . Using the i , and the notations defined in the previous section, we use all H(ψ ωj i ) to form β ωH,snap ωi span of all these basis functions VH,snap is the local snapshot space for ωi . ωi Now, we give the local spectral problem for VH,snap for the construction of the ωi offline space VH,off . The local spectral problem is to find real number λ and v ∈ ωi , such that VH,snap ωi ai (v, w) = λsi (v, w), ∀w ∈ VH,snap , (6.23) ωi where ai and si are symmetric positive definite bilinear operators defined on VH,snap × ωi VH,snap . Motivated by the analysis, we choose

ai (v, w) =

ωi

si (v, w) =

ωi

κ−1 v · w, κ−1 v · w + div(v)div(w).

Construction of basis functions. Energy stable (minimizing) oversampling ωi Let E i ∈ E H be a coarse edge. We consider the same snapshot space VH,snap defined in the previous section by the oversampling idea. Next, we will construct a local spectral problem and use it to define an offline ωi . For this, we will use the traces ψ ωj i | Ei · m i on E i defined above and space VH,off construct a minimum energy extension of it in the union of snapshot spaces Ui :=  +,ωk V E k ⊂ωi H,snap , where the union is taken for all coarse edges E k that are in ωi .

ωi ∈ Ui such that the following hold Specifically, we will find ψ j

ωi · m i = ψ ωi · m i , on E i , ψ j j ωi

κ−1 |ψ|2 . ψ j = argminψ∈Ui ωi+

(6.24) (6.25)

We notice that this construction is well-defined. We also notice that the above process ωi to Ui , and we use the notation v → Ai (v) to denote defines a mapping from VH,snap this mapping. ωi for the construction of the Now, we give the local spectral problem for VH,snap ωi offline space VH,off . The local spectral problem is to find real number λ and v ∈ ωi , such thatc VH,snap ai (v, w) = λsi (v, w),

ωi ∀w ∈ VH,snap ,

(6.26)

ωi where ai and si are symmetric positive definite bilinear operators defined on VH,snap × ωi VH,snap . Motivated by the analysis [95], we choose

6.3 Basis construction

169

ai (v, w) =

ωi+

si (v, w) =

ωi

κ−1 Ai (v) · Ai (w), κ−1 v · w + div(v)div(w).

Remark that the eigenvalues of this spectral problem is always bounded above by 1. We let λωj i be the eigenvalues of (6.26) arranged in ascending order, and ωj i be the corresponding eigenfunctions. We will use the first li eigenfunctions to construct our ωi . We note that it is important to keep the eigenfunctions with small offline space VH,off ωi . eigenvalues in the offline space. The global offline space VH,off is the union of all VH,off

6.3.2 Multiscale basis functions in GMsDGM We will consider the system (6.2). Our focus is the construction of basis functions. We will again consider various strategies including oversampling and energy minimizing oversampling. Construction of basis functions. Non-oversampling In this section, we discuss the construction of the snapshot space for GMsDGM in a non-oversampling case. The global snapshot space is a sum of local snapshot spaces. Each local snapshot space is constructed on a coarse element. For each coarse element K i ∈ T H , we define snapshot functions ψ ωj i , such that −∇ · (κ∇ψ ωj i ) = 0, ψ ωj i

=

δ ωj i ,

in

Ki ,

on

∂ Ki ,

(6.27)

where δ ωj i is piecewise linear on ∂ K i with respect to the fine grid such that δ ωj i has the value one at the j-th fine-grid node and value zero at all the remaining fine-grid nodes. We denote the number of local snapshots by Ni and the space spanned by these ωi ωi . Next, we will define the local spectral problem for VH,snap by functions by VH,snap ωi finding real number λ and v ∈ VH,snap , such that ai (v, w) = λsi (v, w)

ωi ∀w ∈ VH,snap ,

(6.28)

ωi ωi × VH,snap . where ai and si are symmetric positive bilinear operators defined on VH,snap Motivated by our analysis, we choose κ∇v · ∇w, ai (v, w) = K i si (v, w) = κvw. ∂ Ki

170

6 Selected global formulations for GMsFEM and energy stable oversampling

We let λωj i be the eigenvalues of (6.28) arranged in ascending order, and ωj i be the corresponding eigenfunctions. We will use the first li eigenfunctions to construct ωi . We note that it is important to keep the eigenfunctions with our offline space VH,off small eigenvalues in the offline space. The global offline space VH,off is the union of ωi . all VH,off Construction of basis functions. Oversampling i , For each K i ∈ T H , we consider an oversampled region K i+ ⊃ K i . We define ψ +,K j such that i −∇ · (κ∇ψ +,K )=0 in K i+ j (6.29) + K i ψ +,K = δj i on ∂ K i+ , j

K+

K+

where δ j i is piecewise linear on ∂ K i+ with respect to the fine grid such that δ j i has the value one at the j-th fine-grid node and value zero at all the remaining fine-grid nodes. We denote the number of local snapshots by Ni+ and the space spanned by +,K i these functions by VH,snap . i are supported in K i+ . We denote the restriction of These basis functions ψ +,K j +,K i Ki ψ j on K i by ψ j . Then we remove the linear dependent component of the functions Ki ψ Kj i by performing POD, and we still call the resulting space as VH,snap . Next, we +,K i will define the local spectral problem by: find real number λ and v ∈ VH,snap , such that +,K i (6.30) ai (v, w) = λsi (v, w) ∀w ∈ VH,snap , +,K i +,K i × VH,snap . where ai and si are symmetric positive bilinear operators defined on VH,snap Motivated by the analysis, we choose

ai (v, w) = si (v, w) =

K i+

∂ Ki

κ∇v · ∇w, κvw.

i i be the eigenvalues of (6.30) arranged in ascending order, and +,K We let λ+,K j j i be the corresponding eigenfunctions. We denote  Kj i as the restriction of +,K on j K i We will use the first li restricted eigenfunctions to construct the offline space Ki VH,off . We note that it is important to keep the eigenfunctions with small eigenvalues Ki . in the offline space. The global offline space VH,off is the union of all VH,off

6.3 Basis construction

171

Construction of basis functions. Energy stable (minimizing) oversampling +,K i constructed in the previous part. Next, we We consider the snapshot space VH,snap Ki . will construct a local spectral problem and use it to define an offline space VH,off Ki Ki For this, we need an energy minimizing extension. For each ψ j ∈ VH,snap , we will

K i ∈ V +,K i , such that the following hold: find ψ j H,snap

K i = ψ K i , on ∂ K i , (6.31) ψ j j

+,K i |2 ,

K i = argmin κ|ψ (6.32) ψ j j K i+

+,K i where the minimum is taken over in the space VH,snap . We notice that this construction is K i

well-defined. Wewill alsousethenotation ψ j = Ai (ψ Kj i ) todenotethesamefunction. Ki Now, we give the local spectral problem for VH,snap for the construction of the Ki offline space VH,off . The local spectral problem is to find real number λ and u ∈ Ki , such that VH,snap Ki ∀q ∈ VH,snap , (6.33) ai (u, q) = λsi (u, q), Ki where ai and si are symmetric positive definite bilinear operators defined on VH,snap × Ki VH,snap . Motivated by our analysis [95], we choose γ κ∇ u · ∇ q+ κ u q, ai (u, q) = h ∂ Ki K+ i γ si (u, q) = κ∇u · ∇q + κ u q. + h ∂ Ki Ki

Remark that the eigenvalues of this spectral problem are always bounded above by 1. We let λ Kj i be the eigenvalues of (6.33) arranged in ascending order, and  Kj i be the corresponding eigenfunctions. We will use the first li eigenfunctions to construct Ki . We note that it is important to keep the eigenfunctions with our offline space VH,off small eigenvalues in the offline space. The global offline space VH,off is the union of Ki . We will analyze the convergence in the next section. all VH,off

6.3.3 Multiscale basis functions in nonconforming GMsFEM Recall that the definition of the nonconforming GMsFEM space is VHN C = { q ∈ VH,off :

[[q]]s = 0, ∀ s ∈ M H (E), ∀ E ∈ E H }, E

where VH,off can be generated using the methods described in Section 6.3.2, or the methods described in [294]. Next, we will discuss the construction of M H .

172

6 Selected global formulations for GMsFEM and energy stable oversampling

Construction of moment space M H We first construct a local space M H (E i ) on each coarse neighborhood ωi or ωi+ associated with a coarse edge E i (see Figure 6.1). In the following, only the case for ωi will be discussed, but the case for ωi+ can be carried out similarly. For each coarse neighborhood ωi , we define ζ ωj i as the solution of −∇ · (κ∇ζ ωj i ) = 0, ζ ωj i

=

δ ωj i ,

in

ωi ,

on

∂ωi .

(6.34)

Randomized boundary condition technique can be used to generate ζ ωj i more efficiently, see, for example, Section 4.8 or [294]. Then, we collect the traces of the functions ζ ωj i on E i , i.e., ζ ωj i | Ei , and perform a POD to them. This will result in a set of linearly independent eigenvectors {skEi }, such that skEi 2Ei = μkEi ,

μ1Ei ≥ μ2Ei ≥ · · · ,

(6.35)

where μkEi are corresponding eigenvalues. The space of spectral moments on E i , M H (E i ), is then defined as the span of the first m Ei dominant modes. That is,   M H (E i ) = span skEi : 1 ≤ k ≤ m Ei . Lastly, the global moment space is given by  MH = M H (E).

(6.36)

E∈E H

Local construction of VHN C While the definition in Section 6.3.3 gives a global characterization for the spaces VHN C , it is more practical to construct VHN C locally. To this end, we focus on one coarse edge E i ∈ E H . Let K j and K  be the coarse elements sharing E i , we will construct a local space that consists of functions with supports in ωi . Recall that l j is Kj , and m E is the dimension of the local the dimension of the local offline space VH,off moment space M H (E). We choose the dimensions of all these local spaces, such that  m E . (6.37) l j + l > m Ei + E  ⊂∂ωi ωi K j Then, define VH,off as the space of all functions ψ in VH,off ∪ VH,off such that K

[ψ] E , s H = 0 ψ, s H = 0

∀ s H ∈ M H (E),

E

E

(6.38) ∀ s H ∈ M H (E  ), ∀ E  ⊂ ∂ωi .

6.3 Basis construction

That is,

173

   ωi K VH,off = ψ∈ VH,off : ψ fulfills (7.38) . K ⊂ωi

ωi is non-empty and Notice that by (6.37) and the Rank-Nullity Theorem, VH,off

  ωi  dim VH,off m E  > 0. ≥ l j + l − m Ei − E  ⊂∂ωi

Then the global space is the union of the local spaces VHN C =



ωi VH,off .

E i ∈E H

6.3.4 Multiscale basis functions in GMsHDG We will consider the system (6.1). We will only focus on basis construction using energy minimizing oversampling approach [95]. Construction of basis functions. Energy stable (minimizing) oversampling Let E i ∈ E H be a coarse edge, and consider the oversampled region ωi+ (see  Ji+ ej, Figure 6.1). We can write the boundary of the oversampled region ∂ωi+ = j=1 where the e j ’s are the fine-grid faces contained in ∂ωi+ and Ji+ is the total number

ωi for the local snapshot space of those fine-grid faces. To construct the basis set β snap ω + i

H,snap , for each j = 1, 2, · · · , Ji , we will solve the following local problem: V i i κ−1 v +,ω + ∇u +,ω = 0, j j

in ωi+ ,

i div(v +,ω ) = f, j

in ωi+ ,

i u +,ω j

=

ω+ δj i ,

on

(6.39)

∂ωi+ .

Notice that the problem (6.39) is solved in the fine mesh defined in ωi+ by the HDG i i and u +,ω , we will also method [336]. Therefore, in addition to the solutions v +,ω j j +,ωi defined on the skeleton of the fine mesh, that is, the fine obtain a solution

uj i i i , u +,ω ,

u +,ω ) as the solution of (6.39). We denote edges. Thus, we will write (v +,ω j j j +,ωi +,ωi +

H,snap . the number of these

u j by Ni and the space spanned by these functions by V +,ωi +,ωi +,ωi +,ω i

We also denote the set of all (v j , u j ,

u j ) by W H,snap Furthermore, the local ωi

H,snap snapshot space V for the edge E i is obtained by restricting the functions of +,ωi

H,snap V in E i and removing dependence. We remark that one can also use other discretizations for (6.39), which will modify some of the descriptions above.

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6 Selected global formulations for GMsFEM and energy stable oversampling

Next, we will construct a local spectral problem and use it to define an offline ωi . For this, we will define a minimum energy extension of functions in space VH,off ω ωi +,ωi i i i i

H,snap

H,snap VH,snap . Let

u ωj i ∈ V . Then we can find (v +,ω , u +,ω ,

u +,ω )∈W such that j j j ωi +,ωi +,ωi +,ωi +,ωi i



u +,ω =

u on E . Then we can find a triple (v , u ,

u ) ∈ W so that the j j j j H,snap j −1 i+,ωi 2 energy ω+ κ |v j | is the smallest among all possible choices with the restriction i

i i =

u ωj i on E i . We denote this v +,ω by Ai (

u i ).

u +,ω j j ωi ωi

H,snap On the other hand, for the given

uj ∈ V , we can define the harmonic extension in the coarse region ωi by solving

κ−1 v ωj i + ∇u ωj i = 0 div(v ωj i ) u ωj i u ωj i

= f =

u ωj i

=0

in K , in K , on E i ,

(6.40)

on ∂ K \E i ,

for each coarse element K ⊂ ωi , by the HDG method. We denote the corresponding u ωj i ). vi by Hi (

ωi

H,snap Finally, we state the spectral problem. We find a real number λ and

u∈V , such that ωi

H,snap u,

q ) = λsi (

u,

q ), ∀

q∈V , (6.41) ai (

ωi

H,snap where ai and si are symmetric positive definite bilinear operators defined on V × ωi

VH,snap , and are defined by u,

q) = κ−1 Ai (

u ) · Ai (

q ), ai (

si (

u,

q) =

ωi+

ωi

κ−1 Hi (

u ) · Hi (

q ).

Remark that the eigenvalues of this spectral problem are always bounded above by 1. We let λωj i be the eigenvalues of (6.41) arranged in ascending order, and ωj i be the corresponding eigenfunctions. We will use the first li eigenfunctions to construct our ωi

H,off . We note that it is important to keep the eigenfunctions with small offline space V ωi

H,off

H,off is the union of all V . eigenvalues in the offline space. The global offline space V

6.4 Numerical results In this section, we will present numerical results for the methods derived in the previous sections. More detailed numerical studies can be found in [95]. In Section 6.4.1, we will show the performance of the mixed GMsFEM for the use of both offline and online basis functions. In particular, we see that, using a few offline basis

6.4 Numerical results

175

functions, we are able to produce numerical solutions with good accuracy. By using a couple of online basis, one can also further improve the accuracy of the solutions. In Section 6.4.2, we present numerical results for GMsDGM. We will also compare the performance of using oversampling and without using oversampling. We observe that using oversampling can improve the results significantly.

6.4.1 Mixed GMsFEM In this section, we will present some numerical results for the mixed GMsFEM. We will consider the coarse-grid size, H = 1/10, and the fine-grid size, h = 1/100. The medium parameters κ used in our simulations are shown in Figure 6.4. We take the source function f = 1 in Ω. All the errors are measured in the norm v2V =

Ω

κ−1 |v|2 . x 10 4 2

10

1.8

20

1.6

30

1.4

40

1.2

50

1

60

0.8

70

0.6

80

0.4

90

0.2

100

20

40

60

80

100

Fig. 6.4 Medium parameter. κ. DOF

DOF = 220 (1 basis)

DOF = 440 (2 basis)

DOF = 660 (3 basis)

DOF = 880 (4 basis)

error

44.96%

37.67%

0.77%

0.24%

Table 6.1 Mixed GMsFEM. Errors for κ with contrast (1e-4). DOF

DOF = 220 (1 basis)

DOF = 440 (2 basis)

DOF = 660 (3 basis)

DOF = 880 (4 basis)

error

16.26%

3.81%

1.38%

0.50%

Table 6.2 Mixed GMsFEM. Errors for κ with contrast (1e+4).

First, we present the numerical results for the offline simulations. In Tables 6.1 and 6.2, the velocity errors are presented. As we observe from these numerical results that

176

6 Selected global formulations for GMsFEM and energy stable oversampling

the solution that uses one basis function per edge does not give a good approximation (we also refer to Figure 6.5 for velocity profiles corresponding to 1, 2, and 3 basis function per edge). On the other hand, we observe that the error becomes small when the permeability in channels and inclusions is high, while the error is still large for the case with very low permeability inclusions. As we choose more basis functions, i.e., three basis functions per edge (which corresponds to 660 degrees of freedom), the error is very small. 300

300 10

200

20 30

100

40

0

50

−100

60 70

−200

80

−300

90 100

20

40

60

80

10

100

30 40

0

50 60

−100

70

−200

80

−300

90 100

100

200

20

20

40

60

80

100

300 10

200

20 30

100

40

0

50

300 10 30

60

−200

70

90 20

40

60

80

100

0

50

−100

100

100

40

70

−300

200

20

60 80

−400

−100 −200

80

−300

90 100

20

40

60

80

100

Fig. 6.5 x-component of velocity. Top-Left: Reference solution. Top-Right: Numerical solution with 1 basis per edge. Bottom-Left: Numerical solution with 2 basis per edge. Bottom-Right: Numerical solution with 3 basis per edge.

Next, we discuss the convergence for online construction. In Tables 6.3 and 6.4, we show the numerical results as multiscale basis functions are computed in the online stage. In general, our objective is to minimize the computational cost in the online stage, and thus we would like to achieve a small error in one online iteration. We observe that if we choose only 1 basis function per edge, the error decrease due to adding online basis functions is insignificant. This is particularly evident for the low permeability inclusion cases. For this permeability field, the error decreases from 46.86% to 17.73%. Even though the error falls below 5% for high permeability cases, we observe that the decay of the error is slow. This is consistent with our observation that one needs a sufficient number of initial offline basis functions in order to achieve fast convergence. We observe that the convergence is fast as we have a 3 or 4 basis

6.4 Numerical results

177

function per edge. We would like to note that the online procedures can easily be implemented in an adaptive way by straightforwardly following [79, 111, 124]. We do not present these results here. DOF

e (1 basis) (%)

e (2 basis)

e (3 basis)

e (4 basis)

1(220) 2(440) 3(660) 4(880) 5(1100) 6(1320)

44.96 16.89 16.06 15.83 12.25 6.34

– 37.67% 0.56% 4.19e-2% 5.35e-4% 4.15e-6%

– – 0.77% 7.48e-2% 1.78e-3% 1.54e-5%

– – – 0.24% 6.98e-3% 7.28e-5%

Table 6.3 Online mixed GMsFEM. Errors for κ with contrast (1e-4). DOF

e (1 basis) (%)

e (2 basis)

e (3 basis)

e (4 basis)

1(220) 2(440) 3(660) 4(880) 5(1100) 6(1320)

16.26 2.58 0.46 0.052 9.03e-4 4.34e-6

– 3.81% 0.41% 2.90e-2% 1.31e-3% 1.50e-5%

– – 1.38% 0.13% 4.24e-3% 4.70e-5%

– – – 0.50% 2.57e-2% 3.72e-4%

Table 6.4 Online mixed GMsFEM. Errors for κ with contrast (1e+4).

6.4.2 GMsDGM In this section, we present some numerical results for our GMsDGM. We take the coarse-grid size as H = 1/10 and the fine-grid size as h = 1/100. The medium parameter is taken to be κ which is shown in Figure 6.4. We take the source function f = 1 in Ω. The performance of the scheme is measured by L 2 norm u L 2 (Ω) and the DG norm uDG . Instead of considering the accuracy of the scheme, which has a similar behavior as that of mixed GMsFEM, we will consider the performance on the use of oversampling. First, we will compare the eigenvalue decay for the energy stable (minimizing) oversampling EMO approach and a traditional non-oversampling approach. For the traditional non-oversampling approach, the spectral problem is defined as κ∇ p · ∇q = λ κ pq (6.42) K

∂K

for each coarse element K , and the spectral problem is solved in the fine-scale space defined on K . One can show that, using ideas in [112], the error for the traditional non-oversampling approach is proportional to (min K λlKi +1 )−1 , where λ Kj is the j-th

178

6 Selected global formulations for GMsFEM and energy stable oversampling

eigenvalue for the spectral problem (6.42). In Figure 6.6, we present the eigenvalues decay for these two methods on a chosen coarse element. For comparison purpose, we rescale the eigenvalues so that the first non-zero eigenvalues for the two approaches are the same. From the figure, we see clearly that the EMO technique provides a method with a much rapid decay of eigenvalues. Furthermore, in Table 6.5, we present the convergence history of the two approaches for using different numbers of basis functions and using the coefficient κ1 . In both the L 2 and the DG norms, we see clearly that the EMO method produces solutions with much smaller errors compared with that of the non-oversampling approach. DOF(# basis)

L 2 error (EMO) (%)

DG error (EMO) (%)

L 2 error (nonoversampling) (%)

DG error (nonoversampling) (%)

400 (4) 600 (6) 800 (8) 1000 (10) 1200 (12)

60.26 44.50 6.03 2.63 1.41

75.70 63.72 23.28 15.48 12.06

51.97 25.63 15.92 12.52 10.48

63.79 46.39 36.48 32.21 29.25

Table 6.5 Error comparison for oversampling and non-oversampling GMsDGM for κ with contrast (1e+4).

5

10

1/(λ−1) EMO 1/λ non−oversample 0

10

−5

10

−10

10

−15

10

0

10

1

10

2

10

3

10

Fig. 6.6 Log-log plot for the eigenvalues for the EMO method and the non-oversampling method.

Chapter 7

Constraint energy minimizing concepts

7.1 Introduction In previous chapters, we see that the GMsFEM’s convergence depends on the eigenvalue decay [101]. However, it is difficult to show a coarse mesh-dependent convergence without using oversampling and many basis functions. In this chapter, we present a basis construction strategy so that the resulting method has a meshdependent convergence with a minimal number of basis functions. The convergence analysis of the GMsFEM suggests that one needs to include eigenvectors corresponding to small eigenvalues in the local spectral decomposition. We note that these small eigenvalues represent the channelized features, as we discussed above. To obtain a mesh-dependent convergence, we borrow some ideas from [256, 318, 345, 347], which consists of using oversampling domains and obtaining decaying local solutions. Some of these approaches [318] use variational multiscale method framework [48, 65, 265, 266, 277, 340] to construct subgrid information (see also [73] for the use of multiscale spaces in variational multiscale methods). For highcontrast problems, the local solutions do not decay in channels and we, therefore, need approaches that can take into account the information in the channels when constructing the decaying local solutions. In this chapter, we introduce an energy minimization concept for basis construction [113]. This idea starts with some auxiliary multiscale basis functions constructed using a spectral problem in each coarse block. This auxiliary space contains the information related to channels and the number of these basis functions is the same as the number of the channels, which is a minimal number of basis functions required to represent high-contrast features. This auxiliary space is used to take care of the non-decaying component of the oversampled local solutions, which occurs in the channels. The construction of multiscale basis functions is done by seeking minimization of a functional subject to a constraint, such that the minimizer is orthogonal (in a certain sense) to the auxiliary space. This allows handling the non-decaying components of the oversampled local solutions. The resulting approach, called CEM-

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_7

179

180

7 Constraint energy minimizing concepts

GMsFEM, contains several basis functions per element and one can also consider adaptivity ([101, 111, 115]) when defining the basis functions. This construction allows obtaining the convergence rate H/Λ in the energy norm, where Λ is the minimal eigenvalue that the corresponding eigenvector is not included in the space. The analysis also shows that the size of the oversampling domain depends on the contrast weakly (logarithmically). To relax the contrast dependence of the oversampling domain size, we propose a modified algorithm. In this algorithm, we use the same auxiliary space; however, the minimization is done by relaxing the constraint. We will present numerical results for two heterogeneous permeability fields. In both cases, the permeability fields contain channels and inclusions with high conductivity values. We select auxiliary basis functions that include all channelized features (i.e., the eigenvectors corresponding to very small and contrast-dependent eigenvalues). Our numerical results show that the error decays as we decrease the coarse mesh size; however, this is sensitive to the oversampling domain size. We present numerical results which show that if there is not sufficient oversampling, the errors are large and contrast dependent. Furthermore, we also present the numerical results for our modified algorithm and show that this contrast dependence is relaxed and the oversampling domain sizes are less sensitive to the contrast. Finally, we will present an online approach for CEM-GMsFEM [115] and show that one can significantly improve the existing online approaches in some cases. We note that this new online approach is different from our previous approach discussed in Chapter 5 since CEM-GMsFEM uses oversampling. In particular, the online basis functions are formulated in the oversampled regions. The error decay can be made very fast (i.e., we obtain a very accurate approximation in one iteration) by choosing larger oversampling regions provided we have a sufficient number of offline basis functions. To our best knowledge, this is the first result of this kind. Moreover, the online approaches can be made adaptive, and adaptive error indicators can be derived.

7.2 Preliminaries First, we fix and remind the notations. We consider the following model equation:   − div κ(x) ∇u = f in Ω ⊂ Rd ,

(7.1)

where κ is a high-contrast multiscale field with κ0 ≤ κ(x) ≤ κ1 . The above equation is subjected to the boundary conditions u = 0 on ∂Ω. Next, the notions of fine and coarse grids are introduced. Let T H be a conforming partition of Ω into finite elements. Here, H is the coarse mesh size and this partition is called the coarse grid. We let Nc be the number of vertices and N be the number of elements in the coarse mesh. We assume that each coarse element is partitioned into a connected union

7.3 Construction of multiscale basis functions

181

of fine-grid blocks and this partition is called T h . Note that T h is a conforming refinement of the coarse grid T H with the mesh size h. It is assumed that the fine grid is sufficiently fine to resolve the medium κ and the solution. An illustration of the fine grid, coarse grid, and oversampling domain is shown in Figure 7.1.

Fig. 7.1 Illustration of the coarse grid, fine grid, and oversampling domain.

We let V = H01 (Ω). Then the solution u of (7.1) satisfies  a(u, v) =

Ω

f v, for all v ∈ V,

(7.2)

 where a(u, v) = Ω κ∇u · ∇v. We will discuss the construction of multiscale basis functions in the next section. We consider Vms to be the space spanned by all multiscale basis functions. Then the multiscale solution u ms is defined as the solution of the following problem, find u ms ∈ Vms , such that  a(u ms , v) =

Ω

f v, for all v ∈ Vms .

(7.3)

The construction of the multiscale basis function will be presented in the next section.

7.3 Construction of multiscale basis functions The computation of the multiscale basis functions is divided into two stages. The first stage consists of constructing the auxiliary multiscale basis functions by using the

182

7 Constraint energy minimizing concepts

concept of the generalized multiscale finite element method (GMsFEM). The next step is the construction of the multiscale basis functions. In this step, constrained energy minimization is performed in the oversampling domain. The resulting method is called constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM).

Auxiliary space Let K i ∈ T H be the i-th coarse block and let V (K i ) = H 1 (K i ). We consider the (i) following local spectral problem: find a real number λ(i) j and a function φ j ∈ V (K i ) such that (i) (i) ∀w ∈ V (K i ), (7.4) ai (φ (i) j , w) = λ j si (φ j , w), where





ai (v, w) =

κ∇v · ∇w, si (v, w) = Ki

 κ vw,

(7.5)

Ki

 c ms N c 2 κ|∇χ ms where  κ = Nj=1 j | and {χ j } j=1 are the standard multiscale finite element (MsFEM) basis functions (see [255]), which satisfies the partition of unity property. We let λ(i) j be the eigenvalues of (7.4) arranged in ascending order. We will use the (i) first li eigenfunctions to construct our local auxiliary multiscale space Vaux , where (i) (i) Vaux = span{φ j | j ≤ li }. The global auxiliary multiscale space Vaux is the sum of N (i) these local auxiliary multiscale spaces, namely, Vaux = ⊕i=1 Vaux . This space is used to construct multiscale basis functions that are φ-orthogonal to the auxiliary space. The notion of φ-orthogonality will be defined next. (i) , the bilinear form si in (7.5) defines For the local auxiliary multiscale space Vaux 1 an inner product with norm vs(K i ) = si (v, v) 2 . These local inner products and norms provide natural definitions of inner product and norm for the global auxiliary multiscale space Vaux , which are defined by s(v, w) =

N 

1

si (v, w), vs = s(v, v) 2 , ∀v ∈ Vaux .

i=1

We note that s(v, w) and vs are also an inner product and norm for the space V . Using the above inner product, we can define the notion of φ-orthogonality in (i) the space V . Given a function φ (i) j ∈ Vaux , we say that a function ψ ∈ V is φ j orthogonal if

(i ) s(ψ, φ (i) j ) = 1, s(ψ, φ j ) = 0, if j = j or i = i. (i) Now, we let πi : L 2 (K i ) → Vaux be the projection with respect to the inner product si (v, w). So, the operator πi is given by

7.3 Construction of multiscale basis functions

πi (u) =

li  si (u, φ (i) j )

φ (i) j , (i) (i) j=1 si (φ j , φ j )

183

∀u ∈ V.

In addition, we let π : L 2 (Ω) → Vaux be the projection with respect to the inner product s(v, w). So, the operator π is given by π(u) = Note that π =

N i=1

li N   si (u, φ (i) j )

φ (i) j , (i) (i) i=1 j=1 si (φ j , φ j )

∀u ∈ V.

πi .

Multiscale basis functions We next present the construction of the multiscale basis functions. For each coarse element K i , we define an oversampled domain K i,m ⊂ Ω by enlarging K i by m coarse grid layers, where m ≥ 1 is an integer. We then define the multiscale basis function ψ (i) j,ms ∈ V0 (K i,m ) by

(i) ψ (i) = argmin a(ψ, ψ) | ψ ∈ V (K ), ψ is φ -orthogonal , (7.6) 0 i,m j,ms j where V (K i,m ) = H 1 (K i,m ) and V0 (K i,m ) = H01 (K i,m ). Our multiscale finite element space Vms is defined by

Vms = span ψ (i) j,ms | 1 ≤ j ≤ li , 1 ≤ i ≤ N . The existence of the solution to the above minimization problem is proved in [113]. The following are the main ideas behind this multiscale basis function construction: • The φ-orthogonality of the multiscale basis functions allows a spatial decay, which is one of the contributing factors of a mesh size dependent convergence. • The multiscale basis functions minimize the energy, which is important, in particular, for the decay. • We note that, if we do not choose an appropriate auxiliary space, this will affect the convergence rate and the decay rate will depend on contrast. In Figure 7.2, we illustrate the importance of the auxiliary space on the decay of multiscale basis function. We consider a high-contrast channelized medium as shown in the left plot in Figure 7.2. In the middle plot of Figure 7.2, we show a multiscale basis with the use of only one eigenfunction in the auxiliary space. We see that the basis function has almost no decay. On the other hand, in the right plot of Figure 7.2, we show a multiscale basis with the use of 4 eigenfunctions, and we see clearly that the basis function has very fast decay outside the coarse block.

184

7 Constraint energy minimizing concepts 5

x 10 10

0

0

0 150

0.1

9

0.1

0.2

8

0.2

0.3

7

0.3

0.4

6

0.4

0.5

5

0.5

0.6

4

0.6

0.7

3

0.7

0.8

2

0.8

0.9

1

0.9

100

0.1

35

0.2

30

0.3 50

0.4

0

0.6

25 20

0.5 15 10 0.7 −50

5

0.8

0

0.9 −100

1

0

0.2

0.4

0.6

0.8

1

1

0

0.2

0.4

0.6

0.8

1

1

0

0.2

0.4

0.6

0.8

1

−5

Fig. 7.2 An illustration of the decay property of multiscale basis functions. Left: a high-contrast medium. Middle: a multiscale basis function using one eigenfunction in each local auxiliary space. Right: a multiscale basis function using 4 eigenfunctions in each local auxiliary space.

An orthogonal decomposition property The local multiscale basis construction is motivated by the global basis construction as defined below. The global multiscale basis function ψ (i) j ∈ V is defined by

(i) ψ (i) j = argmin a(ψ, ψ) | ψ ∈ V, ψ is φ j -orthogonal .

(7.7)

Our multiscale finite element space Vglo is defined by

Vglo = span ψ (i) j | 1 ≤ j ≤ li , 1 ≤ i ≤ N . This global multiscale finite element space Vglo satisfies a very important orthogonality property, which is used in the convergence analysis. We define V˜ as the null space of the operator π , namely, V˜ = {v ∈ V | π(v) = 0}. Then for any ψ (i) j ∈ Vglo , we have ˜ a(ψ (i) j , v) = 0, ∀v ∈ V . ⊥ Thus, V˜ ⊂ Vglo . Consequently Vglo ⊂ V˜ ⊥ . Notice that the restriction map π : V˜ ⊥ → Vaux is injective. Therefore, we have dim{V˜ ⊥ } ≤ dim{Vaux } < ∞. By the wellposedness of the construction of the global basis functions, we have dim{Vglo } = ⊥ dim{Vaux } ≥ dim{V˜ ⊥ } and thus V˜ ⊥ = Vglo . Hence, we have V˜ = Vglo . Thus, we have ˜ V = Vglo ⊕ V . In Figure 7.3, we illustrate the decay of the global basis function.

Some convergence results We state some main convergence results for the error between the solution  u and the multiscale solution u ms . First we define the energy norm  · a by ua2 = Ω κ|∇u|2 , and define

7.4 Numerical results

185 5

x 10 10

0

0

0

0.1

9

0.1

0.2

8

0.2

600

0.1

35

0.2

30

400 0.3

7

0.3

0.4

6

0.4

0.3 200

25

0.4 20

0.5

5

0.5

0.6

4

0.6

0.5

0

15 0.6 −200 0.7

3

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1

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−5

Fig. 7.3 An illustration of the decay property of global multiscale basis functions. Left: a highcontrast medium. Middle: a multiscale basis function using one eigenfunction in each local auxiliary space. Right: a multiscale basis function using 4 eigenfunctions in each local auxiliary space.

Λ = min λl(i) . i +1 1≤i≤N

Then we have the following error bound in the energy norm. Theorem 7.1. Let u be the solution of (7.2) and u ms be the multiscale solution of κ1 )), we have (7.3). Assume that the number of oversampling layer m = O(log( H κ0 u − u ms a ≤ C H Λ− 2 κ − 2 f  L 2 (Ω) . 1

1

We also have the following L 2 norm error estimate. Theorem 7.2. Let u be the solution of (7.2) and u ms be the multiscale solution of κ1 )), we have (7.3). Assume that the number of oversampling layer m = O(log( H κ0 κ − 2 (u − u ms ) L 2 (Ω) ≤ C H 2 Λ−1 κ − 2 f  L 2 (Ω) . 1

1

7.4 Numerical results In this section, we will present two numerical examples with two different highcontrast media to demonstrate the convergence of our proposed method. We take the domain Ω = (0, 1)2 . For the first numerical example, we consider the medium parameter κ as shown in Figure 7.4 and assume the fine mesh size h to be 1/400. That is, the medium κ has a 400 × 400 resolution. In this case, we consider the contrast of the medium to be 104 , where the value of κ is large in the red region. The convergence history with various coarse mesh sizes H are shown in Table 7.1. In all these simulations, we take the number of oversampling layers to be approximately 4 log(1/H )/ log(1/10). Form Table 7.1, we can see the energy norm error converges in first order with respect to H and the L 2 norm error converges in second order with respect to H . The first-order convergence in the energy norm matches our theoretical bound.

186

7 Constraint energy minimizing concepts 10000 20

9000

40

8000

60

7000

80

6000

100

5000

120

4000

140

3000

160

2000

180

1000

200

50

100

150

200

Fig. 7.4 The medium κ for the test case 1. # basis per element

H

# oversampling coarse layers

L 2 error

Energy error

3 3 3 3

1/10 1/20 1/40 1/80

4 6 7 8

2.62% 0.51% 0.11% 0.0015%

15.99% 7.04% 3.31% 0.17%

Table 7.1 Numerical results with varying coarse grid size H for the test case 1.

We emphasize that, in this example, we use 3 basis functions per coarse region. The reason of this is that the eigenvalue problem on each coarse region has 3 small eigenvalues, and according to our theory, we need to include the first three eigenfunctions in the auxiliary space. As our theory shows, for a certain contrast value, one needs to use a large enough oversampling size in order to obtain the desired convergence order. Moreover, for a fixed contrast value, the results can be improved as the oversampling size increases. On the other hand, for a fixed oversampling size, the performance of the scheme will deteriorate as the medium contrast increases. This is confirmed by our numerical evidence shown in Table 7.2. Layer \ Contrast

1e+3

1e+4

1e+5

1e+6

3 4 5

48.74% 19.12% 3.70%

73.22% 15.99% 4.19%

87.83% 26.68% 7.17%

91.14% 58.67% 19.24%

Table 7.2 Comparison of various number of oversampling layers and different contrast values for test case 1.

In our second test case, we consider the medium parameter κ defined in Figure 7.5. In this case, the medium has a contrast value of 106 , and the fine-grid size h is 1/200.

7.4 Numerical results

187 5

x 10 10

0 0.1

9

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1

Fig. 7.5 The medium κ for the test case 2. # basis per element

H

# oversampling coarse layers

L 2 error

Energy error

4 4 4

1/10 1/20 1/40

6 8 10

1.55% 0.02% 0.0042%

11.10% 0.59% 0.23%

Table 7.3 Numerical results with varying coarse grid size H for the test case 2.

We will show the convergence history in Table 7.3 using different choices of coarse mesh sizes. For all simulations, the number of oversampling layer is approximately 6 log(1/H )/ log(10), and we use 4 multiscale basis functions per coarse block since there are 4 small eigenvalues for some coarse blocks. From Table 7.3, we can see that the method achieves the theoretical convergence rate. In Table 7.4, we compare the performance of the method with different choices of oversampling layers and contrast values. We see that, for a fixed choice of oversampling layer, the error increases moderately with respect to the contrast value. On the other hand, for a fixed contrast value, the error will improve as the number of oversampling layers increases. We also see that the error will be small once enough oversampling layer is used. Layer \ Contrast

1e+3

1e+4

1e+5

1e+6

4 5 6

21.88% 6.96% 1.82%

47.58% 22.49% 4.33%

65.94% 33.09% 6.49%

81.47% 47.19% 11.09%

Table 7.4 Comparison of various number of oversampling layers and different contrast values for test case 2.

188

7 Constraint energy minimizing concepts

7.5 Relaxed CEM-GMsFEM In this section, we consider a relaxed version of our method. In particular, we relax the constraint in the minimization problem (7.6). Instead of (7.6), we solve the following un-constrained minimization problem: find ψ (i) j,ms ∈ V0 (K i,m ), such that

(i) (i) ψ (i) = argmin a(ψ, ψ) + s(π ψ − φ , π ψ − φ ) | ψ ∈ V (K ) . (7.8) 0 i,m j,ms j j This minimization problem is equivalent to the following variational formulation: (i) (i) a(ψ (i) j,ms , v) + s(π(ψ j,ms ), π(v)) = s(φ j , π(v)),

∀v ∈ V0 (K i,m ).

(7.9)

We remark that the spaces Vms and V˜ are defined as before. We also remark that the condition V = Vglo ⊕ V˜ holds. As predicted by the theory [113], the relaxed version of the method is more robust with respect to contrast. About error bounds, results in Theorem 7.1 and Theorem 7.2 are obtained by using the basis functions in (7.8).

Numerical Results In this section, we present numerical results to show the performance of the relaxed version of the method. We will consider two test cases, which are the same as those considered in Section 7.4. First, in Table 7.5, we show the errors for the first test case using different choices of coarse mesh sizes. We clearly see that the method gives the predicted convergence rate since we have included enough eigenfunctions in the auxiliary space. More importantly, by comparing to the similar test case in Table 7.1, we see that the relaxed version is able to improve the number of oversampling layers. In particular, we see that one needs fewer oversampling layers and obtains much better results. # basis per element

H

# oversampling coarse layers

L 2 error

Energy error

3 3 3 3

1/10 1/20 1/40 1/80

3 4 5 6

0.33% 0.047% 0.010% 0.0015%

3.73% 1.17% 0.47% 0.15%

Table 7.5 Numerical result for the test case 1 with the relaxed method.

In Table 7.6, we show the performance of the relaxed version with respect to the relation between contrast values and the number of oversampling layers. From the results, we see that the relaxed version needs a much smaller number of oversampling layers in order to achieve a good result. In particular, for a given oversampling layer, it can handle a much larger contrast value. We performed a similar computation for the test case 2 and obtain the same conclusion. For the numerical results, see Table 7.7 and Table 7.8.

7.6 Construction of online basis functions

189

Layer \ Contrast

1e+4

1e+6

1e+8

1e+10

3 4 5

3.73% 3.72% 3.72%

3.89% 3.72% 3.72%

11.99% 3.73% 3.73%

65.19% 5.14% 3.72%

Table 7.6 Comparison for the test case 1 with the relaxed method. # basis per element

H

# oversampling coarse layers

L 2 error

Energy error

4 4 4

1/10 1/20 1/40

4 6 7

0.11% 0.021% 0.0042%

1.50% 0.57% 0.23%

Table 7.7 Numerical result for the test case 2 with the relaxed method. Layer \ Contrast

1e+6

1e+8

1e+10

3 4 5

5.00% 1.50% 1.50%

41.11% 1.50% 1.50%

83.26% 11.88% 1.50%

Table 7.8 Comparison for the test case 2 with the relaxed method.

7.6 Construction of online basis functions In this section, we will introduce an online enrichment method for CEM-GMsFEM. We will present the construction of online basis functions. For error estimates, we refer the reader to [115]. To begin, we define a residual functional r : V → R. Let u ms ∈ Vms be a numerical solution computed by solving (7.3). The residual functional r is defined by  r (v) = a(u ms , v) − f v, ∀v ∈ V. Ω

We will also consider local residuals. For each coarse node xi , we define ωi be the set of coarse blocks having the vertex xi . For each coarse neighborhood ωi , we define the local residual functional ri : V → R by ri (v) = r (χi v), ∀v ∈ V. The local residual ri gives a measure of the error u − u ms in the coarse neighborhood ωi . The construction of the online basis function is related to the local residual ri . (i) whose Using the local residual ri , we will construct an online basis function φon + support is an oversampled region ωi , which is obtained by extending ωi by a few (i) ∈ V0 (ωi+ ) is obtained coarse blocks. More precisely, the online basis function βms by solving (i) (i) , v) + s(π(βms ), π(v)) = ri (v), ∀v ∈ V0 (ωi+ ), a(βms

(7.10)

190

7 Constraint energy minimizing concepts

where V0 (ωi+ ) = H01 (ωi+ ). We can perform the above construction for each ri , or for some selected ri (with i ∈ I for an index set I ) based on an adaptive criterion. We remark that the above online basis is obtained in the local region ωi+ . This is the result of a localization result for the corresponding global online basis function (i) ∈ V defined by βglo (i) (i) a(βglo , v) + s(π(βglo ), π(v)) = ri (v), ∀v ∈ V.

(7.11)

After constructing the online basis functions, we can enrich our multiscale space by adding these online basis to the multiscale space, namely, Vms = Vms + (i) }. Using this multiscale finite element space, we can compute a new spani∈I {βms numerical solution by solving the equation (7.3). We can repeat the process to enrich our multiscale space until the residual norm is smaller than a given tolerance. Next, we present the precise online adaptive enrichment algorithm. Online adaptive enrichment algorithm (1) . This is the space obtained by using the offline We first choose an initial space Vms basis functions constructed in Section 7.3. We also choose a real number θ such that 0 ≤ θ < 1. This number determines how many online basis functions are needed (m) and a in each online iteration. Then, we will generate a sequence of spaces Vms (m) sequence of multiscale solutions u ms obtained by solving (7.3). (m) is given. We will perform the folFor each m = 1, 2, . . . , we assume that Vms (m+1) : lowing procedures to obtain the new multiscale space Vms (m) (m) Step 1: Find the multiscale solution in the space Vms . That is, find u (m) ms ∈ Vms , such that (m) a(u (m) ms , v) = ( f, v), for all v ∈ Vms .

(7.12)

Step 2: Compute the local residuals z i (v), where z i (v) = a(u (m) ms , v) − ( f, v), ∀v ∈ V0 (ωi ). r (v) . We re-numerate the va indices of ωi , such that δ1 ≥ δ2 ≥ · · · . Choose the first k regions, so that

Define δi = z i a ∗ where z i a ∗ = supv∈V0 (ωi ) N 

δi2 < θ

i=k+1

N 

δi2 .

(7.13)

i=1

Step 3: Compute the local online basis functions. For each 1 ≤ i ≤ k and coarse (i) region ωi , find βms ∈ V0 (ωi+ ), such that (i) (i) , v) + s(π(βms ), π(v)) = ri(m) (v) ∀v ∈ V0 (ωi+ ), a(βms

where ri(m) (v) = a(u (m) ms , χi v) −

 Ω

f χi v.

7.7 Numerical results using online basis functions

191

Step 4: Enrich the multiscale space. Let (m+1) (m) (i) = Vms + span1≤i≤k {βms }. Vms

In the following, we present a convergence result for this online adaptive enrichment method. Theorem 7.3. Let u be the solution of (7.1) and let {u (m) ms } be the sequence of multiscale solutions obtained by our online adaptive enrichment algorithm. Then we have  2 u − u (m+1) a2 ≤ C(1 + Λ−1 ) E + θ u − u (m) ms ms a , where E decays to zero exponentially as the number of layers  used in ωi+ increases and C is a constant. Remark 7.4. We note that the convergence rate depends on two terms E and θ . By using the number of oversampling layers  = O(log(κ1 /(H κ0 )), the term E tends to zero exponentially. Thus, the factor θ dominates the convergence rate. One can choose θ to obtain a desired convergence rate. We will also confirm this with some numerical examples. This is an improvement over the online method in [111], where the convergence rate is (C1 + C2 θ ) with 0 < C1 < 1.

7.7 Numerical results using online basis functions In this section, we present some numerical results to demonstrate the convergence of the online CEM-GMsFEM. The medium parameter κ in the equation (7.1) is chosen to be the function shown in Figure 7.4. The fine mesh size h is taken to be 1/200, while the coarse mesh size H in this example is 1/10. In all our results, we take the number of oversampling layers  = 2. We will illustrate the performance of 1 our method by using two different source terms f 1 = ((x − 0.5)2 + (y − 0.5)2 )− 4 3 and f 2 = ((x − 0.5)2 + (y − 0.5)2 )− 4 . We will test the performance by considering uniform enrichment and by using the online adaptive enrichment algorithm presented in Section 7.6. In Table 7.9, we present the L 2 error and the energy error for the case f 1 with uniform enrichment, that is θ = 0. From the first two online iterations, we observe very fast convergence of the method. Next, we will consider some adaptive results for this case. In Table 7.10, we present the error decay by using our online adaptive enrichment algorithm with θ = 0.95. That is, we only add basis for regions that account for the largest 5% of the residual. We remark that DOF stands for the total number of degrees of freedom. From the table, we observe that the convergence rate in the energy norm is 0.9154, which is close to 0.95. These results confirm our assertion that the convergence rate can be controlled by the user-defined parameter θ . We remark that the convergence rate is computed by taking the maximum of 2 a2 /u − u (m) all u − u (m+1) ms ms a . In Table 7.11, we present the adaptive result with

192

7 Constraint energy minimizing concepts

θ = 0.1. That is, we add basis for regions that account for the largest 90% of the residual. From the table, we observe that the convergence rate in the energy norm is 0.0589, which is close to 0.1. This result also confirms our prediction. Moreover, we note that the adaptive approach allows adding a very few online basis functions to reduce the error to 1%. Number of offline Online iteration basis

Oversampling layers

L 2 error

Energy error

3 3 3

2 2 2

0.37% 6.75e-05% 1.57e-08%

4.71% 1.28e-03% 2.64e-08%

0 1 2

Table 7.9 Using source term f 1 and uniform enrichment. Number of offline DOF basis

Oversampling layers

L 2 error

Energy error

3 3 3 3

2 2 2 2

0.37% 0.14% 0.073% 0.033%

4.71% 2.21% 1.12% 0.57%

300 311 339 368

Table 7.10 Using source term f 1 and online adaptivity with θ = 0.95. Convergence rate is 0.9154. Number of offline DOF basis

Oversampling layers

L 2 error

Energy error

3 3 3 3

2 2 2 2

0.37% 0.073% 0.014% 2.93e-03%

4.71% 1.09% 0.21% 0.051%

300 341 407 470

Table 7.11 Using source term f 1 and online adaptivity with θ = 0.1. Convergence rate is 0.0589.

Now, we consider the second source term f 2 . In Table 7.12, we present the error decay using uniform enrichment. We observe very fast decay in error from this table. Next, we test the performance using adaptivity. In Table 7.13 and Table 7.14, we present the error decays with θ = 0.95 and θ = 0.1, respectively. We observe that the convergence rates in these two cases are 0.9338 and 0.09, respectively. This confirms that the user-defined parameter is useful in controlling the convergence rate of our adaptive method. Moreover, we note that the adaptive approach allows adding a very few online basis functions to reduce the error to 1%. Furthermore, in Figure 7.6, we show the number of online basis functions added in the computational

7.7 Numerical results using online basis functions

193

domain. For θ = 0.95, we will add a small number of basis functions in each iteration. We observe that the basis functions are added near the singularity of the source f 2 and along the high-contrast channel. For θ = 0.1, more basis functions are added throughout the domain with a faster convergence rate. We still observe that more basis are added near the singularity of f 2 and along the high-contrast channel in κ. Number of offline Online iteration basis

Oversampling layers

L 2 error

Energy error

3 3 3

2 2 2

1.06% 6.43e-05% 1.57e-08%

11.70% 1.51e-03% 4.25e-08%

0 1 2

Table 7.12 Using source term f 2 and uniform enrichment. Number of offline DOF basis

Oversampling layers

L 2 error

Energy error

3 3 3 3

2 2 2 2

1.06% 0.13% 0.062% 0.031%

11.70% 1.95% 0.99% 0.51%

300 309 324 347

Table 7.13 Using source term f 2 and online adaptivity with θ = 0.95. Convergence rate is 0.9338. Number of offline DOF basis

Oversampling layers

L 2 error

Energy error

3 3 3 3

2 2 2 2

1.06% 9.27e-03% 3.65e-04% 5.31e-05%

11.70% 0.13% 6.44e-03% 9.89e-04%

300 391 513 578

Table 7.14 Using source term f 2 and online adaptivity with θ = 0.1. Convergence rate is 0.09.

Finally, we present a test case with a more singular source term f 3 = −∇ · (κ∇(x y)), shown in Figure 7.7, where the reference solution is also presented. In Table 7.15, we present the error decay with uniform enrichment and observe the same type of exponential decay as the earlier examples. We also observe that the error is relatively large when no online basis function is used. In Table 7.16, we present the results with the online adaptive enrichment algorithm with θ = 0.1. We see that the numerically computed convergence rate is 0.0771, which is close to the parameter θ .

194

7 Constraint energy minimizing concepts 8

1

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Fig. 7.6 Number of online basis functions for the source f 2 : Left: θ = 0.95. Right: θ = 0.1.

6

x 10 0.2

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Fig. 7.7 Left: Reference solution for the source f 3 . Right: the source term f 3 .

Number of offline Online iteration basis

Oversampling layers

L 2 error

Energy error

3 3 3

2 2 2

30.01% 0.0066% 4.45e-07%

82.57% 0.0030% 1.22e-07%

0 1 2

Table 7.15 Using source term f 3 and uniform enrichment. Number of offline DOF basis

Oversampling layers

L 2 error

Energy error

3 3 3 3

2 2 2 2

30.01% 8.68% 4.87% 4.46%

82.57% 22.06% 5.41% 1.50%

300 356 378 392

Table 7.16 Using source term f 3 and online adaptivity with θ = 0.1. Convergence rate is 0.0771.

Chapter 8

Non-local multicontinua upscaling

8.1 Introduction In this chapter, we design new upscaled models based on CEM-GMsFEM and discuss the relation between multiscale and upscaled models. The degrees of freedom in CEM-GMsFEM or in multiscale methods do not have physical meanings since they are coordinates in the multiscale space. In order to obtain an upscaled model, one needs to design multiscale basis functions such that the degrees of freedom represent the average of the solutions. This allows relating multiscale methods to upscaled models and one does not need to use basis functions in further updated. In this chapter, we will discuss some of these methods, where the basis functions are constructed using some physics-based constraints. The resulting upscaled models share some similarities with traditional multicontinuum approaches. While choosing different constraints in multiscale methods, one can obtain various upscaled models. This is a key part, which allows connecting multiscale methods and upscaling approaches. To be more precise, we seek the solution U in the basis U=



j

( j)

Ui Φi ,

i, j ( j)

where j denotes the j-th coarse-grid block and i denotes the i-th continuum K i  ( j) within the j-th coarse-grid block. If K (l) Φm = δim δ jl , with δ being the Kronecker i  j symbol, then Ui = K ( j) U . As a result, the coarse-grid model is an equation for the i solutions averaged over each continuum. We present our framework for flows in fractured media. In order to modify the multiscale approach presented in [113], we first assume that one knows each separate fracture network within a coarse-grid block. This is one of the drawbacks of our method; however, such cases occur in many applications. To begin, we follow the general concept of CEM-GMsFEM and simply use constant functions in each fracture © Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_8

195

196

8 Non-local multicontinua upscaling

network and the matrix as auxiliary functions. Because the fracture has zero width, this procedure needs to be carefully formulated. Next, we solve local problems in the oversampled region subject to the constraint that the local solutions vanish in fractures and the matrix in a suitable sense. This class of constraints is important for the localization. The local problem formulated for each continuum (either fracture network or the matrix phase) simply minimizes the local energy subject to the constraint that the local solution “vanishes” in other continua except the one for which it is formulated for. More precisely, for the continua j in a block K , we minimize the energy of the local oversampled solution such that it is orthogonal to all continua except the continuum j with respect to an appropriate inner product. Then, we use these local solutions to compute the upscaled equation. Because the local calculations are done in an oversampled domain, the transmissibilities are nonlocal and extend to the oversampled region, which is O(log(H )) layers around the target coarse block. We show that one can obtain an accurate solution independent of the contrast and the scales. The resulting upscaled equation is written in a discrete form as  i, j ( j) (i) Tmn (u n − u (i) m ) = qm , j,n i, j

where Tmn are non-local transmissibilities for different continua m and n, and i, j i, j correspond to different coarse blocks. The transmissibilities Tmn are defined in oversampled regions and the non-local dependence of them are investigated. Note that the proposed approach modifies the framework developed in [113] to derive the nonlocal multiple continuum upscaled models. The resulting method is called non-local multicontinua (NLMC) upscaling. Non-local upscaling is not new in porous media [135, 166, 188, 279, 404, 415], particularly for transport equations. However, even in elliptic equations, one can obtain non-local upscaling results. A recent paper [216] is also worth mentioning, where the authors derive non-local upscaling for problems without high contrast. In this chapter, some numerical results are presented, where we compare our proposed upscaled model and the fine-grid models. Both averages and downscaled quantities are studied. Our numerical results show that one can achieve a good accuracy using a small localization and choosing several basis functions (continua) per coarse element. More importantly, because the local functions are constants within fractures and the matrix, our variables are physically meaningful and denote average pressures in each continuum.

8.2 Preliminaries We consider the single-phase flow equation − div(κ(x)∇u) = g, in D

(8.1)

8.2 Preliminaries

197

subject to some suitable boundary conditions. To fix our notations, we consider the zero Neumann boundary condition ∇u · n = 0. Here, u is the pressure of the flow, g is the source term, and κ(x) is a heterogeneous field with high contrast. We will mostly focus on applications to fractured media. First we introduce some notations. For the fractured media, the domain D can be divided into two sets of regions, that is  di D f,i , (8.2) D = Dm i

where m and f corresponds to matrix region and fracture regions respectively, and di is the aperture of fracture D f,i . The permeability in the matrix is κm , and the permeability in the i-th fracture is denoted by κi . It is well known that the permeabilities of matrix and fractures can differ by orders of magnitude. We remark that the permeability of the matrix is homogeneous in our numerical examples presented later, our methodology can be applied to the case with heterogeneous matrix without any modification. Our aim is to give the main ideas of our multicontinua upscaling idea with the presence of fracture, and derive an upscaled system so that the solution has physical meaning, that is, it is the mean pressure on each continuum. The case with heterogeneous matrix is a direct extension of our approach, in terms of both theory and construction of the upscaled system. Let V = H 1 (D). The variational form of (8.1) is to find u ∈ V such that a(u, v) = (g, v), ∀v ∈ V,

(8.3)

where  a(u, v) =

κm ∇u · ∇v + Dm



 i

κi ∇ f u · ∇ f v, (g, v) = D f,i

gv D

and ∇ f denotes the gradient along fracture. Next, we recall the notations of fine and coarse grids. Denote by T H a coarse-grid partition of the domain D with mesh size H , and T H is a coarse mesh containing many fine-scale features. By conducting a conforming refinement of T H , we define a fine mesh T h of D with mesh size h. Typically, we assume that 0 < h  H < 1, and that the fine-scale mesh T h is sufficiently fine to fully resolve the small-scale information and fractures of the domain. Let {K i | i = 1, · · · , N } be the set of coarse element in T H , where N is the number of coarse blocks. For each K i , the oversampled region is denoted by K i+ , which is an oversampling of K i with a few layers of coarse blocks. An illustration of the fine and coarse meshes, as well as an oversampling region are shown in Figure 8.1. We remark that our proposed method provides a novel upscaled formulation, which can be used independent of the fine-grid discretizations. The fine grid is only used for calculating upscaled quantities.

198

8 Non-local multicontinua upscaling

Fig. 8.1 An illustration of coarse and fine meshes in fractured a medium, where K i denotes a coarse block, K i1 denotes one layer oversampling of K i .

8.3 The non-local multicontinua upscaling 8.3.1 Multicontinua functions In this section, we introduce an important part of our method, that is, the representation of each continuum using local basis functions. First, we will present the main ideas. Then we will present a simplified construction, designed specifically for fractured media. General spectral setup We recall the local spectral decomposition based on GMsFEM. Our general setup starts with an auxiliary space. Consider a coarse element K i . Let V (K i ) be the restriction of V on K i . We will find eigenvalues λ(i) k and corresponding eigenfunctions φk(i) ∈ V (K i ) satisfying (i) ai (φk(i) , v) = λ(i) k si (φk , v), ∀v ∈ V (K i ),

where ai and si are bilinear operators on V (K i ) × V (K i ), and are defined by   κm ∇u · ∇v + κ j ∇ f u · ∇ f v, ai (u, v) =  si (u, v) =

Dm ∩K i

Dm ∩K i

κuv ˜ +

 j

j

D f, j ∩K i

D f, j ∩K i

κ˜ j uv,

(8.4)

8.3 The non-local multicontinua upscaling

199

  where the definition of κ˜ m = j κm |∇χ j |2 and κ˜ j = i κ j |∇χi |2 are motivated by the analysis in [113], and χ j denotes the standard multiscale partition of unity functions. We arrange the eigenvalues of (8.4) ascendingly, and then select the first li eigenfunctions corresponding to the small eigenvalues to construct the auxiliary basis functions. The span of these auxiliary basis functions will form an auxiliary space, (i) := span{φk(i) , 1 ≤ k ≤ li }, where 1 ≤ i ≤ N . We note that the auxiliary space Vaux needs to be chosen appropriately, that is, all basis functions corresponding to small eigenvalues (representing the channels) have to be included in the space. We can now construct the target multiscale basis ψ (i) j,ms using the auxiliary space (i) Vaux and the constraint energy minimization concept (see [113] for details). We let Ii be the index set containing all coarse block indices with K ⊂ K i+ . To construct + ( ) the required basis, we find ψ (i) j,ms ∈ V0 (K i ) and μ ∈ Vaux , with ∈ Ii , by solving  ∈Ii

a (ψ (i) j,ms , w) +



s (w, μ ) = 0, ∀w ∈ V0 (K i+ ),

∈Ii

s (ψ (i) j,ms , ν)

(8.5) =

s (φ (i) j , ν),

∀ν ∈

( ) Vaux ,

∀ ∈ Ii ,

+ + 1 (i) where φ (i) j ∈ Vaux is a basis in auxiliary space and V0 (K i ) = H0 (K i ). The resulting basis functions form the multiscale space Vms := span{ψ (i) j,ms , 1 ≤ j ≤ li , 1 ≤ i ≤ N }, which will be used to find the multiscale solution. Since the auxiliary space contains basis functions which capture the high-contrast features, it has been proved that the convergence of this method is independent of the contrast and the convergence rate is in order of the coarse mesh size for appropriate oversampling size. Note that this framework is general and can work for complex heterogeneities and multicontinuum case. For a simplified case, for example, when the fracture networks are known, we can construct some simplified basis functions with constraint energy minimization. The details are presented in the next section.

Simplified basis functions representing continua In this section, we discuss the construction of simplified basis for fractured media, following the general concepts presented above. We aim to construct simplified basis functions which have spatial decay property and can separate each continuum automatically. With these simplified basis, non-local (restricted to oversampled regions) transfer and effective properties can be constructed. The main idea behind this construction is to use constants for each separate fracture network within each coarse block and a constant for each matrix. This simplified construction of auxiliary space uses minimal degrees of freedom in each continuum. As a result, we will obtain an upscaled equation with a minimal size. A major drawback of our construction is that it assumes that we know the separate fracture networks and assign a constant. Some of physical applications can identify separate fracture networks and thus our assumption is valid for many cases. Next, we remind a description of basis construction.

200

8 Non-local multicontinua upscaling ( j)

Consider an oversampling region K i+ of a coarse block K i , we write F ( j) = { f m } as the set of discrete fractures inside any coarse element K j ⊂ K i+ , and let L j be the ( j) number of fractures in F ( j) . Also, we let f 0 be the matrix part within K j ⊂ K i+ . For a given coarse element K i and a continuum m, m = 0, 1, · · · , L i , following the constraint energy minimizing framework, the target basis functions ψm(i) ∈ V0 (K i+ ) are constructed by solving the following local problem on the fine grid     ( j) (i) ( j) (μ0 v+ μm v) = 0, ∀v ∈ V0 (K i+ ), a(ψm , v) +  

Kj

K j ⊂K i+

Kj

( j)

fm

m≤L j

ψm(i) = δi j δ0m , ∀K j ⊂ K i+ ,

( j) fm

(8.6)

ψm(i) = δi j δnm , ∀ f m( j) ∈ F ( j) , ∀K j ⊂ K i+ , ( j)

where μm are constants, which are part of the unknowns in (8.6). Finally, the multiscale space for fractured media is Vms = span{ψm(i) , 0 ≤ m ≤ L i , 1 ≤ i ≤ N }. We note this definition separates the matrix and fractures, and each basis corresponds to a continuum. Let φ0(i) , φl(i) (l = 1, · · · , L i ; i = 1, · · · , N ) be functions that satisfy the following conditions   φ0(i) = δi j , φ0(i) = 0, 

Kj

Kj

φl(i) = 0,



( j)

fm

( j) fm

φl(i) = δi j δml .

The number of fracture continua in the coarse block K i is denoted by L i . We can see that, φ0(i) has average 1 in the matrix continua of coarse element K i , and it has average 0 in other coarse blocks K j ⊂ K i+ as well as any fracture inside K i+ . As for φl(i) , it has average 1 on the l-th fracture continuum inside the coarse element K i , and average 0 in other fracture continua as well as the matrix continua of any coarse block K j ⊂ K i+ .   Define the subspace V1 (K i+ ) := {v ∈ V0 (K i+ )| K v = 0, fm( j) v = 0, ∀K ⊂ K i+ , (i) ∀ j, 1 ≤ m ≤ L j }. Let G loc : V → V1 (K i+ ) be a localized operator such that (i) a(G loc (u), v) = a(u, v), ∀v ∈ V0 (K i+ ),

where V0 (K i+ ) = {v ∈ V (K i+ )|v = 0 on ∂ K i+ }. We define ψm(i):= φm(i) − G loc (φm(i) ), and note that the multiscale basis ψm(i) has a spatial decay, and K j ψm(i) =    (i) (i) (i) ( j) ( j) K j φm , f m ψm = f m φm . In Figure 8.2, we present a comparison between eigenfunctions constructed in Section 8.3.1, and the simplified basis constructed in this section for a same coarse block K . We note that the support of the eigenbasis is in K , and the support of the simplified basis is the oversampled region K + . We can see from the top of Figure 8.2 that,

8.3 The non-local multicontinua upscaling

201

Fig. 8.2 Top: spectral eigenfunctions. Bottom: simplified basis.

the first eigenfunction is constant in the coarse block K , and the second eigenfunction is constant on the fracture within coarse block K . For the simplified basis at the bottom of Figure 8.2, we observe that the first basis represents the matrix, and the second one represents the fracture. This indicates the relation between our simplified basis and the eigenfunctions obtained from the spectral problem. Again, we note that the simplified functions assume that one knows separate fracture networks and uses minimal degrees of freedom to setup a coarse system, which represent the average pressures.

8.3.2 Transmissibility computations General spectral basis In the general heterogeneous case, we have constructed the constaint energy mini(i) , 1 ≤ m ≤ li , 1 ≤ i ≤ N } as shown in (8.5). The transmissimization basis {ψm,ms bility matrix can be constructed by calculating (i, j)

(i) ( j) Tmn,loc = a(ψm,ms , ψn,ms ),

(8.7)

where m, n denote the m- or n-th basis in a coarse block, i, j denote the indices of coarse blocks.

202

8 Non-local multicontinua upscaling

Simplified basis For the fractured media, where we assume the fractured networks are known, we constructed the simplified basis {ψm(i) , 0 ≤ m ≤ L i , 1 ≤ i ≤ N } by solving (8.6). We define Tloc by (i, j) (8.8) Tmn,loc = a(ψm(i) , ψn( j) ). We note that m, n denotes different continua, and i, j are the indices for coarse blocks. This construction shows that we can get non-local (in the oversampled regions) transfer and effective properties for multicontinuum.

8.3.3 Approximation using local multiscale basis Using the transmissibility defined in (8.7) and (8.8), our problem is to find the approximation solution uT such that   (i, j) (i) Tmn,loc ([u T ](nj) − [u T ](i) (8.9) m ) = gm . n

j

With a simplification of indices, we write Tloc in the following form ⎡ ⎤ t11 t12 . . . t1n ⎢t21 t22 . . . t2n ⎥ ⎢ ⎥ ⎢ .. .. . . .. ⎥ , ⎣. . . . ⎦ tn1 tn2 . . . tnn

(8.10)

N where n = i=1 (1 + L i ), and 1 + L i means the one matrix continua plus the number of discrete fractures in coarse block K i , and N is the number of coarse blocks. The system (8.9) can then be expressed as in the matrix form ⎞⎛ ⎞ ⎛ (1) ⎞ ⎛  − j t1 j t ... t1n [u T ](1) g0 0 12 (1) ⎟ (1) ⎟ ⎟ ⎜ ⎜ ⎜ t21 − t . . . t [u ] 2 j 2n j ⎟ ⎜ T 1 ⎟ ⎜ g1 ⎟ ⎜ = A T · uT = ⎜ ⎟⎜ ⎟ ⎜ . ⎟ . (8.11) .. .. .. .. .. ⎠⎝ ⎠ ⎝ .. ⎠ ⎝ . . . . .  (L ) N N) tn2 . . . − j tn j tn1 [u T ] N g (L N Since the summation of each row in A T is zero, this indicates our scheme ensures the mass conservation.

8.4 Time-dependent problem We also consider the time-dependent single-phase flow and use spatial upscaling derived above. In particular, we consider ∂u − div(κ∇u) = g, in D. (8.12) ∂t

8.5 Numerical results

203

The fine-scale solution can be found using the standard finite element scheme, with backward Euler method for time discretization: u n − u n−1 , v) + (κ∇u n , ∇v) = (g, v). (8.13) ( dt In matrix form, we have M f u n + A f u n = b f + M f u n−1 ,

(8.14)

where M f and A f are fine scale mass and stiffness matrices respectively, b f is the right-hand side vector. For the coarse scale approximation, we will solve MT u T n + A T u T n = bT + MT u T n−1 , (8.15) where A T is defined in (8.11) and MT is an approximation of coarse scale mass matrix. We note that both A T and MT are non-local and defined for each continuum. One can write the non-local upscaled equation as   i, j d (i) i, j ( j) (i) um + Mmn Tmn (u n − u (i) m ) = gm . dt j,n j,n

8.5 Numerical results 8.5.1 Steady state case In this section, we present some representative numerical examples. First, we use the fractured media as shown in the left of Figure 8.3. The permeability of the matrix is κm = 1, and the permeability of the fractures are κ f = 102 . The source term in the right-hand side of the equation is piecewise constant functions with f = 102 for 0 ≤ x ≤ 0.2, 0.3 ≤ y ≤ 0.4, and f = −102 for 0 ≤ x ≤ 0.2, 0.7 ≤ y ≤ 0.8.

Fig. 8.3 Left: A fractured media. Right: source term.

204

8 Non-local multicontinua upscaling

The degrees of freedom for fine-scale approximation are 22642. Let u f be the fine-scale solution. We define the average of fine-scale solution u¯ such that u| ¯ Ki =   (i) = (i) u. These are the average pressures, which are computed with our u, u| ¯ fm Ki fm approach. We plot u and u¯ in Figure 8.4.

Fig. 8.4 Left: fine-scale solution. Right: average of fine scale solution.

Forthecoarsescaleapproximation,wetakethecoarsemeshsizeas H = 1/10, 1/20 respectively for numerical simulations. When H = 1/10, the coarse scale degrees of freedom is 282. When H = 1/20, the coarse scale degrees of freedom is 927. We present the local solutions constructed. We take the example for the coarse mesh with H = 1/20 and consider a coarse block K i with one discrete fracture in it. The two local solutions are shown in Figure 8.5 when we use two oversampling layers, and in Figure 8.6 when we use six oversampling layers, respectively. From the figures, we notice that with two layers of oversampling, the local distribution has a decay property, and the local solution almost vanish outside the oversampling region with six layers. This indicates that one can localize the effects.

Fig. 8.5 From left to right: A coarse block K i with two oversampling layers K i2+ . Local solution w.r.t matrix. Local solution w.r.t. the fracture.

We present the error u T − u

¯ L 2 in Table 8.1 and Table 8.2 for H = 1/10 and H = 1/20, respectively. From the numerical results, we observe a good convergence comparing u T with the averaged fine-scale solution. It can be seen that, with 2 layers of

8.5 Numerical results

205

Fig. 8.6 From left to right: A coarse block K i with six oversampling layers K i6+ . Local solution w.r.t matrix. Local solution w.r.t. the fracture.

oversampling, the error u T − u

¯ L 2 is 1.46% for H = 1/10. For the case H = 1/20, 4 layers of oversampling gives an error of 0.008%. This indicates the upscaled equation in our modified method can use small local regions. We plot the upscaled solutions using different size of oversampling region and compare them with the averaged finescale solution for the case H = 1/20. The results are presented in Figure 8.7. It shows that we can obtain very good accuracy with 4 layers of oversampling. Ar ea per centage(%)

Over sampling level

u T − u

¯ L 2 (%)

6.5 18.5 36.5 60.5 100

1 2 3 4 global

21.97 1.46 0.015 0.0008 4.57e-10

Table 8.1 Coarse mesh size 1/10. Upscaling errors when oversampling with 1,2,3,4 layers of coarse blocks. Last row shows the error when using global domain for the local computations.

Ar ea per centage(%)

Over sampling level

u T − u

¯ L 2 (%)

1

40.63

2

22.95

4

0.008

6

0.0007

global

6e-5

3.25 9.25 30.25 63.25 100 Table 8.2 Coarse mesh size 1/20. Upscaling errors when oversampling with 1,2,4,6 layers of coarse blocks. Last row shows the error when using global domain for local computations.

Next, we present numerical results for the transmissibility matrix Tloc . We display (i, j) Tmn,loc for two different coarse blocks K i in Figure 8.8 and Figure 8.10 in the global

206

8 Non-local multicontinua upscaling

Fig. 8.7 Corse mesh size 1/20. Upper left: coarse scale solution using global domain for local computations. Upper right: coarse scale solution with oversampling size 1. Lower left: coarse scale solution with oversampling size 2. Lower right: coarse scale solution with oversampling size 4.

domain. In the left of these two figures, we plot the transmissibility between the element K i and its neighboring elements for the matrix continua. In the right of the figures, we show the transmissibility for the fracture continua between K i and neighboring elements. We notice that the regions of influence are almost within the 4 layers of oversampling region, which is in accordance with our numerical results. Figure 8.9 shows a one-dimensional plot of transmissibility between an element K i and other coarse elements in the region marked by the black box in Figure 8.8 (left). We remark that the transmissibility matrix computed using global domain is exact, which can be used as a reference. From the subplot on the top of Figure 8.9, we note that the transmissibility between K i and other coarse blocks along the slab decays fast. It can be seen from the lognormal plot on the bottom of Figure 8.9, that with one layer of oversampling, the values in the transmissibility matrix has large errors, however, with four layers of oversampling, the transmissibility is quite accurate. Remark 8.1. Our method can also be used in three dimension with no difficulty in terms of the formulation of the scheme and the theory behind. The number of oversampling level is based on our theory proved in the reference [113]. According

8.5 Numerical results

207

Fig. 8.8 Using global domain for local computation. Left: Transmissibility between an element K i and neighboring elements for matrix. Right: Transmissibility between K i and neighboring elements for fractures. The dotted white lines denote fractures in the domain.

Fig. 8.9 Transmissibility between an element K i and elements in the cross section marked by the black box in Figure 8.8 (left). Top: plot of exact values. Bottom: log plot of absolute values.

208

8 Non-local multicontinua upscaling

Fig. 8.10 Using the global domain for local computations. Left: Transmissibility between an element K i and neighboring elements for matrix. Right: Transmissibility between K i and neighboring elements for fractures. The dotted white lines denotes fractures in the domain.

to that, the number of oversampling levels in three dimension is the same as that of two dimension. In particular, we showed that the number of layers is proportional to log(1/H ) and the logarithm of the contrast of the media.

8.5.2 Time-dependent case In this example, we present numerical results for the time-dependent case. The source term and geometry are the same as in the previous section, shown in Figure 8.3. The simulation runs for a total time of T = 1.0, we present the error between the coarse scale solution and average fine-scale solution at time instances t = 0.1, 0.5, and 1.0. The results show good accuracy of the proposed method (Figure 8.11 and Table 8.3). Ar ea per centage(%)

Oversampling

t = 0.1

t = 0.5

t = 1.0

1

40.055

61.027

62.741

2

1.065

1.035

1.035

4

0.076

0.009

0.008

6

0.072

0.002

0.0007

global

0.072

0.001

0.0006

3.25 9.25 30.25 63.25 100 Table 8.3 Error u T − u

¯ L 2 (%). Coarse mesh size 1/20. Upscaling errors when oversampling with 1, 2, 4, 6 layers of coarse blocks. Last row shows the error when using global domain for local computation.

8.6 Coupled GMsFEM-NLMC at different resolutions

209

Fig. 8.11 Oversampling size 4. Top: Average of fine-scale solution at t = 0, 0.5, 1.0. Bottom: Coarse scale solution at t = 0, 0.5, 1.0.

8.6 Coupled GMsFEM-NLMC at different resolutions The method presented in the previous sections are two level multiscale methods. In this section, we present a three-level multiscale method based on both the GMsFEM and the NLMC with the aims of taking advantage of both methodologies and reducing offline computational costs. Overall speaking, the proposed technique is the threelevel scheme (see Figure 8.12) described as follows: • fine-grid model for fractured porous media, • intermediate grid model based on the NLMC method, • coarse-grid approximation using the GMsFEM. Our method starts with a fine-grid discretization for the system involving fractured porous media. In the next step, based on the fine-grid model, we construct an NLMC method using an intermediate grid. We note that the system resulting from the NLMC method gives solutions that have physical meaning, namely, mean values on local continua. We remark that by an intermediate grid, we mean that the grid size is between the fine and the coarse grids. In order to enhance locality, the grid size of the intermediate grid needs to be relatively small, and this motivates using such an intermediate grid. However, the resulting NLMC upscaled system has a relatively large dimension. This motivates a further step of dimension reduction. In particular,

210

8 Non-local multicontinua upscaling

we will apply the idea of GMsFEM to the NLMC system to obtain a final reduced model.

Fig. 8.12 Concept of three-level scheme.

To illustrate the main concepts, we use (8.12) as an example. First, we begin with the fine-grid discretization. This gives the equation (8.14), which is written as follows M f u n + A f u n = b f + M f u n−1 , where M f and A f are fine scale mass and stiffness matrices respectively, b f is the right-hand side vector. Next, on an intermediate grid, we construct the NLMC method. This gives the equation (8.15), which is written as follows MT u T n + A T u T n = bT + MT u T n−1 , where A T , defined in (8.11), and MT are intermediate scale stiffness and mass matrices respectively. Finally, on the coarse grid, we will construct the required reduced model based on the GMsFEM idea. The resulting model can be written as MC u C n + AC u C n = bC + MC u C n−1 , where MC and AC are coarse scale mass and stiffness matrices respectively, and these matrices are defined as MC = RC MT RCT and AC = RC A T RCT . The matrix RC represents the GMsFEM basis functions, which will be discussed next. We will describe the construction of the GMsFEM basis functions {ψkω } which is supported in a coarse neighborhood ω, where k represents the numbering of the basis functions. Let TC = ∪i Θi be the coarse grid and assume that each coarse element is

8.6 Coupled GMsFEM-NLMC at different resolutions

211 C

N a connected union of fine grid and intermediate grid blocks. We use {xi }i=1 to denote C of coarse nodes. We the vertices of the coarse mesh TC , where N is the number   define the coarse neighborhood of the node xi by ωi = ∪ j Θ j | xi ∈ Θ j . For each coarse neighborhood ωi , we solve the following local spectral problem within ωi

Aωi Ψ i = λi S ωi Ψ i , where the matrix Aωi is the restriction of the matrix A T in the coarse neighborhood ωi . Moreover, the matrix S ωi is defined as follows: (S ωi ) jk =

 κ j |K j |, j = k, 0, j = k,

where the indices j and k are the indices of all intermediate cells within ωi . Moreover, |K j | is the area/volume of the intermediate cell K j and κ j is the mean value of κ on K j . We choose eigenvectors Ψki (k = 1, ..., Mi ) corresponding to the smallest Mi eigenvalues and set {ψki = χi Ψki }, for 1 ≤ i ≤ N C and 1 ≤ k ≤ Mi to be the required GMsFEM basis functions. Here χi are the standard linear partition of unity functions and Mi denotes the number of eigenvectors that are chosen for each coarse node i. We enumerate the basis {ψki } using single index and define RCT = [ψ1 , . . . , ψ M ] , N C Mi is the size of the coarse-grid system. where M = i=1 We refer [399] for an application of coupled GMsFEM and NLMC methods for flows in fracture media and related works.

Chapter 9

Space-time GMsFEM

9.1 Introduction Many multiscale processes vary over multiple space and time scales. These space and time scales are often tightly coupled. For example, flow processes in porous media can occur on multiple time scales over multiple spatial scales and these scales can be non-separable. In many applications, the heterogeneities change due to the time can be significant and it needs to be taken into account in reduced-order models. In this chapter, we discuss an application of the GMsFEM to space-time heterogeneous problems following [105]. Some well-known approaches for handling separable spatial and temporal scales are homogenization techniques [186, 276, 351, 358], as discussed earlier. In these methods, one solves local problems in space and time. We remind a well-known case of the parabolic equation ∂ u − div(κ(x, x/α , t, t/β )∇u) = f, ∂t

(9.1)

subject to smooth initial and boundary conditions, where  is a small scale. In (9.1), the spatial scale is α and the temporal scale is β . As we discussed earlier that the homogenized equation has the same form as (9.1), but with the smooth coefficients κ∗ (x, t). One can compute the coefficients using the solutions of local spacetime parabolic equations in the periodic cell. This localization is possible thanks to the scale separation. The local problems may or may not include time-dependent derivatives depending on the interplay between α and β since the cell problems are independent of . One can extend this homogenization procedure to numerical homogenization type methods [8, 211, 331], where one solves the local parabolic equations in each coarse block and in each coarse time step. To compute the effective property, one averages the solutions of the local problems. These approaches work well in the scale separation cases, but do not provide accurate approximations when there is no scale separation. © Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_9

213

214

9 Space-time GMsFEM

Previous approaches in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. The extension of the GMsFEM discussed in this chapter is a significant contribution because of (1) the parabolic nature of cell solutions, (2) extra degrees of freedom associated with space-time cells, and (3) local boundary conditions in space-time cells. In our approach, we construct snapshot spaces in space-time local domains. We construct the snapshot solutions by solving local problems. We can construct a complete snapshot space by taking all possible boundary conditions; however, this can result in very high computational cost. For this reason, we use randomized boundary conditions (Section 4.8) for local snapshot vectors by solving parabolic equations subject to random boundary and initial conditions. We compute only a few more than the number of basis functions needed. Computing multiscale basis functions employs local spectral problems. These local spectral problems are in space-time domain. Using space-time eigenvalue problems control the errors associated with ∂u/∂t. We discuss several choices for local spectral problems and present a convergence analysis of the method. We present several numerical examples. We consider the numerical tests with the conductivities that contain high contrast and these high conductivity regions move in time. These are challenging examples since the high-conductivity heterogeneities vary significantly during one coarse-grid time interval. If only using spatial multiscale basis functions, one will need a very large dimensional coarse space. In the numerical results, we use oversampling and randomized snapshots. The numerical results show that one can achieve a small error by selecting a few multiscale basis functions. The numerical results confirm the convergence analysis. We also discuss online multiscale basis functions (Chapter 5) for space-time heterogeneous problems, where we use the residual information and construct new multiscale basis functions adaptively. We would like to choose a number of offline basis functions such that with only 1–2 online iterations, we can substantially reduce the error. This requires a sufficient number of offline basis functions, with the offline basis function construction typically derived by the analysis. Based on the previous results for time-independent problems, we show that one needs a sufficient number of offline basis functions to reduce the error substantially. In the numerical results, we observe a similar phenomenon, i.e., the error decreases rapidly in 1–2 online iterations.

9.2 Space-time GMsFEM 9.2.1 Preliminaries and motivation Let Ω be a bounded domain in R2 with a Lipschitz boundary ∂Ω, and [0, T ] (T > 0) be a time interval. We consider the following parabolic differential equation

9.2 Space-time GMsFEM

215

∂ u − div(κ(x, t)∇u) = f ∂t u=0

in Ω × (0, T ), on ∂Ω × (0, T ),

u(x, 0) = β(x)

(9.2)

in Ω,

where κ(x, t) is a time-dependent heterogeneous media (for example, a timedependent high-contrast permeability field), f is a given source function, β(x) is the initial condition. The main objective is to develop space-time multiscale model reduction within GMsFEM and use the time-dependent parabolic equation as an example. The proposed methods can be used for other models that require spacetime multiscale model reduction. The proposed method follows the space-time finite element framework, where the time-dependent multiscale basis functions are constructed on the coarse grid. Therefore, compared with the time-independent basis structure, it gives a more efficient numerical solver for the parabolic problem in complicated media. Before introducing the method, we define the mesh of the domain first. Let T h be a partition of the domain Ω into fine finite elements where h > 0 is the fine mesh size. Then we form a coarse partition T H of the domain Ω such that every element in T H is a union of connected fine-mesh grid blocks, that is, ∀K j ∈ T H , K j = ∪ F∈I j F for some I j ⊂ T h . Next, let T T = {(Tn−1 , Tn )|1 ≤ n ≤ N } be a coarse partition of (0, T ) where 0 = T0 < T1 < T2 < · · · < TN = T and we define a fine partition of (0, T ), T t , by refining the partition T T . To fix the notations, we will use the standard conforming piecewise linear finite element method in space for the computation of the fine-scale solution. For time, we use piecewise linear finite element basis functions that are continuous within each coarse time interval. One can use discontinuous Galerkin couplings. We define the finite element space Vh with respect to T h × (0, T ) as Vh = {v ∈ L 2 ((0, T ); C 0 (Ω)) | v = φ(x)ψ(t) where φ| K ∈ Q 1 (K ) ∀K ∈ T h , ψ|τ ∈ C 0 (τ ) ∀τ ∈ T T and ψ|τ ∈ P1 (τ ) ∀τ ∈ T t }, then the fine-scale solution u h ∈ Vh is obtained by solving the following variational problem  0

T

 Ω

∂u h v+ ∂t

 0

T

 Ω

κ∇u h · ∇v +  0

T

 Ω

where [·] is the jump operator such that

N −1   n=0



fv+

Ω

Ω

[u h (x, Tn )]v(x, Tn+ ) =

β(x)v(x, T0+ ), ∀v ∈ Vh ,

(9.3)

216

9 Space-time GMsFEM



[u h (x, Tn )] = u h (x, Tn+ ) − u h (x, Tn− ) [u h (x, Tn )] = u h (x, T0+ )

for n ≥ 1, for n = 0.

We assume that the fine mesh size h is small enough so that the fine-scale solution u h is close enough to the exact solution. We remark that the convergence of this type of scheme is analyzed, for example, in [272]. We will use the space-time finite element method to solve problem (9.2) on the coarse grid. That is, we find u H ∈ VH such that  0

T

 Ω

∂u H v+ ∂t



T

 Ω

0

κ∇u H · ∇v + 

T 0

N −1   n=0



 Ω

fv+

Ω

Ω

[u H (x, Tn )]v(x, Tn+ ) =

β(x)v(x, T0+ ),

(9.4) ∀v ∈ VH ,

where VH is the multiscale finite element space which will be introduced in the following subsections. The computational cost for solving equation (9.4) is huge since we need to compute the solution u H in the whole time interval (0, T ) at one time. In fact, if we assume the solution space VH is a direct sum of the spaces only containing the functions defined on one single coarse time interval (Tn−1 , Tn ), we can decompose the problem (9.4) into a sequence of problems and find the solution u H in each time interval sequentially. The coarse space will be constructed in each time interval and we will have N VH(n) , VH = ⊕n=1

where VH(n) only contains the functions having zero values in the time interval (0, T ) except (Tn−1 , Tn ), namely ∀v ∈ VH(n) , v(·, t) = 0 for t ∈ (0, T )\(Tn−1 , Tn ). (n) The equation (9.4) can be decomposed into the following problem: find u (n) H ∈ VH (n) (where VH will be defined later) satisfying  Tn   Tn   ∂u (n) H + + v+ κ∇u (n) · ∇v + u (n) H H (x, Tn−1 )v(x, Tn−1 ) ∂t Tn−1 Ω Tn−1 Ω Ω  Tn   (n) + = fv+ g (n) (9.5) H (x)v(x, Tn−1 ), ∀v ∈ V H , Tn−1

where

Ω

Ω

 g (n) H (·)

=

− u (n−1) (·, Tn−1 ) H β(·)

for n ≥ 1, for n = 0.

Then, the solution u H of the problem (9.4) is the direct sum of all these u (n) H ’s, that (n) N is u H = ⊕n=1 u H .

9.2 Space-time GMsFEM

217

Next, we motivate the use of space-time multiscale basis functions by comparing it to space multiscale basis functions. In particular, we discuss the savings in the reduced models when space-time multiscale basis functions are used compared to space multiscale basis functions. We denote {tn1 , · · ·, tnp } are p fine time steps in (Tn−1 , Tn ). When we construct multiscale basis functions, the solution can be represented as  space-time ωi = c ψ (x, t) in the interval (Tn−1 , Tn ). In this case, the number of coeffiu (n) l,i l,i l H cients cl,i is related to the size of the reduced system in space-time interval. On the other hand, if we use only space multiscale basis functions, we need to construct these multiscale basis functions at each fine time instant tn j , denoted by ψlωi (x, tn j ). The solution u H spanned by these basis functions will have a much larger dimension because each time instant is represented by multiscale basis functions. Thus, performing space-time multiscale model reduction can provide a substantial CPU savings. Inthefollowingsection,wewilldiscussspace-timemultiscalebasisfunctions.First, we will construct multiscale basis functions in the offline mode without using the residual. In Section 9.4, we will discuss online space-time multiscale basis construction.

9.2.2 Construction of offline basis functions Snapshot space Let ω be a given coarse neighborhood in space. We omit the coarse node index to simplify the notations. The construction of the offline basis functions on coarse ω,(Tn−1 ,Tn ) . The snapshot space time interval (Tn−1 , Tn ) starts with a snapshot space VH,snap ω,(T

,T )

n−1 n VH,snap is a set of functions defined on ω × (Tn−1 , Tn ) and contains all or most necessary components of the fine-scale solution restricted to ω × (Tn−1 , Tn ). A spectral problem is then solved in the snapshot space to extract the dominant modes in the snapshot space. These dominant modes are the offline basis functions and the resultω,(Tn−1 ,Tn ) ing reduced space is called the offline space. There are two choices of VH,snap that are commonly used. The first choice is to use all possible fine-grid functions in ω × (Tn−1 , Tn ). This snapshot space provides an accurate approximation for the solution space; however, this snapshot space can be very large. The second choice for the snapshot spaces consists of solving local problems for all possible boundary conditions. In particular, we define ψ ωj as the solution of

∂ ω ψ − div(κ(x, t)∇ψ ωj ) = 0 in ω × (Tn−1 , Tn ), ∂t j ψ ωj (x, t) = δ j (x, t) on ∂ (ω × (Tn−1 , Tn )) .

(9.6)

Here δ j (x, t) is a fine-grid delta function and ∂ (ω × (Tn−1 , Tn )) denotes the boundaries t = Tn−1 and on ∂ω × (Tn−1 , Tn ). In general, the computations of these snapshots are expensive since in each local coarse neighborhood ω, O(Mn∂ω ) number of

218

9 Space-time GMsFEM

local problems are required to be solved. Here, Mn∂ω is the number of fine grids on the boundaries t = Tn−1 and on ∂ω × (Tn−1 , Tn ). A smaller yet accurate snapshot space is needed to build a more efficient multiscale method. We can take an advantage of randomized oversampling concepts [71] and compute only a few snapshot vectors, which will reduce the computational cost remarkably while keeping required accuracy. Next, we introduce randomized snapshots. Firstly, we introduce the notation for oversampled regions. We denote by ω + the oversampled space region of ω ⊂ ω + , defined by adding several fine- or coarse-grid ∗ , Tn ) as the left-side oversampled time region layers around ω. Also, we define (Tn−1 for (Tn−1 , Tn ). In the following, we generate inexpensive snapshots using random ∗ , Tn ). That is, boundary conditions on the oversampled space-time region ω + × (Tn−1 instead of solving Equation (9.6) for each fine boundary node on ∂ (ω × (Tn−1 , Tn )), we solve a small number of local problems imposed with random boundary conditions ∂ +,ω + ∗ ψ − div(κ(x, t)∇ψ +,ω j ) = 0 in ω × (Tn−1 , Tn ), ∂t j  +  ∗ ψ +,ω j (x, t) = rl on ∂ ω × (Tn−1 , Tn ) , where rl are independent identically distributed (i.i.d.) standard Gaussian random ∗ ∗ and on ∂ω + × (Tn−1 , Tn ). vectors on the fine-grid nodes of the boundaries t = Tn−1 ∗ + Then the local snapshot space on ω × (Tn−1 , Tn ) is +,ω,(T

VH,snap n−1

,Tn )

ω ω = span{ψ +,ω j (x, t)| j = 1, · · ·, L + pbf },

ω is the where L ω is the number of local offline basis we want to construct in ω and pbf buffer number. Later on, we use the same buffer number for all ω’s and simply use the notation pbf . In the following sections, if we specify one special coarse neighborhood ωi , we use the notation L i to denote the number of local offline basis. With these snapshots, we follow the procedure in the following subsection to generate offline basis functions by using an auxiliary spectral decomposition.

Offline space To obtain the offline basis functions, we need to perform a space reduction by appropriate spectral problems. Motivated by the convergence analysis, we adopt the following spectral problem on ω + × (Tn−1 , Tn ): +,ω,(T ,T ) Find (φ, λ) ∈ VH,snap n−1 n × R such that +,ω,(T

An (φ, v) = λSn (φ, v), ∀v ∈ VH,snap n−1

,Tn )

,

where the bilinear operators An (φ, v) and Sn (φ, v) are defined by

(9.7)

9.2 Space-time GMsFEM

An (φ, v) = 

1 2 Tn

 





ω+

φ(x, Tn )v(x, Tn ) +

ω+

ω+

Tn−1

Sn (φ, v) =

219

ω+

φ(x, Tn−1 )v(x, Tn−1 ) +

κ(x, t)∇φ · ∇v,

(9.8) 

φ(x, Tn−1 )v(x, Tn−1 ) +



Tn Tn−1

ω+

κ+ (x, t)φv,

where the weighted function

κ+ (x, t) is defined by

κ+ (x, t) = κ(x, t)

Nc 

|∇χi+ |2 ,

i=1 Nc {χi+ }i=1 is a partition of unity associated with the oversampled coarse neighborhoods + Nc {ωi }i=1 and satisfies |∇χi+ | ≥ |∇χi | on ωi where χi is the standard multiscale basis function for the coarse node xi (that is, with linear boundary conditions for cell problems). More precisely,

−div(κ(x, Tn−1 )∇χi ) = 0, in K ∈ ωi , χi c = gi , on ∂ K ,

(9.9)

for all K ∈ ωi , where gi is a continuous function on ∂ K and is linear on each edge of ∂ K . To avoid cumbersome notations, we omit superindex ms for χ, which refers to multiscale basis functions. ω ω We arrange the eigenvalues {λ+,ω j | j = 1, 2, · · · L + pbf } from (9.7) in the ω ascending order, and select the first L eigenfunctions, which are corresponding +,ω,eig +,ω,eig , · · ·, Ψ L ω }. to the first L ω ordered eigenvalues, and denote them by {Ψ1 Using these eigenfunctions, we can define φ˜ +,ω j (x, t) =

ω L ω + pbf



+,ω,eig

(Ψ j

)k ψk+,ω (x, t),

j = 1, 2, · · ·, L ω ,

k=1 +,ω,eig )k (Ψ j

+,ω,eig

denotes the k-th component of Ψ j , and ψk+,ω (x, t) is the where ∗ + snapshot basis function computed on ω × (Tn−1 , Tn ) as in the previous subsection. Then we can obtain the snapshots φ˜ ωj (x, t) on the target region ω × (Tn−1 , Tn ) by restricting φ˜ +,ω j (x, t) onto ω × (Tn−1 , Tn ). Finally, the offline basis functions on ω × (Tn−1 , Tn ) are defined by φωj (x, t) = χφ˜ ωj (x, t), where χ is the standard multiscale basis function from (9.9) for a generic coarse neighborhood ω. We also define the local offline space on ω × (Tn−1 , Tn ) as ω,(T

VH,offn−1

,Tn )

= span{φωj (x, t)| j = 1, · · ·, L ω }. (T

,T )

n−1 n Note that one can take VH(n) in (9.5) as VH(n) = VH,off = span{φωj i (x, t)|1 ≤ i ≤ N Nc , 1 ≤ j ≤ L i }. As a result, VH = VH,off = ⊕n=1 VH(n) .

220

9 Space-time GMsFEM +,ω,eig

+,ω,eig

Remark 9.1. For the convenience, we also denote by {Ψ1 , · · ·, Ψ L ω + pbfω } all the eigenfunctions from (9.7) corresponding to the ordered eigenvalues, and define φ˜ +,ω j (x, t)

ω L ω + pbf

=



+,ω,eig

(Ψ j

)k ψk+,ω (x, t),

ω j = 1, 2, · · ·, L ω + pbf .

k=1 ∗ , Tn ) can be rewritten as We note that the snapshot space on ω + × (Tn−1 +,ω,(T

VH,snap n−1

,Tn )

ω ω = span{φ˜ +,ω j (x, t)| j = 1, · · ·, L + pbf },

and the snapshot space on ω × (Tn−1 , Tn ) can be written as ω,(T

n−1 VH,snap

,Tn )

ω = span{φ˜ ωj (x, t)| j = 1, · · ·, L ω + pbf },

where each φ˜ ωj (x, t) is the restriction of φ˜ +,ω j (x, t) onto ω × (Tn−1 , Tn ). By collecting all local snapshot spaces on each ω × (Tn−1 , Tn ), we can obtain the snapshot space (Tn−1 ,Tn ) VH,snap on Ω × (Tn−1 , Tn ). The local offline space can be rewritten as ω,(T

VH,offn−1

,Tn )

= span{χφ˜ ωj (x, t)| j ≤ L ω }.

9.2.3 Error estimates In this section, we will give a brief account of the convergence theory. To start, we define two norms  · 2V (n) and  · 2W (n) by  Tn    1 1 + u2V (n) = κ|∇u|2 + u 2 (x, Tn− ) + u 2 (x, Tn−1 ), 2 Ω 2 Ω Tn−1 Ω  Tn u2W (n) = u2V (n) + u t (·, t)2H −1 (κ,Ω) , Tn−1

where

 −1 u H(κ,Ω) =

sup v∈H01 (Ω)

(

Ω



uv 1

Ω

κ|∇v|2 ) 2

.

In [105], the following quasi-optimal error estimate is proved. Lemma 9.2. Let u h be the fine-scale solution from Equation (9.3), u H be the multiscale solution from Equation (9.5). We have the following estimate  Cu h − w2W (n) for n = 1, 2 u h − u H V (n) ≤ 2 2 C(u h − wW (n) + u h − u H V (n−1) ) for n > 1, for any w ∈ VH(n) .

9.3 Numerical results for offline GMsFEM

221

To estimate the error of the multiscale solution, we only need to find a function w in VH(n) such that u h − wW (n) is small. The following is the main convergence result. Theorem 9.3. Let u h be the fine-scale solution from Equation (9.3), u H be the multiscale solution from Equation (9.5). Let u˜ h = argminv∈V (Tn−1 ,Tn ) {u h − vW (n) } H,snap

and we denote u˜ h =

 i

u h − u H 2V (n)



ci, j φ˜ ωj i . There holds

 1 + 2 M(D E F + 1) u˜ h,i V (n) (ω+ ) i i λ+,ω L i +1 i χi u˜ h,i with u˜ h,i =

j

(9.10)

+u h − u˜ h 2W (n) + u h − u H 2V (n−1) , where M = max K {M K } with M K is the number of coarse neighborhoods ωi ’s which have nonempty intersection with K ,   2 2 2 2 ωi κ|∇χi | v + ωi κχi |∇v|   D = max{Di } with Di = supv∈H01 (Ω) , 2 2 ωi κ|∇v| + ωi κv   κ|∇w|2 + Ω κw 2 , E = supw∈H01 (Ω) Ω  2 Ω κ|∇w| 1 F = max{Fi } with Fi = , min x∈ωi {|χi+ (x)|2 }  ˜ +,ωi u˜ + h,i = j ci, j φ j and the local norm  · V (n) (ωi+ ) is defined by  v2V (n) (ω+ ) = i

Tn Tn−1

 ωi+

κ|∇v|2 +

1 2

 ωi+

v 2 (x, Tn− ) +

1 2

 ωi+

+ v 2 (x, Tn−1 ).

9.3 Numerical results for offline GMsFEM In this section, we present a representative numerical example to show the performance of the proposed method. In particular, we solve Equation (9.2) using the spacetime GMsFEM to validate the effectiveness of the proposed approaches. The space domain Ω is taken as the unit square [0, 1] × [0, 1] and is divided into 10 × 10 coarse blocks consisting of uniform squares. Each coarse block is then divided into 10 × 10 fine blocks consisting of uniform squares. That is, Ω is partitioned by 100 × 100 square fine-grid blocks. The whole time interval is [0, 1.6] (i.e., T = 1.6) and is divided into two uniform coarse time intervals and each coarse time interval is then divided into 8 fine time intervals. We also use a source term f = 1 and impose a continuous initial condition β(x, y) = sin(πx) sin(π y). We employ a space-time dependent high-contrast permeability field κ(x, t) to examine the method. We first

222

9 Space-time GMsFEM

Fig. 9.1 High-contrast Permeability Field 1. Left: the permeability at the initial time. Right: the permeability at the final time.

solve for u h from Equation (9.3) to obtain the fine-grid solution. Then we solve for the multiscale solution u H using the space-time GMsFEM. To compare the accuracy, we will use the following error quantities: ⎛ T

⎞1/2 2 0 u H (t) − u h (t) L 2 (Ω) ⎠ , e1 = ⎝ T 2 0 u h (t) L 2 (Ω)

 T  e2 =

0

2 Ω κ|∇(u H (t) − u h (t))| T  2 0 Ω κ|∇u h (t)|

1/2 .

(9.11)

Since we are using the technique of randomized oversampling in the computation of the snapshot space, we would like to introduce the concept of snapshot ratio, which is calculated as the number of randomized snapshots divided by the number of the full snapshots on one coarse neighborhood ωi . Here, the number of the full snapshots refers to the number of functions δi (x, t) from Equation (9.6). In the following experiment with 100 × 100 fine-grid mesh, this number of the full snapshots on snap each coarse neighborhood is calculated by n total = 21 × 21 + 40 × 8 = 761. We use a high-contrast permeability field κ(x, t), which is translated uniformly after every other fine time step. High-contrast permeability fields at the initial and final time steps are shown in Figure 9.1. Next, we consider applying the space-time GMsFEM to Equation (9.2) and solve for the multiscale solution u H . Recall the procedures that are described in the Section 9.2, where we need to construct the snapshot spaces in the first place. The number of local offline basis that will be used in each ωi , denoted by L i , and the buffer number pbf needs to be chosen in advance since they determine how many local snapshots are used. Then we can construct the lower dimensional offline space by performing space reduction on the snapshot space. In the experiments, we use the same buffer number and the same number of local offline basis for all coarse neighborhood ωi ’s. First, we fix L i = 11 for all ωi ’s and examine the influences of various buffer numbers on the solution errors e1 and e2 . The results are displayed in the left table of Table 9.1. It is observed that when increasing the buffer numbers, one can get more accurate solutions, which is as expected. But the error decays very slowly,

9.3 Numerical results for offline GMsFEM

223

which indicates that using different buffer numbers doesn’t affect the convergence rate too much. Based on this observation, it is not necessary to choose a large buffer number in order to improve convergence rate. Then we consider the choice of L i , the number of eigenbasis in a neighborhood. With the fixed buffer number pbf = 8, we examine the convergence behaviors of using different L i ’s. Relative errors of multiscale solutions are shown in the right table of Table 9.1. We observe that with a fixed buffer number, the relative errors are decreasing as using more offline basis. To see a more quantitative relationship between the relative errors and the values of L i as well as being inspired by the result in Theorem 9.3, we inspect the values of 1/Λ∗ and the corresponding squared errors (see Table 9.2 and Figure 9.2), where Λ∗ = minωi λωLii +1 and {λωj i } are the eigenvalues associated with the eigenbasis computed by spectral problem (9.7) in each ωi . We note that when plotting Figure 9.2, we don’t use the values of case L i = 2, because in this case as in the case with one basis function per node, the method does not converge as we do not have sufficient number of basis functions. We note that the two curves in Figure 9.2 track each other somewhat closely. This indicates that 1/Λ∗ ’s and e22 ’s are correlated and we calculate for the correlation coefficient to be corr coe f (1/Λ∗ , e22 ) = 0.9778. Observing the dimensions of the offline spaces Voff , one can see that compared with the traditional fine-scale finite element method, the proposed space-time GMsFEM uses much fewer degrees of freedom while achieving an accurate solution. Also, by inspecting the snapshot ratios, one can see that the use of randomization can reduce the dimension of snapshot spaces substantially. We would like to comment that oversampling technique is necessary for the randomization. For example, in the case L i = 6 and pbf = 8, if without oversampling the errors e1 and e2 are 11.19% and 88.42%, respectively, which are worse than the errors obtained with oversampling. pbf Snapshot ratio 1 0.0158 4 0.0197 8 0.0250 12 0.0302 20 0.0407 30 0.0539 40 0.0670

e1 6.18% 5.66% 5.17% 5.16% 4.71% 4.35% 4.23%

e2 53.90% 48.04% 45.86% 43.83% 41.14% 38.68% 37.60%

L i dim(Voff ) Snapshot ratio 2 162 0.0131 6 486 0.0184 10 810 0.0237 20 1620 0.0368 30 2430 0.0499 40 3240 0.0631 50 4050 0.0762

e1 17.03% 8.11% 6.97% 4.81% 3.29% 2.28% 1.54%

e2 129.14% 62.59% 54.85% 41.18% 31.64% 24.43% 18.45%

Table 9.1 First permeability field. Left: errors with the fixed number of offline basis L i = 11. Right: errors with the fixed buffer number pbf = 8.

224

9 Space-time GMsFEM L i 1/Λ∗ 2 6 10 20 30 40 50

0.2734 0.0120 0.0085 0.0061 0.0053 0.0048 0.0042

e12

e22

2.90% 0.66% 0.49% 0.23% 0.11% 0.05% 0.02%

166.78% 39.17% 30.08% 16.96% 10.01% 5.97% 3.40%

Table 9.2 1/Λ∗ values and errors.

−3

12

x 10

0.4 0.35

10

0.3

9

0.25

8

e22

1/Λ*

11

7

0.2 0.15

6

0.1

5

0.05

4

5

10

15

20

25

Li

30

35

40

45

50

0

5

10

15

20

25

Li

30

35

40

45

50

Fig. 9.2 Left: 1/Λ∗ vs L i ; Right: e22 vs L i .

More numerical results can be found in [105].

9.4 Residual-based online adaptive procedure As we observe in the previous examples, the offline errors do not decrease rapidly after several multiscale functions are selected. In these cases, online basis functions can help to reduce the error and obtain an accurate approximation of the fine-scale solution [111]. The use of online basis functions gives a rapid convergence. Next, we will derive a framework for the construction of online multiscale basis functions. We use the index m ≥ 1 to represent the online enrichment level. At the enrichment level m, we use VHm to denote the corresponding space-time GMsFEM space and u mH the corresponding solution obtained in (9.5). The sequence of functions {u mH }m≥1 will converge to the fine-scale solution. We emphasize that the space VHm can contain both offline and online basis functions, and define VH0 = Voff . We will construct a strategy for getting the space VHm+1 from VHm . Next we present a framework for the construction of online basis functions. By online basis functions, we mean basis functions that are computed during the iterative process using the residual. This is the contrary to offline basis functions that are

9.4 Residual-based online adaptive procedure

225

computed before the iterative process. The online basis functions for enrichment level m + 1 are computed based on some local residuals for the multiscale solution u mH . Thus, we see that some offline basis functions are necessary for the computations of online basis functions. In the numerical examples from the following section, we will also see how many offline basis functions are needed in order to obtain a rapidly converging sequence of solutions. For brevity, we denote the left-hand side of (9.5) by a(u (n) H , v) and the right-hand (n) (n) N side F(v). That is, the solution u H = ⊕n=1 u H where u H satisfies (n) a(u (n) H , v) = F(v), ∀v ∈ V H .

Consider a given coarse neighborhood ωi . Suppose that at the enrichment level m, we N φ(n) need to add an online basis function φ ∈ Vh in ωi . Then the required φ = ⊕n=1 (n) satisfies that φ is the solution of a(φ(n) , v) = R (n) (v), ∀v ∈ Vh , where R (n) (v) = F(v) − a(u m(n) H , v) is the online residual at the coarse time interval [Tn−1 , Tn ]. In the following, we would like to form a residual-based online algorithm in each coarse time interval [Tn−1 , Tn ], see Algorithm 1. For simplicity, we will omit the time index (Tn−1 , Tn ) or (n) on the spaces and solutions in this description. We consider Nc enrichment on non-overlapping coarse neighborhoods. Thus, we divide the {ωi }i=1 into P non-overlapping groups and denote each group by {ωi }i∈I p , p = 1, ..., P. We denote by M the number of online iterations. Algorithm 1 Residual based online algorithm (T

n−1 Initialization: Offline space V H0 = VH,off

,Tn )

, offline solution u 0H = u H .

for m=0 to M: do for p=1 to P: do (1) On each ωi (i ∈ I p ), compute residual R m (v) = a(u mH , v) − F(v), v ∈ Vh . (2) For each i, solve a(φi , v) = R m (v), ∀v ∈ Vh . (3) Set VHm = VHm ∪ {φi |i ∈ I p }. (4) Solve for a new u mH ∈ VHm satisfying a(u mH , v) = F(v), ∀v ∈ VHm . end for Set VHm+1 = VHm , and u m+1 = u mH . H end for

To further improve the convergence and efficiency of the online method, we can adopt an online adaptive procedure. In this adaptive approach, the online enrichment is performed for coarse neighborhoods that have a cumulative residual that is θ

226

9 Space-time GMsFEM

fraction of the total residual. More precisely, assume that the V (n) norm of local residuals on {ωi |i ∈ I p }, denoted by {ri |i ∈ I p }, are arranged so that r p1 ≥ r p2 ≥ r p3 ≥ · · · ≥ r p J , where we suppose I p = { p1 , p2 , p3 , · · ·, p J }. Instead of adding {φi |i ∈ I p } into VHm at Step 1 in Algorithm 1, we only add the basis {φ1 , · · ·, φk } for the corresponding coarse neighborhoods such that k is the smallest integer satisfying k J r 2pi ≥ θi=1 r 2pi . i=1

In the examples below, we will see that the proposed adaptive procedure gives a better convergence and is more efficient.

9.5 Numerical results for online GMsFEM In this section, we present numerical examples to demonstrate the performance of the proposed online method in solving Equation (9.2). To implement the space-time online GMsFEM, we will first choose a fixed number of offline basis functions for every coarse neighborhood, and calculate the resulting offline space Voff . Then we conduct the online process by following Algorithm 1. In this experiment, we use the same space-time domain and mesh (coarse and fine), the same source term f and initial condition β(x, y), the same definitions of relative errors e1 and e2 , as in Section 9.3. The permeability field κ(x, t) is chosen as the high-contrast permeability field 1. The buffer number in the computation of snapshot space is chosen to be 8. First, we implement the space-time online GMsFEM by choosing different numbers of offline basis functions (L i = 1, 2, 3, 4, 5) on every coarse neighborhood. The relative errors of online solutions are presented in Tables 9.3 and 9.4. Note that in the first column, we show the number of basis functions used for each coarse neighborhood ωi , and the degrees of freedom (DOF) of multiscale space on each coarse time interval which are the numbers in parentheses, after online enrichment. For example, 2(162) in the first column means that after online enrichment, 2 multiscale basis are used on each ωi and the DOF of multiscale space on each coarse time interval is 162. And if we initially choose L i = 1, then it means 1 online iteration is performed, which add 1 online basis to each ωi . If L i = 2 initially, then it means we do not perform any online iteration and 2 multiscale basis are offline basis functions. By observing each column, one can see that the errors decay fast with more online iterations being performed. This is observed for both e1 and e2 when L i ≥ 4. This suggests that in this specific setting, we can get a fast online convergence with 4 offline basis chosen on each ωi . After a small number of online iterations, the relative errors decrease to a significantly small level. Our numerical results using different contrasts suggest that the number of offline basis functions used to guarantee a fast online convergence rate is related to the high contrast of the permeability field.

9.5 Numerical results for online GMsFEM

227

D O F e1 (L i = 1) e1 (L i = 2) e1 (L i = 3) e1 (L i = 4) e1 (L i = 5) 1(81) 2(162) 3(243) 4(324) 5(405) 6(486) 7(567)

97.57% 93.20% 44.24% 15.37% 8.65% 5.15% 2.58%

96.71% 23.22% 6.53% 3.69% 1.71% 3.11e-1%

21.27% 7.17e-1% 10.20% 2.06e-1% 2.58e-1% 5.20% 5.41e-2% 1.75e-2% 1.06e-1% 5.54e-3% 6.12e-4% 2.99e-3%

Table 9.3 Relative online errors e1 , with the different numbers of offline basis functions. High contrast = 106 .

D O F e2 (L i = 1) e2 (L i = 2) e2 (L i = 3) e2 (L i = 4) e2 (L i = 5) 1(81) 2(162) 3(243) 4(324) 5(405) 6(486) 7(567)

138% 113% 84.93% 82.48% 69.15% 51.17% 37.93%

114% 139% 82.08% 51.13% 34.00% 7.81%

104% 11.43% 73.50% 3.29% 4.78% 48.26% 1.01% 3.53e-1% 1.86% 1.05e-1% 9.89e-3% 4.75e-2%

Table 9.4 Relative online errors e2 , with the different numbers of offline basis functions. High contrast = 106 .

Next, we perform online adaptive basis construction procedure with θ = 0.7. The numerical results for using 3, 4, and 5 offline basis per coarse neighborhood are shown in Table 9.5. Notice that “M1 + M2” in the DOF columns means M1 degrees of freedom are used on the first coarse time interval and M2 degrees of freedom on the second coarse time interval. To compare the behaviors of online processes with and without adaptivity, we plot out the log values of e2 against DOFs. See Figure 9.3. We observe that to achieve a certain error, fewer online basis functions are needed with adaptivity. This indicates that the proposed adaptive procedure gives us better convergence and is more efficient.

228

9 Space-time GMsFEM 3 offline basis DOF e2 243+243 104% 323+322 10.57% 403+392 1.49% 480+465 9.81e-2% 552+533 4.24e-3%

4 offline basis DOF e2 324+324 73.50% 399+401 3.56% 468+466 2.13e-1% 541+529 1.03e-2% 611+601 5.00e-4%

5 offline basis DOF e2 405+405 48.26% 471+473 1.95% 533+536 1.21e-1% 599+603 6.81e-3% 670+669 3.41e-4%

Table 9.5 Relative online adaptive errors e2 with different numbers of offline basis functions.

Fig. 9.3 Adaptivity v.s. no adaptivity.

Chapter 10

Multiscale methods for perforated domains

10.1 Introduction One important class of multiscale problems occurs in perforated domains. In these problems, differential equations are formulated in perforated domains. These domains can be considered the outside of inclusions or connected bodies of various sizes. Due to the variable sizes and geometries of these perforations, solutions to these problems have multiscale features. One solution approach involves posing the problem in a domain without perforations but with a very high contrast penalty term representing the domain heterogeneities [234, 269, 270, 402]. However, the void space can be a small portion of the whole domain, and thus it is computationally expensive to enlarge the domain substantially. Problems in perforated domains [372], as other multiscale problems, require some type of model reduction. The main difference in developing multiscale methods for problems in perforated domains is the complexity of the domains and that many portions of the domain are excluded in the computational domain. This poses a challenging task. For typical upscaling and numerical homogenization (e.g., [247, 372]), the macroscopic equations do not contain perforations and one computes the effective properties. In multiscale methods, the macroscopic equations are numerically derived by computing multiscale basis functions [66, 102, 293]. Several multiscale methods have been developed for problems in perforated domains. The use of GMsFEM for solving multiscale problems in perforated domains is motivated by recent works [75, 247, 293, 325]. In this regard, we would like to mention recent works by Le Bris and his collaborators [293], where accurate multiscale basis functions are constructed. These approaches differ from numerical homogenization and approaches that use RVE [160]. We will consider the use of the Generalized Multiscale Finite Element (GMsFEM) Framework. Using snapshot spaces is essential in problems with perforations, because the snapshots contain necessary geometry information. In the snapshot space, we perform local spectral decomposition to identify multiscale basis functions. In

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_10

229

230

10 Multiscale methods for perforated domains

this chapter, we will present analytical results for the Stokes equations. Similar results can be obtained for the elliptic and the elasticity equations. One can accelerate convergence by computing online multiscale basis functions using a residual as discussed in Chapter 5. As we discussed earlier that for new multiscale basis functions, we solve local problems using the global residual information. Online basis functions are computed adaptively and only added in regions with largest residuals. We design online basis functions for multiscale perforated problems. It is important that adding an online basis function decreases the error substantially and one can reduce the error in one iteration. For this reason, constructing online basis functions must guarantee that the error reduction is independent of small scales and contrast. Constructing online basis functions follows a rigorous analysis. As before, we show that if a sufficient number of offline multiscale basis functions are chosen, one can substantially reduce the error. This reduction is related to the eigenvalue that the corresponding eigenvector is not included in the coarse space. The analysis for the offline procedure starts with the proof of the inf-sup condition, which shows the well-posedness of the scheme. Then, we derive an a-posteriori error bound for the GMsFEM. This bound shows that the error of the solution is bounded by a computable residual and an irreducible error. This irreducible error is a measure of approximating the fine-scale space by the snapshot space. We show that the convergence rate depends on the number of offline basis functions. In the numerical examples, we consider a perforated domain with inclusions of various sizes. We have considered elliptic, elasticity, and Stokes equations and only report the results for elasticity and Stokes equations. The numerical results for the offline basis consist of adding multiscale basis functions where we observe that the error decreases as we increase the number of basis functions. However, the errors (especially those involving solution gradients) can still be large. For this reason, online basis functions are added, which can rapidly reduce the error. We summarize some of the quantitative results below. • For elasticity equations without adaptivity, we observe that, by using 4 offline basis functions per coarse neighborhood, we can achieve 7.4% error in L 2 norm, while the error is 26% in H 1 norm. The results of the offline computations are similar for two different geometries. • For Stokes equations without adaptivity, we observe that, by using 3 offline basis functions per coarse block, we can achieve 0.8% error in L 2 norm, while the error is 7.7% in H 1 norm. All errors are for the velocity field. The results of the offline computations are better in case of many inclusions. • For online simulations, we observe that the error decreases rapidly as we add one online basis functions. The error keeps decreasing fast as we increase the number of online basis functions; however, we are mostly interested in the error decay when one basis function is added. We observe that the error decreases much faster if we have more than 1 initial offline basis function. For example, the error decreases only 4 times if one basis function is chosen, while the error decreases more than

10.2 Preliminaries

231

10 times if 4 initial basis functions are selected (see Table 10.2, Table 10.3 and Table 10.4 for the Stokes case and second geometry). • We observe that one can effectively use adaptivity to reduce the computational cost in online simulations. The adaptive results show that we can achieve better accuracy for the same number of online basis functions.

10.2 Preliminaries 10.2.1 Problem setting In this section, we present the underlying problem as stated in [102] and the corresponding fine-scale and coarse-scale discretization. Let Ω ⊂ Rd (d = 2, 3) be a bounded domain covered by inactive cells (for Stokes flow and Darcy flow) or active cells (for elasticity problem) B  . We will consider the d = 2 case, though the results can be extended to d > 2. We use the superscript  to denote quantities related to perforated domains. The active cells are where the underlying problem is solved, while inactive cells are the rest of the region. Suppose the distance between inactive cells (or active cells) is of order . Define Ω  := Ω\B  . See Figure 10.1 for an illustration of the perforated domain. We consider the following problem defined in a perforated domain Ω  L (w) = f, in Ω  , ∂w = 0, on ∂Ω  ∩ ∂B  , w = 0 or ∂n w = g, on ∂Ω ∩ ∂Ω  ,

(10.1) (10.2) (10.3)

where L denotes a linear differential operator, n is the unit outward normal to the boundary, f and g denote given functions with sufficient regularity. We will focus on the Dirichlet problem, namely, w = 0 in (10.2).

Fig. 10.1 Illustration of a perforated domain.

232

10 Multiscale methods for perforated domains

Denote by V (Ω  ) the appropriate solution space, and V0 (Ω  ) = {v ∈ V (Ω  ), v = 0 on ∂Ω  }. The variational formulation of Problem (10.1)–(10.3) is to find w ∈ V (Ω  ), such that L (w), vΩ  = ( f, v)Ω 

for all v ∈ V0 (Ω  ),

where ·, ·Ω  denotes a specific for the application inner product over Ω  for either scalar functions or vector functions and and ( f, v)Ω  is the L 2 inner product. Some specific examples for the above abstract notations are given below. Laplace: For the Laplace operator with homogeneous Dirichlet boundary conditions on ∂Ω  , we have L (u) = −Δu,

(10.4)

and V (Ω  ) = H01 (Ω  ), L (u), vΩ  = (∇u, ∇v)Ω  . Elasticity: For the elasticity operator with a homogeneous Dirichlet boundary condition on ∂Ω  , we assume the medium is isotropic. Let u ∈ (H 1 (Ω  ))2 be the displacement field. The strain tensor ε(u) ∈ (L 2 (Ω  ))2×2 is defined by 1 ε(u) = (∇u + ∇u T ). 2 Thus, the stress tensor σ(u) ∈ (L 2 (Ω  ))2×2 relates to the strain tensor ε(u) such that σ(u) = 2με + ξ∇ · u I, where ξ > 0 and μ > 0 are the Lamé coefficients. We have L (u) = −∇ · σ,

(10.5)

where V (Ω  ) = (H01 (Ω  ))2 and L (u), vΩ  = 2μ(ε(u), ε(v))Ω  + ξ(∇ · u, ∇ · v)Ω  . Stokes: For Stokes equations, we have   ∇ p − Δu  , (10.6) L (u , p) = ∇ ·u where μ is the viscosity, p is the fluid pressure, u represents the velocity, V (Ω  ) = (H01 (Ω  ))2 × L 20 (Ω  ), and   (∇u, ∇v)Ω  −(∇ · v, p)Ω   L (u , p), (v , q)Ω  = . (∇ · u, q)Ω  0 We recall that L 20 (Ω  ) contains functions in L 2 (Ω  ) with zero average in Ω  . We will show the results for elasticity and Stokes equations. The results for Laplace have similar convergence analysis and computational results as those for elasticity equations, so we will omit them here.

10.2 Preliminaries

233

10.2.2 Coarse- and fine-grid notations For the numerical approximation of the above problems, we first introduce the notations of fine and coarse grids. It follows similar constructions as before with the exception of domain geometries, which can intersect the boundaries of coarse regions. Let T H be a coarse-grid partition of the domain Ω  with mesh size H . Notice that, the edges of the coarse elements do not necessarily have straight edges because of the perforations (see Figure 10.2). By conducting a conforming refinement of the coarse mesh T H , we can obtain a fine mesh T h of Ω  with mesh size h. Typically, we assume that 0 < h  H < 1, and that the fine-scale mesh Th is sufficiently fine to fully resolve the small-scale information of the domain, and T H is a coarse mesh containing many fine-scale features. Let Nc and Ne be the number of nodes and edges in the coarse grid, respectively. We denote by {xi |1 ≤ i ≤ Nc } the set of coarse nodes, and {E j |1 ≤ j ≤ Ne } the set of coarse edges.

Fig. 10.2 Illustration of coarse elements and coarse neighborhoods.

For all three model problems, we define a coarse neighborhood ωi for each coarse node xi by (10.7) ωi = ∪{K j ∈ T H ; xi ∈ K¯ j }, which is the union of all coarse elements having the node xi . For the Stokes problem, additionally, we define a coarse neighborhood ωm for each coarse edge E m by ωm = ∪{K j ∈ T H ; E m ∈ K¯ j },

(10.8)

which is the union of all coarse elements having the edge E m . On the triangulation Th , we introduce the following finite element spaces: Vh := {v ∈ V (Ω  )|v| K ∈ (P k (K ))l for all K ∈ Th }, where P k denotes the polynomial of degree k( k = 0, 1, 2), and l( l = 1, 2) indicates either a scalar or a vector. Note that for the Laplace and elasticity operators, we choose k = 1, i.e., piecewise linear function space as the fine-scale approximation space; for

234

10 Multiscale methods for perforated domains

Stokes problem, we use (P 2 (K ))2 for fine-scale velocity approximation and P 0 (K ) for fine-scale pressure approximation. We use Q h to denote the space for pressure. We will then obtain the fine-scale solution u ∈ Vh by solving the following variational problem L (u), vΩ  = ( f, v)Ω  , for all v ∈ Vh (10.9) for Laplace and elasticity, and obtain the fine-scale solution (u, p) ∈ Vh × Q h by solving the following variational problem L (u, p), (v, q)Ω  = (( f, 0), (v, q))Ω  ,

for all (v, q) ∈ Vh × Q h

(10.10)

for the Stokes system. These solutions are used as reference solutions to test the performance of the schemes.

10.2.3 Outline of GMsFEM Here, we briefly outline the main goal of this chapter. First, we will get a set of basis ω ω functions {φi l }, such that each φi l is supported in some coarse neighborhood ωl . Once the bases are constructed, we define the coarse function space as ω

VH,off := span{φi l }. In the offline stage of GMsFEM, we seek an approximation u H = VH,off , which satisfies the coarse-scale offline formulation, L (u H ), vΩ  = ( f, v)Ω  ,

for all v ∈ VH,off .

M i,l

ω

ci,l φi l in (10.11)

Here, the bilinear forms L (u H ), vΩ  are as defined before, and ( f, v)Ω  is the L 2 inner product. All these calculations are performed outside perforations. Online stage. We will also present the online basis construction following Chapter 5. At the m and u mH the corresponding GMsFEM space and enrichment level m, denote by VH,off solution, respectively. The online basis functions are constructed based on the residm the local residual uals of the current  multiscale  solution u H , as before. We compute  m  Ri = ( f, v)ωi − L (u H ), v ω in each coarse neighborhood ωi . For the coarse neighi borhoods where the residuals are large, we add one or more basis functions by solving L (φion ) = Ri . Adding the online basis in the solution space, we will get a new coarse function m+1 . The new solution u m+1 is calculated in this approximation space. This space VH,off H iterative process is stopped when some error tolerance is achieved. The accuracy of the GMsFEM relies on the coarse basis functions. We shall present the construction of suitable basis functions in both offline and online stages for the differential operators defined above.

10.3 The construction of offline and online basis functions

235

10.3 The construction of offline and online basis functions In this section, we describe the construction of offline and online basis for the elasticity problem and Stokes problem.

10.3.1 Elasticity problem In this section, we will consider the elasticity problem (10.5) with a homogeneous Dirichlet boundary condition. Snapshot space The snapshot space for the elasticity problem consists of extensions of the fine-grid functions δkh in ωi . Here δkh = 1 at the fine node xk ∈ ∂ωi \B  , δkh = 0 at other fine nodes x j ∈ ∂ωi \B  , and δkh = 0 in ∂B  . Let Vhi be the restriction of the fine-grid i ⊂ Vhi be the set of functions that vanish on ∂ωi . We will find space Vh in ωi and Vh,0 i i  u k ∈ Vh on ωi by solving the following problems on a fine grid   i 2με(u ik ) : ε(v) + ξ∇ · u ik ∇ · v d x = 0, ∀v ∈ Vh,0 , (10.12) ωi

with boundary conditions (i)  u ik = (δ (i) j , 0) or (0, δ j ) on ∂ωi .

u ik = 0 on ∂ωi ∩ B  ,

∂u = 0, Note that for the elasticity operator with Neumann boundary conditions ∂n we will use Neumann boundary conditions instead of the local problems proposed above. We will collect the solutions to the above local problems to generate the ω snapshot space. Let ψk i := u ik and define the snapshot space by ω

ω

i VH,snap = span{ψk i : 1 ≤ k ≤ Ji , 1 ≤ i ≤ Nc },

where Ji is the number of snapshot basis in ωi , and Nc is the number of nodes. To  Nc simplify notations, let Msnap = i=1 Ji and write VH,snap = span{ψi } : 1 ≤ i ≤ Msnap }.

Offline space This section is devoted to the construction of the offline space via a spectral decomposition. We will consider the following eigenvalue problems in the space of snapshots:

236

10 Multiscale methods for perforated domains 

ω

ω



ω

Aωi Φk i = λk i S ωi Φk i ,

(10.13)

where 

ω

ω

ωi

ωi

ωi

Aωi = ai (ψmi , ψn i ) = S

  ω ω ω ω 2με(ψmi ) : ε(ψn i ) + ξ∇ · ψmi ∇ · ψn i , 

= si (ψm , ψn ) =

ωi

ω

ωi

(10.14)

ω

(ξ + 2μ)ψmi · ψn i .

We assume that the eigenvalues are arranged in increasing order. To simplify notaω tions, we write λik = λk i . To generate the offline space, we choose the smallest Mi eigenvalues from Eq. (10.13) and the corresponding eigenvectors, then our offline basis can be con ω ω ω ω structed by setting φ˜ k i = j Φk ji ψ j i , for k = 1, . . . , Mi , where Φk ji are the coorω ω dinates of the vector Φ i . The offline space is defined as the span of χi φ˜ i , namely, k

k

ω

VH,off = span{φl i : 1 ≤ l ≤ li , 1 ≤ i ≤ Nc }, where li is the number of snapshot basis in ωi , and {χi } is a set of partition of unity functions for the coarse grid. One can take {χi } as the standard hat functions  N or li standard multiscale basis functions. To simplify notations further, let M = i=1 and write VH,off = span{φi : 1 ≤ i ≤ M}.

Online adaptive method By the offline computation, we construct multiscale basis functions that can be used for any input parameters to solve the problem on the coarse grid. In the earlier works [111, 124], the online method for the diffusion equation with heterogeneous coefficients has been proposed. In this section, we consider the construction of the online basis functions for the elasticity problem in perforated domains and present an adaptive enrichment algorithm. We use the index m ≥ 1 to represent the enrichment level. The online basis functions are computed based on some local residuals m m , where we use VH,off to denote the for the current multiscale solution u mH ∈ VH,off corresponding space that can contain both offline and online basis functions. m+1 m = VH,off + span{φon } be the new approximate space that constructed by Let VH,off i on on the i-th coarse neighborhood ωi . For each coarseadding online basis φ ∈ Vh,0 i , such grid neighborhood ωi , we define the residual Ri as a linear functional on Vh,0 that    i Ri (v) = 2με(u mH ) : ε(v) + ξ∇ · u mH ∇ · v d x, ∀v ∈ Vh,0 f vd x − . ωi

ωi

10.3 The construction of offline and online basis functions

237

The norm of Ri is defined as ||Ri ||(Vhi )∗ = sup

i v∈Vh,0

|Ri (v)| 1

ai (v, v) 2

,



 where ai (v, v) = ω 2με(v) : ε(v) + ξ∇ · v∇ · v d x. i For the computation of this norm, according to the Riesz representation theorem, we can first compute φon as the solution to the following problem   2με(φon ) : ε(v) + ξ∇ · φon ∇ · v d x =  ωi

f v dx −

  ωi

ωi

2με(u mH )

: ε(v) + ξ∇ ·

u mH ∇



(10.15)

· v d x, ∀v ∈

i Vh,0

1

and take ||Ri ||(Vhi )∗ = ai (φon , φon ) 2 . For the construction of the adaptive online basis functions, we use the following error indicators to access the quality of the solution. In those non-overlapping coarsegrid neighborhoods ωi with large residuals, we enrich the space by finding online i using Eq. (10.15). basis φon ∈ Vh,0 • Indicator 1. The error indicator based on local residual ηi = ||Ri ||2(V i )∗ .

(10.16)

h

• Indicator 2. The error indicator based on local residual and eigenvalue  i −1 ||Ri ||2(V i )∗ . ηi = λlωi +1

(10.17)

h

Now we present the adaptive online algorithm. We start with enrichment iteration number m = 0 and choose θ ∈ (0, 1). Suppose the initial number of offline basis functions is lim (m = 1) for each coarse-grid neighborhood ωi , and the multiscale m (m = 1). For m = 1, 2, ...: space is VH,off m , such that • Step 1. Find u mH in VH,off

   m m 2με(u H ) : ε(v) + ξ∇ · u H ∇ · v d x = ωi

ωi

m f v, ∀v ∈ VH,off .

• Step 2. Compute error indicators (ηi ) for every coarse-grid neighborhoods ωi and sort them in decreasing order η1 ≥ η2 ≥ ... ≥ η N . • Step 3. Select coarse-grid neighborhoods ωi , where enrichment is needed. We take the smallest k, such that Nc k

θ ηi ≤ ηi . i=1

i=1

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10 Multiscale methods for perforated domains

• Step 4. Enrich the space by adding online basis functions. For each ωi , where i = i by solving (10.15). The resulting space is denoted by 1, 2, ..., k, we find φon ∈ Vh,0 m+1 VH,off . We repeat the above procedure until the global error indicator is small or we have a certain number of basis functions.

10.3.2 Stokes problem In the above section, we presented the online procedure for the elasticity equations. In this section, we present the constructions of snapshot, offline, and online basis functions for the Stokes problem. Snapshot space Snapshot space is a space that contains an extensive set of basis functions that are solutions to local problems with all possible boundary conditions up to fine-grid resolution (randomized snapshots based on Section 4.8 can be used). To get snapshot functions, we solve the following problem on the coarse neighborhood ωi : find (u li , pli ) (on a fine grid), such that   i i ∇u l : ∇vd x − pli div(v)d x = 0, ∀v ∈ Vh,0 , ωi

ωi



ωi

 qdiv(u li )d x

=

ωi

(10.18) cqd x, ∀q ∈

Q ih ,

with boundary conditions u li = (0, 0), on ∂B  , u li = (δli , 0) or (0, δli ), on ∂ωi /∂B  , where function δli is a piecewise constant function, such that it has value 1 on el Si and value 0 on other fine-grid edges. Notice that ωi /∂B  = ∪l=1 el , where el are the fine-grid edges and Si is the number of these fine-grid edges on ωi /∂B  . In (10.18), i we define Vhi and Q ih as the restrictions of the fine-grid space in ωi and Vh,0 ⊂ Vhi  be functions that vanish on ∂ωi . We remark that the constant c in (10.18) is chosen by compatibility condition, c = |ω1 | ∂ω /∂B u li · n i ds. We emphasize that, for the i i Stokes problem, we will solve (10.18) in both node-based coarse neighborhoods (10.7) and edge-based coarse neighborhoods (10.8). The collection of the solutions of the above local problems generates the snapshot ω space, ψl i = u li in ωi : ω

ω

i = {ψl i : 1 ≤ l ≤ 2Si , 1 ≤ i ≤ (Ne + Nc )}, VH,snap

where we recall that Ne is the number of coarse-grid edges and Nc is the number of coarse-grid nodes.

10.3 The construction of offline and online basis functions

239

Offline space We perform a space reduction in the snapshot space through the use of a local spectral problem in ωi . The purpose of this is to determine the dominant modes in the snapshot space and to obtain a small dimension space for the approximation of the solution. We consider the following local eigenvalue problem in the snapshot space ω



ω

ω



Aωi Φk i = λk i S ωi Φk i , where



ω

ω



ω

ω

(10.19)

Aωi = ai (ψmi , ψn i ) S ωi = si (ψmi , ψn i ) 

and ai (u, v) =

 ωi

∇u : ∇vd x,

and

si (u, v) =

ωi

|∇χi |2 u · v d x

and χi will be specified later. Note that the above spectral problem is solved in the local snapshot space corresponding to the neighborhood domain ωi . We arrange the eigenvalues in increasing order, and choose the first Mi eigenvalues and take the ω corresponding eigenvectors Φk i , for k = 1, 2, ..., Mi , to form the basis functions,   ωi ωi ωi ωi ωi = i.e., φ k j Φk j ψ j , where Φk j are the coordinates of the vector Φk . We define 



ωi ωi , k = 1, 2, ..., 2Si }. H,off V = span{φ k

(10.20)

For the construction of conforming offline space, we need to multiply the functions    ωi , φ ωi ) by a partition of unity function χi . We remark that the partition of ωi = (φ φ k x1 ,k x2 ,k unity functions {χi } are defined with respect to the coarse nodes and the mid-points of coarse edges. One can choose {χi } as the standard multiscale finite element basis. However, upon multiplying by the partition of unity functions, the resulting basis functions do not have constant divergence anymore, which affects the stability of the scheme. To resolve this problem, we solve two local optimization problems in every coarse element K ij ⊂ ωi :    ω  min ∇φx1i ,k 

ω

L 2 (K ij )

ω

such that div(φx1i ,k ) =

1 |K ij |





∂ K ij

ωi , 0) · n i ds, in K i (χi φ j x1 ,k (10.21)

ω

 i , 0), on ∂ K i , and with φx1i ,k = (χi φ j x1 ,k    ω  min ∇φx2i ,k 

ω

L 2 (K ij )

such that div(φx2i ,k ) =

1 |K ij |





∂ K ij

ωi ) · n i ds in K i , (0, χi φ j x2 ,k (10.22)

240

10 Multiscale methods for perforated domains 









ω ωi ), on ∂ K i . We write that φωi = H(χi φ ωi ) and φωi = with φx2i ,k = (0, χi φ x2 ,k x1 ,k x1 ,k x2 ,k j  ωi ), where H(v) is the Stokes extension of the function v. H(χi φ x2 ,k Combining them, we obtain the global offline space ω

ω

ω

i = span{φx1i ,k and φx2i ,k : VH,off

1 ≤ i ≤ (Ne + Nc ) and 1 ≤ k ≤ Mi }.

Using a single index notation, we can write Nu VH,off = span{φi }i=1 ,

 Ne +Nc where Nu = i=1 Mi . This space will be used as the approximation space for the velocity. For coarse approximation of pressure, we will take Q off to be the space of piecewise constant functions on the coarse mesh. Online adaptive method Similar to Section 10.3.1, we will define the online velocity basis for the Stokes problem. For each coarse-grid neighborhood ωi , we define the residual Ri as a linear functional on V i , such that  Ri (v) =

 ωi

f · v dx −



ωi

∇u mH : ∇vd x +

ωi

p mH div(v)d x, ∀v ∈ V i , (10.23)

where (u mH , p mH ) is the multiscale solution at the enrichment level m, and V i = (H01 (ωi ))2 . The norm of Ri is defined as |Ri (v)| . (10.24) ||Ri ||(V i )∗ = sup v∈V i v H 1 (ωi ) We will then use indicators (10.16) and (10.17) for the adaptive enrichment method. i For the computation of online basis φion ∈ Vh,0 , we solve the following problem:   i ∇φion : ∇vd x − p on div(v)d x = Ri (v), ∀v ∈ Vh,0 , ωi



ωi

ωi

div(φion ) q

(10.25) d x = 0, ∀q ∈ Q off .

The adaptivity procedure follows the one presented in Section 10.3.1.

10.4 Numerical results Inthissection,weshowsimulationresultsusingtheframeworkofonlineadaptiveGMsFEM presented in Section 10.2.3 for elasticity equations and Stokes equations. We set Ω = [0, 1] × [0, 1] and use the perforated domain as illustrated in Figure 10.3, where

10.4 Numerical results

241

theperforatedregionsB  arecircular,suchthatthecircularinclusionshavevarioussizes and extremely small inclusions are set around some big ones. We have also used perforated regions of other shapes instead and obtained similar results. The computational domain is discretized coarsely using uniform triangulation, where the coarse mesh size 1 for the elasticity problem and H = 15 for the Stokes problem. Furthermore, H = 10 nonuniform triangulation is used inside each coarse triangular element to obtain a finer discretization. An Example of this triangulation is displayed also in Figure 10.3. First we will choose a fixed number of offline basis (initial basis) for every coarse neighborhood, and obtain corresponding offline space Voff , which is also denoted by 1 . Then, we perform the online iterations on non-overlapping coarse neighborVH,off m , m ≥ 1. We will add online basis both with hoods to obtain enriched space VH,off adaptivity and without adaptivity and compare the results. All the errors shown below are in percentage.

Fig. 10.3 A heterogeneous perforated medium used in the simulations.

10.4.1 Elasticity equations in perforated domain We consider the elasticity operator (10.5). We use zero displacements u = 0 on the inclusions, u 1 = 0, σ2 = 0 on the left boundary, σ1 = 0, u 2 = 0 on the bottom boundary and σ1 = 0, σ2 = 0 on the right and top boundaries. Here, u = (u 1 , u 2 ) and σ = (σ1 , σ2 ). The source term is defined by f = (107 , 107 ), the elastic modulus is given by E = 109 , Poisson’s ratio is ν = 0.22, where Eν E , ξ= . μ= 2(1 + ν) (1 + ν)(1 − 2ν)

242

10 Multiscale methods for perforated domains

We use the following error quantities to measure the performance of the online adaptive GMsFEM ||e|| L 2 = eu  L 2 (Ω  ) =

(ξ + 2μ)(u − u H ) L 2 (Ω  ) , (ξ + 2μ)u L 2 (Ω  ) 

||e|| H 1 = eu  H 1 (Ω  ) =

L (u − u H ), u − u H Ω  , L (u), uΩ 

where u and u H are the fine and coarse solutions, respectively, and L (u), vΩ  = 2με(u), ε(v)Ω  + ξ∇ · u, ∇ · vΩ  . The fine-scale solution and coarse-scale solution corresponding to the perforated domain in Figure 10.3 are presented in Figure 10.4. Fine solutions are shown on the left of the figure, coarse offline solutions are in the middle and online solutions are on the right. In Table 10.1, we present the convergence history when the problem is solved with one, two, and four initial bases in the left, middle, and right column, respectively. Each column shows the error behavior when the online method is applied without adaptivity, with adaptivity using Indicator 1 (see (10.16)) and with adaptivity using Indicator 2 (see (10.17)). Comparing the offline and fine solutions, we notice that some features of offline solution in the interior of the domain are missing, and the errors around the boundary are large. However, the online solution fixes these problems well and show much better accuracy. Looking at Table 10.1, we observe that as we select more initial basis, the error decreases faster after one online iteration. For example, when one online iteration is applied without adaptivity, the H 1 error reduces around 8 times if we use one initial basis, yet it reduces around 12 times if we use two initial basis. Considering the convergence behavior of the online method with adaptivity against the online method without adaptivity, we see that the adaptivity is important. For instance, in a similar DOF of 950 in the case of one initial basis used, the H 1 error 0.002% without adaptivity, while it is only 10−5 % with adaptivity.

10.4 Numerical results

243

DO F ||e|| L 2 ||e|| H 1 (# iter) Not adaptive 286 35.818 59.688 448 (1) 2.128 7.241 610 (2) 0.147 0.654 772 (3) 0.020 0.111 934 (4) 0.002 0.011 Adaptive, ηi2 = ri2 286 35.818 59.688 454 (3) 0.922 4.149 638 (7) 0.033 0.208 776 (10) 0.003 0.018 968 (14) 6.785e-05 0.0004 −1 Adaptive, ηi2 = ri2 λi+1 286 35.818 59.688 438 (3) 1.397 5.701 618 (7) 0.053 0.313 746 (10) 0.005 0.036 936 (14) 0.0002 0.001

DO F ||e|| L 2 (# iter) Not adaptive 386 15.492 548 (1) 0.758 710 (2) 0.067 872 (3) 0.009 1034 (4) 0.0009 Adaptive, ηi2 = ri2 386 15.492 556 (3) 0.504 706 (6) 0.037 890 (10) 0.001 1042 (13) 6.663e-05 −1 Adaptive, ηi2 = ri2 λi+1 386 15.492 544 (3) 0.587 746 (7) 0.017 874 (10) 0.001 1058 (14) 3.874e-05

DO F ||e|| L 2 (# iter) Not adaptive 648 9.345 810 (1) 0.585 972 (2) 0.058 1134 (3) 0.006 1296 (4) 0.0006 Adaptive, ηi2 = ri2 648 9.345 830 (3) 0.373 974 (6) 0.022 1158 (10) 0.0007 1308 (13) 4.453e-05 −1 Adaptive, ηi2 = ri2 λi+1 648 9.345 818 (3) 0.307 962 (6) 0.024 1146 (10) 0.0008 1332 (14) 3.429e-05

||e|| H 1

39.054 3.325 0.379 0.059 0.005 39.054 2.416 0.239 0.008 0.0004 39.054 2.698 0.098 0.009 0.0003

||e|| H 1

29.298 2.427 0.355 0.043 0.003 29.298 1.595 0.148 0.005 0.0003 29.298 1.498 0.154 0.007 0.0002

Table 10.1 Elasticity problem in the perforated domain (Figure 10.3). One (upper left), two (upper right), and four (lower) offline basis functions (θ = 0.7).

244

10 Multiscale methods for perforated domains

Fig. 10.4 Elasticity problem in the perforated domain (Figure 10.3). Comparison of solutions in: Fine scale (left) D O F = 19734, Coarse scale offline, D O F = 386 (middle), Coarse scale online without adaptivity, D O F = 548 (right). Top: u 1 . Bottom: u 2 .

10.4.2 Stokes equations in perforated domain In the final example, we consider the Stokes operator (10.6) with zero velocity = (0, 0) on ∂Ω. For the fine-scale approximation u = (0, 0) on ∂Ω  ∩ ∂B  and ∂u ∂n of the Stokes problem, we use P2 elements for velocity and piecewise constants for pressure. To improve the accuracy of multiscale solutions, we have enriched velocity spaces by adding online velocity basis. The errors will be measured in relative L 2 and H 1 norms for velocity and L 2 norm for pressure ||eu || L 2 = eu  L 2 (Ω  ) = ||e p || L 2 (Ω  ) =

u − u H  L 2 (Ω  ) u L 2 (Ω  )

, ||eu || H 1 = eu  H 1 (Ω  ) =

u − u H  H 1 (Ω  ) u H 1 (Ω  )

,

 p¯ − p H  L 2 (Ω  ) ,  p ¯ L 2 (Ω  )

where (u, p) and (u H , p H ) are fine-scale and coarse-scale solutions, respectively, for velocity

and pressure, and p¯ is the cell average of the fine-scale pressure, that is, p¯ = |K1 | K  p for all K i ∈ T H . i i We present numerical results for the Stokes problem for the perforated domain depicted in Figure 10.3. The solutions are shown in Figure 10.5. In the figure, the

10.5 Convergence results

245

x1 -component and x2 -component of the velocity solution are shown in the first and second rows, and the pressure solution is presented in the third row. The three columns contain the fine-scale, coarse-scale offline, and coarse-scale online solutions. In both cases, we observe that the offline velocity solution is not able to capture the low values at the corners of the domain. Some features between inclusions also do not appear correctly in the offline solution. For example, in Figure 10.5, for the domain that has big inclusions with some extremely small inclusions around, we see bad behavior of the offline solution. The low values of the velocity solution in the x2 component along the right boundary are almost missing in the offline solution. The offline velocity solutions in both components around inclusions are still very poor. These observations highlight the advantage of the online method. We performed other tests for different perforated domains, and the results also suggest that the online method is quite necessary. Now, we turn our attention to velocity L 2 (Ω  ), H 1 (Ω  ) errors and pressure 2 L (Ω  ) error presented in Table 10.2, Table 10.3 and Table 10.4. We consider different numbers of the initial basis on each coarse neighborhood. In this case, we observe that the online approach works better if we start with a more initial basis. For example, the velocity H 1 error is 41.558% with one initial velocity basis, and reduces to 9.499% after adding one online basis. However, it reduces from 23.471% to 3.498% with two initial basis without online enrichment. This implies that it is better to start with two or more initial basis in order to see that the more the online basis are used, the smaller the errors become. We also observe that the online approach with adaptivity reduces errors faster. Compared to the two indicators, we see that the first error indicator (see (10.16)) for the adaptive online method gives better results for any number of initial basis. One can also find that the pressure error also reduces significantly when we only enrich the velocity space.

10.5 Convergence results The result in [111] has shown the convergence for the online adaptive method applied to elliptic problems, and the same results can be applied to elasticity problem. In this section, we will present the convergence theory for the adaptive online GMsFEM for Stokes problem. For proofs of these results, see [119]. First, the following inf-sup condition for the approximation of Stokes problem using offline GMsFEM is proved. This ensures that the method, with both offline and online basis functions, is well-posed. Lemma 10.1. For all p ∈ Q H,off , there is a constant Cinfsup > 0, such that

sup u∈VH,off

div(u) p ≥ Cinfsup  p L 2 (Ω  ) . u H 1 (Ω  ) Ω

(10.26)

246

10 Multiscale methods for perforated domains

Fig. 10.5 Stokes problem. Fine-scale and multiscale solutions for velocity and pressure (u 1 (Top), u 2 (Middle) and p (Bottom)) in perforated domain (Figure 10.3). Left: fine-scale solution, DOF = 113936. Middle: multiscale solutions using 1 multiscale basis function for velocity, DOF = 452, velocity L 2 error is 15.264%. Right: multiscale solutions after 2 online iterations without adaptivity, D O F = 524, velocity L 2 error is 0.484%.

Now, we will present the convergence result of the online adaptive enrichment scheme for the Stokes problem. First, we define a reference solution by (u, p) ∈ (H01 (Ω  ))2 × Q off which solves L (u, p), (v, q)Ω  = (( f, 0), (v, q))Ω  , for all (v, q) ∈ (H01 (Ω  ))2 × Q off . (10.27) Notice that the solution of (10.27) and the solution of (10.10) have a difference proportional to the coarse mesh size H . We also define a snapshot solution by (u, ˆ p) ˆ ∈ VH,snap × Q off which solves

10.5 Convergence results

247 ||eu || L 2 (%)

||eu || H 1 (%)

||e p¯ || L 2 (%)

15.264

41.558

49.675

488 (1)

1.879

9.499

8.360

524 (2)

0.474

3.610

2.298

596 (4)

0.247

2.452

0.736

740 (8)

0.143

1.371

0.181

489 (3)

0.576

4.613

2.700

522 (5)

0.293

3.029

0.577

597 (10)

0.185

1.775

0.407

739 (20)

0.138

1.277

0.089

DOF (# iter) 452 Not adaptive

Adaptive, ηi2 = ri2

−1 Adaptive, ηi2 = ri2 λi+1

518 (2)

0.476

3.614

2.313

552 (3)

0.298

2.914

0.699

620 (5)

0.208

1.965

0.438

757 (11)

0.136

1.305

0.147

Table 10.2 Stokes problem for perforated domain (Figure 10.3). One offline basis functions (θ = 0.7). ||eu || L 2 (%)

||eu || H 1 (%)

||e p¯ || L 2 (%)

5.290

23.417

13.435

730 (1)

0.310

3.498

0.617

766 (2)

0.116

1.727

0.143

838 (4)

0.066

0.950

0.046

982 (8)

0.028

0.485

0.019

736 (3)

0.128

1.811

0.160

765 (5)

0.079

1.184

0.058

840 (10)

0.042

0.651

0.029

988 (20)

0.015

0.305

0.009

DOF (# iter) 694 Not adaptive

Adaptive, ηi2 = ri2

−1 Adaptive, ηi2 = ri2 λi+1

762 (2)

0.116

1.729

0.143

796 (3)

0.087

1.278

0.113

865 (5)

0.055

0.819

0.035

1001 (9)

0.022

0.376

0.014

Table 10.3 Stokes problem for perforated domain (Figure 10.3). Two offline basis functions (θ = 0.7).

248

10 Multiscale methods for perforated domains ||eu || L 2 (%)

||eu || H 1 (%)

||e p¯ || L 2 (%)

0.799

7.684

1.641

972 (1)

0.051

0.893

0.108

1008 (2)

0.017

0.390

0.014

1080 (4)

0.009

0.249

0.004

1224 (8)

0.004

0.136

0.001

Adaptive, ηi2 = ri2 974 (3) 0.021

0.466

0.031

1003 (5)

0.011

0.289

0.009

1081 (10)

0.006

0.177

0.003

1233 (20)

0.003

0.097

0.001

−1 Adaptive, ηi2 = ri2 λi+1 972 (1) 0.051

0.893

0.108

1006 (2)

0.017

0.390

0.143

1108 (5)

0.007

0.208

0.004

1244 (9)

0.003

0.122

0.002

DO F (# iter) 936 Not adaptive

Table 10.4 Stokes problem for perforated domain (Figure 10.3). Four offline basis functions (θ = 0.7).



 L (u, ˆ p), ˆ (v, q) Ω  = (( f, 0), (v, q))Ω  , for all (v, q) ∈ VH,snap × Q off . (10.28)

We notice that the difference u − u ˆ H 1 (Ω  ) represents an irreducible error. Furthermore, standard finite element analysis shows that u − u H  H 1 (Ω  ) ≤ u −  u H  H 1 (Ω  )

(10.29)

for any  u H ∈ VH,off . Next, we have the following a-posteriori error bound for the offline GMsFEM (10.11). The notation a  b means that there is a generic constant C > 0, such that a ≤ Cb. Theorem 10.2. Let u be the reference solution defined in (10.27), uˆ be the snapshot solution defined in (10.28) and u H be the multiscale solution satisfying (10.11). Then, we have Nu

   1 uˆ − u H 2 1  ≤ Cs 1 + i,off Ri 2V ∗ , (10.30) H (Ω ) λli +1 i=1 where li is the number of offline basis functions used for the coarse neighborhood i,off ωi , and λ j is the j-th eigenvalue for the coarse neighborhood ωi . The constant Cs is the maximum number of coarse neighborhoods corresponding to coarse blocks. Moreover, we have

10.5 Convergence results

249

u − u H 2H 1 (Ω  ) ≤ 2Cs

Nu

 1+ i=1

1 i,off λli +1

2  Ri 2V ∗ + 2 u − uˆ  H 1 (Ω  ) .

(10.31)

We recall that the norm of the local residual Ri is defined in (10.24). We define a modified norm as |Ri (v)| ||Ri ||(V0i )∗ = sup , (10.32)  v∈V0i v H 1 (ωi )

where V0i ⊂ V i and the vectors v ∈ V0i satisfies Ω  div(v) q = 0 for all q ∈ Q H,off . It is easy to show that ||Ri ||(V0i )∗ ≤ ||Ri ||(V i )∗ . In the next theorem, we see that the convergence of the online adaptive GMsFEM for the Stokes problem. The theorem states that the method is convergent up to an irreducible error u − u ˆ H 1 (Ω  ) with enough number of offline basis functions. Theorem 10.3. Let u be the reference solution defined in (10.27), uˆ be the snapshot solution defined in (10.28) and u mH be the multiscale solution of (10.11) in the enrichment level m. Assume that li offline basis functions for the coarse neighborhood ωi are used as initial basis for the online procedure. Suppose that one online basis is added to a single coarse neighborhood ωi . Then, there is a constant D, such that u −

2 u m+1 H  H 1 (Ω  )

 ≤ (1 + δ3 )(1 + δ2 ) 1 + δ1 − θCs−1

i,off

λli +1 i,off

λli +1 + 1

uˆ − u mH 2H 1 (Ω  )

+ Du − u ˆ 2H 1 (Ω  ) ,

(10.33) where δ1 , δ2 , δ3 > 0 are arbitrary and D depends only on δi , i = 1, 2, 3. In addition, θ is the relative residual defined by Nu  Ri 2(V i )∗ . θ = ||Ri ||2(V i )∗ 0

i=1

We remark that, in order to obtain rapid convergence, one needs to choose li large is large. In this case, the quantity λli,off /(λli,off + 1) is close to enough so that λli,off i +1 i +1 i +1 one. Then, (10.33) shows that the resulting online adaptive enrichment procedure has a rapid convergence. Theorem 10.3 gives the convergence of the online adaptive enrichment procedure when one online basis is added at a time. One can also add online basis in nonoverlapping coarse neighborhoods. Using the same proof as Theorem 10.3, we obtain the following result. Theorem 10.4. Let u be the reference solution defined in (10.27), uˆ be the snapshot solution defined in (10.28) and u mH be the multiscale solution of (10.11) in the enrichment level m. Assume that li offline basis functions for the coarse neighborhood ωi are used as initial basis to the online procedure. Let S be the index set for the non-overlapping coarse neighborhoods where online basis functions are added. Then, there is a constant D such that

250

10 Multiscale methods for perforated domains i,off



2 −1 u − u m+1 H  H 1 (Ω  ) ≤ (1 + δ3 )(1 + δ2 ) 1 + δ1 − θC s min j∈S

λl +1 j i,off

λl +1 + 1 j



2 uˆ − u m H  H 1 (Ω  )

+ Du − u ˆ 2H 1 (Ω  ) ,

(10.34) where δ1 , δ2 , δ3 > 0 are arbitrary and D depends only on δi , i = 1, 2, 3. In addition, θ is the relative residual defined by θ=

Ri 2(V i )∗

Nu 

0

i∈S

Ri 2(V i )∗ .

i=1

We remark that the above result suggests that adding more online basis functions at each iteration will speed up the convergence. Lastly, we remark that the convergence for the pressure can be obtained using the inf-sup condition (10.26).

Chapter 11

Multiscale stabilization

11.1 Introduction In this chapter, we discuss multiscale stabilization following [69] on the example of multiscale convection-dominated diffusion with a high Peclet number. In these systems, besides finding a reduced order model, one needs to stabilize the system to avoid large errors [355]. Stabilization of multiscale methods for convection-diffusion cannot simply use a modified diffusion and requires more sophisticated techniques. In this chapter, we discuss an application of the GMsFEM in developing a PetrovGalerkin method (e.g., [147, 337, 338]). We consider a convection-diffusion equation in the form − ∇ · (κ∇u) + b · ∇u = f

(11.1)

with a high Peclet number, where κ is a diffusion tensor and b is the velocity vector [205, 355]. Both fields are characterized by multiscale spatial features. GMsFEM generates a dimensionally reduced space on a coarse grid that approximates the solution space by introducing local snapshot spaces and appropriate local spectral decompositions. However, a direct application of these approaches for singularly perturbed problems, such as convection-dominated diffusion, faces difficulties due to the poor stability of these schemes. Simplified stabilization techniques on a coarse grid are not efficient. We use the discontinuous Petrov-Galerkin (DPG) techniques following [80, 143– 145, 147, 433] to stabilize the system. We start with a stable fine-scale finite element discretization that fully resolves all scales of the underlying equation Au = f.

(11.2)

The system is written in a mixed framework using an auxiliary variable as follows: Rw + Au = f,

(11.3)

A w

(11.4)

T

= 0.

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_11

251

252

11 Multiscale stabilization

The variable w plays the role of a test function and the matrix R is related to the norm in which we seek to achieve stability. We assume that the fine-scale system gives w = 0, that is, it is discretely stable. In multiscale methods, one approximates the solution using a dimensionally reduced subspace for u. More precisely u≈



z iu φi ,

or

u ≈ Φz u .

i

The resulting system also needs a dimensionally reduced test space, w≈



z iw ψi

or

w ≈ Ψ zw .

i

The stabilization of (11.2) requires appropriate Φ and Ψ . We discuss the design of these spaces in the following. Within the DPG framework, one can achieve stability by choosing test functions w with global support [45, 146]. In this chapter, we discuss a novel test space that guarantees stability for singularly perturbed problems such as convection-dominated diffusion in a multiscale media with a high Peclet number. To generate a multiscale space for w, we use the theory for GMsFEM for mixed problems. We start by constructing a local snapshot space that approximates the global test functions. The snapshot spaces are augmented with local bubble functions. To reduce the dimension of this space, making the construction independent of the Peclet number, we propose a set of local spectral problems. In these local spectral problems, we use minimum energy snapshot vectors [79] and perform a local spectral decomposition with respect to the A A T norm. Our objective is to find a dimensionally reduced approximation, w N , of w, such that w − w N  is small. We can show that the approximation property of the test space is important to achieve stability (cf. [136]). We note that the least squares approach [55, 56, 202, 267] can also be used to achieve stability in the natural norm. Contrary to the traditional least squares approach, the proposed method minimizes the residual with some special weights related to the test functions. We discuss how to construct online basis [79], which uses residual information. Online basis functions speed-up convergence at a cost proportional to the number of added multiscale test functions, which are computed by solving local problems. In [79], we developed online basis functions for flow equations. As we discussed earlier that, by adding online basis functions, the error reduces by a factor of 1 − Λmin , where Λmin is the smallest eigenvalue for which the corresponding eigenvector is not included in the coarse test space. We observe this behavior in our numerical simulations. Our construction differs from [79]. We present a numerical example of a multiscale transport problem. More detailed studies can be found in [69]. In particular, we consider a heterogeneous velocity field and a constant diffusion such that the resulting Peclet number is high. In the example, we consider how the appropriate error (which is based on our stabilization) behaves as we increase the number of test functions. We observe that one needs several test

11.2 Preliminaries

253

functions per coarse degree of freedom to achieve an error close to the projection error of the solution of the span of the coarse degrees of freedom. The number of test functions does not change as we increase the Peclet number. We also show the performance of using online test functions.

11.2 Preliminaries We consider the following problem: −∇ · (κ∇u) + b · ∇u = f, in Ω u = 0, on ∂Ω where κ and b are highly heterogeneous multiscale spatial fields with a large ratio maxΩ (b)/ minΩ (κ). The weak formulation of this problem is to find u ∈ V = H01 (Ω), such that a(u, v) = l(v), ∀v ∈ V, where  a(u, v) = l(v) =



Ω Ω

κ∇u · ∇v + (b · ∇u)v, f v.

We start with a fine-grid (resolved) discretization of the problem and define u h to be the fine-grid finite element solution in the fine-grid space Vh , Ah and f h are the stiffness matrix and the source vector on the fine grid. Note that we have Ah u h = f h . We introduce an auxiliary variable (a test variable) and re-write the system in mixed form. In particular,  we consider the following problem. Find (u hP G , wh ) ∈  PG Vh × Vh , such that u h = φi (u h )i and wh = φi (wh )i solve 

Ah AhT Ah AhT 0



wh u hP G



 =

fh 0

 ,

where {φi } denotes the basis functions for fine-grid discretization. Since det (Ah ) = 0, we have u hP G = u h and wh = 0. Therefore, these two problems have the same solution. Our objective is to find a dimensionally reduced coarse approximation for wh , which can guarantee that the corresponding u hP G is a good approximation to u h . As before, we use T H to denote a conforming partition of the computational domain Ω. The set T H is the coarse grid and the elements of T H are called coarse elements. We only consider rectangular coarse elements to simplify the discussion and illustrations. Let Nc be the number of nodes in the coarse grid T H , and let

254

11 Multiscale stabilization

Fig. 11.1 Illustration of coarse neighborhoods and elements. Red designates a coarse element. Green designates two neighboring elements that share a common edge (used to construct test functions). Blue designates the union of all coarse elements that share a common vertex (used to construct trial functions).

{xi | 1 ≤ i ≤ Nc } be the set of nodes in the coarse grid (or coarse nodes for short). For each coarse node xi , we define a coarse neighborhood ωi by ωi =

 {K j ∈ T H ; xi ∈ K j }.

(11.5)

We use two neighboring elements sharing a common edge to construct the test functions. An example of this region is depicted in green in Figure 11.1.

11.3 Generalized multiscale finite element method for Petrov-Galerkin approximations Next, we discuss the construction of the multiscale basis functions for the trial space Voff and the test space Woff . In particular, we show that one needs a good approximation for wh in order to achieve discrete stability. We introduce the snapshot space and then the local spectral decomposition used to construct the multiscale basis functions. We define Au := −∇ · (κ∇u) + b · ∇u, and

A∗ u := −∇ · (κ∇u) − ∇ · (b u).

11.3 Generalized multiscale finite element method for Petrov-Galerkin approximations

255

11.3.1 Construction of the multiscale trial space Snapshot space We solve a local problem with specifically designed boundary conditions to construct the snapshot basis functions. For each coarse neighborhood ωl , we define a set of snapshot functions ψiωl , such that A(ψiωl ) := −∇ · (κ∇ψiωl ) + b · ∇ψiωl = 0, ψiωl (x j ) = δi j ,

in ωl , on ∂ωl ,

(11.6) (11.7)

where δi j is the discrete delta function defined on ∂ωl with respect to the fine grid. ωl = span{ψiωl }. The The local snapshot space for the trial space is defined by VH,snap snapshot functions and multiscale basis functions (offline space) are defined in the union of coarse elements that share a common vertex. We use ΨlT to denote the ωl . Here Vh (ωl ) is the change of basis matrix from the fine-grid space Vh (ωl ) to VH,snap restriction of Vh in ωl . Eigenproblem To construct the offline trial space, we solve the following eigenproblem: ωl l l )T Aωsnap v j = λ j Msnap vj, (Aωsnap

(11.8)

where l = ΨlT Aωh l Ψl Aωsnap

ωl Msnap = ΨlT Mhωl Ψl

and (λ j , v j ) is the j-th eigen-pair. In the above definition, Aωh l and Mhωl are the restrictions of the fine-scale stiffness matrix Ah and the fine-scale mass matrix Mh in ωl . We order the eigenvalues in increasing order and we use the first m eigenfunctions as the offline trial basis functions. Specifically, we define ξl, j = Φl v j and Voff = span{χl ξl, j |1 ≤ j ≤ m, 1 ≤ l ≤ Nc }, where {χl } is the partition of unity.

11.3.2 Construction of the multiscale test space Snapshot space The snapshot space for the test space consists of three components and is denoted 1 2 3 + W H,snap + W H,snap . Next, we will give the constructions for as W H,snap = W H,snap 1 2 3 1 W H,snap , W H,snap and W H,snap . In each coarse block K k , we define W H,snap (K k ) as 1 W H,snap (K k ) := {ψ snap ∈ Vh,0 (K k ) | A∗ (ψ snap ) = ξl, j in K k for some ξl, j ∈ VH,off }

256

11 Multiscale stabilization

where Vh (K k ) is the restriction of Vh in K k and Vh,0 (K k ) is the subspace of Vh (K k ) 1 (K k ) contains functions that containing functions that vanish on ∂ K k . The space Wsnap are solution of the adjoint problem on K k with a source term ξl, j and zero Dirichlet 1 1 1 boundary condition. The space W H,snap , is defined as W H,snap = ⊕k W H,snap (K k ). The 1 space W H,snap is considered as the space of multiscale bubble functions. We remark that we obtain perfect test functions (with perfect stabilization) if the above local problems are solved on the whole domain. 2 is defined as follows. For each coarse block K k , we The second space W H,snap define 2 W H,snap (K k ) := {ψ snap ∈ Vh (K k ) | A∗ (ψ snap ) = 0 in K k and ψ snap is linear on E ∈ ∂ K k }.

2 2 2 The space W H,snap is defined by W H,snap = ⊕k W H,snap (K k ) ∩ C 0 (Ω). Note that this space is similar to the classical multiscale finite element space. 3 . For each coarse edge E k , we define Finally, we give the definition for W H,snap snap K (E k ) as the set of all coarse blocks having the edge E k . Then, we find ψi,k ∈ Vh (K (E k )), such that

A∗ ψi,k := −∇ · (κ∇ψi,k ) − ∇ · (bψi,k ) = 0 in each K ∈ K (E k ), snap

snap

snap

snap ψi,k (x j ) snap ψi,k |∂ K (Ek )\Ek

=

δi0j

for all x j ∈ E k ,

= 0.

(11.9) (11.10) (11.11)

In the above system, δi0j is the discrete delta function defined on E k with respect 3 (E k ) = to the fine mesh and is zero on the boundary of E k . Then we define Wsnap snap 3 3 span{ψi,k }, and Wsnap = ⊕k Wsnap (E k ). We remark that Wsnap = Vh . Lemma 11.1. For each u ∈ V off , there exists a test function φ ∈ W snap , such that a(v, φ) = (u, v), ∀v ∈ Vh ,

(11.12)

where (u, v) denotes the L 2 inner product. The proof can be found in [69]. Eigenproblem 3 is proportional to Among the three parts of the test space, the dimension of Wsnap the number of fine-grid blocks, and thus proportional to the Peclet number of the problem. Consequently, our objective is to reduce the degrees of freedom associated 3 1 2 . Both the dimensions of Wsnap and Wsnap are proportional to the number with Wsnap of coarse-grid degrees of freedom. We consider two different eigenvalue problems 3 . to construct the offline test space for Wsnap 3 The first eigenvalue problem for Wsnap (E k ): In this eigenvalue problem, we will use the edge values of the snapshot solutions.

11.3 Generalized multiscale finite element method for Petrov-Galerkin approximations

257



 K (E k )

(A T v)(A T ψ j ) = λ

vψ. Ek

The eigenvalues go to ∞ as we refine the fine mesh. 3 The second eigenvalue problem for Wsnap (E k ): This eigenvalue problem is motivated by [79], where we construct minimum energy snapshot solutions and perform a local spectral decomposition using the same norms. More precisely   T T ˜ (A v)(A ˜ (A T v)(A T ψ j ), ψj) = λ K (E k )

K (E k )

 T ˜ ˜ where ψ˜ = argminψ∈{v∈W { K (Ek ) (A T ψ)(A ψ)}. In this case, the eigen3 ˜ snap |v| E k =ψ| E k } values are always smaller than 1. We will arrange the eigenvalues of the above spectral problems in increasing order, and choose the first L k eigenfunctions as the offline test basis functions. The 3 . The final test space Woff is defined span of these basis functions is denoted as Woff 3 1 2 by Wsnap ⊕ Wsnap ⊕ Woff .

11.3.3 Global coupling We can use the above trial and test spaces to obtain a reduced system for the multiscale solution. In particular, the multiscale solution is computed by solving   T    T Θ fh wms Θ Ah AhT Θ Θ T Ah Ξ = . (11.13) PG 0 u ms 0 Ξ T AhT Θ The columns of Θ consist of the computed multiscale test functions while the columns of Ξ consist of the computed multiscale trial functions.

11.3.4 Summary of the procedures for the offline method We summarize the procedures for the offline method as follows: snap

Step 1: For each coarse neighborhood ωl , we find φi,l satisfying the local problem in snap (11.6). Next, we define the local snapshot space by V snap (ωl ) = span{φi,l }. Step 2: For each coarse neighborhood ωi , we solve the eigenproblem (11.8) in the V snap (ωl ) and select the first m eigenfunctions, ξl,i , as the offline trial basis functions. The trial offline space is defined by Voff = span{χl ξl, j |1 ≤ j ≤ m, 1 ≤ l ≤ Nc }, where {χl } is the partition of unity. Step 3: In each coarse block K k , we solve the local adjoint problem by setting RHS 1 to be the trial basis functions. We define Wsnap (K k ) as 1 Wsnap (K k ) := {ψ snap ∈ Vh,0 (K k ) | A∗ (ψ snap ) = ξl, j in K k for some ξl, j ∈ Vo f f }.

1 1 1 Next, we define Wsnap as Wsnap = ⊕k Wsnap (K k ).

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11 Multiscale stabilization

2 Step 4: We define Wsnap as follows. For each coarse block K k , we define 2 Wsnap (K k ) := {ψ snap ∈ Vh (K k ) | A∗ (ψ snap ) = 0 in K k and ψ snap is linear on E ∈ ∂ K k }.

2 2 2 The space Wsnap is defined by Wsnap = ⊕k Wsnap (K k ) ∩ C 0 (Ω). snap Step 5: For each edge E k , we find ψi,k ∈ Vh (K (E k )) satisfying (11.9). Then we snap 3 define Wsnap (E k ) = span{ψi,k }. 3 (E k ) and select Step 6: For each edge E k , we solve the eigenproblem in the Wsnap the first L k eigenfunctions, μi,k , as the offline trial basis functions. We define 3 3 1 as Woff = span{μi,k }. The final test space Woff is defined by Wsnap ⊕ Woff 3 2 Wsnap ⊕ Woff . Step 7: We solve the global system (11.13) by using the spaces Voff and Woff .

11.3.5 Discussion Next, we discuss the approximation properties for the test space and how the definition of this space affects the discrete stability of the resulting method. To simplify the discussion, we introduce some notation. Let N and M be the dimensions for the test and trial spaces, respectively. Thus, we can write   T   T  w N ,M ΘN fh Θ N Ah AhT Θ N Θ NT Ah Ξ M = . (11.14) T T u NP G,M ΞM Ah Θ N 0 0 For simplicity, we denote by Θ∞ the snapshot matrix that contains all snapshot vectors in the test space and similarly for the trial space Ξ∞ . Therefore, the following statements are satisfied: • w∞,∞ = 0. • u ∞,M is a projection of u ∞,∞ onto Ξ M u ∞,M = Ξ M u ∞,∞ . • Our objective is to find the smallest possible N and M0 , such that u NP G,M − PG PG PG  u ∞,∞ − u ∞,M  for any M, when M > M0 . u ∞,∞ • The inf-sup condition for our discrete saddle-point problem can be written as Θ NT Ah Ξ M T sup ≥ Cin f sup (Ξ M MΞ M )1/2 . (11.15) T T 1/2 (Θ A A Θ ) Θ N h h N The inf-sup condition implies that Cin f sup = inf

u=Ξ M q

 AT Θ N (u)l 2 . ul 2

Next, we take u = A T z. Then, the projection of u onto A T Θ N is  AT Θ N (u) = A T Θ N ((A T Θ N )T A T Θ N )−1 (A T Θ N )T A T z = A T Θ N (Θ NT A A T Θ N )−1 Θ NT A A T z.

(11.16)

11.3 Generalized multiscale finite element method for Petrov-Galerkin approximations

259

We define the Θ projection in the B norm to be Θ,B (z) = Θ(Θ T BΘ)−1 Θ T Bz. Thus, T A A T Θ )−1 Θ T A A T Θ (Θ T A A T Θ )−1 Θ T A A T z  A T Θ N (u)2 = z T A A T Θ N (Θ N N N N N N N T A A T Θ )−1 Θ T A A T z = z T A A T Θ N (Θ N N N

= Θ N ,A A T (z)2A A T .

(11.17)

Also, ul 2 = z A AT . Thus, Cin f sup =

inf

u=A T z,u=Ξ

Mq

Θ N ,A AT (z) A AT . z A AT

(11.18)

If the inf-sup is satisfied, then we have PG PG w N ,M − w∞,∞  + u NP G,M − u ∞,∞   w N ,M − w∞,∞  +  u P G N ,M − u ∞,∞  PG . = 0 + u P G ∞,M − u ∞,∞

(11.19)

PG PG PG − u ∞,∞   u P G ∞,M − u ∞,∞ . w∞,M − 0 + u ∞,M

(11.20)

From here, we have

Because N = ∞, u P G ∞,M =  u P G ∞,M , we get PG u P G ∞,M − u ∞,∞ . w∞,M  

• The discrete inf-sup condition can be shown if for any z (e.g., z = A−T Ξ M q), there exists z 0 in the space spanned by Θ N (i.e., z 0 = Θ N zr ), such that z − z 0  A AT ≤ δz A AT ,

(11.21)

for some δ < 1. In multiscale methods (in particular, in our works [79, 108]), we reduce the error in z − z 0  A AT by selecting appropriate multiscale spaces (as those used herein). In addition, this procedure can be done adaptively. Thus, by selecting a sufficient number of multiscale basis functions, we can reduce the error z − z 0  A AT and achieve the stability sought. We do not have rigorous error estimates, but study this problem numerically. We emphasize that we need good approximation properties in the test space (as in [136]), which is due to the primal formulation and the choice of z in (11.16).

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11 Multiscale stabilization

11.3.6 Online test basis construction (residual-driven correction) One can use residual information to construct online basis functions. Online basis functions use global information and thus accelerate convergence. In [79], we discuss the online basis construction for flow equations using a mixed formulation. We use the local residual to construct an online basis function locally in each non-overlapping coarse-grid region ωi . The offline solution in the fine-scale test space (w∞,M , u ∞,M ) ∈ Vh × Vo f f satisfies      fh Ah AhT Ah Ξ M w∞,M = (11.22) T T ΞM Ah 0 u ∞,M 0 and the multiscale solution (w N ,M , u N ,M ) ∈ Wo f f × Vo f f satisfies    T   T w N ,M ΘN fh Θ N Ah AhT Θ N Θ NT Ah Ξ M . = T T ΞM Ah Θ N 0 u N ,M 0

(11.23)

The above motivates the following local residual operator Ri , which is defined as Ri : Vh (ωi ) → R by

 Ri (v) = v T (Ah AhT )(i) Θ N w N ,M + (Ah )(i) Ξ M u N ,M − f h and the local residual norm, Ri  is defined by |Ri (v)| Ri  = sup , v∈Vh (ωi ) v T (Ah AhT )(i) v T where (Ah AhT )(i) and A(i) h are local sub-matrices of A h A h and A h , which correspond to the coarse-grid subdomain ωi . Next, we use the local residual to construct the local test basis, φ(i) on ∈ Vh (ωi ), such that v T (Ah AhT )(i) φ(i) on = Ri (v), ∀v ∈ Vh (ωi ).

In [79], we show that if online basis functions are constructed using the second eigenE E ), where Λmin is the value problem, then the error will decrease at a rate (1 − min E Λmin 3 minimum of the eigenvalues of the spectral problem defined on Wsnap (E) corresponding to eigenfunctions not chosen as basis. Consequently, using online basis functions, E > 0. we can achieve the discrete inf-sup stability in one iteration provided min E Λmin Online test basis enrichment algorithm (1) , First, we choose an offline trial space, Voff and an initial offline test space, Woff by fixing the number of basis functions for each coarse neighborhood. Next, we (m) and compute the multiscale solution construct a sequence of online test spaces Woff (m) , u (m) (wms ms ) by solving Eq. (11.23). The test space is constructed iteratively for m = 1, 2, 3, . . . , by the following algorithm:

11.4 Numerical results

261

(m) Step 1: Find the multiscale solution in the current space. Solve for (wms , u (m) ms ) ∈ (m) Woff × Voff , such that  (m) T   (m)   (m) T  (m) (m) T (Θoff ) Ah (m) (Θoff ) Ah AhT Θoff wms (Θoff ) f h off . = (m) T T (m) u (m) 0 (Ξoff ) Ah Θoff 0 ms

Step 2: For each coarse region ωi , compute the online basis, φ(i) on ∈ Vh (ωi ), such that v T (Ah AhT )(i) φ(i) on = Ri (v), ∀v ∈ Vh (ωi ). Step 3: Enrich the test space by setting (m+1) (m) = Woff + span{φ(i) Woff on }.

We remark that in each iteration, we perform the above procedure on nonoverlapping coarse neighborhoods; see [79].

11.4 Numerical results In this section, we present representative numerical examples. In all our examples, {χi } isamultiscalepartitionofunity.Ineachcoarsespace,wecomparethel2 projectionerror and the L 2 error for the multiscale solution. For simplicity, we refer to “the multiscale error” as the error between the multiscale solution and the exact solution, and “the projection error” as the error between the exact solution and its L 2 projection onto the span of the coarse trial space. We also assume κ is a constant and b is a multiscale field. In particular, the velocity fields contain oscillations and cells (eddies, separatrices, and/or layers) within a single coarse block of the discretization, and thus we do not have a single streamlined direction per coarse block. The method can easily handle multiscale diffusion coefficients. The fine-grid problem is always chosen such that the local Peclet number is about 1 ensuring a stable fine discretization. All coarse discretizations have a Peclet number at least an order of magnitude larger than 1. We analyze the performance of the trial and test spaces proposed in the previous section. We pay special attention to the effect of eigenvalue problem on the performance of the discrete system and discuss this for each example.

Example 1: offline test basis In this example, we take the velocity field to be κ = 1,

 √ sin(18 2π y) , b = 200 0 f = 1,



262

11 Multiscale stabilization #basis (trial, test) L 2 (projection error) (1,1) 20.12% (1,3) 11.25% (1,5) 4.03% (1,7) 3.93%(3.93%) (3,1) 19.99% (3,3) 13.02% (3,5) 3.31% (3,7) 3.23%(3.23%) (5,1) 13.93% (5,3) 9.14% (5,5) 2.90% (5,7) 2.74%(2.70%)

Table 11.1 Errors for test space derived using Eigenproblem 2 for Example. Coarse and fine mesh sizes are H = 1/10 and h = 1/200, respectively. The projection errors are in parentheses.

which corresponds to solving the flow equations with a channelized permeability field. The numerical results are presented in Table 11.1 (the mesh sizes for the coarse and fine spaces are H = 1/10 and h = 1/200). We observe that the multiscale error is 20.12% for one trial function per coarse block and one test function per coarse interface, while the error reduces to the projection error of 3.93% when we select 7 test functions interface. As before, for 5 trial functions per coarse block, it takes 7 test functions per coarse interface to reduce the error to the projection error from 13.93%. For this discrete problem setup, the smallest eigenvalue is 0.9952 for 7 test functions per edge when minimal energy functions are used (Eigenproblem 2 in Section 11.3.2).

Example 2: online test basis enrichment In this section, we present some numerical results, which use online test basis functions to stabilize the system. In Table 11.2, we show the convergence history for the online test basis enrichment for Example 1. In these cases, with only one iteration, the multiscale error becomes similar to the projection error. In the second iteration, the multiscale error converges to the projection error.

11.4 Numerical results

263 #basis (trial, test) #iter L 2 (projection error) 0 20.12% (1,1) 1 3.93% 2 3.92%(3.92%) 0 11.15% (1,3) 1 3.92% 2 3.92%(3.92%) 0 13.93% (5,1) 1 3.24% 2 2.72%(2.70%) 0 9.14% (5,3) 1 2.74% 2 2.70%(2.70%)

Table 11.2 Error evolution as online basis functions are added to the system (test space derived using Eigenproblem 2 for Example). Coarse and fine mesh sizes are H = 1/10 and h = 1/200, respectively. The projection errors are in parentheses.

Chapter 12

GMsFEM for selected applications

In this chapter, we present some selected applications of GMsFEM. More applications (e.g., for poroelasticity [67, 68]) can be found in the literature. The notations used in the chapter are application-specific and can differ from those used in the rest of the book.

12.1 Multiscale methods for elasticity equations In this section, we discuss the application of GMsFEM to the steady state elasticity equation in heterogeneous media ∂ (ci jkl (x)ekl (u)) = f j (x), ∂xi

(12.1)

∂u l k + ∂x ) and ci jkl (x) is a multiscale field with a high contrast. where ekl (u) = 21 ( ∂u ∂xl k We will present several choices for snapshot spaces, offline spaces, and global coupling. For the snapshot space, we consider two choices. The first one consists of all fine-grid functions in each coarse patch, and the second one consists of harmonic extensions. We consider several choices for the local spectral decomposition including the oversampling approach. To couple multiscale basis functions constructed in the offline space, we consider two methods, conforming Galerkin (CG) approach and discontinuous Galerkin (DG) approach based on symmetric interior penalty method for (12.1).

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_12

265

266

12 GMsFEM for selected applications

We present numerical results where we study the continuous and discontinuous Galerkin methods using various snapshot spaces as well as with and without the use of oversampling. We consider highly heterogeneous coefficients that contain high contrast. Our numerical results show that the proposed approaches allow approximating the solution accurately with a few degrees of freedom. In particular, when using the snapshot space consisting of harmonic extension functions, we obtain better convergence results. In addition, oversampling methods and the use of snapshot spaces constructed in the oversampled domains can substantially improve convergence.

12.1.1 Preliminaries In this section, we present the general framework of GMsFEM for linear elasticity in high-contrast media. Let Ω ⊂ R2 (or R3 ) be a bounded domain representing the elastic body of interest, and let u = (u 1 , u 2 ) be the displacement field. The strain tensor (u) = (i j (u))1≤i, j≤2 is defined by (u) = where ∇u = (

1 (∇u + ∇u T ), 2

∂u i )1≤i, j≤2 . In the component form, we have ∂x j ∂u j  1  ∂u i , 1 ≤ i, j ≤ 2. + i j (u) = 2 ∂x j ∂xi

We assume the medium is isotropic. Thus, the stress tensor σ(u) = (σi j (u))1≤i, j≤2 is related to the strain tensor (u) in the following way: σ = 2μ + λ∇ · u I, where λ > 0 and μ > 0 are the Lamé coefficients. We assume that λ and μ have highly heterogeneous spatial variations with high contrasts. Given a forcing term f = ( f 1 , f 2 ), the displacement field u satisfies the following: − ∇ · σ = f,

in Ω

(12.2)

∂σi2  = fi , ∂x2

in Ω, i = 1, 2.

(12.3)

or in component form −

 ∂σ

i1

∂x1

+

For simplicity, we will consider the homogeneous Dirichlet boundary condition u = 0 on ∂Ω.

12.1 Multiscale methods for elasticity equations

267

As before, we let T H be a standard coarse-grid triangulation of the domain Ω, where H > 0 is the mesh size (see Figure 1.8). The set of all coarse-grid edges is denoted by E H and the set of all coarse-grid nodes is denoted by S H . We also use Nc to denote the number of coarse-grid nodes, and N to denote the number of coarse-grid blocks. In addition, we let T h be a fine-grid conforming refinement of the triangulation T H . Let Vh be a finite element space defined on the fine grid. The fine-grid solution u h can be obtained as (12.4) a(u h , v) = ( f, v), ∀v ∈ Vh , where     2μ(u) : (v) + λ∇ · u ∇ · v d x, ( f, v) = a(u, v) = Ω

and (u) : (v) =

2 

i j (u)i j (v),

i, j=1

f ·v =

2 

Ω

f i vi .

f · v d x (12.5)

(12.6)

i=1

For each vertex xi ∈ S H in the coarse grid, we define the coarse neighborhood ωi by ωi =

 {K j : K j ⊂ T H , xi ∈ K j }.

That is, ωi is the union of all coarse-grid blocks K j having the vertex xi . A snapshot ωi is constructed for each coarse neighborhood ωi . The spectral problem is space VH,snap solved in the snapshot space, and eigenfunctions corresponding to dominant modes are used as the final basis functions. To obtain conforming basis functions, each of these selected modes will be multiplied by a partition of the unity function. The ωi , which is called the offline space for the i-th resulting space is denoted by VH,off coarse neighborhood ωi . The global offline space VH,off is then defined as the linear ωi , for i = 1, 2, · · · , Nc . The CG coupling can be formulated as span of all these VH,off to find u CG ∈ V such that H,off H a(u CG H , v) = ( f, v), ∀v ∈ V H,off .

(12.7)

The DG coupling can be constructed in a similar fashion (see Section 6.2.3). A Ki is constructed for each coarse-grid block K i . This space is snapshot space VH,snap Ki the offline space VH,off for the i-th coarse-grid block. The global offline space VH,off Ki is then defined as the linear span of all these VH,off , for i = 1, 2, · · · , N . The DG coupling can be formulated as follows: find u DG ∈ V H,off such that H aDG (u DG H , v) = ( f, v), ∀v ∈ V H,off , where the bilinear form aDG is defined as

(12.8)

268

12 GMsFEM for selected applications

aDG (u, v) = a H (u, v) −

    { σ(u) n E } · [[v]] + { σ(v) n E } · [[u]] ds+ E∈E H

E

 γ { λ + 2μ}}[[u]] · [[v]] ds h E E∈E H (12.9)

with a H (u, v) =



a HK (u, v), a HK (u, v) =

K ∈T H

 

 2μ(u) : (v) + λ∇ · u∇ · v d x,

K

(12.10) where γ > 0 is a penalty parameter, n E is a fixed unit normal vector defined on the coarse edge E, and σ(u) n E is a matrix-vector product. Note that, in (12.9), the average and the jump operators are defined in the classical way (see Section 6.2.3). We briefly repeat them here. Consider an interior coarse edge E ∈ E H and let K + and K − be the two coarse-grid blocks sharing the edge E. For a piecewise smooth function G, we define { G}} =

1 + (G + G − ), 2

[[G]] = G + − G − ,

on E,

where G + = G| K + and G − = G| K − , and we assume that the normal vector n E is pointing from K + to K − . For a coarse edge E lying on the boundary ∂Ω, we define { G}} = [[G]] = G,

on E,

where we always assume that n E is pointing outside of Ω. For vector-valued functions, the above average and jump operators are defined component-wise.

12.1.2 Construction of multiscale basis functions This section is devoted to the construction of multiscale basis functions. We will present the constructions for both the CG and DG couplings.

Basis functions for CG coupling We begin with the construction of local snapshot spaces. Let ωi be a coarse neighborhood, i = 1, 2, · · · , Nc . We will define two types of local snapshot spaces. The first type of local snapshot space is

12.1 Multiscale methods for elasticity equations

269

ωi ,1 VH,snap = Vh (ωi ), ωi ,1 where Vh (ωi ) is the restriction of the conforming space to ωi . Therefore, VH,snap contains all possible fine-scale functions defined on ωi . The second type of local snapshot space contains all possible harmonic extensions. Next, let Vh (∂ωi ) be the restriction of the conforming space to ∂ωi . Then we define the fine-grid delta function δk ∈ Vh (∂ωi ) on ∂ωi by  1, l = k δk (xl ) = 0, l = k,

where {xl } are all fine-grid nodes on ∂ωi . Given δk , we find u k1 and u k2 by −∇ · σ(u k1 ) = 0,

in ωi

u k1 = (δk , 0)T , and

−∇ · σ(u k2 ) = 0,

on ∂ωi

in ωi

u k2 = (0, δk )T ,

on ∂ωi .

(12.11)

(12.12)

The linear span of the above harmonic extensions is our second type of local snapshot ωi ,2 ωi ωi ,1 . To simplify the notations, we will use VH,snap to denote VH,snap or space VH,snap ωi ,2 VH,snap when there is no need to distinguish the two types of spaces. Moreover, we write ωi = span{ψkωi , k = 1, 2, · · · , M i,snap }, VH,snap ωi where M i,snap is the number of basis functions in VH,snap and ψkωi as before refers to local snapshots. For the construction of conforming multiscale basis functions, we need a partition of unity function χi for the coarse neighborhood ωi . One choice of a partition of unity function is the coarse-grid hat functions Φi , that is, the piecewise bilinear function on the coarse grid having value 1 at the coarse vertex xi and value 0 at all other coarse vertices. The other choice is the multiscale partition of unity function, which is defined in the following way. Let K j be a coarse-grid block having the vertex xi . Then we consider

−∇ · σ(ζi ) = 0,

in K j

ζi = (Φi , 0)T ,

on ∂ K j .

(12.13)

270

12 GMsFEM for selected applications

i = (ζi )1 . The values of Φ i on Then we define the multiscale partition of unity as Φ the other coarse-grid blocks are defined similarly. Based on the analysis, we define the spectral problem as     2μ(u) : (v) + λ∇ · u ∇ · v d x = ξ κu ˜ · v d x, (12.14) ωi

ωi

where ξ denotes the eigenvalue and κ˜ =

Nc  (λ + 2μ)|∇χi |2 .

(12.15)

i=1

The above spectral problem (12.14) is solved in the snapshot space. We let (ηk , ξk ) be the eigenfunctions and the corresponding eigenvalues (in a discrete setup). Assume that ξ1 ≤ ξ2 ≤ · · · ≤ ξ M i,snap . Then the first L i eigenfunctions will be used to construct the local offline space. We define i,snap M ωi ηlk ψkωi , l = 1, 2, · · · , L i , (12.16) φl = k=1

where ηlk is the k-th component of ηl . The local offline space is then defined as ωi = span{χi φlωi , l = 1, 2, · · · , L i }. VH,off

Next, we define the global continuous Galerkin offline space as ωi , i = 1, 2, · · · , Nc }. VH,off = span{VH,off

Basis functions for DG coupling We discuss two types of snapshot spaces as in the CG case. The first type of local snapshot space is all possible fine-grid bilinear functions defined on K i . The second Ki for the coarse-grid block K i is defined as the type of local snapshot space VH,snap linear span of all harmonic extensions. Specifically, given δk , we find u k1 and u k2 by −∇ · σ(u k1 ) = 0,

in K i

u k1 = (δk , 0)T , and

−∇ · σ(u k2 ) = 0,

on ∂ K i

in K i

u k2 = (0, δk )T ,

on ∂ K i .

(12.17)

(12.18)

12.1 Multiscale methods for elasticity equations

271

Ki The linear span of the above harmonic extensions is the local snapshot space VH,snap . We also write Ki = span{ψki , k = 1, 2, · · · , M i,snap }, VH,snap ωi where M i,snap is the number of basis functions in VH,snap and ψki is the k-th basis function. We define the spectral problem as  

ξ λ + 2μ u · v ds, (12.19) 2μ(u) : (v) + λ∇ · u∇ · v d x = H ∂ Ki Ki

where ξ denotes the eigenvalues and λ + 2μ is the maximum value of { λ + 2μ}} on ωi ∂ K i . The above spectral problem (12.19) is again solved in the snapshot space VH,snap . i,snap We let (ηk , ξk ), for k = 1, 2, · · · , M be the eigenfunctions and the corresponding eigenvalues. Assume that ξ1 ≤ ξ2 ≤ · · · ≤ ξ M i,snap . Then the first L i eigenfunctions will be used to construct the local offline space. Indeed, we define i,snap M ωi ηlk ψki , l = 1, 2, · · · , li , (12.20) φl = k=1

where ηlk is the k-th component of ηl . The local offline space is then defined as Ki = span{ψli , l = 1, 2, · · · , li }. VH,off

The global offline space is also defined as Ki VH,off = span{VH,off , i = 1, 2, · · · , N }.

Oversampling technique For the harmonic extension snapshot case, we solve equation (12.11) and (12.12) in ωi+ (oversampled region) instead of ωi for CG case, and solve equation (12.17) and (12.18) in K i+ instead of K i for DG case. We denote the solutions as ψk+,ωi and ψk+,K i , and their restrictions on ωi and K i as ψkωi and ψkK i , respectively. Using the +,snap single index notation, we write the snapshot functions ψk+,ωi and ψk+,K i as ψ j and snap the snapshot functions ψkωi and ψkK i as ψ j . We reorder these functions according to eigenvalue behavior and write   +,snap +,snap snap snap + and Rsnap = ψ1 , . . . , ψ Msnap , = ψ1 , . . . , ψ Msnap Rsnap where Msnap denotes the total number of functions kept in the snapshot space.

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12 GMsFEM for selected applications

For CG, we define the following spectral problems in the space of snapshot:

or where A = [akl ] =

T + + Rsnap A Rsnap Ψk = ζ(Rsnap )T M + Rsnap Ψk ,

(12.21)

+ + + + )T A+ Rsnap Ψk = ζ(Rsnap )T M + Rsnap Ψk , (Rsnap

(12.22)

 

A+ = [akl+ ] =

ωi



M + = [m + kl ] =

snap

2μ(ψk 

ωi+



ωi+

snap

) : (ψl

+,snap

2μ(ψk

+,snap

κψ ˜ k

snap

) + λ∇ · ψk +,snap

) : (ψl +,snap

· ψl

snap

∇ · ψl

+,snap

) + λ∇ · ψk

 d x, +,snap

∇ · ψl

 d x,

d x,

where κ˜ is defined through (12.15). The local spectral problem for DG coupling is defined as

or

+ + + + (Rsnap )T A+ Rsnap Ψk = ζ(Rsnap )T M1+ Rsnap Ψk

(12.23)

+ + + + (Rsnap )T A+ Rsnap Ψk = ζ(Rsnap )T M2+ Rsnap Ψk

(12.24)

in the snapshot space, where    +,snap +,snap +,snap +,snap 2μ(ψk d x, ) : (ψl ) + λ∇ · ψk ∇ · ψl A+ = [akl+ ] = K i+

 1 +,snap +,snap { λ + 2μ}}ψk · ψl d x, H K i+  1 +,snap +,snap M2+ = [m + ] = { λ + 2μ}}ψk · ψl ds. 2,kl H ∂ K i+ M1+ = [m + 1,kl ] =

After solving the above local spectral problems, we form the offline space as in the no oversampling case; see Section 12.1.2 for CG coupling and Section 12.1.2 for DG coupling.

12.1.3 Numerical result In this section, we present numerical results for CG-GMsFEM and DG-GMsFEM with two models. We consider different choices of snapshot spaces such as local-finegrid functions and harmonic functions and use different local spectral problems such as no-oversampling and oversampling described in the previous section. For the first model, we consider the medium that has no-scale separation and features such as high conductivity channels and isolated inclusions. Young’s modulus E(x) is depicted in ν 1 E(x), μ(x) = 2(1+ν) E(x), and the Poisson ratio ν is Figure 12.1, λ(x) = (1+ν)(1−2ν)

12.1 Multiscale methods for elasticity equations

273

taken to be 0.22. For the second example, we use the model that is used in [222] for the simulation of subsurface elastic waves (see Figure 12.2). In all numerical tests, we use constant force and homogeneous Dirichlet boundary conditions. In all tables below, Λ∗ represents the minimum discarded eigenvalue of the corresponding spectral problem. We note that the first three eigenbasis (which correspond to the first three smallest eigenvalues) are constant and linear functions, therefore we present our numerical results starting from the fourth eigenbasis in all cases. In the following, the dimension of a solution represents the total number of basis used for the finite element space.

Fig. 12.1 Young’s modulus (Model 1).

0

0

GPa

GPa

3.0

Depth km

2.5 2.0 1.5

4

1.0

6

2 Depth km

2

5 4

4 3

0.5 2 0

6

6 0

2 4 Distance km

Fig. 12.2 Left: λ. Right: μ (Model 2).

6

0

2 4 Distance km

6

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12 GMsFEM for selected applications

Before presenting the numerical results, we summarize our numerical findings. • We observe a fast decay in the error as more basis functions are added in both CG-GMsFEM and DG-GMsFEM. • We observe the use of multiscale partition of unity improves the accuracy of CGGMsFEM compared to the use of piecewise bilinear functions. • We observe an improvement in the accuracy (a slight improvement in CG case and a large improvement in DG case) when using oversampling for the examples we considered and the decrease in the snapshot space dimension.

Numerical results for Model 1 with conforming GMsFEM (CG-GMsFEM) For the first model, we divide the domain Ω = [0, 1] × [0, 1] into 10 × 10 coarsegrid blocks; inside each coarse block, we use 10 × 10 fine-scale square blocks, which results in a 100 × 100 fine-grid blocks. The number of basis functions used to get the reference solution is 20402. We will show the performance of CG-GMsFEM with the use of local fine-scale snapshots and harmonic extension snapshots. Both bilinear and multiscale partitions of unity functions (see Section 12.1.2) will be considered. For each case, we will provide the comparison using oversampling and no-oversampling. For the error measure, we use relative weighted L 2 norm error and weighted H 1 norm error to compare the accuracy of CG-GMsFEM, which are defined as

eL 2

(λ + 2μ)(u CG H − u h ) L 2 (Ω) = , eH 1 = (λ + 2μ)u h  L 2 (Ω)

CG a(u CG H − uh , u H − uh ) , a(u h , u h )

where u CG H and u h are CG-GMsFEM defined in (12.7) and fine-scale CG-FEM solution defined in (12.4), respectively. Tables 12.1 and 12.2 show the numerical results of using local fine-scale snapshots with piecewise bilinear function and multiscale functions as a partition of unity, respectively. As we observe, when using more multiscale basis, the errors decay rapidly, especially for multiscale partition of unity. For example, we can see that the weighted L 2 error drops from 24.9% to 1.1% in the case of using bilinear function as a partition of unity with no oversampling, while the dimension increases from 728 to 2672. If we use multiscale partition of unity, the corresponding weighted L 2 error drops from 8.4% to 0.6%, which demonstrates a great advantage of multiscale partition of unity. Oversampling can help improve the accuracy as our results indicate. The local eigenvalue problem used for oversampling is Eq. (12.22). Next, we present the numerical results when harmonic extensions are used as snapshots in Tables 12.3 and 12.4. We can observe similar trends as in the local finescale snapshot case. The errors decrease as the number of basis functions increase. The L 2 error is less than 1% when about 13% of degrees of freedom is used. Simi-

12.1 Multiscale methods for elasticity equations

275

larly, the oversampling method helps to improve the accuracy. In this case, the local eigenvalue problem used for oversampling is Eq. (12.21).

Dimension

728 1214 1700 2186 2672

1/Λ∗ eL 2 eH 1 without with without with without with oversampling oversampling oversampling oversampling oversampling oversampling 1.3e+07 3.1e+06 7.0e+05 1.8e+00 9.9e-01

1.4e+07 5.6e+06 2.7e+06 1.7e+06 1.4e+06

0.249 0.048 0.027 0.018 0.011

0.215 0.047 0.024 0.016 0.010

0.444 0.220 0.162 0.133 0.105

0.409 0.213 0.153 0.123 0.099

Table 12.1 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, piecewise bi-linear partition of unity functions are used. The case with local fine-scale snapshots.

Dimension

728 1214 1700 2186 2672

1/Λ∗ eL 2 eH 1 without with without with without with oversampling oversampling oversampling oversampling oversampling oversampling 6.9e+06 5.8e+00 2.1e+00 1.3e+00 9.4e-01

6.2e+06 3.2e+06 1.2e+06 5.9e+05 1.0e+01

0.084 0.031 0.015 0.009 0.006

0.110 0.028 0.012 0.008 0.005

0.254 0.166 0.111 0.088 0.071

0.274 0.160 0.105 0.083 0.066

Table 12.2 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, multiscale partition of unity functions are used. The case with local fine-scale snapshots.

Dimension

728 1214 1700 2186 2672

1/Λ∗

eL 2

eH 1

without with without with without with oversampling oversampling oversampling oversampling oversampling oversampling 1.3e+07 2.1e+06 2.8e+05 1.2e+00 5.8e-01

1.2e+07 5.5e+06 3.2e+06 9.8e+05 2.1e+04

0.254 0.047 0.024 0.016 0.008

0.218 0.048 0.022 0.015 0.010

0.446 0.218 0.153 0.124 0.102

0.418 0.217 0.148 0.122 0.099

Table 12.3 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, piecewise bi-linear partition of unity functions are used. The case with harmonic snapshots.

276

12 GMsFEM for selected applications

Dimension

728 1214 1700 2186 2672

1/Λ∗

eL 2

eH 1

without with without with without with oversampling oversampling oversampling oversampling oversampling oversampling 7.0e+06 5.5e+00 1.9e+00 1.0e+00 7.1e-01

7.2e+06 3.2e+06 1.5e+06 2.5e+05 1.7e+00

0.087 0.034 0.015 0.009 0.007

0.112 0.032 0.013 0.008 0.006

0.259 0.174 0.115 0.090 0.075

0.291 0.169 0.112 0.089 0.074

Table 12.4 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, multiscale partition of unity functions are used. The case with harmonic snapshots.

Numerical results for Model 1 with DG-GMsFEM In this section, we consider numerical results for DG-GMsFEM discussed in Section 12.1.2. To show the performance of DG-GMsFEM, we use the same model (see Figure 12.1) and the coarse- and fine-grid settings as in the CG case. We will also present the result of using both harmonic extension and eigenbasis (local fine-scale) as snapshot space. To measure the error, we define broken weighted L 2 norm error and H 1 norm error

  DG DG K ∈T H K σ(u H − u h )) : ε(u H − u h )) d x   , eH 1 = K ∈T H K σ(u h ) : ε(u h ) d x where u DG H and u h are DG-GMsFEM defined in (12.8) and fine-scale DG-FEM solution, respectively. We note that the dimension of the reference solution u h here is 24200. In Table 12.5, the numerical results of DG-MsFEM with local fine-scale functions as the snapshot space are shown. We observe that DG-MsFEM shows a better approximation compared to CG-MsFEM if oversampling is used. The error decreases more rapidly as we add basis. More specifically, the relative broken L 2 error and H 1 error decrease from 14.1%, 52.5% to 0.2% and 5.8%, respectively, while the degrees of freedom of the coarse system increase from 728 to 2696, where the latter is only 13.2% of the reference solution. The local eigenvalue problem used for oversampling is Eq. (12.23). Table 12.6 shows the corresponding results when harmonic functions are used to construct the snapshot space. We observe a similar error decay trend as local finescale snapshots are used. Oversampling can help improve the results significantly. Although the error is very large when the dimension of the coarse system is 728 (4 multiscale basis is used), the error becomes very small when the dimension reaches 1728 (9 multiscale basis is used). The local eigenvalue problem used for oversampling here is Eq. (12.24). We remark that oversampling can not only help decrease the error but also decrease the dimension of the snapshot space greatly in the periodic case.

12.1 Multiscale methods for elasticity equations Dimension

728 1184 1728 2184 2696

1/Λ∗

277 eL 2

eH 1

without with without with without with oversampling oversampling oversampling oversampling oversampling oversampling 4.9e-03 3.0e-03 2.1e-03 1.2e-03 1.0e-03

1.5e-03 8.5e-04 5.6e-04 3.5e-04 2.7e-04

0.281 0.118 0.108 0.073 0.056

0.141 0.019 0.012 0.007 0.002

0.554 0.439 0.394 0.348 0.300

0.525 0.209 0.145 0.096 0.058

Table 12.5 Relative errors between DG-MsFEM solution and the fine-scale DG-FEM solution. The case with local fine-scale snapshots.

Dimension

728 1184 1728 2184 2696

1/Λ∗

eL 2

eH 1

without with without with without with oversampling oversampling oversampling oversampling oversampling oversampling 2.9e-01 1.6e-01 1.0e-01 7.1e-02 6.3e-02

1.6e-01 6.5e-02 5.4e-02 3.9e-02 2.8e-02

0.285 0.193 0.114 0.081 0.043

0.149 0.076 0.009 0.004 0.002

0.557 0.515 0.432 0.326 0.231

0.528 0.366 0.155 0.078 0.060

Table 12.6 Relative errors between DG-MsFEM solution and the fine-scale DG-FEM solution. The case with harmonic snapshots.

Numerical results for Model 2 The purpose of this example is to test a method for an earth model that is used in [222]. The domain for the second model is Ω = (0, 6000)2 (in meters) which is divided into 900 = 30 × 30 square coarse-grid blocks; inside each coarse block, we generate 20 × 20 fine-scale square blocks. The reference solution is computed through standard CG-FEM on the resulting 600 × 600 fine grid. We note that the dimension of the reference solution is 722402. The numerical results for CG-MsFEM and DG-MsFEM are presented in Tables 12.7 and 12.8, respectively. We observe relatively low errors compared to the high contrast case and the error decrease with the dimension increase of the offline space. Both coupling methods (CG and DG) show very good approximation ability.

278

12 GMsFEM for selected applications dimension

1 Λ∗

eL 2

eH 1

6968 8650 10332 12014

4.9e+00 4.5e+00 3.9e+00 3.6e+00

3.1e-03 2.7e-03 2.5e-03 2.2e-03

5.4e-02 5.2e-02 4.9e-02 4.7e-02

Table 12.7 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, piecewise bi-linear partition of unity functions are used. The case with local fine-scale snapshots.

dimension

1 Λ∗

eL 2

eH 1

7200 9000 10800 12600

6.3e-06 6.0e-06 4.6e-06 4.5e-06

4.1e-03 4.0e-03 3.8e-03 3.1e-03

7.1e-02 6.6e-02 6.3e-02 5.9e-02

Table 12.8 Relative errors between DG-MsFEM solution and the fine-scale CG-FEM solution. The case with local fine-scale snapshots.

12.2 Multiscale methods for multi-phase flow and transport In this section, we show the application of GMsFEM to two-phase flow and transport problems and present simulation results. Specifically, we consider the two-phase flow problem with zero Neumann boundary condition −η(S)κ∇ p = v, div v = f, v · n = 0, where η(S) =

in Ω in Ω on ∂Ω,

κr w (S) κr o (S) + μw μo

and κr w (S) = S 2 , κr o (S) = (1 − S)2 , μw = 1, μo = 5.

12.2 Multiscale methods for multi-phase flow and transport

279

The saturation equation is given by St + v · ∇ F(S) = r, where F(S) =

κr w (S)/μw . κr w (S)/μw + κr o (S)/μo

The above flow equation is solved by the mixed GMsFEM, and the saturation equation is solved on the fine grid by the finite volume method. Let Sin be the value of S on the fine element τi at time tn , where tn = t0 + nΔt, t0 is the initial time, and Δt is the time step size chosen according to CFL condition. Then, Sin satisfies  Sin+1 − Sin + F( S n )(v · n) = ri |τi |, (12.25) |τi | Δt ∂τi where ri is the average value of r on τi and  S n is the upwind flux. Similar results are obtained when using implicit time discretization for the saturation equation. In our simulations, we take f to be zero except for the top-left and bottom-right fine-grid elements, where f takes the values of 1 and −1, respectively. Moreover, we set the initial value of S to be zero. For the source r , we also take it as zero except for the top-left fine element where r = 1. For the simulations of two-phase flow, the mixed GMsFEM (see Section 6.2.2) is used for the flow equations. The multiscale basis functions are computed at time zero using the unit mobility. These multiscale basis functions for the velocity field are used without any modification to compute the fine-scale velocity field. The finescale velocity field is further used to update the saturation file. We note that at each time step, the flow equation is solved on a coarse grid, which provides a substantial computational saving.

1

10000

0.9

9000

0.8

8000

0.7

7000

0.6

6000

0.5

5000

0.4

4000

0.3

3000

0.2

2000

0.1

1000

0

0 0

0.2

0.4

0.6

(a) κ 1

0.8

1

(b) κ 2 in log 10 scale

Fig. 12.3 Two permeability fields in the numerical experiments.

280

12 GMsFEM for selected applications

In the first example, we consider the case when κ = κ1 (see Fig. 12.3(a)). In Figs. 12.4, 12.5, 12.6, and 12.7, the saturation plots, shown from left to right, refer to the simulations at different times, namely t = 1000, 3000, and 5000. The saturation plots in Fig. 12.4 are obtained by using the fine-scale velocity v f in (12.25). We denote these saturations by S f . Similarly, the saturation plots in Figs. 12.5, 12.6, and 12.7 are obtained by using the multiscale velocity vo in (12.25). Overall speaking, we observe error reductions from using 1 basis functions per edge to 5 basis functions per edge. In particular, for t = 1000, the relative error reduces from 9.3% to 2.6% when using 5 basis functions per edge, and for t = 5000, the relative error reduces from 5.5% to 1.3% when using 5 basis functions per edge.

(a) t = 1000

(b) t = 3000

(c) t = 5000

Fig. 12.4 Saturation solution obtained by using v f in (12.25).

In the second numerical example, we show the performance of our method when applying it to a more realistic permeability field. We pick the top layer of the SPE10 permeability field (see Fig. 12.3(b)) in the following set of experiments. The model is again the water and oil two-phase flow equations presented above. The permeability field is originally 220 by 60, and we project it into a fine grid of resolution 220 by 220. Then, the coarse grid is set to be 11 by 11, which means the local grid is 10 by 10 in each coarse block. The saturation plots are depicted in Figs. 12.8, 12.9, 12.10, and 12.11. In this example, we observe that, at first glance, the multiscale saturation solution looks similar to the fine solution if we use 1 multiscale basis function

12.2 Multiscale methods for multi-phase flow and transport

281

(a) Relative L2 error = (b) Relative L2 error = (c) Relative L2 error = 9.3% 5.9% 5.5% Fig. 12.5 Saturation solution obtained by using vo (10 × 10 coarse grid, 1 basis per coarse edge) in (12.25).

(a) Relative L2 error = (b) Relative L2 error = (c) Relative L2 error = 2.8% 1.6% 1.6% Fig. 12.6 Saturation solution obtained by using vo (10 × 10 coarse grid, 3 basis per coarse edge) in (12.25).

(a) Relative L2 error = (b) Relative L2 error = (c) Relative L2 error = 2.6% 1.4% 1.3% Fig. 12.7 Saturation solution obtained by using vo (10 × 10 coarse grid, 5 basis per coarse edge) in (12.25).

per edge. However, if we take a closer look, we notice some missing features in the water front. When we use 3 or 5 basis functions per coarse edge, these features can be recovered correctly. This shows the importance of these additional multiscale basis functions. More quantitatively, we observe more error reductions from using 1 basis functions per edge to 5 basis functions per edge compared with the previous examples. In particular, for t = 1000, the relative error reduces from 18.8% to 3.6%

282

12 GMsFEM for selected applications

when using 5 basis functions per edge. Likewise, for t = 5000, the relative error reduces from 20.7% to 5.3% when using 5 basis functions per edge.

(a) t = 1000

(b) t = 3000

(c) t = 5000

Fig. 12.8 Saturation solution obtained by using v f in (12.25).

(a) Relative L2 error = (b) Relative L2 error = (c) Relative L2 error = 18.8% 25.4% 20.7% Fig. 12.9 Saturation solution obtained by using vo (11 × 11 coarse grid, 1 basis per coarse edge) in (12.25).

Fig. 12.10 Saturation solution obtained by using vo (11 × 11 coarse grid, 3 basis per coarse edge) in (12.25).

12.3 Multiscale methods for acoustic wave propagation: Mixed formulation

283

Fig. 12.11 Saturation solution obtained by using vo (11 × 11 coarse grid, 5 basis per coarse edge) in (12.25).

12.3 Multiscale methods for acoustic wave propagation: Mixed formulation The purpose of this section is to present an application of GMsFEM to the wave equation. This work is reported in [122]. There are previous works on GMsFEM for the wave equation based on the second-order formulation and a discontinuous Galerkin framework [30, 110, 223]. These methods give accurate simulations of waves in coarse meshes, but lack energy conservation. To develop a scheme with energy conservation, the wave equation in the pressure-velocity formulation (called the mixed formulation) is considered. Some ideas from the earlier work on mixed GMsFEM for high contrast flows [123] are used. However, the method in [123] cannot be used directly for the wave equation since it is based on a piecewise constant approximation for pressure, which is not accurate for the wave equation, and the velocity basis functions give a mass matrix that is not block-diagonal. To derive a new GMsFEM with block-diagonal mass matrix and energy conservation, a staggered mesh is used [96–100], where it is shown that such an idea can give a numerical scheme with block-diagonal mass matrix and energy conservation. In addition, this idea can give a smaller dispersion error [78]. In [94], we have applied a staggered coarse mesh in the numerical upscaling framework for the wave equation, where only one multiscale basis function is used for each coarse region. A mixed GMsFEM based on a staggered mesh is developed, giving block-diagonal mass matrix, energy conservation, and systematic enrichment of multiscale basis functions. In the new method, multiscale basis functions for both the pressure and the velocity are constructed. The construction follows the general methodology of GMsFEM by solving local spectral problems. An advantage of using the mixed formulation is that perfectly matched layers can be used in conjunction with our method easily, and we will illustrate this in our numerical experiments.

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12 GMsFEM for selected applications

12.3.1 Problem description Let Ω ⊂ R2 be the computational domain. We consider the wave equation in mixed formulation ∂v + ∇u = 0 ∂t ∂u ρ +∇ ·v = f ∂t κ

in Ω

(12.26)

in Ω

(12.27)

with homogeneous Dirichlet boundary condition u = 0 on ∂Ω. We assume that the bulk modulus κ−1 and the density ρ are highly oscillatory. The aim is to construct multiscale basis functions, which provide accurate and efficient approximations of the pressure u and the velocity v on a coarse grid. The homogeneous Dirichlet boundary condition is chosen to simplify the discussions. The method can be easily applied to other types of boundary conditions as well as perfectly matched layers. Moreover, the extension of our method to the three-dimensional case is straightforward. First, we remind the triangulation of the domain Ω. Let T0H be a coarse-grid initial coarse triangulation of the domain. For each triangle in T0H , we refine it into three triangles by connecting the centroid to the three vertices. The multiscale basis functions are defined in this coarse mesh T H . We use E p to denote the set of all edges in the initial triangulation and use E p0 ⊂ E p to denote the set of all interior edges lying in E p . Moreover, we use Ev to denote the set of new edges formed by the above division process. Note that the set of all edges E H of the coarse mesh is E H = E p ∪ Ev , and the set of all interior edges E H,0 of the coarse mesh is E H,0 = E p0 ∪ Ev . The fine mesh T h is obtained by refining T H in a conforming way. We use E h to denote the set of edges in the fine mesh T h . An illustration of the above definitions is shown in Figure 12.12. In particular, the solid lines are the edges of the initial triangulation T0H . By our construction, these lines also represent edges in E p . Moreover, the dash lines represent the new edges formed by the subdivision process. Note that the coarse mesh T H is defined as the union of all the new triangles obtained from the subdivision process. The wave equations (12.26)–(12.27) can be discretized on the fine mesh T h by the standard Raviart-Thomas finite element (RT0) method. Let (Vh , Q h ) be the standard RT0 space for (v, p) with respect to the fine mesh T h . Then, the RT0 method reads: find vh ∈ Vh and u h ∈ Q h such that 

 ∂vh ·w− κ u h ∇ · w = 0, ∀w ∈ Vh , ∂t Ω Ω   ∂u h q+ ρ q∇ · vh = f q, ∀q ∈ Q h . ∂t Ω Ω Ω

(12.28) (12.29)

Notice that the RT0 scheme (12.28)–(12.29) does not have a block-diagonal matrix. We will present a modified scheme for (12.26)–(12.27) based on the above RT0

12.3 Multiscale methods for acoustic wave propagation: Mixed formulation

285

Fig. 12.12 An illustration of the subdivision process and the definition of T H .

method (12.28)–(12.29). The resulting method has the advantage that the mass matrix is block-diagonal. We will use similar ideas as in [96, 97]. The main idea is to decouple the degrees of freedom for the velocity on the subset E h ∩ E p0 of fine-grid edges, which are the fine-grid edges lying in E p0 . We do not enforce the continuity of the normal components of velocity on the fine-grid edges in E h ∩ E p0 . We will introduce additional pressure variables in order to penalize the normal jumps of velocity on these edges. h is defined by using Vh and h be the decoupled velocity space. That is, V We let V by decoupling the normal components of velocity on the fine-grid edges in E h ∩ E p0 . More precisely, h = { v ∈ L 2 (Ω)2 |  v |τ = aτ + bτ x, ∀τ ∈ T h ;  v · n is continuous on E h ∩ Ev }. V h ⊂ L 2 (Ω) as follows. Let Kh ⊂ T h We will introduce an additional pressure space Q be the set of fine mesh elements having a non-empty intersection with E p . We let KhE ⊂ Kh be a subset containing fine mesh elements with a non-empty intersection h by the with E ∈ E p . See Figure 12.13 for an illustration. Then we define q ∈ Q following conditions: 1 0 h • q|  τ ∈ P (τ ) and q|e ∈h P (e), where τ ∈ K and e is the edge of τ lying in E p ; • τ q = 0 for all τ ∈ K ; • q is continuous on all edges e ∈ E h ∩ E p0 .

h = Q h + Q h . h have supports in Kh . Then we define Q Note that functions in Q Note that we can impose the homogeneous Dirichlet boundary condition p = 0 in h,0 . To construct Q h,0 , we h , and we denote the resulting space by Q the space Q

286

12 GMsFEM for selected applications

Fig. 12.13 The definition of KhE . The elements of KhE are shown as shaded triangles.

h,0 ⊂ Q h by Q h,0 = {q ∈ Q h : q|e = 0, ∀e ∈ E p ∩ ∂Ω}. first define a subspace Q   Then we define Q h,0 = Q h + Q h,0 . We remark that the increase in the dimension by h is the same as the increase in the dimension by enriching decoupling Vh to form V  the space Q h to form Q h . h and u h ∈ The modified numerical scheme is stated as follows. We find vh ∈ V h,0 such that Q  Ω



κ

 ∂vh (I ) ·w− ph ∇ · w + ∂t Ω

 ∂u h q (I ) ∇ · vh − q+ ∂t Ω Ω ρ





e e∈E h ∩E 0p

 e∈E h ∩E 0p



e

(B) h , ph [w · n] = 0, ∀w ∈ V

q (B) [vh · n] =

 Ω

h,0 , f q, ∀q ∈ Q

(12.30) (12.31)

h , respecwhere q (I ) and q (B) denote the components of q in the spaces Q h and Q h . The solution (vh , u h ) of (12.30)–(12.31) is considered as the tively, for any q ∈ Q reference solution. In the following sections, we will construct multiscale solution (v H , u H ) that gives good approximation of (vh , u h ) and derive the corresponding error bound. For an error bound of the reference solution (vh , u h ), one can apply the technique in [96, 97] to show that v − vh V + u − u h  P ≤ Ch, where C depends on the regularity of the exact solution (v, p).

12.3 Multiscale methods for acoustic wave propagation: Mixed formulation

287

12.3.2 Multiscale basis functions In this section, we will introduce multiscale basis functions and give the constructions of the multiscale approximation spaces VH and Q H for approximating vh and u h , respectively. We emphasize that the basis functions and the corresponding multiscale method are defined with respect to the coarse mesh T H . For a coarse element K ∈ T H , we define Vh (K ) as the restriction of Vh in K and Q h (K ) as the restriction of Q h in K . In addition, Vh,0 (K ) is the subspace of Vh (K ) and contains vector fields whose normal components are zero on ∂ K . Below, we will give the definitions for the spaces Q H and VH . Our generalized multiscale finite element method reads: find (v H , u H ) ∈ VH,off × Q H,off such that 

 ∂v H ·w− κ u H ∇ · w = 0, ∀w ∈ VH,off , ∂t Ω Ω    ∂u H q+ ρ q∇ · v H = f q, ∀q ∈ Q H,off . ∂t Ω Ω Ω

(12.32) (12.33)

There are totally three sets of basis functions. The first set of basis functions can be considered as a generalization of the RT0 element. The second set of basis functions gives enrichments of the normal component of velocity across coarse-grid edges. The third set of basis functions corresponds to the standing modes within coarse elements with zero boundary conditions.

The first basis set (1) and Q (1) We use VH,off H,off to denote the first set of basis functions for the velocity v and the pressure u. The space Q (1) H,off is taken as the piecewise constant space with (1) respect to the coarse mesh T H . We will define the space VH,off as follows. For each H coarse edge E ∈ E , we define ω E as the union of all coarse elements having the edge E. For each E ∈ E H , we define one basis function φ(1) E whose support is ω E . (1) (1) is defined by finding (φ , u ) ∈ V (K ) × Q h (K ) such that The basis φ(1) h E E E



 K

Ei κ φ(1) ·w−

u (1) E ∇ · w = 0, ∀w ∈ Vh,0 (K ), K   q∇ · φ(1) = c E q, ∀q ∈ Q h (K ) E K

(12.34) (12.35)

K

for each K ⊂ ω E , with the following boundary condition:  φ(1) E

·n =

1, on E, 0, on ∂ K \E,

(12.36)

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where n denote the unit normal vectors on edges. In (12.35), the constant c E = |E|/|K |. The space VH(1) is defined by   (1) H . (12.37) = span φ(1) VH,off E : E ∈E (1) We remark that scheme (12.32)–(12.33) with VH,off = VH,off and Q H,off = Q (1) H,off can be regarded as a generalization of the classical RT0 method.

The second basis set We use VH(2) to denote the second set of basis functions for the velocity v. We will see below that this corresponds to enrichments of velocity with respect to coarse-grid edges. Notice that, in the second basis set, we only enrich the approximation space for the velocity. Let E ∈ E H . Note that E ∩ E h defines a partition for E by fine-grid edges. We define D E to be the space of piecewise constant functions on E with respect to the partition E ∩ E h . Let D E be the subspace of D E containing functions with zero mean. We write D E = span{δ E,i : i = 1, 2, · · · , N E }, where N E is the dimension of D E . For each E ∈ E H , we define a set of basis functions φ(2) E, j whose (2) (2) support is ω E . We first find (ψ E,i , q E,i ) ∈ Vh (K ) × Q h (K ) such that   (2) κ ψ (2) · w − q E,i ∇ · w = 0, ∀w ∈ Vh,0 (K ), (12.38) E,i K K  q∇ · ψ (2) (12.39) E,i = 0, ∀q ∈ Q h (K ) K

for each K ⊂ ω E , with the following boundary condition:  δ E,i , on E, (2) ψ E,i · n = 0, on ∂ K \E. (2) on E by We next define a snapshot space VE,snap   (2) . VE,snap = span ψ (2) : ∀ δ ∈ D E,i E E,i

(12.40)

(12.41)

(2) We remark that the snapshot space VE,snap is large and dimension reduction is necessary. To obtain a reduced dimension space, we use the following spectral problem. (2) and λ ∈ R such that Find φ ∈ VE,snap   (2) (φ · n) (w · n) = λ κ φ · w, ∀w ∈ VE,snap . (12.42) E

ωE

We arrange the eigenvalues in increasing order, that is, λ E,1 < λ E,2 < · · · < λ E,N E . For each coarse edge E, we take the first n E eigenfunctions φ(2) E, j . We then define

12.3 Multiscale methods for acoustic wave propagation: Mixed formulation

  (2) . VE,off = span φ(2) : j = 1, 2, · · · , n E E, j

289

(12.43)

Finally, the space VH(2) is defined as (2) = VH,off



(2) VE,off .

(12.44)

E∈E H

We will call the basis functions resulting from the first and the second basis sets the boundary basis.

The third basis set We use VH(3) and Q (3) H to denote the third set of basis functions for the velocity v and the pressure u. We will see below that these basis functions correspond to some standing modes within coarse elements. Let K ∈ T H be a given coarse element. We consider the following spectral problem: find ψ ∈ Vh,0 (K ), p ∈ Q h (K ) and μ ∈ R such that   κψ · w − u∇ · w = 0, ∀w ∈ Vh,0 (K ), (12.45) K K  q∇ · ψ = μ ρ u q, ∀q ∈ Q h (K ) (12.46) K

K



subject to the zero mean condition K p = 0. Notice that the spaces Vh,0 (K ) and Q h (K ) are considered as the snapshot spaces. We arrange the eigenvalues in increasing order, that is, μ K ,1 < μ K ,2 < · · · < μ K ,M K , where M K + 1 is the dimension of (3) Q h (K ). We will take the first m K eigenfunctions (φ(3) K , j , u K , j ) as basis. We define

and

  VK(3),off = span φ(3) K , j : j = 1, 2, · · · , m K

(12.47)

  (3) Q (3) K ,off = span u K , j : j = 1, 2, · · · , m K .

(12.48)

Then we define (3) = VH,off

 K ∈T

H

VK(3),off

and

Q (3) H,off =

 K ∈T

Q (3) K ,off .

(12.49)

H

(1) (2) (3) (3) + VH,off + VH,off and Q H,off = Q (1) Finally, we can take VH,off = VH,off H,off + Q H,off as the approximation spaces in the scheme (12.32)–(12.33). We will call the basis functions resulting from the third basis set the interior basis.

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12 GMsFEM for selected applications

12.3.3 The mixed GMsFEM As discussed in the previous section, we can define the mixed GMsFEM for the wave (1) + equation by the system (12.32)–(12.33) together with the choices VH,off = VH,off (2) (3) (1) (3) VH,off + VH,off and Q H,off = Q H + Q H as the approximation spaces. However, the resulting scheme does not have a block-diagonal mass matrix, and is therefore slow in the time stepping process. The reason for getting a non-diagonal mass matrix is (1) (2) and VH,off have overlapping supports on two adjacent that the basis functions in VH,off coarse elements. The basis functions for the velocity are coupled through the coarsegrid edges E H . In the following, we will modify these spaces so that the velocity basis functions have disjoint supports. One important feature is that we will only decouple the basis functions on the coarse-grid edges belonging to E p0 only. We do not decouple the velocity basis functions for all coarse edges. The resulting velocity basis functions have disjoint supports on triangles of the initial coarse mesh T0H ; see Figure 12.12. (1) Let E ∈ E p0 be a coarse edge. We recall that the basis functions in the space VH,off (2) and VE,off have supports in ω E and their normal components have common values on (1) (2) H,off E,off E. We will decouple this continuity, and call the resulting space as V and V , respectively. In particular, the normal component of velocity does not necessarily have a single value on E. Globally, we define (2) H,off = V



(2) E,off . V

(12.50)

E∈E H

We will introduce a new pressure space in order to penalize the normal continuity of velocity on E ∈ E p0 . Let E ∈ E p . This new pressure space is denoted by Q (2) E,off , and its dimension is n E + 1. We recall that KhE ⊂ Kh is the subset of fine elements having non-empty intersection with E; see Figure 12.13. The basis functions of Q (2) E,off have (2) h supports on K E . Each q ∈ Q E,off is defined by the following conditions: • q|τ ∈ P 1 (τ ) and q|e ∈ P 0 (e), where τ ∈ KhE , and e is the edge of τ lying in E; • q is continuous on E, and q = φ · n on E for some φ ∈ VH(1) ∪ VH(2) ; • τ q = 0 for all τ ∈ KhE . We write

    (1) + span u (2) Q (2) E,off = span u E E, j : j = 1, 2, · · · , n E ,

(2) (2) where u (1) E = 1 on E and u E, j = φ E, j · n on E. We define

(2) Q H,off =

 E∈E p

Q (2) E,off .

(12.51)

12.3 Multiscale methods for acoustic wave propagation: Mixed formulation

291

Next, we discuss the boundary condition. We consider Dirichlet boundary condi(2) tion p = 0 for pressure. We will need to modify the space Q H,off for the boundary (2)  condition. We define Q H,0 by  (2) (2) Q E,off , (12.52) Q H,0 = E∈E p0 (1) H,off + where the sum is taken over all interior edges E ∈ E p0 . We can take VH,off = V (2) (3) (1) (2) (3) H,0 + Q H,off . Since the space VH,off is not H,off + VH,off and Q H,off = Q H,off + Q V in H (div), we will replace the variational form (12.32)–(12.33) by the following. We find (v H , u H ) ∈ VH,off × Q H,off such that

 Ω

κ

∂v H ·w− ∂t

 Ω

ρ

∂u H q+ ∂t



(I )

pH ∇ · w +

Ω

 Ω

 

q (I ) ∇ · v H −

 E∈E 0p

(B)

E

E∈E 0p

u H [w · n] = 0, ∀w ∈ VH,off ,

q (B) [v H · n] = E

(12.53)

 Ω

f q, ∀q ∈ Q H,off ,

(12.54)

(I ) (2) to denote the where we use q (B) to denote the component of q in Q H,0 and q (1) (3) component of q in Q H,off + Q H,off , for any q ∈ Q H,off . Equations (12.53) and (12.54) give our mixed GMsFEM for the wave equation (12.26)–(12.27). For the time discretization, we will apply the leap-frog scheme to (12.53)-(12.54). The velocity v H are approximated at times tn = nΔt, and the pressure u H are approxn+ 1

imated at times tn+ 21 = (n + 21 )Δt, n = 0, 1, · · · . We let v nH and u H 2 be the approximate solutions at times tn and tn+ 21 , respectively. The leap-frog scheme reads:  Ω

κ

n v n+1 H − vH ·w− Δt



n+ 23

Ω

ρ

uH

 Ω

q+

∇ ·w+

  E∈E 0p



n+ 21

− uH Δt

(I,n+ 21 )

pH

Ω

q (I ) ∇ · v n+1 H −

(B,n+ 21 )

E

uH

  E∈E 0p

E

[w · n] = 0, ∀w ∈ VH,off ,

q (B) [v n+1 H · n] =

(12.55)

 Ω

f (tn+1 , ·)q, ∀q ∈ Q H,off ,

(12.56) (B,n+ 21 )

where u H

n+ 21

denotes the component of u H

(I,n+ 2 ) (2) in Q denotes the H,0 and u H 1

n+ 1

(3) component of u H 2 in Q (1) H,off + Q H,off . We remark that the stability estimate for Δt can be obtained by standard techniques and the inverse estimate.

12.3.4 Numerical results In this section, we present some numerical results to show the performance of our method. In our simulations, the computational domain Ω = [0, 1]2 , ρ = 1, and the source term f is chosen as the Ricker wavelet

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12 GMsFEM for selected applications

f (x, t) = g(x)(t − 2/ f 0 )e−π

2

f 02 (t−2/ f 0 )2

, g(x) = δ −2 e((x−0.5)

2

+(y−0.5)2 )/δ 2

,

where f 0 is the central frequency and δ = 2h measures the size of the support of the source. We assume that the initial conditions are zero. In the first two examples, we will consider the performance of our mixed GMsFEM for the propagation of the above point source with two types of heterogeneities and with the homogeneous Dirichlet boundary condition. In our third example, we will apply the perfectly matched layer (PML) [50] to simulate wave propagation in an unbounded domain containing a heterogeneous medium.

A layered stochastic coefficient In the first example, we consider κ−1 as a layered stochastic medium shown in Figure 12.14. We will first construct the initial mesh T0H for the domain Ω = [0, 1] × [0, 1]. Todoso,wefirstdefineauniformtriangularcoarsemeshonΩ withmeshsize H = 1/8. This triangular mesh is obtained by first dividing the domain into uniform squares and then dividing each square into two triangles using the diagonal. This process gives the initial mesh T0H . Next, to construct the fine mesh T h , each coarse triangular element in T H is sub-divided into a union of uniform triangular fine mesh blocks with mesh size h = 1/64 in the standard way. Notice that each coarse triangular element in T H is divided into 64 fine triangles. We choose the source frequency f 0 = 20, and compute the solution at the time T = 0.2. The reference solution at time T = 0.2 is shown in Figure 12.15. We apply our mixed GMsFEM (12.53)–(12.54) for this problem with various choices of a number of basis functions. In Table 12.9, we present the relative errors of the pressure u in terms of the Q norm for various choices of the number of basis functions. In particular, we use 3 to 6 basis functions per coarse edge for boundary basis and use 4 to 16 basis functions per coarse element for interior basis. We observe excellent performance of our method. For reference, the fine-grid solver has 24576 unknowns for the pressure u and 28928 unknowns for the velocity v. From Table 12.9, we see that a very small dimensional approximation space can give an error below 5%. For instance, with the use of 4 boundary basis functions per coarse edge and the use of 12 interior basis functions per coarse element, the dimensions of the spaces VH and Q H are 7680 and 5312, respectively (Table 12.10). # boundary basis\# interior basis 3 4 5 6

4 13.22% 12.76% 12.97% 13.01%

8 8.75% 5.76% 5.64% 5.65%

12 7.84% 3.65% 3.31% 3.31%

16 7.63% 2.95% 2.47% 2.46%

Table 12.9 Convergence history for various choices of number of basis functions for the first example.

12.3 Multiscale methods for acoustic wave propagation: Mixed formulation

293

Fig. 12.14 A stochastic coefficient for the first example. −3

x 10

1

15 0.9 0.8 10 0.7 0.6 5 0.5 0.4 0 0.3 0.2 −5 0.1 0

0

0.2

0.4

0.6

0.8

1

Fig. 12.15 The reference solution for the first example. #boundary basis\#interior basis 3 4 5 6

4 13 19 26 47

8 24 31 42 56

12 38 51 60 74

16 58 75 81 95

Table 12.10 Computational time for various choices of number of basis functions for the first example. (Computational time of reference solution: 228s)

The Marmousi model In the second example, we consider κ−1 as a part of the Marmousi model shown in Figure 12.16. We assume that the domain Ω = [0, 1] × [0, 1] is partitioned as the first example with coarse mesh size H = 1/16 and fine mesh size h = 1/256. We

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12 GMsFEM for selected applications

take the source frequency f 0 = 20, and compute the solution at the time T = 0.2. The reference solution and the multiscale solution using 6 boundary basis functions per coarse edge and 12 interior basis functions per coarse element at the time T = 0.2 are shown in Figure 12.17, and we observe very good agreement. In Table 12.11, we present the relative errors of the pressure u in terms of the Q norm for various choices of the number of basis functions. In particular, we use 3 to 6 basis functions per coarse edge for boundary basis and use 4 to 12 basis functions per coarse element for interior basis. We again observe excellent performance of our method. For reference, the finegrid solver has 98304 unknowns for the pressure u and 135296 unknowns for the velocity v. From Table 12.9, we see that with the use of 6 boundary basis functions per coarse edge and the use of 12 interior basis functions per coarse element, the relative error for the pressure is 8.59%, and the dimensions of the spaces VH and Q H are 36864 and 22848, respectively.

Fig. 12.16 The Marmousi model for the second example.

Fig. 12.17 Left: The reference solution for the second example with f 0 = 20. Right: The corresponding multiscale solution for the second example using 6 boundary basis functions per coarse edge and 12 interior basis functions per coarse element.

12.3 Multiscale methods for acoustic wave propagation: Mixed formulation # boundary basis\#interior basis 3 4 5 6

4 46.66% 38.11% 40.35% 41.58%

8 45.64% 23.83% 13.66% 12.91%

295

12 46.02% 23.39% 10.84% 8.59%

Table 12.11 Convergence history for various choices of number of basis functions for the second example with f 0 = 20.

To further test the performance of our method, we take a higher frequency source term with f 0 = 50 and compute the solution at T = 0.16. The reference solution and the corresponding multiscale solution with 8 boundary basis functions per coarse edge and 20 interior basis functions per coarse element are shown in Figure 12.18, and we observe very good agreement. In addition, the relative errors of the pressure u in terms of the Q norm for various choices of the number of basis functions are presented in Table 12.12. In particular, we use 2 to 8 basis functions per coarse edge for boundary basis and use 4 to 20 basis functions per coarse element for interior basis. From this table, we observe excellent performance of our method (Tables 12.13 and 12.14).

Fig. 12.18 Left: The reference solution for the second example with f 0 = 50. Right: The corresponding multiscale solution for the second example using 8 boundary basis functions per coarse edge and 20 interior basis functions per coarse element.

# boundary basis\#interior basis 2 4 6 8

4 125.75% 91.55% 100.46% 101.04%

8 128.19% 58.00% 36.11% 40.67%

12 129.09% 61.29% 18.65% 16.95%

16 129.29% 62.87% 16.95% 9.41%

20 129.37% 63.48% 17.04% 6.92%

Table 12.12 Convergence history for various choices of number of basis functions for the second example with f 0 = 50.

296

12 GMsFEM for selected applications #boundary basis\#interior basis 2 4 6 8

4 87 172 298 417

8 179 293 476 593

12 299 443 599 795

16 452 626 804 1020

20 638 881 1028 1278

Table 12.13 Computational time for various choices of number of basis functions for the second example with f 0 = 50. (Computational time of reference solution: 8340s)

#boundary basis\#interior basis 2 4 6 8

4 164.80% 114.44% 120.43% 120.66%

8 170.44% 70.44% 45.73% 50.57%

12 173.50% 73.74% 25.56% 22.71%

16 174.27% 74.55% 23.03% 13.30%

20 174.29% 75.22% 22.99% 10.23%

Table 12.14 Convergence history for various choices of number of basis functions for the second example with f 0 = 50 and delta source.

The Marmousi model with PML In the third example, we present an application of our method with the use of PML. We assume that the computational domain Ω = [0, 1]2 and the medium is part of the Marmousi model shown in Figure 12.16. We take the source with f0 = 20. The coarse and fine mesh sizes for the computational domain Ω are H = 1/8 and h = 1/64, respectively. To absorb the outgoing waves, we use a PML with a width of 10 finegrid blocks. We apply our mixed GMsFEM within the computational domain Ω and the standard fine-grid method (12.28)–(12.29) within the artificial layer. In Figure 12.19, we present the multiscale scale solutions at different times T . We see that the PML is able to absorb the outgoing waves without much artificial reflection.

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport 1

0.02

0.9

1

0.02

0.9 0.015

0.8 0.7

0.015 0.8 0.7

0.01

0.01 0.6

0.6 0.005

0.5 0.4

0.005

0.5 0.4

0

0 0.3

0.3 −0.005

0.2 0.1

−0.005

0.2 0.1

−0.01

−0.01 0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

0.6

0.8

1

1

1

0.9

0.9

0.015

0.015 0.8

0.8 0.7

0.01

0.7

0.01

0.6

0.6 0.005

0.5 0.4

0

0.005

0.5 0.4

0 0.3

0.3 0.2

−0.005

0.2

−0.005

0.1

0.1 0

297

−0.01 0

0.2

0.4

0.6

0.8

1

0

−0.01 0

0.2

0.4

0.6

0.8

1

Fig. 12.19 The multiscale solution at different times T for the third example. Top-Left: T = 0.24; Top-Right: T = 0.28; Bottom-Left: T = 0.32; Bottom-Right: T = 0.36.

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport Shale gas transport is an active area of research due to a growing interest in producing natural gas from source rocks. The shale systems have added complexities due to the presence of organic matter, known as kerogen. The kerogen brings in new fluid storage and transport qualities to the shale. A number of authors, e.g., Loucks et al. (2009), Sondergeld et al. (2010), and Ambrose et al. (2012), [21, 314, 381], have previously discussed the physical properties of the kerogen using scanning electron microscopy (SEM) and showed the co-existence of nanoporous kerogen and microporous conventional inorganic rock materials. Gas transport in the kerogen typically develops at low Reynolds number and relatively high Knudsen number values. Under these conditions, it is expected that the transport is not driven by laminar (Darcy) flow dominantly but instead by the pore diffusion and other molecular transport mechanisms such as Knudsen diffusion and the adsorbed phase (or surface) diffusion. The latter introduces nonlinear processes at the pore scale that occur in heterogeneous pore geometry. Some types of upscaled models are needed to represent these complex processes for reservoir simulations. In large-scale simulations, the complex pore-scale transport needs to be coupled to the transport in fractures. This brings additional difficulty in multiscale simulations. In particular, the multiscale simulations of the processes describing the interaction between the fracture and the matrix require reduced-order model approaches that work for problems with high contrast and without scale separation. The objective

298

12 GMsFEM for selected applications

of this chapter is to discuss the development of such approaches for describing the fracture and the matrix interaction by taking the upscaled matrix model following the previous work [15]. In the previous work [15], a set of macroscopic models that take into account the nanoporous nature and nonlinear processes of the shale matrix is proposed. The multiscale approaches proposed in [15] are limited to representing the features that have scale separation. To represent the fracture network and the interaction between the fracture network and the matrix, we use the framework of Generalized Multiscale Finite Element Method (GMsFEM). To represent the fractures on the fine grid, we use Discrete Fracture Model (DFM) [425]. The fine grid is constructed to resolve the fractures. For the coarse grid, we choose a rectangular grid. The GMsFEM framework uses these fine-scale models in computing the snapshot space and the offline space. The nonlinear models are handled with GMsFEM by locally updating multiscale basis functions. We remark that the study of flows in fractured media has a long history. Some modeling techniques on the fine grid include the Discrete Fracture Model (DFM), Embedded Fracture Model (EFM) [298, 302, 315], the single-permeability model, the multiple-permeability models [37, 149, 237, 298, 316, 365, 420, 421], and hierarchical fracture models [298]. Though these approaches are designed for finescale simulations, a number of these approaches represent fractures at a macroscopic level. For example, multiple-permeability models represent the network of connected fractures macroscopically by introducing several permeabilities in each block. The EFM [298, 302, 315] models the interaction of fractures with the fine-grid blocks separately for each block. The proposed method using the GMsFEM framework constructs multiscale basis functions by appropriately selecting local snapshot space and the local spectral problems for the underlying nonlinear problem. We discuss adaptivity issues and how to add multiscale basis functions in some selected regions. To reduce the computational cost associated with constructing the snapshot space, we use randomized boundary conditions (Section 4.8). We also use online multiscale basis functions, which are constructed during the simulation using the residual. The use of online multiscale basis functions can reduce the error substantially. These basis functions are used if the offline basis functions cannot reduce the error below a desired threshold. We present numerical results for some representative examples. In these examples, we use nonlinear matrix and fracture models. The numerical results show that the coarse-scale models with fewer degrees of freedom can be used to get an accurate approximation of the fine-scale solution. In particular, only 10% degrees of freedom are needed to obtain an accurate representation of the fine-scale solution. Geomechanical contribution to the permeability term is added, where the permeability depends on the pressure. Furthermore, we demonstrate the use of online basis functions and how they can reduce the error.

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport

299

12.4.1 Model problem We study nonlinear gas transport in fractured media. Similar equations arise in other models, where one considers a free gas in the tight reservoirs. We consider general equations ∂c = div(bm (c, x)∇c) in Ω, (12.57) am (c) ∂t where c is the amount of free gas and a(c) and b(c) contain terms related to storage and adsorption coefficients. In [14], the authors consider the nonlinear terms having the forms am (c) = φ + (1 − φ)γ

κ ∂F ∂F , bm (c, x) = φD + (1 − φ)γ Ds + φ RT c, ∂c ∂c μ (12.58)

where γ is a parameter, which is unity in kerogen and is equal to Vgrain,k /Vgrain in the inorganic material (Vgrain,k is grain volume and Vgrain is kerogen grain volume). Diffusivity D and porosity φ are defined for the free fluid in the inorganic matrix and in the kerogen as follows:   φk in kerogen Dk in kerogen , φ= . D= Di in inorganic matrix φi in inorganic matrix For the free gas we have the ideal gas assumption. The Darcy law of free gas flow in an inorganic matrix is used with permeability κ and gas viscosity μ. For the sorbed gas, we can use Langmuir or Henry’s isotherms F = F(c). In [301, 425], the authors discuss a general framework, where the equations also include nonlinear diffusivity due to adsorbed gas in a shale formation. In [297], the nonlinear terms appear due to barotropic effects. The nonlinear flows also contain components that are due to diffusion in the fractures. One needs additional equations for modeling fractures. The fractures have high conductivity. We will use a general equation of the form a f (c)

∂c = div(b f (c, x)∇c) ∂t

(12.59)

to describe the flow within fractures. In [14], the authors use κf RT c, a f (c) = φ f , b f (c, x) = μ where φ f and κ f are the fracture porosity and permeability. These problems are solved on a fine grid using DFM as will be described in Section 12.4.2. In many shale gas examples, the matrix heterogeneities can be upscaled and the resulting upscaled equation has the form (12.57). However, the interaction between matrix and fractures requires some type of multiscale modeling approach, where the effects of the fractures need to be captured more accurately. Approaches such as multicontinuum [301] are often used, but these approaches use idealized assumptions

300

12 GMsFEM for selected applications

on fracture distributions. We use multiscale basis functions to represent fracture effects. We consider arbitrary fracture distributions in the context of nonlinear flow equations. The overall model equations will be solved on a coarse grid, as before (see Figure 1.8). T H is a usual coarse-grid conforming partition of the computational domain Ω into finite elements (triangles, quadrilaterals, tetrahedra, etc.). The fine-grid partition is denoted by T h .

12.4.2 Fine-scale discretization To discretize the system on fine grid, we use the finite element method and DFM for fractures. To solve Problem (12.57) using the finite element method (FEM), we need a fine-grid discretization to capture the fractures. These computations can be expensive. We apply the discrete fracture network (DFM) model for modeling flows in fractures [341]. In the discrete-fracture model, the aperture of the fracture appears as a factor in front of the one-dimensional integral for the consistency of the integral form. This is the main idea of the discrete-fracture model, which can be applied in any complex configuration for fractured porous media. To demonstrate it, we consider the two-dimensional problem of Equation (12.59). We simplify the fractures as the lines with small apertures. Thus, a one-dimensional element is needed to describe fractures in the discrete-fracture model. The system of equations (12.57) will be discretized in a two-dimensional form for the matrix and in a one-dimensional form for the fractures. The whole domain Ω can be represented by (12.60) Ω = Ωm ⊕i di Ω f,i , where m and f represent the matrix and the fracture of the permeability field κ, respectively. Here, di is the aperture of the i-th fracture and i is the index of the fractures. Note that Ωm is a two-dimensional domain and Ω f,i is a one-dimensional domain (Figure 12.20). Then Equations (12.57) and (12.59) can be written as follows (for any test function v): m(

  ∂c ∂c ∂c , v) + a(c, v) = am (c) v d x + di a f (c) v d x+ ∂t ∂t ∂t Ωm Ω f,i i    + bm (c, x)∇c · ∇v d x + di bm (c, x)∇c · ∇v d x = 0. Ωm

i

Ω f,i

(12.61) We use d x for both integrals in matrix and fracture regions, though fracture regions are low dimensional. To solve (12.61), we will first linearize the system. We will use the following linearization:

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport

301

Fig. 12.20 Fine grid and fractures.

 +

Ωm

 cn+1 − cn cn+1 − cn n+1 m( am (cn ) , v) + a(c , v) = v d x+ τ τ Ωm  cn+1 − cn v d x+ a f (cn ) di τ Ω f,i i  n n+1 bm (c , x)∇c · ∇v d x + di b f (cn , x)∇cn+1 · ∇v d x = 0. i

Ω f,i

(12.62) The standard fully implicit finite difference scheme is used for the approximation with time step size τ and superscripts n, n + 1 denote previous and current time levels. This is a first-order in time and an unconditionally stable linearization. the standard Galerkin finite element method, we write the solution as c =  NFor f 0,fine , where φi0,fine are the standard linear element basis functions defined i=1 ci φi h on T and N f denotes the number of the nodes on the fine grid. Equation (12.62) can be presented in matrix form: cn+1 − cn + An cn+1 = 0, (12.63) τ where M n is the mass matrix given by   0,fine 0,fine n n M = [m i j ] = am (c )φi φ j d x + di a f (cn )φi0,fine φ0,fine d x, j Mn

Ωm

Ω f,i

i

n

and A is the stiffness matrix given by 

An = [ai j ] =

Ωm

bm (cn , x)∇φi0,fine · ∇φ0,fine d x + di j

 i

Ω f,i

b f (cn , x)∇φi0,fine · ∇φ0,fine d x. j

Hence, at each time step we have the following linear problem:

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12 GMsFEM for selected applications

Q n cn+1 = M n cn ,

(12.64)

where Q n = (M n + τ An ). This fine-scale discretization yields matrices of the size Nf × Nf.

12.4.3 Coarse-grid discretization using GMsFEM: Offline spaces We use multiscale basis functions to represent the solution space. We consider the continuous Galerkin (CG) formulation, though other discretizations can also be used. We denote the basis functions by φωk i , which is supported in ωi , and the index k represents the numbering of these basis functions. In turn, the CG solution will be sought as  cki (t)φωk i (x). c H (x, t) = i,k

Once the basis functions are identified, the CG global coupling is given through the variational form ∂c H , v) + a(c H , v) = 0, for all v ∈ VH,off , (12.65) m( ∂t where VH,off is used to denote the space spanned by multiscale basis functions and   am c v d x + di a f c v d x, m(c, v) = Ωm

 a(c, v) =

Ωm

Ω f,i

i

bm ∇c · ∇v d x + di

 Ω f,i

i

b f ∇c · ∇v d x.

Let Vh be the conforming finite element space with respect to the fine-scale partition T h . We assume c ∈ Vh is the fine-scale solution satisfying ∂c , v) + a(c, v) = 0, v ∈ Vh . (12.66) ∂t We now present the construction of the offline basis functions and the corresponding ω spectral problems for obtaining a space reduction. The snapshot space, VH,snap , is the space of all fine-scale basis functions or the solutions of some local problems with variouschoicesofboundaryconditions.Weusethefollowingκ-harmonicextensionsto form a snapshot space. For each fine-grid function, δ hj (x), which is defined by δ hj (x) = δ j,k , ∀ j, k ∈ Jh (ωi ), where Jh (ωi ) denotes the fine-grid boundary node on ∂ωi . For simplicity, we omit the index i. Given a fine-scale piecewise linear function defined on ∂ω (here, ω is a generic coarse element), we define ψ ωj by the following variational problem:   ω,snap , v) = bm ∇ψ ωj · ∇v d x + d j b f ∇ψ ωj · ∇v d x = 0 in ω, a(ψ j m(

ωm

j

ω f, j

(12.67)

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport

303

and ψ ωj = δ hj (x) on ∂ω, ω = ωm ⊕ j d j ω f, j . For brevity of the notation, we now omit the superscript for ω, yet it is assumed throughout this section that the offline space computations are localized to respective coarse subdomains. Let li be the number of functions in the snapshot space in the region ω, and VH,snap = span{ψ j : 1 ≤ j ≤ li }, for each coarse subdomain ω. Denote  snap snap  . Rsnap = ψ1 , . . . , ψli ω , we perform a dimension reduction of In order to construct the offline space VH,off the snapshot space using an auxiliary spectral decomposition. The analysis in [174] motivates the following eigenvalue problem in the space of snapshots: off off Aoff Ψkoff = λoff k S Ψk ,

(12.68)

where  off ] = Aoff = [amn

ωm

bm ∇ψm · ∇ψn d x + d j

ω f, j

j

 off ] = S off = [smn



ωm

bm ψm ψn d x + d j

 j

ω f, j

T AR b f ∇ψm · ∇ψn d x = Rsnap snap ,

T SR b f ψm ψn d x = Rsnap snap ,

where Aoff and S off denote analogous fine-scale matrices. To generate the offline ω eigenvalues from Equation (12.68) and space, we then choose the smallest Moff form the corresponding eigenvectors in the space of snapshots by setting φk = li off ω off Ψ ψ (for k = 1, . . . , M ), where Ψ j off j=1 k j k j are the coordinates of the vector off Ψk . To construct an appropriate solution space and variational formulation that for a continuous Galerkin approximation, we begin with an initial coarse space VHinit = N span{χi }i=1 . Recall that N denotes the number of coarse neighborhoods. Here, χi are the standard multiscale partition of unity functions defined by   a(χi , v) = bm ∇χi · ∇v d x + d j b f ∇χi · ∇v d x = 0 K ∈ ω ωm

j

ω f, j

χi = gi on ∂ K , for all K ∈ ω, where gi is a continuous function on ∂ K and is linear on each edge of ∂ K . We then multiply the partition of unity functions by the eigenfunctions in the ωi to construct the resulting basis functions offline space Voff ωi , φωk i = χi ψkωi for 1 ≤ i ≤ N and 1 ≤ k ≤ Moff

(12.69)

304

12 GMsFEM for selected applications

ωi where Moff denotes the number of offline eigenvectors that are chosen for each coarse node i. We note that the construction in Equation (12.69) yields continuous basis functions due to the multiplication of offline eigenvectors with the initial (continuous) partition of unity. Next, we define the continuous Galerkin spectral multiscale space as ωi }. (12.70) VH,off = span{φωk i : 1 ≤ i ≤ N and 1 ≤ k ≤ Moff Nc Using a single index notation, we may write VH,off = span{ψi }i=1 , where Nc = N ωi i=1 Moff denotes the total number of basis functions in the space Voff . We also construct an operator matrix

  R0T = ψ1 , . . . , ψ Nc , where ψi are used to denote the nodal values of each basis function defined on the fine grid.  We seek c H (x) = i ci φi (x) ∈ VH,off such that m(

∂c H , v) + a(c H , v) = 0 for all v ∈ Voff . ∂t

(12.71)

We note that variational form in (12.71) yields the following linear algebraic system: Q n0 c0n+1 = M0n c0n , (12.72) where c0 denotes the nodal values of the discrete CG solution, and Q n0 = R0 Q n R0T and M0n = R0 M n . Numerical result We present numerical results for the coarse-scale solution using offline basis functions. The basis functions of the offline space are constructed following the procedure described above. Note that the basis functions are constructed only once at the initial time and used for generating the stiffness matrix and the right-hand side. We consider the solution of the problem with constant and nonlinear matrixfracture coefficients in (12.62). As constant coefficients (see the previous section) representing matrix and fracture properties, we use the following: am = 0.8, bm = 1.3 · 10−7 and a f = 0.001, b f = 1.0. For nonlinear matrix-fracture coefficients, we use (12.58) and κf RT c, a f (c) = φ f , b f (c, x) = μ

(12.73)

where Dk = 10−7 [m2 /s], Di = 10−8 [m2 /s], φ = 0.04, T = 413 [K], μ = 2 · 10−5 [kg/(m s)], and for fractures k f = 10−12 [m2 ], φ f = 0.001.

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport

305

As for permeability κ in (12.58), we use constant κ = κ0 and stress-dependent model κ = κm (see [221, 410]) with 



κm = κ0 1 −

pc − α p p1

M 3

,

where κ0 = 10−18 [m2 ], p = RT c, pc = 109 [Pa], p1 = 1.8 · 109 [Pa], α = 0.5, and M = 0.5. For the sorbed gas, we use the Langmuir model F(c) = cμs

s , (1 + sc)2

where s = 0.26 · 10−3 and cμs = 0.25 · 10−5 [mol/m3 ].

Fig. 12.21 Coarse and fine grids. Coarse-grid configuration contains 50 cells, 85 facets, and 36 vertices. Fine-grid configuration contains 7580 cells, 11470 facets, and 3891 vertices.

Fig. 12.22 Coarse and fine grids. Coarse-grid configuration contains 200 cells, 320 facets, and 121 vertices. Fine-grid configuration contains 13036 cells, 19694 facets, and 6659 vertices.

306

12 GMsFEM for selected applications

The equation is solved with Dirichlet boundary condition c(x, t) = 5000 on the = 0 on other boundaries. left boundary and Neumann boundary conditions ∂c(x,t) ∂n The domain Ω has a length of 60 m in both directions. We calculate the concentration for tmax = 5 years with the time step τ = 10 days. As for the initial condition, we use c(x, t = 0) = 10000 [mol/m3 ]. For the numerical solution, we construct two structured coarse grids with 36 nodes (Figure 12.21) and with 121 nodes (Figure 12.22). As for fine grids, we use unstructured grids, which resolve the existing fractures.

Fig. 12.23 Solution with constant matrix-fracture coefficients on coarse (right) and on fine (left) grids for t=1, 3, and 5 years (from top to bottom).

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport

307

Fig. 12.24 Coarse-scale (right) and fine-scale (left) solutions for t=5 year for the case of nonlinear matrix-fracture coefficients with κ = κ0 .

Fig. 12.25 Coarse-scale (right) and fine-scale (left) solutions for t=5 year for the case of nonlinear matrix-fracture coefficients with κ = κm .

In Figure 12.23, we show the pressure distribution for three concrete time levels t = 1, 3, and 5 years. For the pressure and concentration, we have the following relationship: p = RT c. Pressure distribution for nonlinear matrix-fracture coefficients in (12.62) is presented in Figures 12.24 and 12.25 for last time level. In these figures, we show fine-scale (reference) and coarse-scale (multiscale) solutions. The coarse-scale solution is obtained in an offline space of dimension 288 (using Moff = 8 multiscale basis functions per coarse neighborhood) and the fine-scale solution is obtained in a space of dimension 3891. Compared to the fine-scale solution on the left with the coarse-scale solution on the right of the figures, we observe that GMsFEM can approximate the fine-scale solution accurately. To compare the results, we use relative weighted errors ||ε||∗ = ||c H − ch ||∗ /||ch ||∗ , using L a2 and Ha1 weighted norms that are defined as

308

12 GMsFEM for selected applications

 ||ε|| L a2 =

1/2 a ε dx 2

Ω

 , ||ε|| Ha1 =

Ω

1/2 (a ∇ε, ∇ε) d x

.

Fig. 12.26 Multiscale basis functions for a subdomain corresponding to the first 3 smallest eigenvalues in the case with constant fracture-matrix properties (after multiplication to the partition of unity function), ψi,k = χi ψkωi ,off , i = 25 and k = 0, 1, 2 (from left to right).

In Table 12.15, we present relative errors (in percentage) for the last time level for constant fracture and matrix properties in (12.62) using coarse grids with 36 and 121 nodes. For the coarse-scale approximation, we vary the dimension of the spaces by selecting a certain number of offline basis functions (Moff ) corresponding to the smallest eigenvalues. In Table 12.15, we recall that Voff denotes the offline space, dim(Voff ) is the offline space dimension, Moff is the number of the multiscale basis functions per coarse neighborhood (we use a similar number of Moff for each ωi ), and cms and ch are the multiscale and reference solutions, respectively. Figure 12.26 presents the multiscale basis functions corresponding to the first 3 smallest eigenvalues in the case with constant fracture-matrix properties in (12.62). These offline basis functions are multiplied by partition of unity functions. When we use Moff = 8 and the case with 36 coarse nodes, the relative L a2 and Ha1 weighted errors are 0.3% and 0.7%, respectively, for the final time level. The dimension of the corresponding offline space is 288 and for reference solution it is 3891. For a coarse grid with 121 nodes, the relative errors are slightly smaller 0.1% and 0.2% for L a2 and Ha1 weighted errors, respectively. The dimension of the corresponding offline space is 968 and for reference solution it is 6659. The relative L a2 and Ha1 errors at different time instants for the cases with 36 and 121 coarse grids are presented in Figures 12.27 and 12.28. As we observe if we take 4 or more basis functions per coarse node, the relative errors remain small. We present relative weighted errors in Tables 12.16 and 12.17 for different number of eigenvectors Moff for the case with nonlinear matrix-fracture coefficients in (12.62). We consider a case with 36 coarse nodes. When we use Moff = 8 and the case with κ = κ0 , the relative L a2 and Ha1 errors are 0.2% and 0.7%, respectively. The dimension of the corresponding offline space is 288 and for reference solution it is 3891. For the case with κ = κm in (12.58), we have 0.4% and 1.0% of relative L a2 and Ha1 errors, respectively. The dimensions of coarse spaces for the corresponding

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport Moff dim(Voff ) 1 2 4 8 12

36 72 144 288 432

λmin 10−9

9.0 4.5 10−8 1.1 10−7 2.2 10−6 0.19

L a2 24.484 12.229 1.068 0.303 0.083

Ha1 84.383 33.923 2.162 0.737 0.258

Moff dim(Voff ) 1 2 4 8 12

121 242 484 968 1452

λmin 10−8

2.5 9.6 10−8 1.6 10−7 0.37 1.23

L a2

Ha1

17.136 3.975 0.651 0.110 0.060

68.989 36.337 3.595 0.246 0.108

309

Table 12.15 Numerical results (relative errors (%) for the final time level). Left: for the case with 36 coarse nodes. Right: for the case with 121 coarse nodes.

Fig. 12.27 Relative L a2 and Ha1 weighted errors (%) for coarse grid in Figure 12.21 with 36 nodes. Constant matrix-fracture properties.

Fig. 12.28 Relative L a2 and Ha1 weighted errors (%) for coarse grid in Figure 12.21 with 121 nodes. Constant matrix-fracture properties.

number of eigenvectors are 72, 144, 288, 432, and 576 for Moff = 2, 4, 8, and 12. We observe that as the dimension of the coarse space (the number of selected eigenvectors Moff ) increases, the respective relative errors decrease. Also, we have similar error behavior as for cases with constant matrix-fracture coefficients. Moreover, we see that the decrease in the relative error is fast initially and one can obtain small errors using only a few basis functions.

310

12 GMsFEM for selected applications Moff dim(Voff ) 1 2 4 8 12

36 72 144 288 432

λmin

L a2

10−9

4.8 2.4 10−8 6.0 10−8 1.1 10−6 0.19

Ha1

21.717 10.772 0.933 0.270 0.123

87.897 38.774 1.947 0.737 0.323

λmin

Moff dim(Voff ) 1 2 4 8 12

121 242 484 968 1452

10−8

2.5 9.6 10−8 1.6 10−7 0.37 1.23

L a2

Ha1

14.333 3.673 0.646 0.110 0.063

64.197 30.510 3.272 0.251 0.159

Table 12.16 Numerical results (relative weighted errors (%) for final time level) for case with κ = κ0 in (12.58). Left: the case with 36 coarse nodes. Right: the case with 121 coarse nodes.

Moff dim(Voff ) 1 2 4 8 12

36 72 144 288 432

λmin

L a2

Ha1

2.0 10−9 1.0 10−8 2.6 10−8 5.1 10−7 0.19

24.484 10.785 1.247 0.432 0.234

92.039 35.874 2.423 1.069 0.712

λmin

L a2

Ha1

2.5 10−8 9.6 10−8 1.6 10−7 0.37 1.23

16.318 3.715 0.645 0.134 0.096

60.267 22.872 3.000 0.386 0.282

Moff dim(Voff ) 1 2 4 8 12

121 242 484 968 1452

Table 12.17 Numerical results (relative weighted errors (%) for the final time level) for case with κ = κm in (12.58). Left: the case with 36 coarse nodes. Right: the case with 121 coarse nodes.

Remark 12.1. In the numerical simulations, we do not use empirical interpolation procedures for approximating the nonlinear functionals a· (c, ·) and b· (c, ·) (see [72] for more details). In the approaches of [72], empirical interpolation concepts [82] are used to evaluate the nonlinear functions by dividing the computation of the nonlinear function into coarse regions, evaluating the contributions of nonlinear functions in each coarse region taking advantage of a reduced-order representation of the solution. By using these approaches, we can reduce the computational cost associated with evaluating the nonlinear functions and consequently making the computational cost to be independent of the fine grid. Remark 12.2. Connected fracture networks in each coarse region are represented by small (dominant) eigenvalues. For example, if a coarse region contains two distinct fracture networks, which are disconnected, then there are two very small eigenvalues and one needs to choose at least two multiscale basis functions corresponding to the eigenvectors of these two small eigenvalues. The proposed procedure can automatically find the disconnected fracture networks. Moreover, using adaptivity criteria, we can select the necessary number of basis functions.

12.4.4 Randomized oversampling GMsFEM Next, we present numerical results for the oversampling and the randomized snapshots (as discussed in Section 4.8) that can substantially save the computational cost

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport

311

for snapshot calculations. In this algorithm, instead of solving local harmonic problems (12.67) for each fine-grid node on the boundary, we solve a small number of harmonic extension local problems with random boundary conditions (see Section 4.8). More precisely, we let ω ,rsnap

ψj i

= r j , x ∈ ∂ωi+ ,

where r j are independent identical distributed standard Gaussian random vectors on the fine-grid nodes of the boundary. When we use randomized snapshots, we only generate a fraction of the snapshot vectors by using random boundary conditions. For snapshot space calculations, we use the extended coarse-grid neighborhood for m = 1, 2, . . . , by ωi+ = ωi + m, where m is width of the fine-grid layer. Here, for example, ωi+ = ωi + 1 means the coarse-grid neighborhood plus all 1 layer of an adjacent fine grid of ωi , and so on (see Figure 12.29 for illustration). Calculations in the oversampled neighborhood domain ωi+ reduce the effects due to the artificial oscillation in random boundary conditions.

Fig. 12.29 Neighborhood domain with oversampling (ωi+ = ωi + m, m = 1, 2, 4, 6) for the coarse grid with 36 nodes.

Numerical results The simulation results are presented in Tables 12.18 and 12.19 for the 36-node coarsegrid case. We use constant matrix-fracture properties; see (12.62). We present the results for the randomized snapshot case for the last time level. In the simulations, we set the oversampling size m = 0, 2, 4, 6 for ωi+ = ωi + m and use different number of multiscale basis functions Moff = 2, 4, 8, and 12. In Table 12.18, we investigate the effects of the oversampling ωi+ = ωi + m, as we increase the number of fine-grid extensions m = 0, 2, 4, and 6. We see that the oversampling helps to improve the results initially, but the improvements are slow

312

12 GMsFEM for selected applications

and larger oversampling domains do not give significant improvement in the solution accuracy. When we use a snapshot ratio of 25.6% (between the standard number of snapshots and the randomized algorithm for ωi+ = ωi + 4), the relative L a2 and Ha1 weighted errors are 0.2% and 0.8% for full snapshots and 0.2% and 0.9% for randomized snapshots. We observe that the randomized algorithm can give similar errors as full snapshots. Table 12.19 shows relative L a2 and Ha1 errors for different number of randomized snapshots Mi . The oversampled region ωi+ = ωi + 4 is chosen, that is, the oversampled region contains an extra 4 fine-grid cell layers around ωi . The numerical results show that one can achieve a similar accuracy when using a fraction of snapshots with randomized algorithms and thus, it can provide substantial CPU savings.

Moff

full snapshots randomized snapshots L a2 Ha1 L a2 Ha1

without oversampling, ωi 100 % 39.7 % 2 12.229 33.923 8.303 33.237 4 1.068 2.162 1.704 4.730 8 0.303 0.737 1.005 2.962 12 0.083 0.258 0.557 1.643

2 4 8 12

with oversampling, ωi+ = ωi + 2 100 % 28.3 % 12.247 33.943 8.921 33.399 1.073 4.237 0.972 3.750 0.261 0.744 0.354 1.003 0.114 0.329 0.219 0.704

2 4 8 12

with oversampling, ωi+ = ωi + 4 100 % 25.6 % 12.216 33.657 9.334 28.213 1.015 4.576 0.626 2.561 0.262 0.841 0.264 0.949 0.114 0.349 0.153 0.441

2 4 8 12

with oversampling, ωi+ = ωi + 6 100 % 22.5 % 12.746 35.899 9.455 27.922 1.013 5.014 0.603 2.377 0.251 0.820 0.277 0.875 0.124 0.369 0.120 0.421

Table 12.18 The results with randomized oversampling for GMsFEM with the number of snapshots Mi = 24 (constant matrix-fracture properties) in every ωi+ = ωi + n, n = 0, 2, 4, 6 for the coarse mesh with 36 nodes (relative errors (%) for final time level).

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport Moff

313

12.8 % (Mi = 12) 17.0 % (Mi = 16) 21.3 % (Mi = 20) 25.6 % (Mi = 24) 29.8 % (Mi = 28) L2 H1 L2 H1 L2 H1 L2 H1 L2 H1

2 8.228 4 1.908 8 0.861 12 -

24.878 4.208 1.777 -

9.449 1.381 0.589 0.313

28.895 3.581 1.563 0.781

7.346 0.779 0.292 0.217

22.774 2.692 1.189 0.581

9.334 0.626 0.264 0.153

28.213 2.561 0.949 0.441

9.335 0.843 0.245 0.110

26.973 4.439 0.894 0.393

Table 12.19 The results for randomized oversampling for GMsFEM with different number of snapshots Mi = 12, 16, 20, 24, 28 in every ωi+ = ωi + 4 (constant matrix-fracture properties) for the coarse mesh with 36 nodes (relative errors (%) for final time level).

12.4.5 Residual-based adaptive online GMsFEM In this section, we consider the construction of the online basis functions that are used in some regions adaptively to significantly reduce the error (c.f. Chapter 5). It is important that the offline space contains some essential features of the solution space. In the numerical simulations, we demonstrate that with a sufficient number of offline basis functions, we can achieve a rapid convergence for the proposed online procedure. First, we derive the error indicator for the error (cn − cnH ) for a time-dependent problem (12.75) in the energy norm. Furthermore, we use the error indicator to develop an enrichment algorithm. The error indicator gives an estimate of the local error on the coarse-grid region ωi and we can then add basis functions to improve the solution. We assume, as before, Vh is the fine-scale finite element space. To find the finescale solution cn+1 ∈ Vh , we solve (as before) m(

cn+1 − cn , v) + a(cn+1 , v) = ( f, v), ∀v ∈ V τ

(12.74)

and for multiscale solution cn+1 ∈ VH,off we have H m(

n cn+1 H − cH , v) + a(cn+1 H , v) = ( f, v), ∀v ∈ V H,off . τ

(12.75)

We define a linear functional r n (v) for the n-th time level by n+1 n r n (v) = τ ( f, v) − m(cn+1 H − c H , v) − τ a(c H , v).

Let ωi be a coarse region and Vi = H01 (ωi ). We define  rin (v) = τ − dj

 ωi

fv−

 f, j

j

ωi

 ωim

n am (cn+1 H − cH ) v d x − τ

n a f (c) (cn+1 H − cH ) v d x − τ d j

ωim

bm ∇cn+1 H · ∇v d x

 j

ωi

f, j

b f ∇cn+1 H · ∇v d x,

314

12 GMsFEM for selected applications f, j

where ωi = ωim ⊕ j d j ωi and m and f represent the matrix and the fracture. We note that the functionals r n and rin are residual and local residual at the n-th time level. The solution at the (n + 1)-th time level (cn+1 H ) is the solution of the elliptic problem of the form n (12.76) aτ (cn+1 H , v) = τ ( f, v) + m(c H , v). We use the following notation: aτ (u, v) = m(u, v) + τ a(u, v). Error estimators for the spatial discretization error take into account the dependence of the elliptic problem (12.76) on the time step parameter τ , and we will use the τ -weighted H1 norm ||v||2τ = τ ||v||a2 + ||v||2m , where ||v||2τ = aτ (v, v), ||v||a2 = a(v, v), ||v||2m = m(v, v). Similarly, we define these norms in a subdomain, e.g., ||v||2τ in a subdomain ω is defined with the notation ||v||2τ ,ω . We define the projection  : V → VH,off by v =

Nc 

χi (Pi v),

i=1

where Pi : V → span{φωk i } is the projection defined by Pi v =

li 

bτωi (v, φωk i )φωk i

k=1

φωk i

with being the dominant eigenvectors corresponding to Rayleigh Quotient aτωi (·, ·)/bτωi (·, ·), where aτωi is the restriction of aτ on ωi and bτωi (v, v) = τ ωi b|∇χi |2 v 2 d x. Let en = cn − cnH be error for nth time level. Using the results in [388], we obtain ||e

n+1

1/2 1/2   N C n 2 ||τ ≤ C + ||ri ||∗,ωi + ||en ||m , Λmin i=1

where Λmin = mini λl(i) and λ(i) are eigenvalues of the Rayleigh Quotient k i+1 ωi ωi aτ (·, ·)/bτ (·, ·). This inequality gives a computable indicator of the error in the τ -weighted H1 norm. In the above inequality, ||rin ||∗,ωi en+1 = cn+1 − cn+1 H is the functional norm with respect to  · τ ,ωi .

12.4 Multiscale methods for flows in fractured media: Applications to shale gas transport

315

Remark 12.3. We note that the analysis in [388] suggests the use of a local eigenvalue problem based on Rayleigh Quotient aτωi (·, ·)/bτωi (·, ·). This eigenvalue problem is “slightly” different from (12.68) that we have used earlier. The numerical simulations show that the use of the eigenvalue problem based on aτωi (·, ·)/bτωi (·, ·) improves the convergence of the offline or online procedures slightly in the numerical examples. We will use the spectral problems based on (12.68) in the numerical simulations as it is independent of time stepping. Next, we consider online basis construction. We use the index m ≥ 1 to represent m the enrichment level. At the enrichment level m, we use VH,off to denote the corresponding space that can contain both offline and online basis functions. We will consider a m+1 m from VH,off . By online basis functions, we mean strategy for getting the space VH,off basis functions that are computed during the iterative process, contrary to offline basis functions that are computed before the iterative process. The online basis functions are . computed based on some local residuals for the current multiscale solution cn+1,(m) H m+1 m = VH,off + span{ϕ} be the new approximate space that is constructed by Let VH,off m+1 adding online basis ϕ ∈ Vi on the ith coarse neighborhood ωi and cn+1,(m+1) ∈ VH,off H be the corresponding GMsFEM solution. Following ideas in Chapter 5, we will find an online basis function ϕ ∈ Vi by solving aτ (ϕ, v) = ri (v), ∀v ∈ Vi ,

(12.77)

where ri (v) is the local residual defined above. For a solution in each time level, we iteratively enrich the offline space by residualbased online basis function. These basis functions are calculated using Equation (12.77) with zero Dirichlet boundary conditions, and the residual norm ||rin ||∗,ωi provides a measure of the amount of reduction in energy error. For the construction of the adaptive online basis functions, we first choose 0 < θ < 1; for each coarse neighborhood ωi , find the online basis ϕi ∈ Vi using equation (12.77). After computing the norm of local residuals, we calculate ηi ηi2 := ri 2∗,ωi , where ri ∗,ωi = φi τ ,ωi . Furthermore, we arrange them in descending order, i.e., η12 ≥ η22 ≥ ... ≥ η 2N . Then, we choose the smallest k such that θ

N  i=1

ηi2 ≤

k 

ηi2 .

i=1

This implies that, for the coarse neighborhood ω j ( j = 1, ..., k), we add the correm . sponding online basis ϕ j to the original space VH,off Numerical results Next, we present numerical results for residual-based online basis functions. We consider a similar problem as in the previous section with the constant matrix-fracture

316

12 GMsFEM for selected applications

properties in (12.62) and iteratively enrich the offline space by online residual basis functions in some selected time steps. The coarse- and fine-grid setups are the same as in Section 12.4.3. Because the re-generation of the matrix R is needed when we add the online basis function, we add them for some selected time steps. We note that, when we add new online basis functions, which are based on current residuals, we remove previously calculated online basis functions and keep them till we update the online basis functions. Computational time can be saved with small dimensional coarse-scale problems (Fig. 12.30). In Table 12.20, we present L a2 and Ha1 errors. We consider three different cases. In the first case (we call it Case 1), online basis functions are added at the first time step and after that in every 30th time step. In the second case (we call it Case 2), online basis functions are replaced at the first five consecutive time steps, and after that the online basis functions are updated in every 30th time step. In the third case (we call it Case 3), online basis functions are replaced at the first ten consecutive time steps, and after that the online basis functions are updated in every 30th time step. More updates initially help to reduce the error due to the initial condition. As we mentioned that the offline space is important for the convergence, and we present the results for different number of initial offline basis functions per coarse neighborhood. We use multiscale basis functions from the offline space as initial basis functions. In Table 12.21, we show errors when online basis functions are replaced at the first five consecutive time steps (as in Case 2), and, afterwards, online basis functions are updated at 10th, 20th, and 30th time step. For the calculations, we use tmax = 5 years with τ = 10 days. Calculations are performed in the coarse grid with 121 nodes for the case with constant matrix-fracture properties. We observe from this table the following facts. • Choosing 4 initial offline basis functions improves the convergence substantially. This indicates that the choice of the initial offline space is important. • Adding online basis functions less frequently (such as at every 30th time step) provides an accurate approximation of the solution. This indicates that the online basis functions can be added only at some selected time steps. Next, we would like to show that one can use online basis functions adaptively and use the adaptivity criteria discussed above. In Table 12.22, we present results for residual-based online basis functions with adaptivity with θ = 0.7. In Figure 12.31, we show errors by time. We observe that applying an adaptive algorithm can reduce errors.

12.5 Non-local multicontinua upscaling for poroelasticity in fractured media

317

Fig. 12.30 Fine-scale solution (right), coarse-scale using 2 offline basis functions (middle), and coarse-scale after two online iterations for some time levels (left) for t = 1, 3, and 5 years (from top to bottom) (constant matrix-fracture coefficients). Fine-scale solution size is 6659. For 2 offline basis functions, the coarse solution dimension is 242 and, after two online iterations, the coarse-scale solution size is 484.

12.5 Non-local multicontinua upscaling for poroelasticity in fractured media In the reservoir simulation, mathematical modeling of the fluid flow and geomechanics in the fractured porous media plays an important role. A coupled poroelastic model can help for a better understanding of the processes in the fractured reservoirs. In this work, we consider an embedded fracture model (EFM) for coupled flow and mechanics problems based on the dual continuumDual continuum approach. The mathematical model is described by the coupled system of equations for displacement and fracture/matrix pressures [369]. Coupling of the fracture and matrix equations is derived from the mass exchange between the two continua (transfer term) and based on the embedded fracture model. For the geomechanical effect, we consider the deformation of the porous matrix due to pressure change, where pressure plays a role of specific source term for deformation [67, 68, 282–284, 288]. Fundamentally, the

318

12 GMsFEM for selected applications DO F (# iter)

L a2

Ha1

Moff = 1 121 17.136 68.989 242 (1) 13.047 43.662 363 (2) 7.275 12.262 Moff = 2 242 3.975 36.337 363 (1) 1.653 6.705 484 (2) 0.889 0.972 Moff = 4 484 0.651 3.595 605 (1) 0.208 0.307 726 (2) 0.171 0.056

DO F (# iter)

L a2

Ha1

Moff = 1 121 17.136 68.989 242 (1) 13.209 42.777 363 (2) 6.603 12.689 Moff = 2 242 3.975 36.337 363 (1) 1.716 7.841 484 (2) 0.546 0.914 Moff = 4 484 0.651 3.595 605 (1) 0.165 0.313 726 (2) 0.105 0.057

DO F (# iter)

L a2

Ha1

Moff = 1 121 17.136 68.989 242 (1) 13.002 42.091 363 (2) 6.125 13.186 Moff = 2 242 3.975 36.337 363 (1) 1.692 8.709 484 (2) 0.449 0.862 Moff = 4 484 0.651 3.595 605 (1) 0.144 0.318 726 (2) 0.076 0.056

Table 12.20 Convergence history (in %) using one, two, and four offline basis functions (Moff = 1, 2, and 4). We add online basis functions for every 30th time step and for N th first steps (Cases 1, 2, and 3). Left: N = 1. Middle: N = 5. Right: N = 10. Here, D O F for the last time step.

DO F (# iter)

L a2

Ha1

Moff = 1 121 17.136 68.989 242 (1) 12.511 41.477 363 (2) 5.910 13.153 Moff = 2 242 3.975 36.337 363 (1) 1.624 8.225 484 (2) 0.378 0.934 Moff = 4 484 0.651 3.595 605 (1) 0.126 0.303 726 (2) 0.048 0.033

DO F (# iter)

L a2

Ha1

Moff = 1 121 17.136 68.989 242 (1) 12.925 42.778 363 (2) 6.275 12.705 Moff = 2 242 3.975 36.337 363 (1) 1.669 8.034 484 (2) 0.474 0.880 Moff = 4 484 0.651 3.595 605 (1) 0.147 0.305 726 (2) 0.080 0.041

DO F (# iter)

L a2

Ha1

Moff = 1 121 17.136 68.989 242 (1) 13.209 42.777 363 (2) 6.603 12.689 Moff = 2 242 3.975 36.337 363 (1) 1.716 7.841 484 (2) 0.546 0.914 Moff = 4 484 0.651 3.595 605 (1) 0.165 0.313 726 (2) 0.105 0.057

Table 12.21 Convergence history (in %) using one, two, and four offline basis functions (Moff = 1, 2, and 4). We add online basis functions for every N th time step and for first 5 time steps. Left: N = 10. Middle: N = 20. Right: N = 30. Here, D O F for last time step.

system of equations is coupled between flow and geomechanics, where displacement equation includes the volume force, which is proportional to the pressure gradient, and the pressure equations include the term which describes the compressibility of the medium.

12.5 Non-local multicontinua upscaling for poroelasticity in fractured media DO F  iter L a2 (# iter) Moff = 1 121 17.136 242 (1) 11 13.209 363 (2) 22 6.603 Moff = 2 242 3.975 363 (1) 11 1.716 484 (2) 22 0.546 Moff = 4 484 0.651 605 (1) 11 0.165 726 (2) 22 0.105

Ha1

DO F

68.989 42.777 12.689

121 243 381

36.337 7.841 0.914

242 376 504

3.595 0.313 0.057

484 635 737



iter

L a2

Moff = 1 17.136 44 4.082 78 2.681 Moff = 2 3.975 44 0.441 77 0.376 Moff = 4 0.651 44 0.110 68 0.098

319

Ha1 68.989 6.418 2.871 36.337 0.720 0.325 3.595 0.044 0.039

Table 12.22 Convergence history (in %) using one, two, and four offline basis functions (Moff = 1, 2, and 4). We add online basis functions for every 30th time step and for first 5 steps. Left: without space adaptivity. Right: with space adaptivity. Here, D O F for last time step.

Due to high permeability, fractures have a significant impact on the flow processes. A common approach to fracture modeling is to model them as lower dimensional problems [137, 159, 214, 320]. The result is a coupled mixed dimensional flow model, where we consider flow in the two domains (matrix and fracture) with mass transfer between them. The fractures are not resolved by grid but are included as an overlaying continuum with an exchange term between fracture and matrix that appears as an additional source (Embedded Fracture Model (EFM)) [238, 389, 390]. This approach is related to the class of the multicontinuum model [41, 152, 409]. Instead of the dual continuum approach, we represent fractures directly using a lower dimensional flow model embedded in a porous matrix domain. In EFM, we have two independent grids for fracture networks and matrix, where simple structured meshes can be used for the matrix. For geomechanics, we derive an embedded fracture model, where each fracture provides an additional source term for the displacement equation. This approach is based on the mechanics with dual porosity model [417, 432]. In this model, we suppose displacement continuity on the fracture interface. For the discrete fracture model, specific enrichment of the finite element space can be used for accurate solution of the elasticity problem with displacement discontinuity [13]. Here, we focus on the fully coupled poroelastic model for embedded fracture model and construct an upscaled model for fast coarse-grid simulations. For the fine-grid approximation, we use the finite volume method (FVM) for flow problem and the finite element method (FEM) for geomechanics. FVM is widely used as discretization for the simulation of flow problems [58, 391]. We use a cell centered finite volume approximation with two-point flux approximation (TPFA) for pressure. FEM is typically used for

320

12 GMsFEM for selected applications

Fig. 12.31 Relative L a2 (left) and Ha1 (right) weighted errors (%) as a function of time for the coarse grid in Figure 12.22 with 121 nodes. The case with constant matrix-fracture properties. Errors using offline basis functions and online basis functions with and without adaptivity. Top: 1 offline basis function. Middle: 2 offline basis functions. Bottom: 4 offline basis functions.

approximating the solid deformation problem. We use a continuous Galerkin method with linear basis functions with an accurate approximation of the coupling term. Fine-grid simulation of the processes in fractured porous media leads to very expensive simulations due to the extremely large degrees of freedom. To reduce the cost of simulations,multiscalemethodsorupscalingtechniquesareused.Inthiswork,weconstruct an upscaled coarse-grid poroelasticity model with an embedded fracture model. Our approach uses the general concept of non-local multicontinua (NLMC) upscaling for flow [120], and significantly generalized it to the coupled flow and mechanics problems. The local problems for the upscaling involve computations of local basis functions via an energy minimization principle and the degrees of freedom are chosen such that they represent physical parameters related to the coupled flow and mechanics problem. We summarize below the main goals of our work: • a new fine-grid embedded fracture model for poroelastic media (coupled system);

12.5 Non-local multicontinua upscaling for poroelasticity in fractured media

321

• a new accurate and computationally effective fully coarse-grid model for coupled multiphysics problem using NLMC whose degrees of freedom have physical meaning on the coarse grid. Non-local multicontinua (NLMC) upscaling for processes in the fractured porous media provides an effective coarse-scale model with physical meaning, and leads to a fast and accurate solver for coupled poroelasticity problems. To capture fine-scale processes at the coarse-grid model, local multiscale basis functions are presented. Constructing the basis functions based on the constrained energy minimization problem in the oversampled local domain is subject to the constraint that the local solution vanishes in other continua except for the one for which it is formulated. Multiscale basis functions have spatial decay properties in local domains and separate background medium and fractures. The proposed upscaled model has only one coarse degree of freedom (DOF) for each fracture network. Numerical results show that our NLMC method for fractured porous media provides an accurate and efficient upscaled model on the coarse grid.

12.5.1 Embedded fracture model for poroelastic medium The proposed mathematical model of a coupled flow and mechanics in a fractured poroelastic medium contains an interacting model for fluid flow in the porous matrix, flow in the fracture network and mechanical deformation. The matrix is assumed to be linear elastic and isotropic with no gravity effects. The mechanical and flow models are coupled through hydraulic loading on the fracture walls and using the effective stress concept [369, 417]. For fluid flow, we consider a mixed dimensional formulation, where we have a coupled problem for fluid flow in the porous matrix in Ω ∈ Rd (d = 2,3), and flow in the fracture network on γ ∈ Rd−1 (see Figure 12.32 for d = 2).

Fig. 12.32 Schematic illustration of the problem with embedded fracture model.

322

12 GMsFEM for selected applications

Porous matrix flow model. Using the mass conservation and Darcy law in the domain Ω: km ∂m + div(ρqm ) = ρ f m , qm = − grad pm , x ∈ Ω, ∂t νf

(12.78)

where m is the fluid mass, pm is the matrix pressure, qm is Darcy velocity, ν f is the viscosity, ρ is the fluid density, and f m is the source term. Due to the motion of the solid skeleton and Biot’s theory, we have the following relationships [131, 282–284]:   1 v (12.79) ( pm − p0 ) + αε , m − m0 = ρ M where subscript 0 means reference state, α is Biot coefficient, M is Biot’s modulus, εv is the volumetric strain (the trace of the strain tensor, εv = tr ε), and 1 1 = Φc f + , M N

1 α − Φ0 1 dρ = , cf = . N Ks ρ dpm

Here, K s is the solid grain stiffness, c f is the fluid compressibility, and Φ is Lagrange’s porosity (also known as reservoir porosity). From equation (12.79), we can express the reservoir porosity change induced by mechanical deformation as Φ − Φ0 = αv +

1 ( pm − p0 ). N

(12.80)

The permeability of the matrix is updated using the current porosity by the power-law relationship  n Φ , (12.81) km = k0 Φ0 where the cubic law with n = 3 is usually used [411, 430]. Therefore by assuming slightly compressible fluids, for the fluid flow in the porous matrix, we have the following parabolic equation:   ∂εv km 1 ∂ pm +α − div grad pm = f m , M ∂t ∂t νf defined in the domain Ω. For the case of fracture porous medium, we should add mass transfer term between matrix and fracture:   ∂εv km 1 ∂ pm +α − div grad pm + L m f = f m , (12.82) M ∂t ∂t νf where for the mass exchange between matrix and fracture, we assume a linear relationship L m f = βm f ( pm − p f ).

12.5 Non-local multicontinua upscaling for poroelasticity in fractured media

323

This mass exchange term occurs only on the fracture boundary. Fracture flow model. For the highly permeable fractures, we use the following reduced dimension model for the fluid flow on γ ∈ R(d−1) [230, 369]: kf ∂(ρb) + div(ρ q f ) − ρL m f = ρ f f , q f = −b grad p f , x ∈ γ, ∂t νf

(12.83)

where b is the fracture aperture, p f is the fracture pressure, and q f is the average velocity of fluid along the fracture plane that can be calculated using the cubic low (k f = b2 ). For the calculation of the fracture aperture b, we can use the follow2 ) ing relation b(t) = zp f (t), where z = 2(1−ν and deformation proportional to the E fracture pressure p f , where η is Poisson’s ratio and E is the elastic modulus [236, 343]. Since   ∂pf ∂b ∂ρ ∂b ∂(ρb) =ρ +b =ρ + bc f , (12.84) ∂t ∂t ∂t ∂t ∂t and by assuming slightly compressible fluids [230]     ∂pf ∂pf ∂b ∂b + bc f ≈ ρ0 + bc f , ρ ∂t ∂t ∂t ∂t div(ρ q f ) ≈ ρ0 div q f , ρL m f ≈ ρ0 L m f , ρ f f ≈ ρ0 f f . Therefore, we have the following equation on fracture γ:   ∂pf kf ∂b + bc f − div b grad p f + L f m = f f , x ∈ γ, ∂t ∂t νf

(12.85)

where, for the mass exchange between matrix and fracture, we assume a linear relationship between the flux and pressure difference, namely L f m = β f m ( p f − pm ). Let β f m = η f β and βm f = ηm β, where β is the transfer term proportional to the matrix permeability, η f and ηm are the geometric factors that will be described in next section. Then we have ∂εv ∂ pm +α − div(bm grad pm ) + ηm β( pm − p f ) = f m , x ∈ Ω, ∂t ∂t (12.86) ∂pf ∂b + − div(b f grad p f ) + η f β( p f − pm ) = f f , x ∈ γ, af ∂t ∂t

am

where am = 1/M, a f = bc f , bm = km /ν f , and b f = bk f /ν f . β is the transfer term proportional to the matrix and fracture probabilities, and η f and ηm are geometric factors that will be defined in the next section. The pressure coupling term expresses the conservation of the flow rate (the fluid that is lost in the fractures goes into the porous matrix). Here, we assume that the fractures have a constant aperture; in the

324

12 GMsFEM for selected applications

general case, we can use b(t) = z p f (t) as a relationship between fracture width and pressure. Mechanical deformation model. The balance of a linear momentum in the porous matrix is given by − div σT = 0, σT = σ − α pm I, x ∈ Ω,

(12.87)

where pm is the matrix pressure, σT is the total stress tensor, and σ is the effective stress [369]. Relation between the stress σ and strain ε tensors is given as σ = λεv I + 2με(u), ε(u) = 0.5(∇u + (∇u)T ), where u is the displacement vector in the porous matrix, and λ and μ are Lame’s coefficients. For incorporating the fracture pressure into the model, we assume negligible shear traction on the fracture walls and consider normal tractions on the fractures [369] with τ f = − p f n f , where n f is the normal vector to the fracture surface. After some manipulation, we obtain the following equation in domain Ω: (12.88) − div (σ − α pm I) +r f p f = 0, x ∈ Ω,  where r f comes from the integration over fracture surface ( γ p f n f ds) and contains direction of the fracture pressure influence. In the presented model, we follow the classic dual porosity model and add fracture pressure effects as an additional source (reaction) term. In a more general case, fractures are modeled by an interface condition, where displacements have discontinuity across a fracture but stress is continuous [13].

12.5.2 Fine-grid approximation of the coupled system Let Th = ∪i ςi be a fine-scale finite element partition of the domain Ω and Eγ = ∪l ιl be the fracture mesh (see Fig. 12.32). The implementation is based on the open-source library FEniCS [312, 313]. We use geometry objects for the construction of the discrete system for the coupled problem. For the approximation of the flow part of the system, we use cell centered finite volume approximation with two-point flux approximation. For displacement, we use the Galerkin method with linear basis functions [427]. In this work, we use the two-dimensional problem for illustration of the robustness of our method. In particular, we consider the following coupled system of equations for displacements (two displacements, u x and u y ) and fluid pressures (fracture and matrix, p f and pm ) ∂εv ∂ pm +α − div(bm grad pm ) + ηm β( pm − p f ) = f m , x ∈ Ω, ∂t ∂t ∂pf ∂b (12.89) af + − div(b f grad p f ) + η f β( p f − pm ) = f f . x ∈ γ, ∂t ∂t − div (σ(u) − α pm I) + r f p f = 0, x ∈ Ω.

am

12.5 Non-local multicontinua upscaling for poroelasticity in fractured media

325

Using an implicit scheme for the approximation of time, a finite volume approximation for pressures, and standard Galerkin method for displacements, we have the following approximation:     εv − εˇ v pm − pˇ m dΩ + dΩ − α div(bm grad pm )dΩ + ηm β( pm − p f )dΩ = f m dΩ, τ τ Ω Ω Ω Ω Ω      p f − pˇ f b − bˇ dγ + dγ − div(b f grad p f )dγ − η f β( pm − p f )dγ = af f f dγ, τ τ γ γ γ γ γ    (σ(u), ε(v))dΩ − (α pm I , ε(v))dΩ + (r f p f , v)dΩ = 0, 

am

Ω

Ω

Ω

(12.90) ˇ are solutions from the previous time step and τ is the given time where ( pˇ m , pˇ f , u) step. Using the two-point flux approximation for pressure equations, we obtain am

 εv − εˇ iv pm,i − pˇ m,i |ςi | + α i |ςi | + Ti j ( pm,i − pm, j ) + βil ( pm,i − p f,l ) = f m |ςi |, ∀i = 1, N mf τ τ j

 p f,l − pˇ f,l bl − bˇl f |ιl | + |ιl | + af Wln ( p f,l − p f,n ) − βil ( pm,i − p f,l ) = f f |ιl |, ∀l = 1, N f , τ τ n

(12.91) where Ti j = bm |E i j |/Δi j (|E i j | is the length of interface between cells ςi and ς j , Δi j is the distance between midpoint of cells ςi and ς j , Wln = b f /Δln (Δln is the distance between points l and n), and |ςi | and |ιl | are the volumes of the cells ςi and ιl . N mf is f

the number of cells in Th , and N f is the number of cells for fracture mesh Eγ . Here, we use ηm = 1/|ςi | and η f = 1/|ιl |. Also, βil = β if Eγ ∩ ∂ςi = ιl and equals zero otherwise. Matrix form. Combining the above schemes, we have the following discrete system of equations for y = ( pm , p f , u x , u y ) in the matrix form: 

 1 M + A y = F, τ

(12.92)

where ⎛

Mm ⎜ 0 M =⎜ ⎝ 0 0

0 Mf 0 0

0 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0 ⎛

⎞ Fm + τ1 Mm pˇ m + τ1 (Bm,x + Bm,y )uˇ ⎟ ⎜ F f + τ1 M f pˇ f ⎟, F =⎜ ⎠ ⎝ 0 0 ⎛

⎞ Am + Q −Q τ1 Bm,x τ1 Bm,y ⎜ −Q A f + Q 0 0 ⎟ ⎟, A=⎜ ⎝ −Bm,x −B f,x Dx Dx y ⎠ −Bm,y −B f,y Dx y Dy

326

12 GMsFEM for selected applications Mm = {m imj }, m imj =

 am |ςi |/τ i = j, , 0 i = j

Am = {Ti j },

f

 a f |ιl |/τ l = n, , 0 l = n

 β i = l, Q = {qil }, qil = , 0 i = l

A f = {Wln },

Fm = { f im },

f

M f = {m ln }, m ln =

f im = f m |ςi |,

f

F f = { fl },

f im = f f |ιl |.

Here, D is the elasticity stiffness matrix 

Dx = [dixj ] =

Ω

σx (ψi ) : εx (ψ j ) dΩ,

Bm,x = [bim,x j ]=

y

D y = [di j ] =



Ω

σ y (ψi ) : ε y (ψ j ) dΩ,

 Ω

(α pm,i , εx (ψ j ))dΩ,

m,y

Bm,y = [bi j ] =

 B f,x =

f,x [bl j ]

=−

xy

Dx y = [di j ] =

 Ω

σx (ψi ) : ε y (ψ j ) dΩ,

 Ω

(α pm,i , ε y (ψ j ))dΩ,

 f,y [bl j ]

(r f p f,l , ψ j )dΩ, B f,y = = − (r f p f,l , ψ j )dΩ, Ω     εx εx y σx σx y and ε = . with linear basis functions ψi and σ = σ yx σ y ε yx ε y We remark that the dimension of fine-grid problem is given by Ω

N f = N mf + N f + 2N vf , f

where N vf is the number of vertices on the fine grid.

12.5.3 Coarse-grid upscaled model for coupled problem Consider a coarse-grid partition T H = {K i } of the domain, where K i is the i-th coarse cell. Let K i+ be the oversampled region for the coarse cell K i obtained by enlarging K i by a few coarse-grid blocks. For our coarse-grid approximation, we will construct multiscale basis functions using the non-local multicontinua method (NLMC) [120]. In general, the construction of the multiscale basis functions starts with an auxiliary space, which is constructed by solving local spectral problems [113], and then we take eigenvectors that correspond to small (contrast dependent) eigenvalues as basis functions. These spectral basis functions represent the channels (high contrast features). Using the auxiliary space, the target multiscale space is obtained by solutions of constraint energy minimization problems in oversampling domain K i+ , subject to a set of orthogonality conditions related to the auxiliary space. More precisely, for each auxiliary basis function, we will find a corresponding multiscale basis function such that it is orthogonal to all other auxiliary basis functions with respect to a weight inner product. Our basis functions have a nice decay property away from the target coarse element. We will use the NLMC method. In the NLMC method, we use a simplified construction that separates continua in each local domain K i (coarse cell). Instead of using an auxiliary space, we obtain the required basis functions by minimizing an energy over an oversampling domain K i+ subject to

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327

the conditions that the minimizer has mean value zero on all fractures and matrix except the fracture or matrix that the basis function is formulated for. The resulting multiscale basis functions have a spatial decay property in local domains and separate background medium and fractures.

Fig. 12.33 Multiscale basis functions on mesh 20 × 20 for local domain K 4 for pressures and displacements. L For the fractures, we write γ = ∪l=1 γ (l) , where γ (l) is the l-th fracture network L j (l) γ j , where and L is the total number of fracture networks. We also write γ j = ∪l=1 (l) (l) γ j = K j ∩ γ is the fracture inside the coarse cell K j and L j is the number of fractures in K j . Again, for the construction of multiscale basis functions, we solve the constrained energy minimization problem in the oversampled local domains subject to the constraint that the local solution has zero mean on other continua except for the one for which it is formulated. For construction for coarse-grid approximation for the coupled problem, we construct a multiscale basis function for ( pm , p f , u x , u y ). For simplicity, we ignore the coupling term between pressure and displacements and find multiscale basis functions for pressure and displacements separately. In general, coupled poroelastic basis functions can be constructed using the coupled poroelastic system and the constrained energy minimization principle. Multiscale basis function for matrix and fracture pressures. To define our multiscale basis functions, we will minimize an energy subject to some constraints. In the following, we will define the constraints. We remark that we will find a set of multiscale basis functions for each coarse cell K i , and these basis functions have support in K i+ . Thus, the following constraints are needed for each K i , and they are defined within K i+ . For each coarse cell K j ∈ K i+ :

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(1) background medium (ψl0 ) : 

 Kj

ψ0i

d x = δi, j ,

γ (l) j

ψ0i ds = 0, l = 1, L j .

We note that these constraints are defined for the matrix part in K i , and they require the resulting basis function to have mean value on each continuum in K i+ except for the continuum corresponding to the matrix part in K i . (2) l-th fracture network in K i (ψli ): 

 Kj

ψli

d x = 0,

γ (l) j

ψli ds = δi, j δm,l , l = 1, L j ,

where L i is the number of fracture networks in K i . We note that these constraints are defined for a fracture network in K i , and they require the resulting basis function to have mean value on each continuum in K i+ except for the continuum corresponding to a specific fracture network in K i . For the construction of the multiscale basis functions, we solve the following local problems in K i+ using an operator restricted in K i+ . This results in solving the following local problems in K i+ : ⎛ K+ ⎞⎛ ⎞ ⎛ ⎞ + + Am i + Q K i −Q K i CmT 0 ψm 0 ⎜ ⎟ K i+ ⎜0⎟ T ⎟ ⎜ψ f ⎟ K i+ ⎜ −Q K i+ Af + Q 0 Cf ⎟⎜ ⎟ = ⎜ ⎟ (12.93) ⎜ ⎝ ⎠ ⎝ Fm ⎠ ⎝ Cm 0 0 0 ⎠ μm μf Ff 0 0 0 Cf with zero Dirichlet boundary conditions on ∂ K i+ for ψm and ψ f . Here, we used Lagrange multipliers μm and μ f to impose the constraints for multiscale basis construction. We remark that we have used the notations ψm , ψ f , μm , μ f to denote the vector representations of the corresponding functions in terms of fine-scale basis. For example, ψm is the vector of coefficients of the matrix pressure expanded in terms of fine-scale basis. We set Fm = δi, j and F f = 0 for construction of multiscale basis function for porous matrix ψ 0 = (ψm0 , ψ 0f ). For multiscale basis function for fracture network, we set Fm = 0 and F f = δi, j δm,l . In Figure 12.33, we depict multiscale basis functions for oversampled region K i+ = K i4 (four oversampling coarse cell layers) on coarse mesh 20 × 20. Multiscale basis function for displacements. The construction is similar to that of pressure. More precisely, we construct a set of basis functions ψ X,i := (ψxX,i , ψ yX,i ) and ψ Y,i := (ψxY,i , ψ Y,i y ), which minimize the energy for elasticity problem operator restrictedintheregion K i+ andsatisfytheconstraintsdescribedbelowforall K j ⊂ K i+ : (1) X-component, ψ X,i :   ψxX,i d x = δi, j , ψ yX,i d x = 0, Kj

Kj

12.5 Non-local multicontinua upscaling for poroelasticity in fractured media

(2) Y-component, ψ Y,i : 

329

 Kj

ψxY,i

d x = 0, Kj

ψ Y,i y d x = δi, j .

For further error reduction, we can add an additional basis function for the heterogeneous source term. This results in solving the following local problems in K i+ : ⎞⎛ ⎞ ⎛ ⎞ ⎛ K+ K+ Dx i Dx yi SxT 0 ψx 0 ⎟⎜ ⎟ ⎜ ⎟ ⎜ K i+ K i+ T ψ ⎜ Dx y D yy 0 S ⎟ ⎜ y ⎟ ⎜ 0 ⎟ y⎟ (12.94) ⎜ ⎝ ⎠ = ⎝ Fx ⎠ ⎝ Sx 0 0 0 ⎠ μx μy Fy 0 Sy 0 0 with zero Dirichlet boundary conditions on ∂ K i+ for ψx and ψ y . We set (Fx , Fy ) = (δi, j , 0) and (0, δi, j ) for the construction of multiscale basis functions for X- and Ycomponents. In Figure 12.33, we depict multiscale basis functions for displacements in oversampled domain K i+ = K i4 . In general, the permeability or elastic coefficients can be heterogeneous, where for high-contrast cases more basis should be used and constrained energy minimization (CEM) GMsFEM can identify important modes [113]. We note that the fracture contributions are divided in each coarse cell and then coupled. Each local fracture network introduces an additional degree of freedom for the current coarse cell. In general, CEM-GMsFEM can be applied, where local spectral problems automatically identify important modes [113]. Coarse-scale coupled system. We first define a projection matrix using the multiscale basis functions ⎞ ⎛ Rmm Rm f 0 0 ⎜R f m R f f 0 0 ⎟ ⎟, R=⎜ ⎝ 0 0 Rx x Rx y ⎠ 0 0 R yx R yy where  T = ψ 0,0 , ψ 1,0 . . . ψ Nc ,0 , Rmm m m m

! Nc ,L Nc 0,L 0 1,L 1 N ,1 , , ψ 1,1 ,...,ψf c ...ψf R Tf f = ψ 0,1 f ...ψf f ...ψf

 T = ψ 0,0 , ψ 1,0 . . . ψ Nc ,0 , Rm f f f f

! Nc ,L Nc 0,L 1,L N ,1 0,1 1,1 . . . ψm 0 , ψm . . . ψm 1 , . . . , ψm c . . . ψm , R Tf m = ψm

 X,N R xTx = ψxX,0 , . . . ψx c

 X,N R xTy = ψ yX,0 , . . . ψ y c

 T = ψ Y,0 , . . . ψ Y,Nc R yx x x

 T = ψ Y,0 , . . . ψ Y,Nc . R yy y y

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In the above definition, ψmi,l is the basis function for matrix pressure corresponding to the coarse block K i and the continuum l. The definition for ψ i,l f is the basis function for fracture pressure corresponding to the coarse block K i and the continuum l. The notation ψmi,l stands for both the function and its vector representation in fine-grid basis. We note that we construct only decoupled multiscale basis functions for flow and mechanics. Coupled construction of the multiscale basis functions can provide better results and will be considered and investigated in future works. Finally, we obtain the following upscaled coarse-grid model 

 1 ¯ ¯ M + A¯ y¯ = F, τ

(12.95)

where A¯ = R A R T , F¯ = R F, y¯ = ( p¯ m , p¯ f , u¯ x , u¯ y ). Here p¯ m , p¯ f , u¯ x , and u¯ y are the average solution on coarse-grid cell for matrix, fracture, and displacement X- and Ycomponents, respectively. For mass matrix, we can use a property of the constructed multiscale basis functions, and obtain diagonal mass matrix by direct calculation on the coarse grid ⎞ ⎛ M¯ m 0 0 0 ⎜ 0 M¯ f 0 0⎟ ⎟ M¯ = ⎜ ⎝ 0 0 0 0⎠ , 0 0 00 where M¯ m = diag{am |K i |}, M¯ f = diag{a f |γi |}. The coarse-grid upscaled model has only one coarse degree of freedom (DOF) for each fracture network and provides an effective coarse-scale model with physical meaning, and leads to a fast and accurate solver for the coupled poroelasticity problem.

12.5.4 Numerical results We present numerical results for the poroelastic model in Ω with a length of 1 m in both directions. We consider two test cases: (1) domain with 30 fractures and (2) domain with 60 fractures. In Figures 12.34 and 12.35, we show computational coarse and fine grids, where the fractures are depicted with red color and fine mesh with blue color. For fracture network, we constructed separate mesh and for domain Ω, we use structured fine mesh. We consider two coarse grids with 400 cells and 1600 cells. The coarse grids are uniform.

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For coupled poroelastic model, we use the following parameters: E Eν • Elastic parameters: μ = 2(1+ν) and λ = (1+ν)(1−2ν) , where E = 10 × 109 , ν = 0.3, and α = 0.1; • Flow parameters am = 10−6 , a f = 10−7 , bm = 10−11 , b f = 10−6 , and β = 10−10 .

Boundary condition for the displacement: u x = 0.0 on the left and right boundaries; u y = 0.0 on the bottom and top. We set a point source at the two coarse cells with q = 0.01 and set initial pressure p0 = 107 . We simulate tmax = 10 years with 50 time steps for multiscale and fine-scale solvers.

Fig. 12.34 Computational grids with 30 fracture lines. First: Coarse grid 20 × 20 with 400 cells. Second: Coarse grid 40 × 40 with 1600 cells. Third: Fine grid for matrix domain Ω with 14641 vertices and 28800 cells (blue). Fine grid for fracture domain γ with 1042 cells (red and white).

Fig. 12.35 Computational grids with 60 fracture lines. First: Coarse grid 20 × 20 with 400 cells. Second: Fine grid for matrix domain Ω with 14641 vertices and 28800 cells (blue). Fine grid for fracture domain γ with 1312 cells (red and white).

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Fig. 12.36 Fine-scale solution for pressure ( p ∗ = ( p − p0 )/ p0 ) and displacements (from left to right) for the different time layers t5 , t15 , and t50 (from top to bottom). Test case with 30 fractures.

K s ep eu x eu y Coarse grid 20 × 20 1 4.740 86.865 82.598 2 0.723 43.721 37.034 3 0.369 6.716 4.668 4 0.359 2.718 2.854

K s ep eu x eu y Coarse grid 40 × 40 1 1.986 96.667 95.454 2 0.191 78.718 74.957 3 0.174 30.550 25.220 4 0.158 4.1302 3.321 6 0.157 1.127 1.233

Table 12.23 Numerical results of relative errors (%) at the final simulation time. D O F f = 59124 and D O Fc = 1393. Test case with 30 fractures.

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333

We use D O Fc to denote the problem size of the coarse-grid upscaled model and D O F f for the fine-grid system size. To compare the results, we use the relative L 2 error between the coarse cell averf ine f ine f ine age of the fine-scale solution p¯ m , u¯ x , u¯ y and upscaled coarse-grid solutions p¯ m , u¯ x , u¯ y e p = || p¯ mf ine − p¯ m || L 2 , eu x = ||u¯ xf ine − u¯ x || L 2 , eu y = ||u¯ yf ine − u¯ y || L 2 ,   ¯ Kf − v¯ K )2 1 K (v K  ¯ 2L 2 = , v ¯ = v f d x, v = p, u x , u y , ||v¯ f − v|| f |K | K ¯ Kf )2 K (v (12.96) for matrix pressure and displacements. Fine-grid solution for computational domain with 30 fractures is presented in Figure 12.36 for the different time instants t5 , t15 , and t50 , where tn = nτ . On the first column of the figure, we depict pressure p ∗ = ( p − p0 )/ p 0 , on the secondand third-row columns – displacements u x and u y . We perform computations on the coarse grid with 400 cells with 4 oversampling layers in the construction of basis functions (K + = K 4 ). In the first column, we depict fine-grid pressure solution, in the second column – reconstructed fine-scale solution from upscaled coarse-grid solution, in the third column – coarse cell average for fine-scale solution, and in the fourth column – coarse cell average for upscaled coarse grid. Fine-grid system has size D O F f = 59124. By performing the NLMC method, we reduce size of the system to D O Fc = 1393. At the final time, we have less than one percent of error for pressure and nearly 2.5% for displacement. In Table 12.23, we present relative errors at the final time for two coarse grids and for different number of oversampling layers for the oversample region K s with s = 1, 2, 3, 4, and 6, where K s is obtained by extending K by s coarse-grid layers. From the numerical results, we observe a good convergence behavior, when we take a sufficient number of oversampled layers. For the coarse mesh with 400 cells, when we take 4 oversampling layers, we have 0.359% relative error for pressure, and for displacement – 2.718% (u x ) and 2.854% (u y ). For the coarse mesh with 1600 cells with 6 oversampling layers, the relative error is 0.157% for pressure, and for displacement – 1.127% (u x ) and 1.233% (u y ). We note that, on the 20 × 20 coarse mesh, the size of the upscaled system is D O Fc = 1393 and for the 40 × 40 coarse mesh, we have D O Fc = 5165. From Table 12.23, we observe that we can use a smaller number of oversampling layers for pressure than for displacements. For pressure, it is enough to take 2 oversampling layers for obtaining errors smaller than one percent for both coarse grids, on the other hand for displacements we should

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take 4 or 6 oversampling layers in coarse grids 20 × 20 and 40 × 40, respectively. Note that, in general, the presented algorithm can work with different number of oversampling layers for pressure and displacement due to coupled construction of the coarse-grid system. Next, we consider a test case with 60 fractures. Fine-grid system has size D O F f = 59394. By performing the NLMC method, we reduce the size of the system to D O Fc = 1484. At final time, we have 0.4217% of error for pressure and for displacement – 2.739% (u x ) and 3.124% (u y ). We obtain similar results with good accuracy for both test cases. Next, we discuss the computational advantages of our approach. The computational time is divided into offline and online stages. In the offline stage (preprocessing), we generate local domains, calculate multiscale basis functions, and generate a coarse-grid system. In the online stage, we solve the coarse-grid problem, with different input parameters (source term, boundary conditions, time steps, etc.). Let D O F f be the size of fine-scale solution y = ( pm , p f , u), then the dimension of the fine-grid coupled problem is D O F f × D O F f . The coarse-grid system size is D O Fc for coupled poroelasticity problem, which depends on the coarse-grid size and the number of local multiscale basis functions. In each local domain (coarsegrid cell), we have a degree of freedom for displacement X and Y components, of matrix pressure, and additional degree of freedom for each fracture network in the current coarse cell. We note that the number of degrees of freedom is similar to the classic embedded fracture model (EFM). For two-dimensional problems ,we have i M = 3 + Mi (( pm , p 1f ... p M f , u x , u y )) degrees of freedoms in local domain, where Mi is the number of the fracturenetworks in coarse cell K i . Therefore, the size of coarse-grid system is D O Fc = K i (3 + Mi ). Let N f be the number of cells for fracture network mesh, and Nc and Nv be the number of cells and vertices on the fine grid for domain Ω. Then for two-dimensional problems with finite element approximation for displacement equation and finite volume approximation for flow problem, we have D O F f = N f + Nc + 2Nv . Then, we can compare the computational cost of solving coarse- and fine-grid problems. For example in a test case with 30 fractures, a coarse solution has D O Fc = 1393 in the coarse grid with 400 cells, where we have 400 and 800 degrees of freedom for matrix pressure and displacement X and Y components, and 193 degrees of freedom for fractures. For the fine-scale system, D O F f = 59124 on fine grid with Nv = 14641 vertices and Nc = 28800 cells. Then, we can obtain an accurate solution for a multiscale solver using only 2.3% from D O F f . We note that the number of Mi in K i and therefore the size of coarse-grid system is independent of fine-grid size and a few basis functions can approximate the fine-scale solution accurately no matter how fine the fine grid is. When we use a classic direct solver, the solution time of the time-dependent coupled fine-grid problem is 81.17 seconds and 5.74 seconds for the coarse grid. We have computational gain in the simulations, because in each time step, the proposed method solves a small coarse-grid system compared to the fine-grid system.

12.6 Multiscale methods for elastic wave propagation in fractured media

335

12.6 Multiscale methods for elastic wave propagation in fractured media Seismic imaging of Earth materials has long provided key insights into geologic structures and processes on scales ranging from the first several meters below the surface to the entire planet. However, the key motivation for our work is that there are still geological problems that require more accurate determination of rock properties than is possible with current seismic imaging and modeling methods. For example, subsurface fractures often control fluid flow, especially in low permeability rock. Various field studies have suggested ways of relating seismic measures to fracture properties [47, 90, 235, 356, 359, 360, 377, 416, 429]. Improving seismic inference of fracture properties has strong potential benefits in many areas, including global climate change, CO2 sequestration, and so on. Seismic reflection data can be effective in monitoring the movement of fluid or gas, since its low bulk modulus and density lead to strong contrasts in elastic properties of rock [327]. One primary difficulty in developing improved seismic measurements is that it is not easy to develop realistic relationships between fractures and elastic properties. Anisotropic stress systems lead to the preferential alignment of compliant cracks and fractures that causes the rock to behave as an elastically anisotropic material, so effective medium models often assume that the fractures are parallel and vertical [132, 264, 374]. This is a reasonable starting point, but natural fractures have much more complicated geometries and distributions [375, 385]. One way to address this is through discrete fracture network (DFN) models, which can have very complex spatial distributions [357]. Such fractures will not be aligned and may have orientations with many directions [227, 263]. Other models assume that fractures are still aligned, but that the geologic history of the area and current stress regime may have created a configuration with two or more directions of such fractures [39, 226], and some approaches aim to relate these fracture systems to effective properties [376]. All of these approaches still assume idealized fracture geometries and statistically uniform spatial distributions within the volume of interest. Given the importance of accurately incorporating fractures into seismic data analysis, there are strong incentives to apply numerical modeling to provide more complete characterizations of fracture networks. However, current practice in modeling elastic wave propagation in complex, three-dimensional media often relies on finite difference methods using regular, Cartesian grids. These algorithms have important limitations. First, in many applications, discretization is typically on the order of 10 to 20 m in each coordinate direction, with results restricted to relatively low frequencies (e.g., 25 Hz; field data will typically include frequencies closer to 30 or 40 Hz) [289, 307, 387, 392]. For fault zone studies, because of the greater depths and lower frequencies, grid dimensions may need to be even larger, though fracture features on scales of 1 m or smaller will still be important. In either case, numerical grids cannot directly represent the important fracture networks. While effective medium theories might be applied, this may not provide accurate and reliable results given the strong assumptions applied by such models [291]. When considering individual,

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larger scale fractures or faults, a useful alternative is to apply the linear slip model (LSM), representing a fracture as a surface with discontinuous displacement [129, 204, 403]. Nonetheless, when fractures have complexities in shape or distribution within a single grid cell, their subtle and scattered influence is not likely to be reliably simulated. Another option is to apply multiple grid methods, which used grids that are locally refined to include small cells in the regions of interest, though there are challenges in suppressing numerical artifacts [88, 289, 290, 307]. Increasing the number of cells also increases computation time. A central research problem is therefore to carry out numerical simulations on coarse grids for fast results, while simultaneously including the influence of finer scale heterogeneity such as fractures. The methods we propose offer an even more powerful and flexible approach. The main idea is to capture the influence of smallscale fractures and other heterogeneities with basis functions computed on a fine grid, and to then apply these bases to simulate wave propagation rapidly on a coarsescale grid. The proposed multiscale approaches reduce the number of unknowns in calculations and thus allow simulations for cases that are not possible with direct numerical methods. In this section, we develop a multiscale model reduction technique for elastic waves in fractured media within the framework of Generalized Multiscale Finite Element Method. We extend this approach for wave propagation in fractured media. The main novelties are as follows. (1) We develop a fine-grid DG approach on unstructured grid that takes into account the effects of the fractures. (2) A coarse-grid approximation within GMsFEM for this fine-grid DG discretization is developed. Both snapshot spaces and spectral problems are designed and implemented. (3) Design of artificial boundary conditions by adding a damping layer (cf., [127, 198]). We present numerical results and consider three different fracture configurations. In the first example, there are a few fractures that are far apart. In the next examples, we add more fractures that are close to each other. We select examples, where the fracture lengths are longer than the coarse-block sizes in order to introduce various length scales. In our numerical simulations, we test different number of offline multiscale basis functions and study the accuracy of the proposed GMsFEM by comparing the coarse-grid solution with the fine-grid solution. In all examples, we observe that the method converges as we increase the number of basis functions and we achieve good accuracy with a fraction of degrees of freedom. The numerical observations are consistent with our previous studies without fractures.

12.6.1 Problem formulation We consider the elastic wave equation in computational domain Ω, ρ

∂2u = div σ + f, t ∈ [0, T ], ∂t 2

(12.97)

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1 (∇u + (∇u)T ), 2 where u = u(x, t) is the displacement, ε is the strain tensor, σ is the stress tensor, f = f (x, t) is a given source vector, ρ is density, and λ and μ are Lame parameters. We will consider (12.97) with homogeneous Neumann boundary conditions and with the following initial conditions: ∂u(x, 0) u(x, 0) = u 0 (x), = u 1 (x). ∂t σ = λ div uI + 2με, ε =

Fig. 12.37 Computational geometry with fractures, Ω.

The above problem is considered in fractured media Ω (see Figure 12.37). For numerical simulations of the elastic wave equation in the fractured media, we apply the linear slip model (LSM). Specifically, we assume that the fractures have a vanishing width across which the tractions are taken to be continuous. Following the linear slip model [261, 373, 431], we have a linear relation between the traction vector and the magnitude of the discontinuity in the displacement field, as follows: [u] = Z τ ,

(12.98)

where [u] is the jump of the displacement field at fracture, τ is the traction vector at the surface of fracture, and Z is the fracture compliance matrix. The compliance matrix is characterized in terms of two parameters. For the fracture with up-down symmetry and rotational symmetry about the normal, the fracture compliance matrix is diagonal ⎤ ⎡ zt 0 0 Z = ⎣ 0 zt 0 ⎦ , 0 0 zn where z n and z t are the normal and tangential compliances, respectively.

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12.6.2 Fine-scale discretization For the fine-grid approximation, we use the DG method that allows for discontinuities in the displacement field to simulate fractures with the linear slip model. Let e be the edge between the elements K 1 and K 2 , then the average and jump of a vector function u of e are given by {u} =

u| K 1 + u| K 2 , [u] = u| K 1 − u| K 2 . 2

(12.99)

Let T h denote a finite element partition of the domain and Γh the set of all the interior faces between the elements T h . Let Γc ⊂ Γh be the subset of all faces where the displacement field is continuous and Γ f ⊂ Γh be the subset of faces that represent fractures. The weak formulation of the elastic wave equation for the Interior Penalty Discontinuous Galerkin method (IP-DGM) in fractured media (see Section 12.1 and [139, 140]) is given by  

  ∂2u vd x + (σ(u), ε(v))d x 2 K ∂t K K ∈T h K ∈T h    {τ (u)}[v]ds − {τ (v)}[u]ds + γ f (λ + 2μ)[u][v]ds − +

ρ

e∈Γc

e

e∈Γ f

e



e∈Γc

Z −1 [u][v]ds =

e

e∈Γc

 

K ∈T h

e

f vd x, v ∈ Vh , K

(12.100) where u ∈ Vh (= H 1 (Ω), γ f is penalty parameter and τ (u) = σ(u)n is the traction vector, and the linear slip condition for fractures (12.98) is imposed weakly [141]. For the time approximation, we use the explicit scheme  

  u n+1 − 2u n + u n−1 ρ vd x + (σ(u n ), ε(v))d x 2 Δt K K K ∈T h K ∈T h    n n {τ (u )}[v]ds − {τ (v)}[u ]ds + γ f (λ + 2μ)[u n ][v]ds − +

e∈Γc

e

e∈Γ f

e



e∈Γc

Z −1 [u n ][v]ds =

e

e∈Γc

 

K ∈T h

e

f vd x. K

(12.101) The explicit scheme is conditionally stable, therefore the time step is taken based on the velocity field. We can write (12.101) in the matrix form Mh

u n+1 − 2u n + u n−1 + K h u n = Fh , Δt 2

(12.102)

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where Mh is the mass matrix and K h is the stiffness matrix. Then, for every time step, we solve the following linear system: Mh u n+1 = (2Mh − Δt 2 K h )u n − Mh u n−1 + Δt 2 Fh .

(12.103)

Because the mass matrix Mh is block-diagonal for the DG method, we can invert the mass matrix before the time loop and obtain an efficient time-marching algorithm. In the computations, the energy of waves needs to be absorbed in artificial boundaries in order to avoid spurious reflections caused by the finite computational domain. We use a first-order absorbing boundary condition [127, 198] u t = −Lσn, x ∈ ∂Ω, t > 0,

(12.104)

⎤ ⎡ √ 0 1/ 2μ + λ 0 1 √ 0 1/ μ 0 ⎦ . L=√ ⎣ √ ρ 0 0 1/ μ

where

We can write (12.104) as follows: − σn = L −1 u t , x ∈ ∂Ω, t > 0, with L

−1

(12.105)

⎡√ ⎤ 2μ + λ 0 0 √ ⎣ √ μ 0 ⎦ 0 = ρ √ μ 0 0

and impose it weakly. Using the second-order central difference scheme for the time derivative in (12.105), we have  

n+1 − u n−1     u n+1 − 2u n + u n−1 −1 u vds + vd x + ρL (σ(u n ), ε(v))d x 2 2Δt Δt K e K e∈Γb K ∈T h K ∈T h    n n − {τ (u )}[v]ds − {τ (v)}[u ]ds + γ f (2λ + μ)[u n ][v]ds

ρ

e∈Γc e

+

 

e∈Γ f

e

e∈Γc e

Z −1 [u n ][v]ds =

e∈Γc e

 

K ∈T h

K

f vd x,

(12.106) where Γb is the subset of boundary faces. For homogeneous Neumann boundary conditions, we have the following matrix form: Mh u n+1 + (Δt 2 K h − 2Mh )u n + Mh u n−1 = Δt 2 Fh ,

(12.107)

and for first-order absorbing boundary conditions, we obtain (Mh +

Δt −1 n+1 Δt −1 n−1 L )u L )u + (Δt 2 K h − 2Mh )u n + (Mh − = Δt 2 Fh . 2 2 (12.108)

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12.6.3 Coarse-scale discretization In this section, we consider the DG-GMsFEM. First, we describe the construction of the snapshot space, and then consider the local spectral problems for the construction of the multiscale space.

Fig. 12.38 Computational mesh with fractures. Red designates fractures, blue designates the coarse cells, and green designates the fine cells.

Let Th and T H be fine and coarse grids of the domain Ω (Figure 12.38) with mesh sizes H >> h > 0. We use N to denote the number of coarse-grid blocks. The set of all coarse-grid edges is denoted by E H . Fine-scale triangulation (as defined above) Th is conforming with coarse-grid edges and fractures (Figure 12.38). Let VH,off be a finite-dimensional function space, which consists of functions that are smooth on each coarse- grid block. For the local basis functions, a snapshot space K is first constructed for each coarse-grid block K ∈ T H . We have two types of VH,snap K ,i K ,b K ,b K local snapshot spaces, VH,snap = VH,snap + VH,snap . The local snapshot space VH,snap for the coarse-grid block K is defined as the linear span of all harmonic extensions and will be defined more precisely later on (see (12.110)). The second local snapshot K ,i K ,i is defined as VH,snap = Vh0 (K ) and will be defined more precisely space VH,snap below (see (12.114)). The global multiscale space VH,off is defined as the linear K , K ∈ T H . The global multiscale space VH will be used as span of all these VH,off the approximation space of our IPDG coupling [110], which can be formulated as follows. Find u H ∈ VH :   n−1 n − 2u + u u n+1 H H , v + a DG (u nH , v) = l(v), v ∈ VH , (12.109) ρ H Δt 2

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341

where a DG (u, v) =

  K ∈T

−

  e

e∈E H

l(v) =

K

H

{τ (u)}[v]ds −

H

  e∈E H

 

K ∈T

(σ(u), ε(v))d x

K

e

{τ (v)}[u]ds + γc

  e

e∈E H

(2λ + μ)[u][v]ds,

f vd x,

where γc = R/ h is the penalty parameter, R > 0. Next,wedescribethelocalsnapshotspaceandlocalspectralproblems.Let K ∈ T H K ,b be the coarse-grid block. The space VH,snap represents a wave field propagating across K ,b grid cells and their boundaries ∂ K . To construct the local snapshot space VH,snap for each fine-grid edge el on the boundary of K , we find ψlK ∈ Vh (K ) by solving −∇ · σ(ψlK ) = 0, x ∈ K

(12.110)

ψlK = gl , x ∈ ∂ K .

In the two-dimensional case, we can set gl = (δl , 0) or gl = (0, δl ) at the el , where δl is the delta function and l = 1, ..., JK with JK being the total number of boundary edges. The functions ψlK defined above are the harmonic extensions. The linear span of the harmonic extensions is the local snapshot space K ,b K = span{ψ1K , ψ2K , ..., ψ2J }. VH,snap K

(12.111)

K ,b K ,b to form a basis function space VH,off , We select a few important modes from VH,snap and the important modes are obtained from the following local spectral problem defined in the local snapshot space:

Aφb = λb Sφb , b and where φb ∈ VH,snap

(12.112)



A = [amn ] = a DG (ψmK , ψnK ), S = [smn ] = s(ψm , ψn ) =

∂K

ρψmK ψnK ds.

K ,b VH,off ,

To construct a reduced space we select the first M b eigenvectors φ1K ,b , φ2K ,b , ..., φ KM,bb corresponding to the first M b smallest eigenvalues, λb1 ≤ λb2 ≤ K ,b by ... ≤ λbM b and define the space VH,off K ,b VH,off = span{φ1K ,b , φ2K ,b , ..., φ KM,bb }.

(12.113)

K ,b are called boundary basis functions. These multiscale basis functions from VH,off

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K ,i The second space VH,snap is defined to capture interior eigenmodes for K . The local K ,i K ,i snapshot space VH,snap for the coarse-grid block K is defined as VH,snap = Vh0 (K ). For the dimension reduction on the snapshot space, we use the following spectral problem to identify the important nodes:

−∇ · σ(φi ) = λi ρφi , x ∈ K φi = 0, x ∈ ∂ K .

(12.114)

We select the first M i eigenvectors φi1 , φi2 , ..., φiM i corresponding to the first M i smallest eigenvalues λi1 ≤ λi2 ≤ ... ≤ λiM i of the above problem (12.114). The space VHi (K ) is spanned by these functions VHi (K ) = span{φ1K ,i , φ2K ,i , ..., φ KM,ii }.

(12.115)

K ,i are called interior basis functions. The multiscale basis functions from VH,off Chung et al. [110] presented a complete stability and convergence analysis of the multiscale method for acoustic wave equation case. We expect a similar behavior for the current algorithm. The error of the presented multiscale method depends on coarse-scale mesh size (H ), size of penalty parameter (γc ), and number of interior and boundary basis functions (Λi = 1/λiM i +1 and Λb = 1/λbM b +1 ). The coarse-scale system can be calculated by projecting the fine-scale matrices onto the coarse grid with a global projection matrix assembled from the calculated multiscale basis functions

R = (R1 , R2 , ..., R N )T ,

R j = [φi1 , φi2 , ..., φiM i , φb1 , φb2 , ..., φbM b ],

(12.116)

where R j is the local projection matrix for coarse elements and N is the number of coarse-grid elements. In the numerical implementation, we first assemble the global matrices Mh , K h , and vector Fh in (12.107). Then using the global projection matrix R, we can define the coarse-scale system n−1 2 n 2 M H u n+1 H + (Δt K H − 2M H )u H + M H u H = Δt FH ,

(12.117)

where M H = R Mh R T is the coarse-scale matrix, K H = R K h R T is the coarse-scale stiffness matrix, and FH = R Fh . After calculation of the coarse-scale solution u n+1 H , T n+1 = R u . we can recover the fine-scale solution, u n+1 H H

12.6.4 Numerical results We present numerical results for the coarse-scale solution using multiscale basis functions. The basis functions of the offline space are constructed following the procedure described above. Note that the basis functions are constructed only once at the initial time and then used for generating the stiffness and mass matrices and the right-hand side vector.

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Fig. 12.39 Computational domains (L x = L y = 1800 m) with different fracture distributions (different fracture spacings d f = 300, 180, and 60 m.). Left: Case 1. Middle: Case 2. Right: Case 3.

For numerical simulations, we use the following parameters. Computational domains with different configurations of fracture distributions are presented in Figure 12.39 and have dimensions L x = L y = 1800 m. On the left of Figure 12.39, we have a low-density configuration with very few distant fractures (Case 1), in the middle of Figure 12.39, we have a medium-density configuration with more fractures (Case 2), and on the right of Figure 12.39, we have a high-density configuration with fractures that are close to each other (Case 3). In the & computational domain, we set α = 3000 & is the p-wave velocity, and β = μρ [m/s] and β = 1800 [m/s], where α = λ+2μ ρ is the s-wave velocity, respectively. As for fracture compliances, we use z n = 10 [m/Pa] and z t = 30 [m/Pa]. We use the following source term: f (x, t) = G(x)P(θ)R(t), where P(θ) = (cos θ, sin θ) is the polar angle of the source force vector with θ = 0, R(t) = (1 − 2a 2 ) exp(−a) is the Ricker wavelet with a = π f 0 (t − 1/ f 0 ), f 0 = 5 Hz, and the spatial function G(x) is defined as point source, G(x) = δ(x − x0 ) with x0 assigned as the center of computational domain. We take penalty parameters γc = 1.5/ h for the coarse-scale solution and γ f = 4/ h for the fine-scale solution. The initial conditions are zero, and we compute the solution at time Tmax = 0.4 s with the number of steps Tcount = 2000. Time parameters are similar for fine- and coarse-scale solutions. We note that the multiscale basis functions can be used for different time parameters, boundary conditions, and source terms. For the numerical solution, we construct structured coarse grids with 900 elements. As for fine grids, we use unstructured grids that resolve the existing fractures (Figure 12.39). In Figure 12.39, for Case 1 with low-density fracture configuration, the fine mesh contains 106322 cells, 159843 facets, and 53522 nodes, for Case 2 with medium-density fracture configuration, we have 106640 cells, 160320 facets, and 53681 nodes, and for Case 3 with high-density fracture configuration, we have 110722 cells, 166443 facets, and 55722 nodes.

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Coarse-scale (multiscale) and fine-scale (reference) solutions in final time are shown in Figures 12.40, 12.41, and 12.42. The coarse-scale solution is obtained in the multiscale space of dimension 72000 (using M i + M b = 20 + 20 multiscale interior and boundary basis functions per coarse element) and the fine-scale solution is obtained in a space of dimension ≈ 6 · 105 . Compared to the fine-scale solution on the left with the coarse-scale solution on the right of the figures, we observe that the DG-GMsFEM can approximate the fine-scale solution accurately. Next, we present the errors in a tabular format. M i + M b D O F ||eu || L 2 (%) ||eu x ||t (%) ||eu y ||t (%) Mi = Mb 10 + 10 36000 12.797 8.506 6.063 15 + 15 54000 8.186 4.630 3.773 20 + 20 72000 6.015 3.737 4.095 25 + 25 90000 5.805 3.586 4.057 M i = 10 10 + 15 45000 8.589 4.854 3.872 10 + 20 54000 6.819 3.837 3.551 10 + 25 63000 5.972 3.576 3.568 10 + 30 72000 5.361 3.510 3.540 M b = 10 15 + 10 45000 12.480 8.332 5.902 20 + 10 54000 12.065 8.004 5.631 25 + 10 63000 11.758 7.881 5.499 30 + 10 72000 11.536 7.740 5.411 Table 12.24 Relative errors for displacements for domain with fractures (Case 1) for final time.

To compare the results, we use relative L 2 error between fine-scale u h and coarsescale solutions u H   (u h − u H , u h − u H ) d x 2 , ||e|| L 2 = K K  K K (u h , u h ) d x and the relative L2 errors for u x1 and u x2 along line x2 = 1200 m by ||eu x ||t , ||eu y ||t . In Tables 12.24, 12.25, and 12.26, we present relative errors (in percentage) for the final time step. For the coarse-scale approximation, we vary the dimension of the spaces by selecting a certain number of interior and boundary basis functions (M i and M b ) corresponding to the smallest eigenvalues. In the tables, we recall that D O F denotes the dimension of the multiscale space VH , while M i and M b are the number of the multiscale basis functions per coarse element. When we use M i + M b = 20 + 20 multiscale basis functions (i.e., 20 boundary basis functions and 20 interior basis functions), the relative L 2 error is 6.015% in Case 1 for the final time step and 5.658%, 6.070% in Cases 2 and 3, respectively. We observe that as the dimension of the coarse space (the number of selected eigenvectors M i and M b ) increases, the

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345

M i + M b D O F ||eu || L 2 (%) ||eu x ||t (%) ||eu y ||t (%) 10 + 10 15 + 15 20 + 20 25 + 25

36000 54000 72000 90000

Mi = Mb 13.206 7.788 5.658 5.323

9.651 4.673 3.300 3.276

6.056 3.322 3.518 3.936

10 + 15 10 + 20 10 + 25 10 + 30

45000 54000 63000 72000

M i = 10 8.212 6.499 5.736 5.168

4.947 3.766 3.333 3.084

3.423 3.112 3.096 3.302

45000 54000 63000 72000

M b = 10 12.761 12.300 11.940 11.699

9.288 8.916 8.691 8.510

5.788 5.521 5.314 5.200

15 + 10 20 + 10 25 + 10 30 + 10

Table 12.25 Relative errors for displacements for domain with fractures (Case 2) for final time. M i + M b D O F ||eu || L 2 (%) ||eu x ||t (%) ||eu y ||t (%) 10 + 10 36000 15 + 15 54000 20 + 20 72000 25 + 25 90000 30 + 30 108000

Mi = Mb 14.804 8.298 6.070 4.607 3.784

13.004 6.015 3.327 2.620 2.404

8.040 4.273 3.083 2.632 2.694

10 + 15 10 + 20 10 + 25 10 + 30

45000 54000 63000 72000

M i = 10 8.916 7.056 6.078 5.115

6.642 4.374 3.733 3.069

4.556 3.433 3.010 2.674

15 + 10 20 + 10 25 + 10 30 + 10

45000 54000 63000 72000

M b = 10 14.086 13.420 13.028 12.764

12.415 11.668 11.342 11.059

7.827 7.441 7.337 7.182

Table 12.26 Relative errors for displacements for domain with fractures (Case 3) for final time.

respective relative errors decrease. Computational times are presented in Table 12.27 and measured in seconds. In the fourth column, we present the online computational time (in seconds). As shown in [110], the number of boundary basis functions has a greater impact on errors than the interior basis functions. In Tables 12.24, 12.25, and 12.26, we

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investigate the influence of the number of boundary basis (M b ) for a fixed number of interior bases (M i = 10). We observe that for the fixed number of the interior basis functions, we can obtain similar errors by adding more boundary basis functions. In these tables, we also presented relative errors for fixed number of the boundary basis (M b = 10); however, for this case, by taking more interior basis functions we cannot reduce errors.

Fig. 12.40 Fine-scale (top) and coarse-scale (bottom) solutions for u m , u x , u y (from left to right) for domain with fractures (Case 1) for final time. D O F f = 637932 and D O Fms = 72000.

Mi + Mb fine-scale 10 + 10 15 + 15 20 + 20 25 + 25 30 + 30

D O F ||eu || L 2 (%) 664332 36000 14.804 54000 8.298 72000 6.070 90000 4.607 108000 3.784

Solve, s 1582.5 87.9 130.4 186.8 247.5 331.4

Table 12.27 Computational time (sec.) for domain with fractures (Case 3) with 2000 time steps.

In addition, we have performed numerical tests to study various effects, which we summarize below.

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347

Fig. 12.41 Fine-scale (top) and coarse-scale (bottom) solutions for u m , u x , u y (from left to right) for domain with fractures (Case 2) for final time. D O F f = 639840 and D O Fms = 72000.

Fig. 12.42 Fine-scale (top) and coarse-scale (bottom) solutions for u m , u x , u y (from left to right) for domain with fractures (Case 3) for final time. D O F f = 664332 and D O Fms = 72000.

• We have studied the effects of the penalty term γc on the accuracy of the proposed method. Our results (cf., [110]) show that an optimal penalty term is of the order of h −1 for the ranges of the parameters considered above. The error is decreasing

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as we increase the penalty; however, very small penalty terms can lead to overpenalization, which can result in larger errors. • We have studied the sensitivity of the error with respect to the frequency f 0 . For larger f 0 , the errors increase (see Table 12.28 and Figure 12.43). However, with the finer coarse-mesh size or larger dimensional multiscale spaces, we can reduce the error. • We have considered various coarse mesh sizes and observed that the error is smaller for finer coarse meshes. In Table 12.29, we show the errors of the coarse-scale solution (with 225 coarse-grid blocks) for the final time for Case 3 and f 0 = 5 Hz. As one can observe, the errors are larger as we coarsen the mesh.

Fig. 12.43 Fine-scale (top) and coarse-scale (bottom) solutions for u m , u x , u y (from left to right) for domain with fractures (Case 3) for final time with f 0 = 10 Hz. D O F f = 664332 and D O Fms = 72000.

Mi + Mb 15 + 15 20 + 20 25 + 25 30 + 30

D O F ||eu || L 2 (%) ||eu x ||t (%) ||eu y ||t (%) 54000 79.608 16.862 16.513 72000 33.741 9.207 9.442 90000 10.811 6.773 7.445 108000 8.237 6.576 7.445

Table 12.28 Relative errors for displacements for domain with fractures (Case 3) for final time with f 0 = 10 Hz. Coarse mesh with 900 cells.

12.7 GMsFEM for stochastic problems using clustering Mi + Mb 15 + 15 20 + 20 25 + 25 30 + 30

349

D O F ||eu || L 2 (%) ||eu x ||t (%) ||eu y ||t (%) 13500 36.134 39.456 27.576 72000 21.196 21.074 13.995 22500 13.686 11.549 7.507 27000 10.412 8.794 5.688

Table 12.29 Relative errors for displacements for domain with fractures (Case 3) for final time with f 0 = 5 Hz. Coarse mesh with 225 cells.

12.7 GMsFEM for stochastic problems using clustering Many multiscale problems have stochastic nature due to missing information at small scales. For example, in porous media applications, the media properties are unknown in many locations and interpolated from some indirect and direct measurements. As a result, engineers can typically produce many realizations of the permeability field. In some cases, the stochastic description of the permeability is parameterized; however, in many cases, one deals with a very large number of permeability realizations. The objective is to perform many simulations and understand the solution as a stochastic quantity. In this section, our objective is to propose a fast method based on coarsening of both spatial and uncertainty space (in terms of realizations) for computing the solution space. The stochastic description of the permeability field contains multiple scales, which do not have apparent scale separation. Moreover, the uncertainties at different scales can be tightly coupled and one needs to upscale their interaction together. For this reason, stochastic upscaled models are often used. Stochastic upscaled models use coarse grids in the physical space and propose an ensemble average of the fine-grid permeability. As a result, one deals with stochastic flow equations on coarse grids. Some commonly used approaches include upscaling a few realizations and making predictions based on these simulations. In our approach, we consider the joint coarsening of both spatial and uncertainty space motivated by many physical applications, where uncertainties and spatial scales are tightly coupled. For representing the spatial scales, we use the generalized multiscale finite element method (GMsFEM) [104, 168, 171, 180, 255]. The main idea of this approach is to use the snapshots and local spectral decomposition for computing a few basis functions. These ideas are used for a few selected realizations to compute

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basis functions that can be used to represent multiple realizations. However, it is obvious that the use of many realizations will introduce many spatial patterns and can require too many multiscale basis functions. This can lead to very expensive computations. For this reason, we use coarsening algorithm in the uncertainty space that can allow using a few multiscale basis functions and possibly basis functions that can be written as a product of spatial and stochastic functions. In general, if the uncertainty space is partitioned uniformly, we expect larger dimensional coarse spaces. To coarsen the uncertainty space, we first define a distance function for realizations in each coarse block. The appropriate distance functions in clustering must include the distance between the solutions as this is a correct measure. One usually has many local solutions. To define the distance between two realizations, we use local solutions with randomized boundary conditions motivated by [71]. Because there are many realizations, we compute these local solutions for selected realizations and then compute their Karhunen-Loève expansion (KLE) [278, 310] to approximate the solutions at all other realizations. Using these approximate solutions, we compute the distance between two realizations. Using this distance and k-mean clustering algorithms [241], we cluster the uncertainty space and define the corresponding coarse blocks. To construct multiscale basis functions in each cluster and a coarse block, we select a realization or an average realization. The latter is used within GMsFEM to construct multiscale basis functions. The basis function is constructed by computing snapshot vectors and performing local spectral decomposition. In this construction, we assume that the permeability fields within each cluster are almost uniform in the appropriate metric as defined for clustering. One can also use other approximations with respect to ω in each cluster, besides piecewise constant, e.g., product basis functions. In the latter, we can take the product of stochastic basis functions and spatial multiscale basis functions for some average realizations. For global coupling of multiscale basis functions, we use continuous Galerkin formulation, though other discretization approaches can be used (e.g., Petrov-Galerkin and Discontinuous Galerkin methods). We consider two global couplings. The first one is a realization-based approach, where we solve the global problem for each realization. The computations for each realization can be performed in parallel. Another approach is to use Galerkin in space and uncertainty space. In this approach, the variational formulation takes into account the average across the realizations. The second approach is cheaper for ensemble-level calculations because it involves fewer global computations. In the following sections, we will give a brief account of our proposed method, following [121].

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351

12.7.1 Preliminaries In this section, we present the problem setting and the fine-grid discretization. Let D be a bounded domain in Rd (d = 2, 3) and Ω be a parameter space in Rm , where the dimension m can be large in practice. We consider the following parameter-dependent second-order linear elliptic equation,  −∇ · (κ(x, ω)∇u(x, ω)) = f (x, ω), (x, ω) ∈ D × Ω, (12.118) u(x, ω) = 0, (x, ω) ∈ ∂ D × Ω, where κ(x, ω) is highly heterogeneous with respect to the physical space D and f (x, ω) ∈ L 2 (D), ∀ω ∈ Ω. In the reservoir simulation, κ(x, ω) is used to model the permeability of the heterogeneous porous medium, which is assumed to be known at every location in the flow domain. Realistically, however, only a handful of permeability measurements may be available (e.g., the SPE10 model). Thus, the uncertainties in the permeability field of the reservoir model could be quite large. How to build an efficient mathematical model to generate permeability is an important research area [364], although this is not the main focus here. Here, we assume that we only obtain a set of permeability fields κ(x, ωi ), i = 1, 2, ..., M. Namely, we assume that the event space Ω is approximated by a discrete set of ωi ’s. As such, the notation realization of the permeability fields means the data of the permeability field obtained by engineers. The weak formulation of the problem (12.118) is to find u(x, ω) ∈ L 2 (Ω; H 1 (D)) such that   Ω

a(ω; u, v) dω =

Ω

l(ω; v) dω,

∀v ∈ L 2 (Ω; H 1 (D)),

(12.119)

where the bilinear form a and the linear functional l are defined as  a(ω; u, v) = κ(x, ω)∇u(x, ω) · ∇v(x, ω)d x, D l(ω; v) = f (x, ω)v(x, ω)d x. D

We remark that the measure dω in (12.119) is assumed to be the uniform measure as in many cases we only have realizations of the permeability field κ(x, ω) (e.g., the SPE10 model in reservoir simulation). In the space L 2 (Ω; H 1 (D)), we define the norm of the tensor space H01 (D) ⊗ L 2 (Ω) as  u2H 1 (D)⊗L 2 (Ω) 0

=

a(ω; u, u) dω.

(12.120)

Ω

12.7.2 Outline of the method In this section, we shall introduce the coarse-grid discretization of the spatial domain D and the parameter space Ωd . One obvious choice of the partition of the product

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space D × Ωd is to construct the partitions of D and Ωd separately and then use the tensor products. However, this approach may not be good in practice. We shall use a more adaptive approach, namely the partition of the parameter space depends on the partition of the spatial domain D. In particular, we shall first define a partition for the domain D. Then for each of the elements in the partition of D, we shall construct a corresponding partition for the set Ωd . Our numerical experiments demonstrate that this new approach is very effective in solving heterogeneous multiscale problems.

Coarse spatial grid We repeat the construction of the coarse grid and related notations for the spatial domain D as some notations will be changed (ω denotes a parameter instead of the coarse neighborhood). We use T H to denote a coarse-grid partition of the domain D and the elements of T H are called coarse elements, where H is the mesh size of the coarse element. We assume that the fine-grid partition T h is a refinement of the coarse-grid partition T H . Let xi , 1 ≤ i ≤ N , be the set of interior nodes in the coarse grid, and N be the number of the interior coarse-grid nodes. For each coarse node xi , the coarse neighborhood Di is defined by Di = ∪{K j ∈ T

H

: xi ∈ K j },

that is, the union of all coarse elements K j ∈ T

H

having the vertex xi .

Coarsening Ωd via clustering algorithm. Next, we discuss the coarsening of Ωd via clustering algorithm. The clustering algorithm will be performed for each neighborhood of Di in general. In simplified cases, one can assume the same clustering for all Di ’s; however, we emphasize that in general cases, one needs to coarsen Ωd for each coarse spatial grid block Di separately. To perform the clustering, we shall use a distance function d i (·, ·) defined on Ωd for the coarse neighborhood Di . The distance function d i (·, ·) is defined in the solution space instead of the sample space. More precisely, for any two elements ωn , ωm ∈ Ωd , we define the distance d i (ωn , ωm ) be the distance of the two functions φn , φm in Di , where φn , φm are solutions of −∇ · (κ(x, ω)∇φ(x, ω)) = 0 with ω = ωn , ωm in Di respectively and with an appropriate boundary condition on ∂ Di . However, solving the multiscale problem to get φn and φm for every pair of elements ωn and ωm in Ωd is expensive. Therefore, we present a simplified way to reduce the computational cost of our clustering algorithm. For each coarse neighborhood Di , we first construct a snapshot space for a subset in Ωd . We denote this subset of Ωd by Ωdsubset . Our objective is to find simplified distance functions in Ωd that can be used for clustering. The main idea is to use randomized harmonic extensions and KL expansion. The main idea of our motivation

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353

is that the local problems with randomized boundary conditions can represent the main modes of local problems that are computed using all boundary conditions. The clustering algorithm is illustrated in Figure 12.44. In the algorithm, we solve some local problems to compute the distance between selected realizations. This distance function is further interpolated using KL expansion and the approximate distance is used for defining the clusters.

Fig. 12.44 Illustration of the clustering algorithm.

We consider the i-th coarse neighborhood Di and a fixed subset Ωdsubset of Ωd . We let Di+ be an oversampled domain for Di , that is, Di ⊂ Di+ . More details about the oversampling technique for the multiscale finite element method can be found in [187] and references therein. In addition, we let Vhi,+ be the subspace of Vh defined by Vhi,+ = {v(x, ω) ∈ L 2 (Ωdsubset ; H 1 (Di+ )) : v(x, ω)|τ ∈ Q 1 (τ ), ∀τ ∈ T h ∩ Di+ , ∀ω ∈ Ωdsubset }, i,+ and the subspace Vh,0 be the subspace of Vhi,+ containing functions with zero trace on ∂ Di+ . Then we construct a set of snapshot functions ψ ij (x), by solving the local problem in the oversampling subdomain Di+ × Ωdsubset . In particular, we find ψ ij (x, ω) ∈ Vhi,+ such that i,+ , ∀ω ∈ Ωdsubset , a(ω; ψ ij (x, ω), v) = l(ω, v), ∀v ∈ Vh,0

(12.121)

subject to the boundary conditions ψ ij (x, ω)|∂ Di+ = R j , j = 1, ..., k i , where R j is a discrete random function defined with respect to the fine-grid boundary point on ∂ Di+ and k i are the number of local problems that one needs to solve in coarse neighborhood Di . In the next step, we perform a KL expansion of the snapshot functions {ψ ij (x, ω)} and obtain the dominant modes in the solution space ψ ij (x, ω) = ψ¯ ij (x) +

 l≥1

pij,l (ω)φli (x),

(12.122)

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where ψ¯ ij (x) =

1



subset |Ωdsubset | Ωd

ψ ij (x, ω) dω is the mean of snapshot functions and

|Ωdsubset | is the number of samples in Ωdsubset . We note that φli (x) ∈ Vh,0 (Di+ ) are defined by Vh,0 (Di+ ) = {v(x) ∈ H01 (Di+ ) : v(x)|τ ∈ Q 1 (τ ), ∀τ ∈ T h ∩ Di+ }. We also note that ψ¯ ij (x) satisfies the same boundary condition as ψ ij (x). Among the set {φli (x), l = 1, 2, ...} of the KL expansion of the snapshot functions in (12.122), we shall restrict them in the coarse neighborhood Di and then take the i . The lowfirst L i dominant parts and form the reduced spatial snapshot space Vsnap i dimensional reduced space Vsnap is defined by i Vsnap = span{φli (x), 1 ≤ l ≤ L i }.

The key idea of our new algorithm is to use this low-dimensional space to solve the local problem for each realization in Ωd . More precisely, for each ω ∈ Ωd , we find a function ψ˜ ij (x, ω) such that  p˜ ij,l (ω)φli (x), x ∈ Di , (12.123) ψ˜ ij (x, ω) = ψ¯ ij (x) + 1≤l≤L i

and

a(ω, ψ˜ ij , v) = l(ω, v),

i ∀v ∈ Vsnap .

We remark that solving the above problems is very efficient due to the small dimeni sion of the solution space Vsnap , which enables us to construct a distance function to define a distance function on Ωd , in Ωd . To be specific, we use local solutions ' denoted as d i (·, ·) : Ωd × Ωd → R+ {0}, by

  i d (ωn , ωm ) = ( p˜ ij,l (ωn ) − p˜ ij,l (ωm ))2 . (12.124) j

1≤l≤L i

We then use the k-means clustering algorithm [241] with the distance function defined in (12.124) to cluster the sampling space Ωd into J i clusters Ω ij , j = 1, ..., J i , such that Ωd = ∪1≤ j≤J i Ω ij . Remark 12.4. We should point out that the essential idea is that the clustering of the sampling space Ωd = ∪1≤ j≤J i Ω ij depends on the spatial location Di . This enables us to efficiently explore the heterogeneities of κ(x, ω) as well as the solution space.

Choosing the number of the snapshot functions. How to choose the number of the boundary conditions ψ ij (x, ω)|∂ Di+ = R j , j = 1, ..., k i , will determine the number of snapshot functions ψ ij (x, ω) obtained from

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355

the cell problem (12.121), thus affecting the performance of our method. Our goal is to choose a small number of effective boundary conditions so that the snapshot functions ψ ij (x, ω) can approximate the solution space well. We find that this is closely related to the range-finding problem. On a discrete level, we can formulate the range-finding problem as follows: for a matrix T , we want to find a matrix Q with independent columns such that its column space accurately approximates the column space of T , i.e., (12.125) ||T − Q Q T T || ≤ , where  is a small parameter and || · || is a matrix norm. Randomized range-finding algorithm is a very efficient method to construct the matrix Q and to capture the column space of T . See [239, 304, 418] for more details about the randomized algorithms for finding low-dimensional structures. To justify the approximation error ||T − Q Q T T ||, we draw r random vectors ω j , j = 1, ..., r . We use T to act on these ω j , i = 1, ..., r , and compute the residual after projecting the images T ω j to the column space of Q, namely ||(I − Q Q T )T ω j ||. One can get the following estimate. Lemma 12.5. Fix a positive integer r and a real number α > 1. Draw an independent family of {ω j , j = 1, 2, ..., r } of standard Gaussian ( vectors. If all the r number of residuals ||(I − Q Q T )T ω j || are smaller than α π2 , then we have that ( ||T − Q Q T T || ≤ α π2 holds, except with a small probability α−r . The proof of the above lemma appears in Section 3.4 of [418]. The good thing about the randomized range-finding algorithm and the error estimation in Lemma 12.5 is that they provide a cheap computational way to estimate the numerical rank and the range space of T . Namely, one does not require access to each entry of the matrix T , but only requires the matrix-vector multiplication. Moreover, Lemma 12.5 provides an efficient posterior error estimation for the range-finding problem. This motivates us to adopt the randomized range-finding idea to determine the number of the boundary conditions k i so that we can reduce the computational cost. In our method, we first make a guess for the number of the boundary conditions, still denoted by k i . Then, we solve (12.121) with these k i random boundary conditions and obtain the snapshot functions {ψ ij (x, ω)}, j = 1, ..., k i . In addition, we apply KL expansion to obtain the reduced basis functions ψ¯ ij (x) and {φli (x)}, which has been introduced in Eq. (12.122). To estimate the approximation property of the reduced basis functions, we randomly generate r coefficients κ(x, ωn ), ωn ∈ Ωdsubset , n = 1, ..., r . For those newly generated coefficients, we solve (12.121) with a reference numerical method, e.g., finite element method, and obtain reference solution ψr e f (x, ωn ), n = 1, ..., r . Alternatively, we expand the solution using the reduced basis functions ψ¯ ij (x) and {φli (x)} (see Eq. (12.123)) and apply the Galerkin method to obtain approximation solution ψnum (x, ωn ), n = 1, ..., r . If all the r error functions ||ψr e f (x, ωn ) − ψnum (x, ωn )||

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12 GMsFEM for selected applications

are smaller than a prescribed threshold , then number of the boundary conditions k i is big enough since the snapshot functions {ψ ij (x, ω)} have already captured the solution space well and they generate a set of optimal reduced basis functions ψ¯ ij (x) and {φli (x)}. Otherwise, we increase the number of the boundary conditions k i and repeat the same numerical tests.

12.7.3 The construction of offline space In this section, we describe the construction of the local offline basis functions. Let Di , i = 1, ..., N and Ω ij , j = 1, ..., J i be a given coarse neighborhood and a cluster of Ωd , (i, j)

respectively. In the constructing process, we first construct a snapshot space Vsnap for (i, j) Di × Ω ij . In the notation Vsnap , the indices i are corresponding to coarse neighborhood Di and the indices j are induced by i when we use the clustering algorithm. (i, j) The construction of the snapshot space Vsnap consists of solving the local problems (i, j) for several random boundary conditions. To construct the offline space Voff , we need (i, j) to solve a local spectral problem in the Vsnap for the dimension reduction. We shall discuss the full description of the construction in the following subsections. Construct the snapshot space (i, j)

The definition of the snapshot space Vsnap is based on κ(ω)-harmonic extensions. Let Bh (Di ) be the set of all fine-grid nodes lying on the boundary of the coarse neighborhood of ∂ Di . For each fine-grid node xi , the discrete delta function δkh (x) is defined by  1, k = l h δk (xl ) = , xl ∈ Bh (Di ). 0 k = l (i, j),snap

Next, we define the k-th snapshot basis function ψk

∈ Vh (Di ) as the solution

of 

(i, j),snap

Di

κ¯ (i, j) (x)∇ψk

· ∇v = 0, ∀v ∈ Vh,0 (Di )

(i, j),snap

ψk

(x) = δkh (x), on ∂ Di ,

(12.126) (12.127)

1  i κ(x, ω)dω). Here, we take the mean of κ(x, ω) within |Ω ij | Ω j a cluster for computing offline spaces. In general, one can use a typical realization or a few realizations in a similar fashion. The main idea is that within each cluster, we assume that the realizations are similar in an appropriate metric (i.e., the distance between randomized snapshots).

where κ¯ (i, j) (x) = (

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357

(i, j)

The dimension of Vsnap is equal to the number of fine-grid nodes lying on ∂ Di . We can use the randomized snapshots with oversampling technique to reduce the (i, j) dimension of Vsnap and therefore reduce the computational cost of solving these snapshot basis functions. Construct the offline space (i, j)

To obtain the offline space Voff , we shall perform a spectral decomposition in the (i, j)(x) (i, j) snapshot space. The spectral decomposition is defined as to find (φk , λk ) ∈ (i, j) Vsnap × R such that   (i, j) i, j (i, j) (i, j) κ¯ (i, j) (x)∇φk (x) · ∇v = λk κ¯ (i, j) (x)|∇χi (x)|2 φk (x)v, ∀v ∈ Vsnap , Di

Di

(12.128) where {χi (x)} is the partition of unity functions for the spatial domain D with respect to the partition {Di } and κ¯ (i, j) (x) is defined the same as in (12.126). We assume the eigenvalues are arranged in ascending order. Then, the offline basis functions for the (i, j) (i, j) coarse neighborhood Di × Ω ij are defined by φ˜ k (x, ω) = χi (x)IΩ ij (ω)φk (x), for 1 ≤ k ≤ M (i, j) , where IΩ ij is the characteristic function of Ω ij . That is, Voff = span{φ˜ k

(i, j)

: ∀i, 1 ≤ j ≤ J i , 1 ≤ k ≤ M (i, j) }.

(12.129)

Global formulation The offline space Voff can be used to solve the multiscale stochastic problem (12.118) for any input parameter ω and any right-hand side function f (x, ω). In particular, the stochastic GMsFEM for the multiscale stochastic problem (12.118) is to find u ms ∈ Voff such that   a(ω; u ms , v) dω = l(ω; v) dω, ∀v ∈ Voff . (12.130) Ωd

Ωd

By solving (12.130) using the Galerkin method, we can obtain the numerical approximation solution u ms to the multiscale stochastic solution. Online basis functions The numerical solution obtained in the offline space has already produced a good approximation to the multiscale stochastic solution. In some cases, however, we need to improve the accuracy of the solution by using some online basis functions, which are constructed in the online stage. To demonstrate the main idea of the online update, we consider a coarse neighborhood Di and the set Ω ij for example. We define the following residual function:

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12 GMsFEM for selected applications

 Ri, j (ω; v) =

 Ω ij

a(ω; u ms , v) dω −

Ω ij

l(ω; v) dω, ∀v ∈ L 2 (Ω ij ; Vh,0 (Di )), (12.131)

where Vh,0 (Di ) = {v(x) ∈ H01 (Di ) : v(x)|τ ∈ Q 1 (τ ), ∀τ ∈ T h ∩ Di }. (i, j)

To construct an online basis function φon ∈ L 2 (Ω ij ; Vh,0 (Di )), we solve the following problem:  Ω ij

j) 2 i a(ω; φ(i, on , v) dω = Ri, j (ω; v), ∀v ∈ L (Ω j ; Vh,0 (Di )).

(12.132)

We assume a piecewise constant approximation in the space Ω ij . The above problem (12.132) leads to j) a(ωk ; φ(i, on (k, ·), v) = Ri, j (ωk ; v), ∀v ∈ Vh,0 (Di ).

(12.133)

(i, j)

The basis function φon is added to the space Voff . In general, we do not need to solve (12.133) for every ωk in the set Ω ij . We only need to choose those ωk such that the residual Ri, j (ωk ; v) are large and update its corresponding reduced basis space.

12.7.4 Numerical results In this section, we perform some numerical experiments to test the performance and accuracy of the proposed cluster-based generalized multiscale finite element method for multiscale elliptic PDEs with random coefficients. As we will demonstrate, our new method could offer accurate numerical solutions with considerable computational savings over traditional stochastic methods. In the results below, we will use two types of error quantities to show the performance of our method. The first type of error is

 

  e1,Ω :=

Ω

|u h − u H |2 d xdω, e2,Ω :=

|

D

D

Ω

(u h − u H )dω|2 d x

which measures the L 2 norm of the error over all realizations. The second type of error is ) ) *M  * M * *  e1,S := + |u h (x, ωi ) − u H (x, ωi )|2 d xdω, e2,S := + | u h (x, ωi ) − u H (x, ωi )dω|2 d x i=1

D

D

i=1

which measures the L 2 norm of the error over M realizations, where M = 10 in our numerical experiments.

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359

Permeability with high-contrast inclusions and channels In this example, we consider permeability fields containing high-contrast inclusions and channels (see Fig. 12.45) and consider the following two-dimensional elliptic PDE with random coefficient, −∇ · (κ(x, ω)∇u(x, ω)) = f (x), (x, ω) ∈ D × Ω u(x, ω) = 0, (x, ω) ∈ ∂ D × Ω.

(12.134) (12.135)

To introduce randomness, we move these high-contrast inclusions at different directions and change their permeability values, while keeping the values at a high range. More precisely, the permeability realizations are generated as κ(x, ω) = κ0 (x − ωv, α(ω)),

(12.136)

where v is a fixed direction, ω is a scalar quantity, and κ0 (x, α(ω)) is a permeability field with α(ω) values in the channels. The formula (12.136) represents the shift of a fixed permeability field with a variable channel conductivity. In Figure 12.45, we depict two realizations, where one can observe κ0 (x, α). In Table 12.7.4, we show numerical results for one of the clusters. In this table, we compare L 2 norms of the mean and realizations between the fine-grid solution and the coarse-grid solution across the space and realizations. In particular, we show the results when choosing all realizations and when selecting 10 random realizations. In our results, we increase the number of basis functions and add more basis functions in each coarse-grid block. The latter can be done adaptively. We observe that the error has been reduced to 8% when 5 basis functions are used, which is similar to the case of using 3 basis functions. We note that this is an error due to the clustering size and cannot be reduced further unless we change the cluster size. In the next tables, we increase the number of clusters. In Table 12.31, we use 3 clusters and increase the number of basis functions per coarse element. We observe from this table that the error decreases as we increase the number of basis functions and, moreover, the error is smaller when using 3 clusters compared to the case of using 1 cluster. Next, in Table 12.32, we use 5 clusters and increase the number of basis functions per coarse element. In this case, we again observe an error decreasing as the number of bases is increased and a smaller error compared to the case with 3 clusters. In this table, we also show an error corresponding to when using GMsFEM basis functions for each realization. The latter is used to identify the error due to GMsFEM coarsegrid discretization. We observe that this error is about 4% compared to 6%, which indicates that our approach requires a few clusters in this example to achieve good accuracy.

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12 GMsFEM for selected applications 4

4

x 10 2

x 10 2

10

1.8

10

1.8

20

1.6

20

1.6

30

1.4

30

1.4

40

1.2

40

1.2

50

1

50

1

60

0.8

60

0.8

70

0.6

70

0.6

80

0.4

80

0.4

90

0.2

90

100 20

40

60

80

100

0

0.2

100 20

40

60

80

100

0

Fig. 12.45 Permeability coefficient κ of two different realizations for case 1. number of basis e1,Ω e2,Ω 1 24.38% 24.19% 3 14.57% 14.22% 5 12.28% 11.86% number of basis e1,S e2,S 1 21.76% 21.59% 3 10.89% 10.57% 5 8.73% 8.39% Table 12.30 Errors for the case 1 using #cluster = 1.

number of basis e1,Ω e2,Ω 1 23.96% 23.76% 3 12.50% 12.06% 5 10.25% 9.75% number of basis e1,S e2,S 1 21.76% 21.59% 3 9.52% 9.15% 5 7.37% 7.01% Table 12.31 Errors for the case 1 using #cluster = 3.

Finally, we present a test case to show the performance of using online basis functions defined in Section 12.7.3. In Table 12.33, we show the numerical results. In our computations, we start the process by using 3 offline basis functions, and this step corresponds to zeroth online iteration. Then we show the errors for the next 3 online iterations, where one online basis is added for each iteration. We observe that the method is able to produce very accurate results after a couple of online iterations. In Figure 12.46, we present the convergence history of the online iteration, namely we show the logarithm of the error versus the number of degrees of freedoms. We observe clearly the exponential decay of the error from this figure.

12.7 GMsFEM for stochastic problems using clustering number of basis e1,Ω 1 23.90% 3 12.26% 5 8.82% number of basis e1,S 1 21.76% 3 9.59%(5.64%) 5 6.31%(3.92%)

361 e2,Ω 23.70% 11.86% 8.35% e2,S 21.59% 9.24%(5.58%) 6.10%(3.88%)

Table 12.32 Errors for the case 1 using #cluster = 5. The error in the parenthesis shows the error when the solution is computed using GMsFEM basis functions corresponding. This represents the error due to spatial multiscale representation.

# online iteration e1,S e2,S 0 11.53% 11.31% 1 3.08% 2.55% 2 2.52% 1.96% 3 1.32% 0.89% # online iteration e1,S e2,S 0 8.40% 8.20% 1 2.55% 1.94% 2 2.11% 1.48% 3 0.93% 0.48% Table 12.33 Online results for using 3 offline basis functions with different numbers of clusters. Top: results for 1 cluster. Bottom: results for 5 clusters.

Permeability with randomness and spatial heterogeneities In this example, we consider the same two-dimensional elliptic PDE (12.134) with homogeneous boundary conditions and a random permeability field κ(x, ω) = e

2+sin(7πx ) sin(8πx )

2+sin(13πx ) sin(11πx )

2+sin(12πx ) sin(14πx )

1

1

1

0.1+ 2+sin(9πx1 ) sin(7πx2 ) ξ1 (ω)+ 2+sin(11πx1 ) sin(13πx2 ) ξ2 (ω)+ 2+sin(15πx1 ) sin(15πx2 ) ξ3 (ω) 2

2

2

, (12.137)

where we have 3 random variables ξi (ω) follow the standard normal distribution N (0, 1) that represent the randomness and spatial heterogeneities. We depict two realizations of this permeability field in Figure 12.47. As in our previous example, we shall vary the number of clusters and the number of basis functions. In Tables 12.34 and 12.35, we consider two cluster sizes with 6 and 10 clusters. In each case, we vary the number of basis functions in each coarse-grid block. We observe from these numerical results that the method based on 6 clusters has provided an accurate result that is very comparable to that obtained from using 10 clusters. In Table 12.35, we present the numerical results when using the multiscale basis functions for each realization (see the numbers in the parentheses). These results show that 10 clusters provide almost the same results compared when using multiscale

362

12 GMsFEM for selected applications −2 5 clusters 1 cluster −4

log(L2 Error)

−6

−8

−10

−12

−14 200

250

300

400 350 Degrees of freedom

450

500

Fig. 12.46 Error decay for online basis functions using 3 offline basis functions with different number of clusters.

basis functions for each realization, and it is sufficient to have only 10 clusters for our multiscale simulations. Remark 12.6. We should point out that the coefficient a(x, ω) in (12.137) is a highly nonlinear functional of ξi (ω) and does not have the affine parameter dependence property. The clustering algorithm helps us automatically explore the heterogeneities of the solution space and constructs the reduced basis functions. Thus, our method can efficiently solve this type of challenging problem.

number of basis e1,Ω e2,Ω 1 17.33% 17.26% 3 11.24% 11.14% 5 8.70% 8.59% number of basis e1,S e2,S 1 16.91% 16.80% 3 10.64% 10.54% 5 8.20% 8.10% Table 12.34 Errors for the case 2 using #cluster = 6.

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363

200 10

400 10

180 20

350

20 160

30

30

300

40

250

140 40 120 50

50 200

100 60

60 80

70

150 70

60 80

100

80 40

90

50

90 20

100

100 20

40

60

80

100

20

40

60

80

100

Fig. 12.47 Permeability coefficient κ of two different realizations for Eq. (12.137).

number of basis e1,Ω e2,Ω 1 17.26% 17.21% 3 10.11% 10.02% 5 8.01% 7.92% number of basis e1,S e2,S 1 16.96% 16.86% 3 9.29% 9.23% 5 7.83%(7.18%) 7.80%(7.13%) Table 12.35 Errors for the case 2 using #cluster = 10.

12.8 GMsFEM for uncertainty quantification in inverse problems Uncertainties in the description of media properties, such as reservoir lithofacies, porosity, and permeability, are major contributors to the uncertainties in reservoir performance forecasting. The uncertainties can be reduced by integrating additional data, especially dynamic ones such as pressure or production data, in subsurface modeling. The incorporation of all available data is essential for the reliable prediction of subsurface properties. The Bayesian approach provides a principled framework for combining prior knowledge with dynamic data in order to make predictions on quantities of interest [271]. However, it poses significant computational challenges largely due to the fact that exploration of the posterior distribution requires a large number of forward simulations. High-contrast flow is a particular example, where the forward model is multiscale in nature and only a limited number of forward simulations can be carried out before becoming prohibitively expensive. In this chapter, we present a framework for uncertainty quantification of quantities of interest based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. GMsFEM provides a hierarchy of approximations to the solution, and the MLMC provides an efficient way to estimate quantities of interest

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12 GMsFEM for selected applications

using samples on multiple levels. Therefore, the framework naturally integrates the multilevel feature of the MLMC with the multiscale nature of the high-contrast flow problem. A notable advantage of the GMsFEM construction is that the flexible coarse space dimension naturally provides a hierarchy of approximations to be used within the MLMC framework. Further, we avoid unnecessary large-dimensional eigenvalue computations for each parameter realization. In particular, the offline stage constitutes a one-time preprocessing step in which the effect of a suitable range of parameter values is embedded into the offline space. In turn, the online stage only requires solving much smaller eigenvalue problems within the offline space, along with the construction of a conforming or nonconforming basis set. Multilevel Monte Carlo (MLMC) was first introduced by Heinrich [244] for highdimensional parameter-dependent integrals and was later applied to stochastic ODEs by Giles [228, 229], and PDEs with stochastic coefficients by Schwab et al. [46] and Cliffe et al. [128]. However, it has not been considered in the ensemble-level method context before. The main idea of MLMC methods is to use a respective number of samples at different levels to compute the expected values of quantities of interest. In these techniques, more realizations are used at the coarser levels with inexpensive forward computations, and fewer samples are needed at the finer and more expensive levels due to the smaller variances. By suitably choosing the number of realizations at each level, one can obtain a multilevel estimate of the expected values at much reduced computational efforts. It also admits the interpretation as hierarchical control variates [382]. Such hierarchical control variates are employed in evaluating quantities of interest using the samples from multilevel distributions. In this work, we couple GMsFEM with the MLMC methods to arrive at a general framework for the uncertainty quantification of the quantities of interest in highcontrast flow problems. Specifically, we take the dimension of the multiscale space to be the MLMC level, where the accuracy of the global coarse-grid simulations depends on the dimension of the multiscale coarse space. The convergence with respect to the coarse space dimension plays a key role in selecting the number of samples at each level of MLMC. Specifically, we take different number of online basis functions to generate the multiscale coarse spaces, running more forward coarse-grid simulations with the smaller dimensional multiscale spaces and fewer simulations with larger dimensional multiscale spaces. By combining these simulation results in a MLMC framework, one can achieve better accuracy at the same cost as the classical Monte Carlo (MC) method. To this end, one needs to assess the convergence of ensemble-level methods with respect to the coarse space dimension, which can be estimated based on a small number of a priori computations. Further, we will consider the use of MLMC jointly with multilevel Markov chain Monte Carlo (MLMCMC) methods following [280]. The main idea of the MLMCMC approach is to condition the quantities of interest at one level (e.g., at a finer level) to that at another level (e.g., at a coarser level). The multiscale model reduction framework provides the mapping between the levels, and it can be used to estimate the expected value. Specifically, for each proposal, we run the simulations at different levels to screen the proposal and accept it conditionally at these levels. In this

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365

manner, we obtain samples from hierarchical posterior corresponding to our multilevel approximations which can be used for rapid computations within a MLMC framework.

12.8.1 Preliminaries Let Ω ⊂ Rd (d = 2, 3) be an open bounded domain, with a boundary ∂Ω. The model equation for a single-phase, high-contrast flow reads: − ∇ · (κ(x; μ)∇u) = f

in Ω

(12.138)

subject to suitable boundary conditions, where f is the source term and u denotes the pressure within the medium. Here, κ(x; μ) is the heterogeneous spatial permeability field with multiple scales and high contrast, where μ represents the dependence on a multidimensional random parameter, typically resulting from a finite-dimensional noise assumption on the underlying stochastic process [36] (see also [224, 422, 423] for related spectral methods based on polynomial chaos expansion for problems with random coefficients). In practice, one can measure the observed data Fobs (e.g., pressure or production), and then condition the permeability field k with respect to the measurements Fobs for predicting quantities of interest. Below, we recall the preliminaries of the Bayesian formulation (likelihood, prior, and posterior) for systematically performing the task. The main objective is to sample the permeability field conditioned on the observed pressure data Fobs . The pressure is an integrated response, and the map from the pressure to the permeability field is not one-to-one. So there may exist many different permeability realizations that equally reproduce the given pressure data Fobs . In practice, the measured pressure data Fobs inevitably contains measurement errors. For a given permeability field κ, we denote the pressure as F(κ), which can be computed by solving the model equation (12.138) on the fine grids. The computed pressure F(κ) will contain also modeling error, which induces an additional source of errors, apart from the inevitable measurement error. By assuming the combined error as a random variable  R , we can write the model as Fobs = F(κ) +  R .

(12.139)

For simplicity, the noise  R will be assumed to follow a normal distribution N (0, σ 2f I ), i.e., the likelihood p(Fobs |κ) is assumed to be of the form 

F(κ) − Fobs 2 p(Fobs |κ) ∝ exp − 2σ 2f

 .

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We will represent the permeability field κ, which includes facies and interfaces, through the now classical Karhunen-Loéve expansion (KLE) [311], which we describe in more detail in Section 12.8.5. We let the vector θ parameterize the permeability field within facies and τ parameterize the velocity in the level set method, respectively. By parameterizing the interfaces with level sets, the permeability field κ is completely determined by θ and τ . Our goal is to generate permeability fields κ consistent with the observed pressure data Fobs . This can be achieved using Bayes’ formula which expresses the posterior distribution π(k) as π(κ) = p(κ|Fobs ) ∝ p(Fobs |κ) p(κ) = p(Fobs |κ) p((θ, τ ) ) = p(Fobs |κ) p(θ) p(τ ), where the last step follows from the standing assumption that the random variables θ and τ are independent. In the expression for the posterior distribution π(κ), p(Fobs |κ) is the likelihood function, incorporating the information in the data Fobs , and p(θ) and p(τ ) are the priors for the parameters θ and τ , respectively, encoding prior knowledge of the permeability fields. In the absence of interfaces, we shall suppress the notation τ in the above formula. Further, we may also incorporate other prior information, e.g., that the permeability field κ is known at some spatial locations corresponding to wells, into the posterior distribution. From a probabilistic point of view, the problem of sampling from the posterior distribution π(κ) amounts to conditioning the permeability fields to the pressure data Fobs with measurement errors, i.e., the conditional distribution p(κ|Fobs ). Generally, this is achieved by Markov chain Monte Carlo (MCMC) methods, especially the Metropolis-Hastings algorithm. The main computational effort of the algorithm lies in evaluating the target distribution π(κ), which enters the computation through the acceptance probability. The map between the permeability κ and the pressure data F(κ) is only defined implicitly by the governing PDE system. Hence, to evaluate the acceptance probability, one needs to solve a PDE system on the fine scale for any given permeability κ. Consequently, its straightforward application is very expensive, which necessitates the development of faster algorithms. In the next section, we describe GMsFEM for the efficient forward simulation to provide a hierarchy of approximations which can be efficiently used for constructing Monte Carlo estimates.

12.8.2 GMsFEM for parameter-dependent problem This procedure differs from the one proposed in Section 12.7. To discretize the model equation (12.138), we will use fine and coarse grids, as before (see Figure 1.8). Let T H be a coarse-grid conforming triangulation of the computational domain Ω into finite elements (triangles, quadrilaterals, tetrahedra, etc.). Each coarse subregion is further partitioned into a connected union of fine-grid blocks. The fine-grid partition will be denoted by T h .

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367

Next, we describe the offline-online computational procedure for the construction of GMsFEM coarse spaces. At the offline stage, one generates the snapshot set, and constructs a low-dimensional offline space by model reduction. At the online stage, for each input parameter μ, one first computes multiscale basis functions and then solves a coarse-grid problem for any force term and boundary condition. We note that the online stage construction is different from the residual-based online multiscale basis construction presented in Chapter 5. Below, we describe the offline and online procedures in more detail. Offline computation ωi on each coarse neighAt the offline stage, we first construct a snapshot space VH,snap borhood ωi in the domain (cf. Figure 1.8). The construction involves solving a set of localized problems for various choices of input parameters. Specifically, we solve the following eigenvalue problems on each ωi : ω ,snap

A(μ j )ψl,ij

ω ,snap

= λl,ij

ω ,snap

S(μ j )ψl,ij

in ωi ,

(12.140)

where {μ j } Jj=1 is a set of parameter values to be specified. Here, we consider only Neumann boundary conditions, but other boundary conditions are also possible. The matrices A(μ j ) and S(μ j ) in (12.140) are fine-scale matrices defined by  A(μ j ) = [a(μ j )mn ] =  S(μ j ) = [s(μ j )mn ] =

ωi

ωi

κ(x; μ j )∇φ0,fine · ∇φ0,fine d x, n m (12.141)  κ(x; μ j )φ0,fine φ0,fine d x, n m

denotes the standard bilinear, and fine-scale basis functions and  k will where be described below, cf. (12.144). We note that (12.140) is the discrete counterpart of the continuous Neumann eigenvalue problem φ0,fine n

κ(x; μ j )ψl,ωij in ωi . −div(κ(x, μ j )∇ψl,ωij ) = λl,ωij  For notational simplicity, we omit the superscript ωi . For each ωi , we keep the first L i eigenfunctions of (12.140) corresponding to the lowest eigenvalues to form the snapshot space ωi = span{ψl,ωij : 1 ≤ j ≤ J, 1 ≤ l ≤ L i }. VH,snap

We then stack the snapshot functions into a matrix   Rsnap = ψ1 , . . . , ψ Msnap , where Msnap = J × L i denotes the total number of snapshots used in the construction.

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12 GMsFEM for selected applications

ωi Next, we construct the offline space VH,off , which will be used to efficiently (and accurately) construct a set of multiscale basis functions for each μ value at the online stage. To this end, we perform a dimensionality reduction of the snapshot space using an auxiliary spectral decomposition. Specifically, we seek a subspace of the snapshot space such that it can approximate any element of the snapshot space in a suitable sense. The analysis in [192] motivates the following eigenvalue problem in the space of snapshots: off off (12.142) Aoff Ψkoff = λoff k S Ψk ,

where the matrices Aoff and S off are defined by  A

off

S

off

=

off [amn ]

=

off [smn ]

=  =

T κ(x; μ)∇ψm · ∇ψn d x = Rsnap A Rsnap ,

ωi

ωi

T  κ(x; μ)ψm ψn d x = Rsnap S Rsnap ,

respectively. Here, κ(x, μ) and  κ(x, μ) are domain-based, parameter-averaged coefficients, and A and S denote fine- scale matrices for the averaged coefficients. To generate the offline space, we choose the smallest Moff eigenvalues to (12.142), and take the corresponding eigenvectors in the space of snapshots by setting φk = j Ψk j ψ j (for k = 1, . . . , Moff ) to form the reduced snapshot space, where Ψk j are the coordinates of the vector Ψk . We then create the offline matrix   Roff = φ1 , . . . , φ Moff to be used in the online computation. Online computation Next for a given input parameter μ value, we construct the associated online coarse ωi (μ) on each coarse subdomain ωi . We seek a subspace of the offline space space VH,on such that it can approximate any element of the offline space in an appropriate sense: on on Aon (μ)Ψkon = λon k S (μ)Ψk ,

(12.143)

where the matrices Aon (μ) and S on (μ) are defined by  Aon (μ) = [a on (μ)mn ] =  S on (μ) = [s on (μ)mn ] =

ωi

ωi

T κ(x; μ)∇ψm · ∇ψn d x = Roff A(μ)Roff , T  κ(x; μ)ψm ψn d x = Roff S(μ)Roff ,

respectively. Note that κ(x; μ) and κ(x; μ) are now parameter-dependent. To generate the online space, we choose the eigenvectors  on corresponding to the smallest Mon on eigenvalues of (12.143), and set φon j Ψk j φ j (for k = 1, . . . , Mon ), where Ψk j k = on are the coordinates of the vector Ψk .

12.8 GMsFEM for uncertainty quantification in inverse problems

369

We note that one can use adaptivity in the parameter space.

Global coupling mechanism To incorporate the online basis functions into a reduced-order global formulation of the original problem (12.138), we begin with an initial coarse space V init (μ) = Nv (Nv denotes the number of coarse nodes). The functions χi are standard span{χi }i=1 multiscale partition of unity functions, as before, defined by

−div κ(x; μ) ∇χi = 0 K ∈ ωi , χi = gi on ∂ K , for each coarse element K ∈ ωi , where gi is a bilinear boundary condition. Next, we define the summed, pointwise energy  κ as, cf. (12.141),  κ=κ

Nv 

H 2 |∇χi |2 ,

(12.144)

i=1

where H is the coarse mesh size. In order to construct the global coarse-grid solution space we multiply the partition of unity functions χi by the online eigenfunctions ωi (μ) to form the basis functions. We define the online φkωi ,on from the space VH,on spectral multiscale space as ωi }, VH,on (μ) = span{χi φωk i : 1 ≤ i ≤ Nv , 1 ≤ k ≤ Mon

(12.145)

Nc and using a single index notation, we write VH,on (μ) = span{ψi }i=1 where Nc denotes the total number of basis functions in the coarse-scale formulation. Using the online basis functions, we define the operator matrix R = [ψ1 , . . . , ψ Nc ], where ψi represents the vector of nodal values of  each basis function defined on the fine grid. To solve (12.138), we seek u(x; μ) = i u i ψi (x; μ) ∈ Von such that



 Ω

κ(x; μ)∇u · ∇vd x =

Ω

f vd x for all v ∈ Von .

(12.146)

The above equation yields the discrete form A(μ)u = F,

(12.147)

 κ(x; μ)∇ψ I · ∇ψ J d x is a coarse stiffness matrix, F := where A(μ) := [a I J ] = Ω  [ fI ] = f ψ I d x is the coarse forcing vector, Pc denotes the vector of unknown Ω

pressure values, and ψ I denotes the coarse basis functions that span Von .

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12 GMsFEM for selected applications

12.8.3 Multilevel Monte Carlo methods As was mentioned earlier, one standard approach for exploring posterior distributions is the Monte Carlo method, especially Markov chain Monte Carlo (MCMC) methods. Here, generating each sample requires the solution of the forward model, which is unfortunately very expensive for many practical problems defined by partial differential equations, including high-contrast flows. Therefore, it is imperative to reduce the computational cost of the sampling step. We shall couple the multilevel Monte Carlo with the multiscale forward solvers to arrive at a general framework for uncertainty quantification of high-contrast flows. MLMC-GMsFEM framework The MLMC approach was first introduced by Heinrich in [244] for finite- and infinitedimensional integration. Later on, it was applied to stochastic ODEs by Giles [228, 229]. More recently, it has been used for PDEs with stochastic coefficients [46, 128]. We now briefly introduce the MLMC approach in a general context, and derive our MLMC-GMsFEM framework for uncertainty quantification. Let X (ω) be a random variable. We are interested in the efficient computation of the expected value of X , denoted by E[X ]. In our calculations, X is a function of the permeability field k, e.g., the solution to (12.138) evaluated at measurement points. To compute an approximation to E[X ], a standard approach is the Monte Carlo (MC) method. Specifically, one first generates a number M of independent realizations of M , and then approximates the expected the random variable X , denoted by {X m }m=1 value E[X ] by the arithmetic mean E M (X ) :=

M 1  m X . M m=1

Now we define the Monte Carlo integration error e M (X ) by e M (X ) = E[X ] − E M (X ). Then the central limit theorem asserts that for large M, the Monte Carlo integration error (12.148) e M (X ) ∼ Var[X ]1/2 M −1/2 ν, where ν is a standard normal random variable, and Var[X ] is the variance of X . Hence, the error e M (X ) in Monte Carlo integration is of order O(M −1/2 ) with a constant depending only on the variance Var[X ] of the integrand X [367]. In this work, we are interested in MLMC methods. The idea is to compute the quantity of interest X = X L using the information on several different levels. Here, we couple the MLMC with the GMsFEM, where the level is identified with the size of the online space. We assume that L is the level of interest, and computing

12.8 GMsFEM for uncertainty quantification in inverse problems

371

many realizations at this level is very expensive. Hence, we introduce levels smaller than L, namely L − 1, . . . , 1, and assume that the lower the level is, the cheaper the computation of X l is, and the less accurate X l is with respect to X L . By setting X 0 = 0, we decompose X L into L 

XL =

(X l − X l−1 ) .

l=1

The standard MC approach works with M realizations of the random variable X L at the level of interest L. In contrast, within the MLMC approach, we work with Ml realizations of X l at each level l, with M1 ≥ M2 ≥ · · · ≥ M L . We write E [X L ] = 



L    E X l − X l−1 , l=1

and next approximate E X l − X l−1 by an empirical mean: Ml   m

1  m E X l − X l−1 ≈ E Ml (X l − X l−1 ) = X l − X l−1 , Ml m=1

(12.149)

where X lm is the mth realization of the random variable X computed at the level l (note that we have Ml copies of X l and X l−1 , since Ml ≤ Ml−1 ). Then the MLMC approach approximates E[X L ] by E L (X L ) :=

L 

E Ml (X l − X l−1 ) .

(12.150)

l=1

We note that the realizations of X l used with those of X l−1 to evaluate E Ml (X l − X l−1 ) do not have to be independent of the realizations of X l used with those of X l+1 to evaluate E Ml+1 (X l+1 − X l ). In our context, the permeability field samples used for computing E Ml (X l − X l−1 ) and E Ml+1 (X l+1 − X l ) do not need to be independent. We would like to mention that the MLMC can also be interpreted as a multilevel control variate, following [382]. Specifically, suppose that X = X L on the level L is the quantity of interest. According to the error estimate (12.148), the error is −1/2 proportional to the product of M L and the variance Var[X L ] of X L . Let X L−1 be a cheaper pointwise approximation, e.g., the finite element approximation on a coarser grid, to X L with a known expected value. Then it is natural to use X L−1 as the control variate to X L [367] and to approximate the expected value E[X L ] by E[X L ] = E[X L − X L−1 ] + E[X L−1 ] ≈ E M L (X L − X L−1 ) + E[X L−1 ].

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12 GMsFEM for selected applications

Here, we approximate the expected value E[X L − X L−1 ] by a Monte Carlo estimate, which, according to the error estimate (12.148), will have a small error, if the approximations X L and X L−1 are close to each other. More generally, with a proper choice of weights, the latter condition can be relaxed to high correlation. In practice, the expected value E[X L−1 ] may be still nontrivial to evaluate. In the spirit of classical multilevel methods, we can further approximate the expected value E[X L−1 ] by E[X L−1 ] ≈ E M L−1 (X L−1 − X L−2 ) + E[X L−2 ], where X L−2 is a cheap approximation to X L−1 . By applying this idea recursively, one arrives at the MLMC estimate as described in (12.150). Now we can give the outline of the MLMC-GMsFEM framework, cf. Algorithm 2. Here, the offline space is fixed and preprocessed. The level of the samples is determined by the size of the online multiscale basis functions. The larger the online multiscale space VH,on is, the higher the solution resolution is, but the more expensive the computation is; the smaller the online multiscale space VH,on is, the cheaper the computation is, but the lower the solution resolution is. The MLMC approach as described above provides a framework for elegantly combining the hierarchy of approximations from GMsFEM, and leveraging the expensive computations on level L to those lower level approximations. In addition, we note that the samples Ml used in the Monte Carlo estimate E Ml (X l − X l−1 ) are identical on every {k m }m=1 Ml two consecutive levels, i.e., the permeability samples {k m }m=1 used in the Monte Carlo estimates on two consecutive levels are nested. Algorithm 2 MLMC-GMsFEM 1. Offline computations – Construct the snapshot space. – Construct a low-dimensional offline space by model reduction. 2. Multilevel online computations for estimating an expectation at level l, 1 ≤ l ≤ L. – – – –

M

Ml m } l−1 ). Generate Ml realizations of the permeability {klm }m=1 (from {kl−1 m=1 m For each realization kl , compute online multiscale basis functions. m Solve the coarse-grid problem for X l . Calculate the arithmetic mean E Ml (X l − X l−1 ) by (12.149).

3. Output the MLMC approximation E L (X ) by (12.150).

Cost analysis In the following, we are interested in the root mean square errors e M L MC (X L ) =

(

E[E[X L ] − E L (X L )2 ],

12.8 GMsFEM for uncertainty quantification in inverse problems

e MC (X L ) =

&

373

E[E[X L ] − E M L (X L )2 ],

for the MLMC estimate E L (X L ) and the MC estimate E M L (X L ), respectively, with an appropriate norm depending on the quantity of interest (e.g., the absolute value for any entry of the permeability coefficient, and the L 2 -norm of the solution). For the error estimation, we will use the fact that for any random variable X and any norm, E[E[X ] − E M (X )2 ] defines a norm on the error E[X ] − E M (X ), and further, there holds the relation E[E[X ] − E M (X )2 ] =

1 E[X − E[X ]2 ]. M

In the analysis, we will be dealing with solutions at different scales. In the MLMC framework, we denote the scale hierarchy by H1 ≥ H2 ≥ · · · ≥ HL . The number of realizations used at the level l for the scale Hl is denoted by Ml . We take M1 ≥ M2 ≥ · · · M L . For the MLMC approach, the error reads e M L MC (X L ) =

(

E[E[X L ] − E L (X L )2 ] ) * L L *   E Ml (X l − X l−1 ))2 ] = +E[(E[ (X l − X l−1 )] − l=1

l=1

) * L *  = +E[( (E − E Ml )(X l − X l−1 ))2 ] l=1

≤

L & 

E[((E − E Ml )(X l − X l−1 ))2 ]

l=1

≤

L  l=1

1 ( E[(X l − X l−1 − E(X l − X l−1 ))2 ], √ Ml

where the second last line follows from the triangle inequality for norms, and the last line follows from (12.148). Next, we rewrite X l − X l−1 = (X l − X ) + (X − X l−1 ), and since Ml ≤ Ml−1 , we deduce

374

12 GMsFEM for selected applications  & & 1 E[(X l − X − E(X l − X ))2 ] + E[(X l−1 − X − E(X l−1 − X ))2 ] √ Ml l=1 & & 1 1 = √ E[(X L − X − E(X L − X ))2 ] + √ E[X 2 ] ML M1  & L−1  1 1 ( + +√ E[((X l − X ) − E(X l − X ))2 ] Ml Ml+1

e M L MC (X L ) ≤

L 

l=1

≤

L 

2 Ml+1

(

2 1 δl + √ M1 Ml+1

l=1

≤

L  l=1

& & 1 E[(X l − X )2 ] + √ E(X 2 ) M1

(

& E[X 2 ],

√ where the second last line follows from the inequality E[((X l − X )− ( ( E(X l − X ))2 ] ≤ E[(X l − X )2 ], a direct consequence of the (bias-variance decomposition, and the assumption M L+1 ≤ M L . Here, we denote E[(X l − X )2 ] as δl . As mentioned in Section 12.8.3, the lower the level l is, the less accurate the approximation X l is with respect to X L , hence we will have δ1 > δ2 > · · · > δ L . To equate the error terms, we choose ⎧  2 ⎨ 1 E[X 2 ], δ Ml = M  L 2 ⎩ δl−1 , δL

l = 1, 2 ≤ l ≤ L + 1,

where M > 1 is a fixed positive integer. Then we end up with e M L MC (X L ) ≤

(2L + 1) δL . M

In principle, for a prescribed error bound  such that e M L MC (X L ) ≤ , one can deduce from the formula the proper choice of the number N of samples, for any given level L.

12.8.4 Multilevel Markov chain Monte Carlo One of the most popular and versatile methods for numerically exploring posterior distributions arising from the Bayesian formulation is the Markov chain Monte Carlo (MCMC) method. The basic idea is to construct a Markov chain with the target distribution as its stationary distribution. However, the sampling step remains very challenging in high-dimensional spaces. One powerful idea of improving the sampling efficiency is preconditioning, first illustrated in [91, 178], and more recently extended in [40]. In the latter work, some theoretical properties, e.g., asymptotic confidence interval, of a multistage version of the two-level algorithm [178] are also established.

12.8 GMsFEM for uncertainty quantification in inverse problems

375

MLMCMC with GMsFEM The standard Metropolis-Hastings algorithm generates samples from the posterior distribution π(κ) = p(κ|Fobs ), cf. Algorithm 3. Here, U(0, 1) is the uniform distribution over the interval (0, 1). As was described in Section 12.8.1, the permeability field κ is determined by the parameters θ and τ . Hence, given the current sample κm , parameterized by its parameters θm and τ m , one can generate the proposal κ by generating the proposal for θ and τ first, i.e., draw θ from distribution qθ (θ|θm ) and τ from distribution qτ (τ |τ m ) (in view of the independence between θ and τ ), for some proposal distributions qθ (θ|θm ) and qτ (τ |τ m ), and then form the proposal for the entire permeability field κ. Algorithm 3 Metropolis-Hastings MCMC 1: Specify κ0 and M. 2: for m = 0 : M do 3: Generate the entire permeability field proposal κ from q(κ|κm ). 4: Compute the acceptance probability γ(κm ) by (12.151). 5: Draw u ∼ U (0, 1). 6: if γ(κm , κ) ≤ u then 7: κm+1 = κ. 8: else 9: κm+1 = κm . 10: end if 11: end for

The transition kernel K r (κm , κ) of the Markov chain generated by Algorithm 3 is given by    m m K r (κ , κ) = γ(κ , κ)q(κ|κ ) + δκm (κ) 1 − γ(κ , κ)q(κ|κ )dκ , m

m

m

where q(κ|κm ) denotes the proposal distribution and γ(κm , κ) denotes the acceptance probability for the proposal κ defined by /  q(κm |κ)π(κ) . γ(κm , κ) = min 1, q(κ|κm )π(κm )

(12.151)

Now we integrate the multilevel idea with the Metropolis-Hastings algorithm and GMsFEM. Like before, we start with the telescopic sum  EπL [FL ] =

FL (x)π L (x)d x 

=

F0 (x)π0 (x)d x +

L  

(Fl (x)πl (x) − Fl−1 (x)πl−1 (x))d x,

l=1

where πl denotes the approximate target distribution at level l, and π0 is our initial level. We note that after the initial level, each expectation involves two measures, πl

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12 GMsFEM for selected applications

and πl−1 , which is different from the case of the MLMC (see [280]). Therefore, we rewrite the integration using a product measure as 

 Fl (x)πl (x)d x − Fl−1 (y)πl−1 (y)dy   = (Fl (x) − Fl−1 (y))πl (x)πl−1 (y)d xd y



(Fl (x)πl (x) − Fl−1 (x)πl−1 (x))d x =

= Eπl ,πl−1 [Fl (x) − Fl−1 (y)]. Therefore, we have Eπ L [FL ] = Eπ0 [F0 ] +

L 

Eπl ,πl−1 [Fl − Fl−1 ].

(12.152)

l=1

The idea of our multilevel method is to estimate each term of the right-hand side of equation (12.152) independently. In particular, we can estimate each term in (12.152) by an MCMC estimator. The first term Eπ0 [F0 ] can be estimated using the standard MCMC estimator in Algorithm 3. We estimate the expectation Eπl ,πl−1 [Fl (x) − Fl−1 (y)] by the sample mean Eπl ,πl−1 [Fl (x) − Fl−1 (y)] ≈

Ml 1  (Fl (xlm ) − Fl−1 (ylm )), Ml m=1

(12.153)

Ml where the samples {(ylm , xlm )}m=1 are drawn from the product measure πl−1 (y) ⊗ πl (x). Next, we describe an efficient preconditioned MCMC method for generating samples from the product measure πl−1 (y) ⊗ πl (x), extending our earlier work [178]. Here, we introduce a multilevel MCMC algorithm by adapting the proposal distribution q(κ|κm ) to the target distribution π(κ) using GMsFEM with different sizes of the online space which we call different levels, cf. Algorithm 4. The process modifies the proposal distribution q(κ|κm ) by incorporating online coarse-scale information. Let Fl (κ) be the pressure/production computed by solving the online coarse problem at level l for a given κ. The target distribution π(κ) is approximated on level l by πl (k), with π(κ) ≡ π L (κ). Here, we have

  ||Fobs − Fl (κ)||2 × p(κ). πl (κ) ∝ exp − 2σl2

(12.154)

In the algorithm, we still keep the same offline space for each level. From level 0 to level L, we increase the size of the online space as we go to a higher level, which means for any levels l, l + 1 ≤ L, samples of level l are cheaper to generate than that of level l + 1. This idea underlies cost reduction using the multilevel estimator. Hence, the posterior distribution for coarser levels πl , l = 0, . . . , L − 1 do not have to model the measured data as faithfully as π L , which in particular implies that by choosing a suitable value of σl2 it is easier to match the result Fl (k) with the observed

12.8 GMsFEM for uncertainty quantification in inverse problems

377

data. We denote the number of samples at level l by Ml , where we will have M0 ≤ · · · ≤ M L . As was discussed above, our quantity of interest can be approximated by the telescopic sum (12.152). We denote the estimator of Eπ0 [F0 ] at the initial level 0 . Then by the MCMC estimator, we have by F M0 1   F0 (x0m ). F0 = M0 m=1

Here, x0m denotes the samples we accepted on the initial level (after discarding the samples at the burn-in period). Similarly, we denote the estimator of the differences l . Then with (12.153), we have Eπl ,πl−1 [Fl (x) − Fl−1 (y)] by Q Ml  l = 1 Q (Fl (xlm ) − Fl−1 (ylm )), Ml m=1 Ml where the samples {(ylm , xlm )}m=1 are drawn from the product measure πl−1 (y) ⊗ πl (x). Finally, denote the estimator of EπL [FL ] or our full MLMCMC estimator by L , then the quantity of interest E πL [FL ] is approximated by F

L = F 0 + F

L 

l . Q

(12.155)

l=1

We refer to [193] for the convergence analysis of MLMCMC.

12.8.5 Numerical results In our numerical examples, we consider permeability fields described by two-point correlation functions, and use Karhunen-Loève expansion (KLE) to parameterize the permeability fields. Then we apply the MLMC and MLMCMC with the GMsFEM algorithms described earlier. First, we briefly recall the permeability parametrization, and then we present numerical results. Permeability parameterization To obtain a permeability field in terms of an optimal L 2 basis, we use the KLE [311]. For our numerical tests, we truncate the expansion and represent the permeability matrix by a finite number of random parameters, which is often adopted in polynomial chaos expansion [422, 423]. We consider the random field Y (x, ω) = log[κ(x, ω)], where ω represents randomness. We assume a zero mean E[Y (x, ω)] = 0, with a known covariance operator R(x, y) = E [Y (x)Y (y)]. Then we expand the random field Y (x, ω) as

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12 GMsFEM for selected applications

Algorithm 4 Multilevel Metropolis-Hastings MCMC m 1: Given κm , draw a trial proposal κ from distribution q(κ|κm 1 ) = q0 (κ|κ1 ) 2: Compute the acceptance probability /  q0 (κm 1 |κ)π1 (κ) ρ1 (κm 1 , κ) = min 1, m m q0 (κ|κ1 )π1 (κ1 )

3: u ∼ U (0, 1) 4: if u < ρ1 (κm 1 , κ) then 5: κm+1 = κ (at the initial level) 1 6: else 7: κm+1 = κm 1 1 (at the initial level) 8: end if 9: for l = 1 : L − 1 do 10: if κ is accepted at level l then 11: Form the proposal distribution ql (on the l + 1th level) by  m m m m m m m (1 − ql (κ|κl+1 ) = ρl (κl+1 , κ)ql−1 (κ|κl+1 ) + δκl+1 ρl (κl+1 , κ)ql−1 (κ|κl+1 )dκl+1 ) 12:

Compute the acceptance probability  0  0 m |κ)π m )π ql (κl+1 πl (κl+1 l+1 (κ) l+1 (κ) m ρl+1 (κl+1 , κ) = min 1, = min 1, m )π m m ) ql (κ|κl+1 πl (κ)πl+1 (κl+1 l+1 (κl+1 )

13: u ∼ U (0, 1) m , κ) then 14: if u < ρl+1 (κl+1 m+1 = κ and go to next level (if l = L − 1, accept κ and set κm+1 = κ). 15: κl+1 L 16: else m+1 m , and break 17: κl+1 = κl+1 18: end if 19: end if 20: end for

Y (x, ω) =

∞ 

Yk (ω)Φk (x),

k=1



with Yk (ω) =

Ω

Y (x, ω)Φk (x)d x.

The functions {Φk (x)} are eigenvectors of the covariance operator R(x, y), and form a complete orthonormal basis in L 2 (Ω), i.e.,  Ω

R(x, y)Φk (y)dy = λk Φk (x),

k = 1, 2, . . . ,

(12.156)

12.8 GMsFEM for uncertainty quantification in inverse problems

379

2 where √ λk = E[Yk ] > 0. We note that E[Yi Y j ] = 0 for all i = j. By denoting ηk = Yk / λk (whence E[ηk ] = 0 and E[ηi η j ] = δi j ), we have

Y (x, ω) =

∞ ( 

λk ηk (ω)Φk (x),

(12.157)

k=1

where Φk and λk satisfy (12.156). The randomness is represented by the scalar random variables ηk . After discretizing the domain Ω into a rectangular mesh, we truncate the KLE (12.157) to a finite number of terms. In other words, we keep only the leading-order terms (quantified by the magnitude of λk ), and capture most of the energy of the stochastic process Y (x, ω). For an N -term KLE approximation YN =

N ( 

λk ηk Φk ,

k=1

the energy ratio of the approximation is defined by N λk EY N 2 = k=1 . e(N ) := ∞ 2 EY  k=1 λk If the eigenvalues {λk } decay very fast, then the truncated KLE with the first few terms would be a good approximation of the stochastic process Y (x, ω) in the L 2 sense. In our examples, the permeability field k is assumed to follow a log-normal distribution with a known spatial covariance, with the correlation function R(x, y) given by  |x − y |2 |x2 − y2 |2  1 1 , (12.158) − R(x, y) = σ 2 exp − 2 2l1 2l22 where l1 and l2 are the correlation lengths in x1 - and x2 -directions, respectively, and σ 2 = E[Y 2 ] is a constant that determines the variation of the permeability field. MLMC In our simulations, we evaluate the performance of the MLMC method in computing the expected values of our quantity of interest F. In particular, we consider the stationary, single-phase flow model (12.138) on the unit square domain Ω = (0, 1)2 with f ≡ 1 and linear boundary conditions. The forward problem is solved with GMsFEM, and the fine grid and coarse grid are chosen to be 50 × 50 and 5 × 5, respectively. The quantity of interest F for this set of simulations is the fine-scale pressure field. We consider the following two Gaussian covariance functions: • Isotropic Gaussian field with correlation lengths l1 = l2 = 0.1 and a stochastic dimension 5; • Anisotropic Gaussian field with correlation lengths l1 = 0.1 and l2 = 0.05, and a stochastic dimension 5.

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12 GMsFEM for selected applications

In both cases, we use the variance σ 2 = 2, and keep N = 5 terms in the final KL expansion where the ηk coefficients are drawn from a normal distribution with zero mean and unit variance. We denote by Fl the fine-scale pressure field at level l in MLMC. The level of our interest is L = 3. As stated in Algorithm 2, we generate Ml realizations at level l of the permeability field, solve the model problems by choosing Nl eigenvalues to generate the online space in GMsFEM, and compute the MLMC approximation of E[FL ] by (12.150). We compare MLMC with the standard MC at the level L of interest with the same amount of cost. Hence, we choose = M

L

Nl2 Ml N L2

l=1

as the number of samples in the standard MC algorithm. We use the arithmetic mean of Mr e f samples of the pressure field as reference and compute the relative L 2 -errors ref

(erMelL MC )[FL ]

=

E Mr e f [FL ] − E L [FL ] L 2 (Ω) ref

E Mr e f [FL ] L 2 (Ω) ref

el (erMC )[FL ] =

E Mr e f [FL ] − E MC  [FL ] L 2 (Ω) M ref

E Mr e f [FL ] L 2 (Ω)

.

For the simulations, use N1 = 4, N2 = 8, and N3 = 16 eigenfunctions for the online space construction. We respectively set the number of samples at each level to be M1 = 128, M2 = 32, and M3 = 8, and equate the computational costs for the MLMC and MC relative error comparisons. With this choice of realizations for MLMC, we  = 20 permeability realizations for the standard MC forward simulations. use M The parameters we have used and the respective relative errors are summarized in Table 12.36. Figure 12.48 illustrates expected pressure fields for different correlation lengths and different methods (MLMC and MC). For both covariances, we observe that the MLMC approach yields errors which are about 1.5 times smaller than those resulting from the MC approach. We note that the gain is larger for the isotropic case than for the anisotropic case.

MLMCMC In our MLMCMC experiments, we also consider the model problem (12.138) on Ω = (0, 1)2 with f ≡ 1 and linear boundary conditions. The prior permeability distribution p(k) is also parameterized by KLE as above. The “observed” data Fobs is obtained by generating a reference permeability field (indicated as reference solution in Figure 12.52), solving the forward problem with GMsFEM, and evaluating the pressure at nine points away from the boundary. The locations of the reference

12.8 GMsFEM for uncertainty quantification in inverse problems

381

Table 12.36 Parameters and errors for the estimates by MLMC vs. MC Isotropic Gaussian Anisotropic Gaussian (N1 , N2 , N3 ) (4, 8, 16) (4, 8, 16) (M1 , M2 , M3 ) (128, 32, 8) (128, 32, 8)  N 16 16  M 24 24 M MCr e f 5000 5000 erMelL MC 0.0431 0.0653 el erMC 0.0802 0.0952 el /er el erMC 1.86 1.45 M L MC

(a) Isotropic Gaussian

(b) Anisotropic Gaussian Fig. 12.48 Pressure field solutions for different methods and correlation lengths. Fig. 12.49 The points where pressure is evaluated.

1

0.5

0

0.5

1

pressures are shown in Figure 12.49. We note that the data generated by GMsFEM is very close to that by the standard finite element method on a refined mesh, with a relative error less than 0.1%, and thus the “inverse crime” is not present.

382

12 GMsFEM for selected applications anisotropic Gaussian

acceptance rate

acceptance rate

isotropic Gaussian 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2 level

3

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2 level

3

Fig. 12.50 Acceptance rate of multilevel sampler with both isotropic and anisotropic trials.

Our proposal distribution is a random walker sampler in which the proposal distribution depends on the previous value of the permeability field and is given by q(κ|κn ) = κn + δn where n is a random perturbation with mean zero and unit variance, and δ is a step size. The random perturbations are imposed on the ηk coefficients in the KL expansion. We consider two examples, one with isotropic Gaussian field of correlation length l1 = l2 = 0.1, the other with anisotropic Gaussian field of correlation lengths l1 = 0.05, l2 = 0.1. For both examples, we use δ = 0.2 in the random walk sampler. We again use the level L = 3, and for each level l we take the same number of KLE terms, N = 5 for the tests. For GMsFEM, we take the number of eigenvalues to generate the online space at each level as N1 = 4, N2 = 8, and N3 = 16. We take our quantities of interest F as the pressure values at the same nine points and use them in order to compute the acceptance probabilities as shown in Algorithm 4. For the MLMCMC examples, we run Algorithm 4 until P4 = 1000 total samples pass the final level of acceptance. We note that 300 initial accepted samples are discarded as burn-in. The acceptance rates of the multilevel sampler are shown in Fig. 12.50. To compute the acceptance rates, we assume that P1 , P2 , and P3 samples are proposed for respective levels L 1 , L 2 , and L 3 . Then, the rate at the l-th level is the ratio Pl+1 /Pl . Most notably, the results in Fig. 12.50 show that the acceptance rate increases as l increases. In particular, for more expensive (larger) levels, we observe that it is much more probable that a proposed sample will be accepted. This is an advantage of the multilevel method, due to the fact that fewer proposals are wasted on more expensive computations. We also show a set of plots in Fig. 12.51 that illustrate the errors E k = Fobs − FL (k), cf. (12.154), of the accepted samples on the finest level. In Fig. 12.52, we plot some of the accepted permeability realizations that have passed all levels of computation. We note that the general shapes of the accepted fields do not necessarily match that of the reference field, reinforcing the notion that the problem is ill-posed due to the fact that a variety of proposals may explain the reference data equally well.

12.9 Other applications

383 −3

x 10

isotropic Gaussian 2.5

2

2

1.5

1.5

error

error

2.5

1

1

0.5

0.5

0

0

400 200 iteration

0

−3 x 10 anisotropic Gaussian

0

400 200 iteration

Fig. 12.51 Plots of iteration versus error with both isotropic and anisotropic trials.

Fig. 12.52 Isotropic MLMCMC accepted realization.

12.9 Other applications We would like to mention that GMsFEM and related methods are used in other applications. Below, we list some of them. • GMsFEM has been developed for thin domain problems, where in some regions, several multiscale basis functions are used. See [397] for details. • The applications of GMsFEM for multiscale problems in permafrost have been considered in a number of papers. See, e.g., [384]. • The application of GMsFEM to modeling of Li-ion batteries is presented in [401]. • GMsFEM has been developed and applied to piezoelectric problems in heterogeneous media [23].

384

12 GMsFEM for selected applications

• The applications of GMsFEM to heterogeneous problems in Cosserat media can be found in [22]. • GMsFEM has been developed for multiscale Allen-Cahn problems in perforated domains [395]. • The applications of GMsFEM to simplified MHD problems are considered in [17]. • The applications of GMsFEM to the Darcy-Forchheimer model are considered in [383]. • GMsFEM and its variant have been studied for flow and heat transport in fractured geothermal reservoirs [398].

Chapter 13

Homogenization and numerical homogenization of nonlinear equations

13.1 Monotone and pseudomonotone operators In this section, we briefly summarize the properties of monotone and pseudomonotone operators. We consider two operators a(x, η, ξ) and a0 (x, η, ξ), η ∈ R and ξ ∈ R d . We assume that these operators satisfy the following conditions: |a(x, η, ξ)| + |a0 (x, η, ξ)| ≤ C (1 + |η| p−1 + |ξ| p−1 ),

(13.1)

(a(x, η, ξ1 ) − a(x, η, ξ2 )) · (ξ1 − ξ2 ) ≥ C |ξ1 − ξ2 | p ,

(13.2)

a(x, η, ξ) · ξ + a0 (x, η, ξ)η ≥ C|ξ| p .

(13.3)

H (η1 , ξ1 , η2 , ξ2 , r ) = (1 + |η1 |r + |η2 |r + |ξ1 |r + |ξ2 |r ),

(13.4)

Denote for arbitrary η1 , η2 ∈ R, ξ1 , ξ2 ∈ R d , and r > 0. We further assume that |a(x, η1 , ξ1 ) − a(x, η2 , ξ2 )| + |a0 (x, η1 , ξ1 ) − a0 (x, η2 , ξ2 )| ≤ C H (η1 , ξ1 , η2 , ξ2 , p − 1) ν(|η1 − η2 |) + C H (η1 , ξ1 , η2 , ξ2 , p − 1 − s) |ξ1 − ξ2 |s ,

(13.5)

where s > 0, p > 1, s ∈ (0, min( p − 1, 1)) and ν is the modulus of continuity, a bounded, concave, and continuous function in R+ , such that ν(0) = 0, ν(t) = 1 for t ≥ 1 and ν(t) > 0 for t > 0. p  is defined by 1/ p + 1/ p  = 1, y = x/. Inequalities (13.1)–(13.5) are the general conditions that guarantee the existence of a solution and are used in homogenization of nonlinear operators [350]. Here p represents the rate of the polynomial growth of the fluxes with respect to gradient and, consequently, it controls the summability of the solution. We do not assume any differentiability with respect to η and ξ in the coefficients. We note that these kinds of equations with the assumptions formulated above arise in many applications, such as nonlinear heat conduction, flow in porous media, etc. (see, e.g., [305, 333, 378, 380]). © Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_13

385

386

13 Homogenization and numerical homogenization of nonlinear equations

13.2 Homogenization We briefly discuss homogenization for general nonlinear elliptic equations. Let u  ∈ 1, p W0 (Ω) be the solution of ∂ ai, (x, u  , ∇u  ) + a0, (x, u  , ∇u  ) = f, (13.6) − ∂xi where ai, (x, η, ξ) and a0, (x, η, ξ), η ∈ R, ξ ∈ Rd satisfy assumptions given by (13.1)–(13.5), which guarantee the well-posedness of the nonlinear elliptic problem (13.6). Here Ω ⊂ Rd is a Lipschitz domain and  denotes the small scale of the problem. There are many application examples that can be described by (13.6). These include, for example, Richards’ equations, Forchheimer equations, nonlinear elasticity, and so on. The homogenization of nonlinear partial differential equations has been studied previously (see, e.g., [350] and the references therein). It can be shown that a solution 1, p u  converges (up to a subsequence) to u in an appropriate norm, where u ∈ W0 (Ω) is the solution of a homogenized equation ∂ ∗ a (x, u, Du) + a0∗ (x, u, Du) = f. (13.7) − ∂xi i The homogenized coefficients can be computed if we make an additional assumption on the heterogeneities, such as periodicity, almost periodicity, or when the fluxes are strictly stationary fields with respect to spatial variables. In these cases, an auxiliary problem is formulated and used in the calculations of the homogenized fluxes, a ∗ and a0∗ . Next, we discuss this. We assume that a and a0 are periodic functions with respect to the spatial variable. Then, the homogenized fluxes are defined as follows:  ∗ a(y, η, ξ + D y Nη,ξ (y)) dy, (13.8) a (η, ξ) = Y

a0∗ (η, ξ) =

 a0 (y, η, ξ + D y Nη,ξ (y)) dy,

(13.9)

Y

where a ∗ and a0∗ satisfy the conditions similar to (13.1)–(13.5). Here Nη,ξ ∈ W per (Y ) is the periodic solution (with average zero) of 1, p

−

∂ ai (y, η, ξ + D y Nη,ξ (y)) = 0 in Y, ∂ yi

(13.10)

where Y is a unit period. We do not present the proof of the homogenization here and refer to, e.g., [350]. In the following, we will present some examples. Example. Linear case. We consider a linear case

13.2 Homogenization

387

ai, (x, η, ξ) = ai j (x/)ξ j , a0 = 0. Then, the cell problem is −

∂ ∂ (ai j (y)(ξ + Nξ (y))) = 0 in Y. ∂ yi ∂yj

From here, we can write Nξ = χ · ξ =



χi ξi ,

i

where χ solves −

∂ ∂ (ai j (y)(δ jl + χl (y))) = 0 in Y. ∂ yi ∂yj

Then

ai∗ (η, ξ) = ai∗j ξ j .

Example. Nonlinear case. We consider a nonlinear case ai, (x, η, ξ) = ai j (x/)b(η)ξ j , a0 = 0. Then, the cell problem is −

∂ ∂ (ai j (y)b(η)(ξ + Nη,ξ (y))) = 0 in Y. ∂ yi ∂yj

From here, we can write Nη,ξ = χ · ξ =



χi ξi ,

i

where χ solves −

∂ ∂ (ai j (δ jl + χl (y))) = 0 in Y. ∂ yi ∂yj 

Then ai∗ (η, ξ)

=

ai j (y)b(η)(ξ j + Y

∂ N )dy = b(η)ai∗j ξ j . ∂yj

Example. One-dimensional nonlinear case. Consider one-dimensional case d x d a( , u) = f. dx  dx The cell problem is given by d d a(y, ξ + Nξ ) = 0. dy dy

388

13 Homogenization and numerical homogenization of nonlinear equations

Assume that the solution of

a(y, ζ) = α

is given by

ζ = ζ(y, α).

Then, we have a(y, ξ + ξ+

d Nξ ) = C = const. dy d Nξ = ζ(y, C). dy

Integrate over Y , to find C

 ζ(y, C)dy.

ξ= Y

Denote the solution of this equation C = a ∗ (ξ). We consider

a(y, ξ) = a(y)|ξ|γ−1 ξ 

for some γ > 1. Then, ζ = sign(ζ)

α a(y)

1/γ .

Then, C = a ∗ (ξ) is the solution of  C 1/gamma dN )( ) ξ= sign(ξ + dy. dy a(y) Y

13.3 Numerical homogenization (computation of effective parameters) 13.3.1 Pre-computing the effective coefficients For simplicity, we consider −

∂ ai (x, u, ∇u) = f. ∂xi

We consider a coarse-grid block K and our goal for each coarse-grid block to compute the effective property. This can be done by solving a local problem

13.3 Numerical homogenization (computation of effective parameters)

389

∂ ai (x, η, ∇ Nη,ξ ) = 0 in K ∂xi and Nη,ξ = ξ · x on ∂ K . Note that as in Section 2.2, we can use Dirichlet boundary conditions (instead of periodic) if there is no periodicity. Then ai∗ (η, ξ) =

1 |K |

 ai (x, η, ∇ Nη,ξ )d x. K

The coarse-grid equation is given by ∂ ∗ a (x, u ∗ , ∇u ∗ ) = f in Ω, ∂xi i u = 0 on ∂Ω. Other boundary conditions. We note that one can use other boundary conditions: (1) Mixed Dirichlet-Neumann boundary condition (2) Periodic boundary condition Look-up tables. Note that as a result of the upscaling, in practice, one generates a table for the values of upscaled quantities (also, called a look-up table). These tables can be constructed for some selected values of the parameters and interpolated inbetween. Remark 13.1 (Convergence). We remark that one can estimate the error in numerical homogenization when the coefficients are periodic (similar to that in Section 2.2). In this case, we can consider ∂ nh ai (x/, η, ξ + ∇x Nη,ξ ) = 0 in K , ∂xi nh = 0 on ∂ K . The numerical homogenized coefficients can be computed where Nη,ξ by  1 ∗,nh nh = ai (x/, η, ξ + ∇x Nη,ξ ). ai (13.11) |K | K



One can show that |ai∗,nh

−

ai∗ |

≤C

 s    p( p−s)  , + H H

(13.12)

where s and p are given by (13.1)–(13.5). Remark 13.2 (Computing the effective coefficients on-the-fly). Look-up tables can be expensive to compute if these tables are computed for each value of η and ξ. For this reason, one can consider computing the effective properties on-the-fly. These approaches are discussed in many papers and we briefly mention them in a simplified example ∂ ∗ ∂ ∗ (a (x, u ∗ , ∇u ∗ ) u ) = f in Ω, ∂xi i j ∂x j

390

13 Homogenization and numerical homogenization of nonlinear equations

with unknown a ∗ that is not pre-computed. We can write an iterative solution procedure (though other iterative procedures can also be considered) ∂ ∗ ∂ (ai j (x, u ∗n , ∇u ∗n ) u n+1 ) = f in Ω. ∂xi ∂x j We assume that as n → ∞, this iterative process converges. The iterative procedure is: • 1. Start with u n . • 2. Compute Ai j (x) = ai∗j (x, u n , ∇u n ) for each coarse grid. • 3. Solve ∂ ∗ ∂ (Ai∗,n u ) = f. j (x) ∂xi ∂x j n+1 • 4. Compute downscaled solution  u ∗n+1 . u ∗n+1 . • 5. Go back to Step 2 with u n+1 =  In each iteration, we require a computation of effective coefficients in each coarse grid block for some selected η = |K1 | K u n and ξ = |K1 | K ∇u n . If one follows this procedure, then the effective properties are computed only for those values of the solution that appear in one simulation. These pre-computed coefficients can be reused for different right-hand sides; however, we may need to compute additional effective parameters. Remark 13.3 (Using subdomains). As in Section 2.2, we can consider using subdomains (see Figure 2.15). In this case, the local problems are solved in small subdomains. Using the solutions in these subdomains, we compute the effective properties, which are used in the upscaled equations.

13.3.2 Parabolic equation One can consider space-time heterogeneities in nonlinear equations. Homogenization for the parabolic equations with space-time heterogeneities are studied in the literature (e.g., [350]). Numerical homogenization for the parabolic nonlinear problems with space-time heterogeneities can be studied (cf. Section 2.2). We consider ∂ ∂ u− ai (x, t, u, ∇u) = f. ∂t ∂xi The effective properties are computed in each K × [tn , tn+1 ] by solving local problem ∂ ∂ N− ai (x, t, η, ∇ N ) = 0, ∂t ∂xi N (t = 0) = ξ · x, and N = ξ · x on ∂ K . In each K , the effective property is com tn+1  puted as 1 ai (x, t, η, ∇ N ). a ∗ (η, ξ) = |K | tn K

13.4 MsFEM for nonlinear problems

391

The numerical homogenized equation ∂ ∗ ∂ ∗ u − a (x, t, u ∗ , ∇u ∗ ) = f. ∂t ∂xi i One can use this within explicit or implicit discretizations.

13.4 MsFEM for nonlinear problems Let T H be a coarse-grid partition of Ω (see Figure 1.8). We denote by T0H standard family of finite dimensional space, which possesses approximation properties, e.g., piecewise linear functions over triangular elements, as defined before. In further presentation, K is a triangular element that belongs to T H . To formulate MsFEM for general nonlinear problems, we will need (1) a multiscale mapping that gives us the desired approximation containing the small-scale information and (2) a multiscale numerical formulation of the equation. We consider the formulation and analysis of MsFEM for general nonlinear elliptic 1, p equations, u ∈ W0 (Ω) − diva(x, u, ∇u) + a0 (x, u, ∇u) = f,

(13.13)

where a(x, η, ξ) and a0 (x, η, ξ), η ∈ R, ξ ∈ R d satisfy the assumptions (13.1)– (13.5). Multiscale mapping. Introducethemapping E Ms F E M : T0H → VH inthefollowing way. For each element v 0H ∈ T0H , v H = E Ms F E M v H is defined as the solution of − diva(x, η v H , ∇v H ) = 0 in K , 0

(13.14)

0 v H = v 0H on ∂ K and η v H = |K1 | K v 0H d x for each K (coarse element). Note that for linear problems, E Ms F E M is a linear operator, where for each v 0H , v H is the solution of the linear problem. Consequently, VH is a linear space that can be obtained by mapping a basis of T0H . This is precisely the construction for linear elliptic equations. Multiscale numerical formulation. Multiscale finite element formulation of the problem is the following. Find u 0H ∈ T0H (consequently, u H (= E Ms F E M u 0H ) ∈ VH ) such that  0 0 f v 0H d x ∀v H ∈ T0H , (13.15) A H u H , v H = Ω

where A H u 0H , v 0H

=

 K

0

K

0

(a(x, η u H , ∇u H ) · ∇v 0H + a0 (x, η u H , ∇u H )v 0H )d x.

(13.16) Note that the above formulation of MsFEM is a generalization of the PetrovGalerkin MsFEM introduced earlier for linear problems.

392

13 Homogenization and numerical homogenization of nonlinear equations

13.4.1 Multiscale finite volume element method (MsFVEM) The formulation of a multiscale finite element (MsFEM) can be extended to a finite volume method. By its construction, the finite volume method has local conservative properties [201] and it is derived from a local relation, namely, the balance equation/conservation expression on a number of subdomains which are called control volumes. The finite volume element method can be considered as a Petrov-Galerkin finite element method, where the test functions are constants defined in a dual grid. Consider a triangle K , and let z K be its barycenter. The triangle K is divided into three quadrilaterals of equal area by connecting z K to the midpoints of its three edges. We denote these quadrilaterals by K z , where z ∈ Z h (K ) are the vertices of K . Also,

we denote Z h = K Z h (K ), and Z h0 are all vertices that do not lie on ΓΩ , where ΓΩ is Dirichlet boundaries. The control volume Vz is defined as the union of the quadrilaterals K z sharing the vertex z (see Figure 13.1).

z Kz

z

K

Vz

zK

K

Fig. 13.1 Left: Portion of triangulation sharing a common vertex z and its control volume. Right: Partition of a triangle K into three quadrilaterals.

The multiscale finite volume element method (MsFVEM) is to find u 0H ∈ T0H (consequently, u H = E Ms F V E M u 0H , such that  −

∂Vz

   0 a x, η u H , ∇u H · n d S +

Vz

   0 a0 x, η u H , ∇u H d x =

f dx Vz

∀z ∈ Z h0 ,

(13.17) where n is the unit normal vector pointing outward on ∂Vz . Note that the number of control volumes that satisfies (13.17) is the same as the dimension of Wh .

13.4.2 Examples of VH Linear case. For linear operators, VH can be obtained by mapping a basis of T0H . Define a basis of T0H , T0H = span(φi0 ), where φi0 are standard linear basis functions. In each element K , we define a set of nodal basis {φi }, i = 1, . . . , n d with n d (= 3)

13.4 MsFEM for nonlinear problems

393

being the number of nodes of the element, satisfying − div(a∇φi ) = 0 in K

(13.18)

and φi = φi0 on ∂ K . Thus, we have VH = span{φi }. Special nonlinear case. For the special case, a(x, u, ∇u) = a(x)b(u)∇u, VH can be related to the linear case. Indeed, for this case, the local problems associated with the multiscale mapping E Ms F E M (see (13.14)) have the form −div(a(x)b(η v H )∇v H ) = 0 in K . 0

Because η v H are constants over K , the local problems satisfy the linear equations, 0

−div(a(x)∇φi ) = 0 in K , and VH can be obtained as it is done for the first example. Vh using subdomain problems. One can use the solutions of smaller (than K ∈ Th ) subdomain problems to approximate the solutions of the local problems (13.14). This can be done in various ways based on a homogenization expansion. For example, instead of solving (13.14), we can solve (13.14) in a subdomain S with boundary conditions v 0H restricted onto the subdomain boundaries, ∂ S. Then the gradient of the solution in a subdomain can be extended periodically to K .

13.4.3 MsFEM for parabolic equations Consider 0 = t0 < t1 < · · · < tm−1 < tm = T and max(ti − ti−1 ) = h t . Denote Ωix,t = [ti , ti+1 ] × Ω, Vi = L p (ti , ti+1 , W 1, p (Q 0 )), and Wi = {u ∈ Vi , Ωt u ∈ L q (ti , ti+1 , W −1,q (Q 0 ))}. The main idea of MsFEM is to approximate A∗ (u h , vh ) using the solution of the local problems. Denote by φi0 (x) a basis in T0H consisting of piecewise linear functions, such that φi0 (x j ) = δi j (δi j is the Kronecker symbol), and x j are the nodal N points of the finite element partition. Consider u H = i=1 θi (t)φi0 (x), where θin+1 = θi (t = tn+1 ) (θi (t) is continuous) are defined by   (θin+1 − θin ) φi0 (x)φ0j (x)d x+ 

tn+1 tn

Ω

i

 Ω

a(x, t, η

u H (tn+1 )

, ∇x v) ·

 ∇x φ0j d xdt

= tn

tn+1

 Ω

Here v is the solution to the local problem and computed as

(13.19) f φ0j d xdt.

394

13 Homogenization and numerical homogenization of nonlinear equations

∂ v = div a(x, t, η u H (tn+1 ) , ∇x v) in K × [tn , tn+1 ], ∂t

(13.20)

where v = u H (tn ) on ∂ K , v(x, t = tn ) = u H (tn ). For further analysis, θ and ζ denote discrete vectors defined at the nodal points. MsFEM introduced above has the following discrete representation: M(θn+1 − θn ) + A H (θn+1 ) = b,

(13.21)

where Mi j = Q 0 φi0 (x)φ0j (x)d x is a mass matrix, A H is defined as above, bi = tn+1 0 tn Q 0 f φi d xdt. (13.21) is solved using Newton’s method or its variations. For the explicit formulation of the numerical homogenization procedure A H (θn+1 ) is replaced by A H (θn ) in (13.21).

13.5 Remark on the analysis of MsFEM for nonlinear problems The analysis of MsFEM for nonlinear PDEs is presented in a number of papers [176, 182, 185]. The analysis assumes the periodic coefficients. In this section, we briefly outline the results. First, we recall that the assumptions (13.1)–(13.5) hold. In [185], we have shown using G-convergence theory that lim lim u 0H − u 0 W 1, p (Ω) = 0,

H →0 →0

0

(13.22)

(up to a subsequence) where u 0 is a solution of the homogenized equation and u 0H is a MsFEM solution given by (13.15). This result can be obtained without any assumption on the nature of the heterogeneities and cannot be improved because there could be infinitely many scales, α(), present, such that α() → 0 as  → 0. Next, we will present the convergence results for MsFEM solutions. In the proof of this theorem, the form of the truncation error (in a weak sense) in terms of the resonance errors between the mesh size and small scale . The resonance errors are derived explicitly. To obtain the convergence rate from the truncation error, one needs some lowerbounds.Underthegeneralconditions,suchas(13.1)–(13.5),onecanprovestrong convergence of MsFEM solutions without an explicit convergence rate (cf. [380]). To convert the obtained convergence rates for the truncation errors into the convergence rate of MsFEM solutions, additional assumptions, such as monotonicity, are needed. Theorem 13.4. Assume a (x, η, ξ) and a0, (x, η, ξ) are periodic functions with respect to x and let u 0 be a solution and u 0H is a MsFEM solution given by (13.15). Moreover, we assume that ∇u H is uniformly bounded in L p+α (Ω) for some α > 0. Then = 0, (13.23) lim u 0 − u  1, p →0

H

0 W (Ω) 0

where H = H ()   and H → 0 as  → 0 (up to a subsequence).

13.5 Remark on the analysis of MsFEM for nonlinear problems

395

Theorem 13.5. Let u 0 and u 0H be the solutions of the homogenized problem and MsFEM (13.15), respectively, with the coefficient a (x, η, ξ) = a(x/, ξ) and a0, = 0. Then    s p p  ( p−1)( p−s)    p−1 p (13.24) ≤C + + H p−1 . u 0H − u 0  1, p W0 (Ω) H H

Chapter 14

GMsFEM for nonlinear problems

14.1 Introduction The interaction between nonlinearities and multiple scales can be complex and nonseparable as discussed in Chapter 13. To discuss some main concepts, in this chapter, we consider the following elliptic equation as an example div(a(x, u, ∇u)) = f.

(14.1)

We assume a(x, ·, ·) is highly heterogeneous with respect to x. From the point of view of the interaction between nonlinearities and the multiple scales, one can distinguish several classes. In some nonlinear problems, the nonlinearities within coarse regions (a computational grid), that induce the change in the heterogeneities, can be parametrized with a low-dimensional parameter, e.g., a(x, u, ∇u) = a0 (x, u)∇u (assuming smoothness and boundedness of a). In this case, the homogenization and numerical homogenization can be simplified as discussed in Chapter 13. For example, if the nonlinearities and heterogeneities are separable in this case, i.e., a(x, u, ∇u) = a(x)b(u)∇u, then, in fact, one can use a linear theory of multiscale methods. The situation is very different when a(x, u, ∇u) = a1 (x, ∇u)∇u. Because ∇u is highly heterogeneous, one cannot use any low-dimensional approximation and linear theories as we discussed in Chapter 13 (see also, e.g., [20, 89, 183, 245, 248, 306, 334, 351] for homogenization and numerical homogenization). This is true even for a separable case a(x, u, ∇u) = a0 (x)b(|∇u|)∇u. These problems require nonlinear cell problems [20, 89, 183, 245, 248, 306, 334, 351]. In this chapter, we focus on a(x, u, ∇u) = a(x, ∇u) and discuss these nonlinear cell problems for GMsFEM. Some common ingredients in multiscale methods for linear problems are that local solutions are calculated and used to form equations on a coarse grid. The extensions of these methods to nonlinear problems (as (14.1)) use nonlinear local problems. For example, in numerical homogenization methods (see Chapter 13), one can use as a local problem in each coarse cell, for the case a(x, u, ∇u) = a(x, ∇u), © Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_14

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14 GMsFEM for nonlinear problems

−div(a(x, ∇φξ )) = 0 with the boundary conditions φ = ξ · x. The homogenized fluxes are computed by averaging the flux a ∗ (ξ) = a(x, ∇φξ ) (see also [20, 89, 245, 248, 306, 334, 351]). Inthischapter,themainobjectiveistodiscusstheGMsFEMfornonlinearproblems. The main idea of GMsFEM for linear problems, as discussed earlier, is to form snapshot spaces and perform local spectral decomposition in the snapshot space. We follow the same general concept and introduce nonlinear eigenvalue problems. We focus on the development of a systematic model reduction using nonlinear a-harmonic functions. The latter is important as it allows capturing the effects of separable scales. Without using nonlinear a-harmonic functions, one cannot, in general, capture the effects of small separable scales. This is in contrast to linear problems, where one can construct one linear basis function per every coarse node that contains the effects of small scales. Using local solutions allows compressing the effects of small scales within a coarse block and we work with a system reduced to the boundaries of coarse cells. In this case, we can also guarantee that our approaches recover homogenization results when there is a scale separation (note that previous approaches [189] cannot guarantee it). The proposed method is in a spirit of hybridization techniques [103, 130, 195] and necessary to eliminate the scale interactions in each block. The snapshot space can be thought as a nonlinear map from the boundaries to the interior. We discuss the use of nonlinear eigenvalue problems, which are motivated by the analysis. The analysis allows removing some of the major assumptions that are used when not using local nonlinear harmonic functions (see [189]). The numerical results are presented for several examples. We consider a highcontrast permeability field in p-Laplacian example with heterogeneous coefficients. The high-contrast permeability field contains several channels and inclusions with high permeability. In the numerical results, we increase the number of local multiscale basis functions and compute the errors. The results show that the error decreases as we increase the number of basis functions and we can approximate the global solution accurately with very few degrees of freedom.

14.2 Preliminaries and motivation 14.2.1 Preliminaries and notations We consider the following heterogeneous p-Laplacian equation − div(a(x, ∇u)) = f (x) in Ω,

u = g on ∂Ω,

(14.2)

where a(x, ∇u) = κ(x)|∇u| p−2 ∇u, p ≥ 2, κ(x) ≥ κ0 > 0 is a high-contrast coefficient (i.e., κmax /κmin is large), f ∈ W −1,q (Ω) (1/ p + 1/q = 1) is an external forcing term, and g ∈ W 1/q, p (Ω) is the Dirichlet boundary data.

14.2 Preliminaries and motivation

399

The corresponding weak formulation is Find u ∈ Wg1, p (Ω) ≡ {v ∈ W 1, p (Ω) : v = g on ∂Ω}, such that   1, p a(x, ∇u) · ∇v = f v, ∀v ∈ W0 (Ω). (P) : Ω

Ω

The well-posedness of (P) is well established, and one can refer to, for example, Glowinski and Marrocco [233] or the account in Ciarlet [126]. Throughout the chapter, we define the energy norm of u ∈ W 1, p (Ω) as  1/ p

u 1, p(Ω) = κ(x)|∇u| p d x . Ω

Next, we describe the finite element approximation of the solution. We let T h be a fine triangulation and denote by Vh = Vh (Ω) the usual finite element space containing continuous piecewise linear functions with respect to T h . We also let Vh,0 (Ω) be the subset of Vh (Ω) containing functions that vanish on ∂Ω. Similar notations, Vh (D), Vh,0 (D), are used for D ⊂ Ω. The discrete fine-scale problem is defined in the following: (P h ) Find u h ∈ Vh (Ω),   such that a(x, ∇u h ) · ∇v = f v, ∀v ∈ Vh,0 (Ω). Ω

Ω

The T H (see Figure 1.8) denotes a coarse-grid partition, where each coarse eleNc ment is comprised of a localized fine mesh. We use {xi }i=1 to denote the vertices of the coarse mesh. Inside each coarse neighborhood ωi (i=1,...,Nc ), we call the collection of the coarse edges with xi being a common vertex a cross.

14.2.2 Motivation In this section, we will present the motivation for our method. First, we introduce the concept of p-harmonic extension. Definition 14.1. Letu ∈ W 1, p (K )( p ≥ 2)beagivenfunctionand K beagivencoarse 1, p element. Let u˜ ∈ W 1, p (K ) be defined so that u˜ − u ∈ W0 (K ), and that u˜ satisfies −div(a(x, ∇ u)) ˜ = 0 in K , ˜ Then u˜ is called the p-harmonic extension of where a(x, ∇ u) ˜ = κ(x)|∇ u| ˜ p−2 ∇ u. u. We denote u˜ := H p (u). Remark 14.2. The p-harmonic extension minimizes the energy norm, i.e.   κ(x)|∇ u| ˜ p d x = min κ(x)|∇v| p d x, K 1, p

1, p

v∈Wu (K )

K

where Wu (K ) = {v ∈ W 1, p (K ) | v = u on ∂ K }.

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Remark 14.3. In this context, all p-harmonic extensions are accomplished coarse element by coarse element. Though we might use the notation H p directly on a larger domain such as coarse neighborhoods ωi or the whole domain Ω, it means that the p-harmonic extension is performed on each coarse element contained in ωi or Ω. Our main idea is solving for the Generalized Multiscale Finite Element solution of Equation (14.2) on the crosses of the coarse mesh and then the solution in the whole domain can be approximated by p-harmonically extending the obtained cross values into the domain. This idea is motivated by the technique of Numerical Homogenization (NH), which is described in the following. Our goal is to show that our proposed GMsFEM recovers NH.

Numerical Homogenization (NH) and Generalized Multiscale Finite Element In this section, we briefly repeat some numerical homogenization concepts and introduce nonlinear GMsFEM. We consider a coarse-grid block K and our goal for each coarse-grid block is to compute the effective property. This is done by solving a local problem −div(a(x, ∇ Nξ )) = 0 in K , with boundary condition Nξ = ξ · x on ∂ K . According to the previous definition, we can write Nξ = H p (ξ · x). Then a ∗ (x, ξ) is defined as 1 a (x, ξ) = |K | ∗

 a(y, ∇ Nξ )dy,

x ∈ K.

K

The coarse-grid equation is given by −div(a ∗ (x, ∇u ∗ )) = f in Ω, with u ∗ = 0 on ∂Ω. Suppose u ∗ =



ck φk , where {φk } are linear basis, then



 a ∗ (x, ∇ ck φk ) · ∇φ j d x Ω    = a∗( ck ∇φk ) · ∇φ j d x,

c) = F N H (

K ∈Ω

K

where c is a vector of the coefficients {ck }. At this step, we denote  constant, then Nξ = H p ( ck ∇φk · x) and



ck ∇φk = ξ =

14.3 The GMsFEM

401

F N H ( c) =

 K ∈Ω

a ∗ (ξ) · ∇φ j d x K

   1  a(x, ∇ Nξ )d x · ∇φ j d x = |K | K K ∈Ω K     1 a(x, ∇ H p ( ck ∇φk · x))d x · ∇φ j d x = Ω |K |  K    1 a(x, ∇ H p ( ck φk ))d x · ∇φ j d x = |K | K Ω = f φ j d x. Ω

Compared with the numerical homogenization, the GMsFEM seeks the approxi mation of the solution in the form i,k χi ckωi φωk i , which solves   a(x, ∇ H p ( χi ckωi φωk i ) · ∇φωj i d x F( c) = Ω

 =

Ω

i,k

f φωj i d x,

Li where {φωk i }k=1 (L i is the number of basis chosen in ωi ) are generalized multiscale Nc basis constructed in each ωi (i = 1, · · ·, Nc ), {χi }i=1 is the set of partition of unity functions. Our main approach is to construct multiscale basis functions in a systematic way and provide a priori error estimate. We see from the above discussion that the GMsFEM can be thought as an extension of NH, where we need to identify appropriate procedures for finding multiscale basis functions. In the following section, we will describe the details of constructing a multiscale basis as well as the partition of unity functions.

14.3 The GMsFEM The goal of our GMsFEM is to find a numerical approximation of the homogenized solution that arises in homogenization as well as employ the degrees of freedom only on the crosses in order to exhibit model reduction. Suppose the GMsFEM solution we   Li Li χi ckωi φωk i ), where {φωk i }k=1 are multiscale basis are seeking for is u H = H p ( i k=1 Nc constructed in each coarse neighborhood ωi , {χi }i=1 is the set of partition of unity functions, then the generalized multiscale finite element formulation for Equation (14.2) is the following. Find c = {ckωi }i,k , such that   Li  a(x, ∇ H p ( χi ckωi φωk i ) · ∇φωj i d x = f φωj i d x, (14.3) Ω

i

k=1

Ω

where a(x, ∇u) = κ(x)|∇u| p−2 ∇u, as defined in Section 14.2.1.

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14 GMsFEM for nonlinear problems

14.3.1 Partition of unity functions To propose our method, we first need to construct a set of of unity functions partition Nc Nc . Each χi has support ωi , and this set satisfies i=1 χi = 1. In addition, χi {χi }i=1 has value 1 at the vertex xi . The following are two commonly used sets of partition of unity functions: • A bilinear partition of unity: χi is defined as usual bilinear basis functions χi0 on ωi , which is equal to 1 at node xi and equal to 0 on ∂ωi . • A multiscale partition of unity (with linear boundary conditions): χi is defined by −div(a(x, ∇χi ) = 0 in K ⊂ ωi , χi = χi0 on ∂ K , for all K ⊂ ωi . We remark that using the second choice of partition of unity functions can, in general, provide better numerical performance.

14.3.2 Multiscale basis Snapshot space Let ωi be a given coarse neighborhood. The construction of the multiscale basis ωi ωi . The snapshot space VH,snap is a functions on ωi starts with a snapshot space VH,snap set of functions defined on ωi and contains all or most necessary components of the fine-scale solution restricted to ωi . A spectral problem is then solved in the snapshot space to extract the dominant modes in the snapshot space. These dominant modes are the offline basis functions and the resulting reduced space is called the offline ωi that are commonly used. space. There are two choices of VH,snap The first choice is to use all possible fine-grid functions in ωi . This snapshot space provides an accurate approximation for the solution space; however, this snapshot space can be very large. The second choice for the snapshot space consists of harmonic extensions. In particular, we denote by Mh (ωi ) the set of all nodes of the fine mesh T h which lie on ∂ωi . For each fine-grid node x j ∈ Mh (ωi ), we construct a discrete delta function δ hj (x) defined on Mh (ωi ) by  1 for k = j h δ j (xk ) = , ∀xk ∈ Mh (ωi ). 0 for k = j Then the j−th snapshot basis function ψ ωj i is defined as the solution of −div(κ(x)∇ψ ωj i ) = 0 in ωi , ψ ωj i = δ hj on ∂ωi .

(14.4)

ωi is equal to the size of Mh (ωi ). We note that one can use The dimension of VH,snap randomized snapshots in conjunction with oversampling to reduce the computational

14.3 The GMsFEM

403

cost associated with the snapshot calculations. We refer to Section 4.8 and [71] for more details. With these snapshots, we follow the procedure in the following subsection to generate offline basis functions by using an auxiliary spectral decomposition.

Offline space The construction of multiscale basis functions for solving p-Laplacian equation in the fashion of p-harmonic extension is based on the design of a proper nonlinear spectral problem which will be solved in the snapshot space. In each coarse neighborhood ωi , we define the following nonlinear eigenvalue problem which can be characterized by the Rayleigh-Ritz method (RRM): ⎧ ωi ω ⎪ ⎨φ1 = c i ,

G ωi (v) ωi ⎪ ⎩φk = arg minωi G ωi (v), v∈X k χ

λω1 i = 0, G ωi (φωi ) λωk i = ωi kωi , G χ (φk )

for k ≥ 2,

(14.5)

ωi ωi where cωi ∈ VH,snap is a constant function in ωi , X kωi is a subspace of VH,snap and

⊥ ωi ωi ωi defined as X k = span{φ1 , · · ·, φk−1 } where the orthogonality ⊥ is defined ωi , the functionals are given by G ωi (v) = with respect to the H 1 norm in VH,snap   p ωi p ωi κ(x)|∇ H p (v)| d x and G χ (v) = ωi κ(x)|∇ H p (χi v)| d x. This nonlinear eigenvalue problem is a standard orthogonal subspace minimization method and is welldefined (see, e.g., [428]). After the eigenvalue problem (14.5) is solved, the eigenfunctions {φωk i }k in each coarse neighborhood ωi will contribute as offline basis (or we call them generalized multiscale basis or eigenbasis) after being multiplied by the associated partition of unity function χi . We choose the first L i eigenfunctions on each ωi and denote the offline space as

VH,off = span{χi φωk i : k = 1, · · ·, L i ; i = 1, · · ·, Nc } ⊆ W 1, p (Ω).   L i ωi Recall that our solution assumes the form of u H = H p ( i k=1 ck χi φωk i ), which   L i ωi means u H is obtained by p-harmonically extending i k=1 ck χi φωk i in each coarse   L i ωi block K , thus only the values of i k=1 ck χi φωk i on each coarse edge matter in coarse this sense. If we consider one coarse neighborhood ωi , for example,  L i ωthe ck i χi φωk i on neighborhood of an interior coarse vertex, it is the restriction of k=1 the 12 coarse edges that will matter in the process of p-harmonic extension. Notice on and beyond the boundary of ωi , that the partition of unity function  L i χωi i vanishes ck χi φωk i on the cross (that is, the inside 4 coarse thus merely the restriction of k=1 edges) makes an influence. Therefore, we can restrict χi φωk i (k = 1, · · ·, L i ) on the (which we call cross basis in this context) cross of ωi and denote the restricted basis   Li ckωi φˆ ωk i ). We denote by φˆ ωk i . Then we can write u H = H p ( i k=1

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14 GMsFEM for nonlinear problems

H,off = span{φˆ ωk i : k = 1, · · ·, L i ; i = 1, · · ·, Nc }. V In this way, we can focus on the degrees of freedom on the crosses and perform spectral decomposition on these crosses.

14.4 Convergence of the method In this section, we will state the main convergence results of our method. For details of the analysis, we refer the readers to [107]. In the following, we use the notation F  G to represent F ≤ CG with a constant C independent of the mesh, contrast, and the functions involved. First of all, we state the following best approximation error estimate. It says that the multiscale solution u H obtained by the GMsFEM is the best one among all other functions in the offline space VH,off . Notice that the error on the right-hand side of the estimate is computed using the p-harmonic extension of the function v H defined on the crosses. Lemma 14.4. Suppose u is the exact solution of Equation (14.2), u H is the GMsFEM solution from Equation (14.3), then for any p ≥ 2, we have q

p−2

u − u H 1, p(Ω)  u − H p (v H ) 1,p p(Ω) u 1,p−1p(Ω) for any v H ∈ VH,off , where 1/ p + 1/q = 1, u 1, p(Ω) = in Section 14.2.1.

 Ω

κ(x)|∇u| p d x

1/ p

is the energy norm defined

In the next lemma, we obtain a convergence estimate for the quantity u − H p (u). Lemma 14.5. Suppose u is the exact solution of Equation (14.2), K is any coarse block of size H , p ≥ 2, then we have 

 κ(x)|∇(u − H p (u))| p d x  H q

| f |q d x,

K

(14.6)

K

where 1/ p + 1/q = 1. We remark that, this local error estimate proved in Lemma 14.5 immediately deduces the global error estimate: p

q

u − H p (u) 1, p(Ω)  H q f L q (Ω) .

(14.7)

Using the result in the previous two lemmas, we can derive the following convergence estimate for our GMsFEM. The idea of the analysis is that, in Lemma 14.4, we can first take v H to be a projection of the solution u in the space VH,off . After that, we can apply Lemma 14.5 and the spectral problem to obtain the desired estimate.

14.5 Numerical implementation and results

405

Theorem 14.6. Suppose u is the exact solution of Equation (14.2), u H is the GMsFEM solution from Equation (14.3), then for any p ≥ 2, we have     1 2 1 p−2 1 1 p( p−1) 1 2 p−1 ( p−1) p−1

u − u H 1, p(Ω)  u 1, p(Ω) H ( p−1)2 f L q (Ω) +

u 1, p(Ω) , Λ∗ (14.8) where Λ∗ = minωi λωLii +1 , {λωj i } are the eigenvalues defined by (14.5) in Section 14.3.2, L i is the number of eigenbasis chosen in each coarse neighborhood ωi .

14.5 Numerical implementation and results In this section, we exhibit the process of implementing the proposed method for p-Laplacian equation. From Glowinski and Marrocco [233], or Ciarlet [126], (P) is equivalent to the following minimization problem: (Q) Find u ∈ Wg1, p (Ω) ≡ {v ∈ W 1, p (Ω) : v = g on ∂Ω}, such that JΩ (u) =

inf 1, p

v∈Wg (Ω)

JΩ (v),

(14.9)

  where JΩ (u) = 1p Ω κ(x)|∇u| p d x − Ω f ud x. It is easily established that JΩ (u) is strictly convex and continuous on Wg1, p (Ω). Besides, JΩ (u) is Gateaux differentiable with    1, p JΩ (u)(w) = κ(x)|∇u| p−2 ∇u · ∇wd x − f wd x ∀w ∈ W0 (Ω). Ω

Ω

Hence, there exists a unique solution to (Q), and (Q) is equivalent to its Euler equation (P). The corresponding discrete problem of (Q) is (Qh ) Find u h ∈ Vh (Ω), such that JΩ (u h ) =

min

v h ∈Vh,0 (Ω)

JΩ (v h ).

(14.10)

The well-posedness of (Qh ) = (P h ) follows in an analogous way to that of (Q) and (P), see Glowinski and Marrocco [233] or Ciarlet [126]. Recall the discussion in Section 14.3.2, we can represent the GMsFEM solution by   L i ωi ωi c φˆ ). For notational brevity, we use a single-index notation u H = H p ( i k=1  Nk k to write u H = H p ( j=1 c j φˆ j ). Then we apply Broyden’s method (which is a QuasiNewton’s method) to solve the minimization problem (Qh ), see Algorithm 5.

14.5.1 Numerical results In this section, we offer a number of representative numerical results to verify the proposedmethodsintheprevioussections.Inparticular,wesolveEquation(14.2)usingthe proposedGMsFEMtovalidatetheeffectivenessoftherespectiveapproaches.Toobtain

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14 GMsFEM for nonlinear problems

Algorithm 5 A Quasi-Newton algorithm  N (0) Initialization: An initial guess c (0) = c j

j=1

and B (0) ∈ R N ×N

for k = 0: do  (0) (1) Compute the gradient vector g (0) ( c(0) ) = ∇ JΩ (H p ( Nj=1 c j φˆj )). (0) (2) Compute the stepsize τ . (3) Set: c (1) = c (0) − τ (0) B (0) g (0) . (4) If c(1) − c (0) < δ, where · is a suitable norm, return. end for for k = 1 to N : do  ˆ c(k) ) = ∇ JΩ (H p ( Nj=1 c(k) (1) Compute the gradient vector g (k) ( j φ j )). (2) Compute the approximation of the inverse of Hessian matrix: B (k) = B (k−1) +

c(k) − c (k−1) )T B (k−1) [( c(k) − c (k−1) ) − B (k−1) ( g (k) − g (k−1) )]( . ( c(k) − c (k−1) )T B (k−1) ( g (k) − g (k−1) )

(3) Compute the stepsize τ (k) . (4) Set: c (k+1) = c (k) − τ (k) B (k) g (k) . (5) If c(k+1) − c (k) < δ, return. end for

benchmark fine-grid solutions we solve (14.2) on the unit square Ω = [0, 1] × [0, 1] using a uniform fine grid of 100 × 100 square finite elements which is divided into 10 × 10squarecoarseelementsuniformly.Wealsouseaforcingterm f = 1andimpose a linear Dirichlet boundary condition u(x, y) = x + y. The high-contrast permeability field κ1 (x) used in our experiments is shown in Figure 14.1, with high-contrast ratio κmax /κmin being 105 . We note that the fine-grid discretization leads to a system of size N f = 10201. As such, we aim to construct a reduced-order system that can accurately approximate the benchmark solutions from the original fine-scale system. × 10

4

10

10

9

20

8

30

7

40

6

50

5

60

4

70

3

80

2

90

1

100 10

20

30

40

50

60

70

80

90

Fig. 14.1 Illustration of the high-contrast permeability field κ1 (x).

100

14.5 Numerical implementation and results

407

Accuracy of GMsFEM using different numbers of basis functions We employ both fine-grid (FEM) and coarse-grid (GMsFEM) methods to solve the model equation (14.2). In comparing the respective approaches, we introduce relative L p errors and relative energy errors, which are defined as

u − u H L p (Ω) × 100%,

u L p (Ω)

u − u H 1, p(Ω) Energy error = × 100%,

u 1, p(Ω) L p error =

(14.11)

where we recall that u denotes the FEM solution and u H denotes the GMsFEM solution. For the first set of experiments, we take p = 3, 4, 5, 6 separately and use different numbers of cross basis (L i for each ωi ) for each fixed value of p. Then we check the relative errors of the GMsFEM solutions. Numerical results are shown in Table 14.1. Note that in the first column of each sub-table, we show the numbers of basis functions used for each coarse neighborhood ωi , and the degrees of freedom (DOF) of offline space which are the numbers in parentheses. To visually observe the accuracy of GMsFEM, we plot the solutions obtained by both FEM and GMsFEM in the case p = 3 using 4 cross basis functions in each coarse neighborhood, see Figure 14.2. By observing the columns in Table 14.1, we can clearly see that for each p, the relative error decays as we use more cross basis functions. We note that as L i increases, the value of (L i + 1)’s eigenvalue increases, and the error bound 1/Λ∗ will correspondingly decrease. In other words, the analysis suggests (and the results validate) that keeping more basis functions for the coarse space construction will indeed yield a decreasing global error. Through a more careful examination, we notice that for each p, when 4 or more than 4 cross basis are chosen in each coarse neighborhood (i.e., L i ≥ 4 for each ωi ), the errors are much smaller. This might suggest that if we use 4 or more than 4 cross basis in each coarse neighborhood, we would get a better convergence. We will explore this in more detail in the following subsection.

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14 GMsFEM for nonlinear problems

L i (DOF)

L p error 9.5211% 6.4469% 5.7582% 0.5204 % 0.4495%

1(81) 2(162) 3(243) 4(324) 5(405)

p=3 Energy error 41.0269% 34.3841% 27.7637% 6.5513% 5.1507%

L i (DOF) 1(81) 2(162) 3(243) 4(324) 5(405)

p=5 L p error Energy error 10.1175% 40.4630 % 7.7140% 34.0488 % 5.1716 % 27.8829 % 0.9379 % 9.9405% 0.8073% 7.9218 %

L i (DOF) 1(81) 2(162) 3(243) 4(324) 5(405)

L i (DOF) 1(81) 2(162) 3(243) 4(324) 5(405)

p=4 L p error Energy error 10.8831% 42.3452 % 6.4747% 32.9286 % 5.1163 % 24.1347 % 0.9182 % 8.5679% 0.8179 % 6.6528 % L p error 8.9522 % 6.9407 % 4.3722 % 1.0653 % 0.9073 %

p=6 Energy error 39.6830 % 30.9199 % 23.8474 % 8.6992 % 7.0817 %

Table 14.1 Relative errors for p = 3, 4, 5, 6 using different numbers of cross basis.

2

2

10

1.8

10

1.8

20

1.6

20

1.6

30

1.4

30

1.4

40

1.2

40

1.2

50

1

50

1

60

0.8

60

0.8

70

0.6

70

0.6

80

0.4

80

0.4

90

0.2

90

0.2

100 10

20

30

40

50

60

70

80

90

100

0

100 10

20

30

40

50

60

70

80

90

100

0

Fig. 14.2 FEM v.s. GMsFEM node-wise solutions, p = 3, DOF = 324.

Correlation between errors and eigenvalues Aside from the accuracy of our proposed method, we are interested in determining how many cross basis (or DOFs) should be used. As we mentioned earlier, there is a “jump” in the relative energy errors when we take 4 cross basis in each coarse neighborhood (i.e., L i = 4 for each ωi , see Table 14.1). Thus, L i = 4 might be a good choice. According to our analysis in Section 14.4, that is probably due to a sudden decrease in the quantity of 1/Λ∗ , where Λ∗ = minωi λωLii +1 , {λωj i } are the eigenvalues defined in (14.5) in Section 14.3.2. To verify this theory, we calculate the corresponding 1/Λ∗ for each L i in the case of p = 3. The results are shown in

14.5 Numerical implementation and results

409

Table 14.2. In this table, we see the jump in Λ∗ and 1/Λ∗ at L i = 4, which explains our earlier inference. Hence, we conclude that the proper number of cross basis is chosen at the spot where there is a sudden increase in the values of Λ∗ (or a sudden decrease in the values of 1/Λ∗ ). We would like to remark that an adaptive method can be employed to determine the best choice of L i for each coarse neighborhood ωi . Moreover, to see a more quantitative relationship between the relative errors and the values of Λ∗ , as well as being inspired by the result in Theorem 14.6, we calculate the cross-correlation coefficient between the relative energy errors and the corresponding 1 1 values of ( Λ1∗ ) p( p−1)2 for the case p = 3. We recall that the quantity ( Λ1∗ ) p( p−1)2 comes from (14.8) in Theorem 14.6. The evaluated cross-correlation coefficient is 0.99. This indicates a linear relationship between the relative energy error and the corresponding 1 ( Λ1∗ ) p( p−1)2 , which verifies our result in (14.8). Li

Λ∗

1/Λ∗

1 886.3223e-6 1.1283e3 2 2.5881e-3 3.8639e2 3 4.4561e-3 2.2441e2 4 1.5519e2 6.4436e-3 5 4.0059e2 2.4963e-3 Table 14.2 Values of Λ∗ and 1/Λ∗ when p = 3.

Numerical tests with other permeability fields To verify that our proposed method is applicable to more situations, we examine other choices of permeability field κ(x). First, we would like to check that the GMsFEM solution errors do not depend on the high-contrast ratio κmax /κmin . To see this, we increase the high-contrast ratio of κ1 (x), which is used in the previous subsections, from 105 to 107 . We denote the new permeability field by κ2 (x). Then we solve Equation (14.2) using both FEM and proposed GMsFEM, and calculate the relative errors and the error bound quantity 1/Λ∗ . Numerical results for p = 3 are shown in Table 14.3. Comparing these results with the top left sub-table in Tables 14.1 and 14.2, we can observe a similar trend inside the columns as well as a slight increase in the values of both relative energy errors and 1/Λ∗ ’s. The jump at L i = 4 still occurs. 1 The cross-correlation coefficient between the relative energy errors and ( Λ1∗ ) p( p−1)2 is calculated to be 0.98.

410

14 GMsFEM for nonlinear problems L i Energy error 1 2 3 4 5

44.1531 % 36.4369 % 27.9932 % 6.7656 % 5.3048 %

1/Λ∗ 1.4234e3 4.0367e2 2.3489e2 6.4924e-3 2.5035e-3

Table 14.3 Relative energy errors and values of 1/Λ∗ using κ2 (x), p = 3.

We also consider a different high-contrast permeability field κ3 (x), see Figure 14.3. We solve Equation (14.2) for p = 3 and the results are presented in Table 14.4. 1 The cross-correlation coefficient between the relative energy errors and ( Λ1∗ ) p( p−1)2 is calculated to be 0.94. We can see that our proposed method works well for this permeability field. 4

x 10

1

18 16

0.8 14 12

0.6

10 8

0.4

6 4

0.2

2 0

0

0.4

0.2

0.6

0.8

1

Fig. 14.3 Illustration of the high-contrast permeability field κ3 (x).

L i Energy error 1 2 3 4 5

47.0772 % 27.6835 % 20.8135 % 4.3282 % 2.6937 %

1/Λ∗ 1.8473e1 4.6391e0 2.6787e0 2.2643e-3 1.0060e-3

Table 14.4 Relative energy errors and values of 1/Λ∗ using κ3 (x), p = 3.

14.5 Numerical implementation and results

411

Comments on the computational cost We note that solving the nonlinear eigenvalue problem (14.5) in each coarse neighborhood is one source of the computational cost. However, this is an offline step, which means when dealing with different forcing terms and boundary conditions we only need to solve this nonlinear eigenvalue problem for a single time. Thus, the computation of this eigenvalue problem will not affect the online cost of our method. We also note that in Algorithm 5, a nonlinear function of the form N (∇u) = κ(x)|∇u| p−2 requires a fine-grid update at each iterative step. In particular, at each iterative step, we must use the fine-scale solution values to construct a gradient and its norm, and we must subsequently multiply the resulting expressions with the original coefficient κ(x) at all fine-grid points in order to update the nonlinear permeability coefficient. This is a fine-grid dependent process that adds an increasing computational cost depending on the size of the fine grid. To decrease this cost, we introduce the discrete empirical interpolation method (DEIM), which allows us to approximate the nonlinear function on the fine grid while only evaluating at a few carefully selected points. In itself, DEIM is a snapshot-based preprocessing procedure in which the dominant spectral behavior of the global nonlinear function is extracted. We refer to [44, 363] for more detailed discussions on the use of DEIM. Moreover, by comparing the degrees of freedoms listed in Table 14.1 with the size of the fine-scale finite element system N f = 10201, we see that we obtain a reduced-size system by applying the generalized multiscale finite element method, which will reduce the computational cost. In this chapter, our objective is to develop a multiscale model reduction using the framework of GMsFEM. We re-cast the problem and use the degrees of freedom defined on the boundaries of coarse elements (cf. hybridization techniques [103, 130, 195]). Our motivation stems from homogenization and the analysis of multiscale methods. Homogenization and numerical homogenization methods rely on nonlinear harmonic extensions of boundary values in order to capture the effects of scales within the domain. Via these local solutions, we can capture the effects of small separable scales. In the linear case, one can use a single basis per coarse element to capture these effects; however, for nonlinear problems, this is not possible because of non-additivity. Moreover, the use of degrees of freedom on the boundaries of coarse elements is important for achieving low-dimensional approximate models. If nonlinear harmonic extensions are not used, one can not estimate the residuals (see [189]). In our framework, we propose a local nonlinear spectral decomposition, which select dominant modes in these nonlinear snapshot spaces. We present convergence and numerical results. Other variants of the GMsFEM for nonlinear problems can be found in the literature, e.g., see [426].

Chapter 15

Nonlinear non-local multicontinua upscaling

15.1 Introduction In the past, many well-known linear and nonlinear upscaling tools have been developed, as mentioned earlier. Single-phase upscaling methods include permeability upscaling [84, 154, 252, 419] and many multiscale techniques [19, 27, 101, 190, 247, 324, 344]). Nonlinear upscaling methods, e.g., known as pseudo-relative permeability approach [42, 84, 292], computes nonlinear relative permeability functions based on single cell two-phase flow computations. It is known that these nonlinear approaches lack robustness and they are processes dependent [165, 167]. To overcome these difficulties, one needs a better understanding of nonlinear upscaling methods, which is our goal. Nonlinear upscaling methods can be traced back to nonlinear homogenization [184, 350], as mentioned earlier. The main idea of nonlinear homogenization is to formulate coarse-grid equations based on nonlinear local problems formulated in each coarse block. In these approaches, the local problems are solved with periodic boundary conditions and some constraints on averages of the solutions of gradients. Our goal is to extend these approaches to problems with high contrast and non-separable scales using NLMC concepts. The main novel components of our approach are introducing multiple macroscopic variables for each coarse-grid block, formulating appropriate local constrained problems that determine the downscaling map; formulating macroscopic equations. In computational mechanics literature, there has been a great deal of research dedicated to nonlinear upscaling methods, which include generalized continuum theories (e.g., [203]), computational continua framework (e.g., [213]), and other approaches. To design the new upscaled model, we use the concept of non-local multicontinuum (NLMC). To extend the concept of NLMC to nonlinear equations, we first identify macroscopic quantities for each coarse-grid block. These variables are typically found via local spectral decomposition and represent the features that cannot be localized (similar to multicontina variables [41, 296, 348, 409]). Next, we consider local problems formulated in the oversampled regions with constraints. These © Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_15

413

414

15 Nonlinear non-local multicontinua upscaling

local problems allow identifying the downscaling map from average macroscopic quantities to the fine-grid variables. By imposing the constraints for each continuum variable via the source term, we define effective fluxes and the homogenized equation. Using the local solutions in the oversampled regions with constraints allows localizing the global downscaled map, which provides an accurate representation of the solution; however, it is expensive as it involves solving the global problem. By using the constraints in the oversampled regions, we can guarantee the proximity between the global and local downscaled maps for a given set of oversampled constraints. The resulting homogenized equation significantly differs from standard homogenization. First, there are several variables per coarse block, which represent each continuum. Secondly, the local problems are formulated in oversampled regions with constraints. Finally, the nonlinear homogenized fluxes depend on all averages in oversampled regions, which bring non-local behavior for the equation. These ingredients are needed to perform upscaling in the absence of scale separation and high contrast. In summary, our nonlinear non-local multicontinua approach has the following steps. • For each coarse-grid block, identify coarse-grid variables (continua) and associated quantities. This is typically done via some local spectral problems and describes the quantities that cannot be localized. • Define local problems in oversampled regions with constraints. The constraints are given for each continua variables over all coarse-grid blocks. Local problems use specific source terms and boundary conditions, which allow localizing the global downscaled map and identifying effective fluxes. This step gives a downscaling from average continua to the fine-grid solution. • The formulation of the coarse-grid problem. We seek the coarse-grid variables such that the downscaled fine-grid solution approximately solves the global problem in a weak sense. The weak sense is defined via specific test functions, which are piecewise constants in each continua. This provides an upscaled model. We consider two types of methods. First, we call linear interpolation and the second is nonlinear interpolation. In the first approach, we seek the solution in the form  j j wi φi , w= i, j j

where φi are multiscale basis functions for coarse cell i and for continua j defined in the oversampled region. This approach is simpler as one uses linear approximation of the solution. However, because of the nonlinearity, the nonlinear approximation is needed (see [182, 184, 185]). In [184], the authors propose such approach for pseudomonotone equations (similar to homogenization). In our problem, we present a nonlinear interpolation (we refer as nonlinear non-local multicontinua). In this

15.2 Preliminaries

415 j

approach, the solution is sought as a nonlinear map, which approximated wi from neighboring cells in a nonlinear fashion, which is defined via local problems. The local problem provides a nonlinear map from coarse-grid macroscopic variables to the fine-grid solution. This mapping is used to construct macroscopic equations. We use non-locality and multicontinua to address the cases without scale separation and high contrast. As one of the numerical examples, we consider two-phase flow and transport, though our approach can be applied to a wide range of problems. Two-phase flow and transport is one of the challenging problems, where many attempts are made to study it. Typical approaches include pseudo-relative permeability approach, also known as multi-phase upscaling, where the relative permeabilities are computed based on local two-phase flow simulations. It is known that these approaches are process dependent. Our approach provides a novel upscaling, which shows that each coarsegrid block needs to contain several average pressures and saturations and, moreover, these coarse-grid relative permeabilities are non-local and depend on saturations and pressures of neighboring cells. This model provides a new way to look at two-phase flow equations and can be further used in different modeling purposes.

15.2 Preliminaries We consider a general nonlinear system of the form ∂t U + G(x, t, U, ∇U ) = g.

(15.1)

In general, U is a vector-valued function and g(x, t) is the source term. G is assumed to be heterogeneous in space (and time, in general), which are resolved on the fine grid (see Figure 15.1 for fine and coarse-grid illustrations). Our objective is to derive coarse-grid equations. First, we discuss some examples for (15.1). Example 1. Nonlinear (pseudomonotone) parabolic equations, where G(x, t, U, ∇U ) := divκ(x, t, U, ∇U ). Example 2. Hyperbolic equations, where G(x, t, U, ∇U ) := v(x) · ∇G(x, U ) and U is a scalar function and v is a vector-valued function. For example, v = −κ(x)∇ p and div(v) = q, as in Darcy’s flow. Example 3. Hamilton-Jacobi equations, where G(x, t, U, ∇U ) := G(x, ∇U ) and U is a scalar function. Example 4. One of our objectives is to address the upscaling of multi-phase flow, which is a challenging problem. In this case, the model equations have the following form. The two-phase flow equations are derived by writing Darcy’s flow for each water (α = w) and oil (α = o) phases and the mass conservation as follows u α = −λr w (Sw )κ(x)∇ p ∇ · (u t ) = q p , ∂t Sα + ∇ · (u α ) = q.

(15.2)

416

15 Nonlinear non-local multicontinua upscaling

 Here, u α is the Darcy velocity of the phase α, u t = α u α , Sα is the saturation of the phase α, and λr α is the relative mobility of the phase α. By denoting the saturation of the water phase via S = Sw , we can write the equations in the following way −div(λ(S)κ∇ p) = q p ∂t S + ∇ · (u f (S)) = q, u = −κ∇ p.

(15.3)

The special case of the two-phase flow consists of single-phase flow −div(κ∇ p) = q p ∂t S + ∇ · (u S) = q, u = −κ∇ p.

(15.4)

15.2.1 Preliminaries. A brief overview of NLMC for linear problems In this section, we will give a brief overview of the NLMC method for linear problems [120] and Chapter 8. Our goal is to summarize the key ideas in linear NLMC (though they are described in Chapter 8) and motivate the ideas in the extension of it to the nonlinear NLMC in the next section. We consider a model parabolic equation with a heterogeneous coefficient, namely, ∂u − div(κ∇u) = g, ∂t

in Ω × [0, T ],

(15.5)

with appropriate initial and boundary condition, where κ is the heterogeneous field, g is a given source, Ω is the physical domain and T > 0 is a fixed time. The NLMC upscaled system is defined on a coarse mesh, T H , of the domain Ω.  H We write T = {K i | i = 1, · · · , N }, where K i denotes the i-th coarse element and N denotes the number of coarse elements in T H . For each coarse element K i , we will identify multiple continua corresponding to various solution features. This can be done via a local spectral problem or a suitable weight function. The upscaled parameter is defined using multiscale basis functions. For each coarse element K i and each continuum within K i , we will construct a multiscale basis function whose support is an oversampled region K i+ , which is obtained by enlarging the coarse block K i by a few coarse-grid layers. See Figure 15.1 for an illustration of coarse grid and oversample region. Now we will specify the definition of continuum. For each coarse block K i , we identify a set of continua which are represented by a set of auxiliary basis functions j φi , where j denotes the j-th continuum. There are multiple ways to construct these j j functions φi . In this framework, the auxiliary basis functions φi are obtained as the dominant eigenfunctions of a local spectral problem defined on K i . These eigenfunctions can capture the heterogeneities and the contrast of the medium. The resulting solution of the upscaled system corresponds to the moments of the true solution with

15.2 Preliminaries

417

Fig. 15.1 Illustration of coarse and fine meshes as well as oversampled regions. The region K + is an oversampled region corresponding to the coarse block K .

respect to these eigenfunctions. In alternative approach, one identifies explicit inforj mation of fracture networks. The auxiliary basis functions φi are piecewise constant functions, namely, they equal one within one fracture network and zero otherwise. The resulting solution of the upscaled system corresponds to the average of the solution on fracture networks, which can be regarded as one of the continua. Finally, one can generalize the previous approach to construct other auxiliary basis functions. In j ( j) particular, we can define φi to be characteristic functions of some regions K i ⊂ K i . ( j) These regions K i can be chosen to reflect various properties of the medium. For ( j) instances, one can choose K i to be the region in which the medium has a certain range of values. j Once the auxiliary basis functions φi are specified, we can construct the required basis functions. The idea generalizes the original energy minimization framework in CEM-GMsFEM. Consider a given coarse element K i and a given continuum j j within K i . We will use the corresponding auxiliary basis function φi to construct j our required multiscale basis function ψi by solving a problem in an oversampled j region K i+ . Specifically, we find ψi ∈ H01 (K i+ ) and μ ∈ Vaux such that 

 j



K i+

κ∇ψi · ∇v +

K i+

μv = 0, ∀v ∈ H01 (K i+ ), (15.6)

ψi φm = δ j δim , ∀K  ⊂ K i+ , j

K

where δim denotes the standard delta function and Vaux is the space spanned by auxiliary basis functions. We remark that the function μ serves as a Lagrange multiplier for the constraints in the second equation of (15.6). We also remark that the basis

418

15 Nonlinear non-local multicontinua upscaling j

function ψi has mean value one on the j-th continuum within K i and has mean value zero in all other continua in all coarse elements within K i+ . Next, we consider a forward Euler method for the time discretization of (15.5). At the n-th time step, the solution vector is denoted by U n , where each component of U n represents the average of the solution  N on a continuum within a coarse element. L i , where L i is the number of continua We note that the size of this vector is i=1 in K i . The upscaled stiffness matrix A T is defined as  (i,) i  (A T ) jm = a(ψ j , ψm ) := κ∇ψ ij · ∇ψm , (15.7) Ω

and the upscaled mass matrix MT is defined as  (MT )(i,) = ψ ij ψm . jm

(15.8)

Ω

Finally, the NLMC system is written as MT (U n+1 − U n ) + Δt A T U n = Δt G n , (15.9)  j where the components of the vector G n are defined as Ω g(·, tn )φi and tn is the time at the n-th time step. We remark that the non-local connections of the continua are coupled by the matrices A T and MT . We also remark that the local computation in (15.6) results from a spatial decay property of the multiscale basis function, see [106, 113, 115] for the theoretical foundation.

15.3 Nonlinear non-local multicontinua model (NLNLMC) 15.3.1 General concept We will first present some general concepts of our nonlinear NLMC. We consider the following nonlinear problem Ut + G(x, U, ∇U ) = g,

(15.10)

where G has a multiscale dependence with respect to space (and time, in general). To formulate the coarse-grid equations in the time interval [tn , tn+1 ], we first intron, j duce coarse-grid variables Ui , where i is the coarse-grid block, j is a continuum representing the coarse-grid variables, and n is the time step (cf. [120]). As we noted that the continuum plays a role of a macroscopic variable, which cannot be localized. For each coarse-grid block i, we need several coarse-grid variables, which will be indexed by j. We consider two types of interpolation for multiscale degrees of freedom. The first approach is called linear interpolation, which constructs approximate solution U Hn at the time tn as

15.3 Nonlinear non-local multicontinua model (NLNLMC)

U Hn =



n, j

419

j

Ui ψi ,

i, j j

where ψi are multiscale basis functions, which are defined via local constrained problems formulated in the oversampled regions. These basis functions are possibly supported in the oversampled regions and the resulting coarse-grid equations will be nonlinear and non-local when projected to the appropriate test spaces. The values n, j of Ui are computed via the variational formulation using an explicit or implicit discretization of time (U Hn+1 , VH ) − (U Hn , VH ) + Δt (G(x, U HL , ∇U HL ), VH ) = Δt (g, VH ), ∀VH , (15.11) where L = n or L = n + 1 depending whether explicit or implicit discretization is used, and VH denotes test functions. Here, (·, ·) denotes usual L 2 inner product. As j for the test space, one can use the multiscale space spanned by ψi or the auxiliary j basis functions φi . This linear approach is simpler to use; however, it ignores the nonlinear interpolation, which is important for nonlinear homogenization and numerical homogenization [184, 350]. Next, we describe the nonlinear approach, which we refer as nonlinear non-local multicontinuum approach. In this case, the solution is sought as a nonlinear intern, j polation of the degrees of freedom Ui defined in the oversampled region. More n, j precisely, we compute a downscale function using the given solution values Ui by solving a constrained problem in the oversampled regions. Mathematically, this n, j downscale function Uhn can be written as a function of all values Ui , namely, Uhn = F(U n ), n, j

where F is a nonlinear map and U n is a vector containing all values {Ui }. The nonlinear map is computed by solving the local problem (cf. (15.14)), and the function Uhn is glued together from local downscaled maps. Once this map is defined, we n, j seek all coarse-grid macroscopic variables {Ui } such that the downscaled fine-grid n solution Uh solves the global problem in a variational setting. The test functions are defined for each degrees of freedom in the form of piecewise constants or piecewise functions, in for example a finite volume or Petrov Galerkin setting. More precisely, the coarse-grid system has the form (Uhn+1 , VH ) − (Uhn , VH ) + Δt (G(x, UhL , ∇UhL ), VH ) = Δt (g, VH ),

(15.12)

for all suitable test functions VH . Here, L = n or L = n + 1 can be used for explicit or implicit discretization. The test functions are chosen to be piecewise polynomials j or auxiliary basis functions φi . We remark that the dimension of the test space is chosen to be the dimension of the coarse-grid macroscopic variables.

420

15 Nonlinear non-local multicontinua upscaling

15.3.2 Nonlinear non-local multicontinuum approach In this section, we give some details of our nonlinear non-local multicontinuum approach, in general, and then present some examples. The illustration is given in Figure 15.2. In later sections, we present a detailed application to two-phase flow equations.

Fig. 15.2 Illustration of the steps.

Step 1. Defining coarse-grid variables. The first step involves defining macroscopic variables. In our cases, we will define them by U n,i j , where i is the coarse-grid block, j is the continua, and n is the time step. These variables in our applications will be defined by prescribing averages to subregions, which can have complex shapes (for example, fracture regions). The equation for U n,i j , in general, will have a form n+1, j

Uj

n, j

j

L

− U j + G i (U ) = 0,

(15.13) j

where L = n or L = n + 1 for explicit or implicit discretizations. The operator G i is determined respect to the given continuum j and coarse element i, and contains L information from the source term and the time step size Δt. In (15.13), we use U L, j to denote the vector consisting of all macroscopic quantities Ui for all i and j. j The operator G i is obtained by solving local problems on an oversampled region corresponding to K i . To define (15.13), we next discuss local problems. j Step 2. Local solves for generic constraints. The computation of G i requires solutions of constrained local nonlinear problems. Here we present a general

15.3 Nonlinear non-local multicontinua model (NLNLMC)

421

formulation. We consider i to be the index for a coarse element K i or a coarse neighborhood ωi , which is defined for a coarse node i by ωi =



{K ∈ T

H

: xi ∈ K }.

The choice of K i or ωi depends on the global coarse-grid discretization. For example, with finite volume or Petrov Galerkin formulation, we choose K i , while for continuous Galerkin formulation, we choose ωi . In this part, we present the steps using ωi . The required local nonlinear problem will be solved on ωi+ , which is an oversampled region obtained by enlarging ωi+ a few coarse-grid layers. We let c := {cm(l) } be a set of scalar values, where m denotes the m-th coarse element in the oversampled region ωi+ and l denotes the l-th continuum within K m ⊂ ωi+ . The local problem is solved by finding a function Nωi (x; c) given by G(x, Nωi (x; c), ∇ Nωi (x; c)) =



(l) μi.m (c)I K m(l) in ωi+

(15.14)

m,l

with constraints

 ω+

Nωi (x; c)I K m(l) (x) = cm(l) .

Here, I K m(l) (x) is an indicator function for the region K m(l) defined for the continua l (l) (c) play the role of Lagrange within coarse block m. We remark that the values μi.m multipliers for the constraints. We notice that Nωi depends on all constraints in the oversampled region. In general, one can also impose constraints on the gradients of Nωi and the constraint function can have a complex form. We remark that the problem (15.14) requires a boundary condition, which is problem dependent. We will discuss it later. Our main message is that the local problems with constraints are solved in oversampled region and involves several coarse-grid variables for each coarse block. For the precise formulation, the values {cm(l) } will be chosen as the macroscopic variables. Discussion on localization. Next, we discuss the idea behind the localization principle stated above (see Figure 15.3 for illustration). For this reason, we first define the global downscaling operator, NG (x; c), which solves the global problem with constraints G(x, NG (x; c), ∇ NG (x; c)) =



(l) μi.m (c)I K m(l) in Ω

(15.15)

m,l

with constraints

 Ω

NG (x; c)I K m(l) (x) = cm(l) .

The boundary conditions are taken as the original global boundary conditions. For our localization, we desire that the local downscaled solution, Nωi (x; c) in K (target

422

15 Nonlinear non-local multicontinua upscaling

Fig. 15.3 Illustration of the localization.

coarse block) defined in (15.14), is approximately the same as NG restricted to K for the same constraint values as the global problem in ωi+ . That is, Nωi (x; c) ≈ NG (x; c) in K , where c values coincide in ωi+ . This property allows solving local nonlinear problems instead of the global problem without sacrificing the accuracy and defining correct upscaled coefficients. This desired property can be shown for monotone elliptic equations using zero Dirichlet boundary conditions on ∂ωi+ . For more complex nonlinear system, one needs to also use appropriate boundary conditions on ∂ωi+ to achieve this property. j Step 3. Defining coarse-grid model. The upscaled flux G i is defined by substituting the downscaled global solution corresponding to the constraints given by c and multiplying it by test functions. For this procedure, one first defines a global fine-grid downscaled field that is constrained. The equation (15.14) defines local finegrid fields constrained to macroscopic variables in oversampled regions. In general, one needs to “glue” together a global downscaled solution, which approximates the global problem in a weak sense. A simple approach to glue together is to use partition of unity functions, χωi for each region ωi . Then, F(U ) =



Nωi χωi .

i

In some discretization scenarios (e.g., finite volume, Discontinuous Galerkin,...), it is not necessary to glue together in order to obtain a global fine-grid field. In the

15.3 Nonlinear non-local multicontinua model (NLNLMC)

423

proposed NLMC approach, our coarse-grid formulation uses “finite volume” type approximation and we will not need to glue local approximations. In this case, F(U ) = Nωi and ∇F(U ) = ∇ Nωi on E i ,

(15.16)

where E i is a coarse edge for the coarse grid. In general, we can write it as F(U ) = Nωi in all local fine-grid blocks, and the following variational formulation (

∂ F(U ), VH ) + (G(x, F(U ), ∇F(U )), VH ) = (g, VH ), ∂t

(15.17)

where VH are test functions. In this setup, (G(x, F(U ), ∇F(U )), VH ) = (G(x, U ), VH ). The above equation can formally be thought as ∂ M(U ) + G(x, U ) = g. ∂t The time discretization of Equation (15.17) can be written as (F(U

n+1

n

L

L

), VH ) − (F(U ), VH ) + Δt (G(x, F(U ), ∇F(U )), VH ) = Δt (g, VH ), (15.18) where L = n or L = n + 1 for explicit or implicit discretizations. In general, the downscaling operator has the following property L

L

L

L

(F(U ), VH ) ≈ (U , VH ), or (F(U ), VH ) = (U , VH )

(15.19)

which simplifies the computations. In the latter situation, the mass matrix M is diagonal. In addition, L (15.20) (F(U ), Vh ) ≈ (UhL , Vh ), L

where UhL is a fine-scale field and U is the corresponding coarse-grid average. The equation (15.20) states that if we use the average of the fine-scale field to reconstruct it, the resulting approximation remains close to the original fine-scale field. j We remark that the operator G i in (15.13) is obtained using (15.18). More precisely, we have j L L G i := Δt (G(x, F(U ), ∇F(U )), VH ) − Δt (g, VH ) by using a test function VH corresponding to region i and continuum j. Next, we present some more concrete constructions for some cases, listed in Section 15.2. Example 1. In this example, we consider a class of nonlinear (pseudomono tone) parabolic equations with G = div k(x, U, ∇U ) , where k is a heterogeneous

424

15 Nonlinear non-local multicontinua upscaling

function in x, and in general, also in U and ∇U . We will consider a continuous Galerkin discretization in space on a coarse grid and explicit Euler discretization in n, j time. Let c = {Ui } be a set of macroscopic variables at the time tn . In particular, n, j Ui denotes the mean value of the solution on continuum j within coarse element i at time tn . On an oversampled region ωi+ , we find a function Nωi on the fine grid such that  (l) μm (c)I K m(l) in ωi+ G(x, Nωi , ∇ Nωi ) = K m(l) ⊂ωi+

subject to the following constraints  ωi



ωi

Nωi (x)I K m(l) (x) = Umn,l , ∀K m(l) ⊂ ωi , Nωi (x)I K m(l) (x) = 0, ∀K m(l) ⊂ ωi+ \ωi .

We remark that μ(l) m plays the role of Lagrange multiplier. The above problem is solved using the Dirichlet boundary condition on ∂ωi+ . We can set Nωi equals to zero on ∂ωi+ , and this choice of motivated by the decay property of multiscale basis functions in CEM-GMsFEM. Then we can define a global fine-scale function by Uhn =



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i

where {χωi } are suitable partition of unity functions, where we notice that the values of Nωi within ωi+ \ωi are not used. Using the global downscale function Uhn and suitable test functions, we can derive a coarse-grid scheme. We note that the upscaled model has the form (15.17). Example 2. In this example, we consider a class of hyperbolic equations, where G(x, t, U, ∇U ) := v(x) · ∇G(x, U ) and U is a scalar function and v is a vectorvalued function. For example, v = −κ(x)∇ p and div(v) = q, as in Darcy’s flow. We will consider a finite volume type discretization in space on a coarse grid and explicit n, j Euler discretization in time. Let c = {Ui } be a set of macroscopic variables at the n, j time tn . Same as above, Ui denotes the mean value of the solution on continuum j within coarse element i at time tn . On an oversampled region K i+ , we find a function N K i on the fine grid such that G(x, ∇ N K i ) =



μ(l) m (c)I K m(l)

in K i+

K m(l) ⊂K i+

subject to the following constraints  K i+

Nωi (x)I K m(l) (x) = Umn,l , ∀K m(l) ⊂ K i+ .

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We remark that μ(l) m plays the role of Lagrange multiplier. The above problem is solved using the standard inflow boundary condition on the inflow part of ∂ K i+ . We n, j can set N K i equals to zero to the value {Ui } chosen by upwinding. Then we can define a global fine-scale function by  N K i χ K i+ , Uhn = i

where {χ K i+ } are suitable partition of unity functions for the overlapping partition {K i+ }. We remark that this global downscale function Uhn preserves the mean values, namely,  1 n, j Uhn = Ui . ( j) ( j) K |K i | i Using the global downscale function Uhn and suitable test functions, we can derive a coarse-grid scheme. For the test functions, we use {I K m(l) (x)}, for all m, l. Hence, we obtain   n+1, j n, j Ui = Ui − Δt G(x, ∇Uhn ) + Δt g, ∀ i, j. (15.21) ( j)

( j)

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We note that this upscaled model has the form (15.17). We also note that the number of unknowns is the same of number of equations. Example 3. For this case, the local problem can be formulated as follows. On an oversampled region K i+ , we find a function N K i on the fine grid such that  μ(l) in K i+ G(x, ∇ N K i ) = m (c)I K m(l) K m(l) ⊂K i+

subject to the following constraints  Nωi (x)I K m(l) (x) = Umn,l , ∀K m(l) ⊂ K i+ . K i+

Similar to Example 2, the upscaled model has the form (15.17) and (15.21). In summary, the local problems involve original local problems with constraints formulated in the oversampled regions. The source term constants represent the homogenized fluxes and their dependence on averages of solutions and gradients are the functional form of the equations.

RVE-based extension of the method The proposed concept can also be used for problems with scale separation. A typical example is a fractured media, where the fracture distributions are periodic or possess some scale separation. In this case, we do not use the oversampling-based local

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15 Nonlinear non-local multicontinua upscaling

problems and simply use local problems in RVE. To be more precise, Step 1 remains the same as before, which involves identifying local multicontinua variables in each coarse-grid block. In Step 2, we solve the local problem (15.14) in RVE subject to the constraints. The main difference is that we use only the constrained in the target coarse-grid block. Thus, there are fewer constraints and the problem is localized to the target coarse block. Once the local solves are defined, via periodicity, we extend the local solution to the ωi and use this extension to compute the effective flux. This construction is similar to e.g., [184] or MsFEM using RVE [176].

15.4 Linear approach In this section, we will focus on the linear approach (c.f. Section 15.3), and present some numerical results.

15.4.1 Linear transport In this section, we will present some numerical results for the linear transport equation with a given velocity. More precisely, we consider ∂t S + ∇ · (u S) = q in Ω, where u ∈ H (div, Ω) is a given divergence-free velocity field with u · n = 0 on ∂Ω. We take Ω = [0, 1]2 , and the velocity field u is shown in Figure 15.4. The coarse grid size H = 1/20, and we use a single continuum model. In Table 15.1, we present a error comparison between our upscaling method and the standard finite volume method. The derivation of the upscaled system follows the ideas in Section 15.2.1. For each coarse element K i , we solve the following system in an oversampled region K i+ : ∇ · (uψi ) + μ = 0

(15.22)

to obtain a basis function ψi . The above system is equipped with the constraints  ψi = δi j . (15.23) Kj

We remark that μ is a piecewise constant function, and plays the role of Lagrange multiplier. We notice that the upscaled system has the form (15.9). In practice, one can add artificial diffusion in (15.22) in order to obtain a stable numerical solution, and use zero Dirichlet boundary condition. We remark that one can use other boundary conditions, see Section 15.3.2. By using the standard finite volume method, the relative L 2 error is 27.78% at the final time T = 1. From Table 15.1, we see that our upscaling method provides much

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better approximations, where “# layer” standards for the number of layers used in the oversampling domains. When # layer equals ∞, it means that the oversampling domain is the whole physical domain. In Figure 15.5, we show the snapshots of the solution at T = 1. From this figure, we observe that the solution computed with CEM provides a good approximation, particularly, in saturated regions. 0.08

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Table 15.1 L 2 relative errors for our upscaling method. The relative error for standard finite volume scheme on the same grid is 27.78%.

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15 Nonlinear non-local multicontinua upscaling

15.4.2 Single-phase flow In this section, we apply our method to a single-phase flow problem which is described as follows: ∂t Sw + ∇ · (u w Sw ) = qw , −∇ · (κ∇ Pw ) = q, u w = −κ∇ Pw ,

(15.24) (15.25) (15.26)

where we assume q is a function independent of time such that we can compute the velocity u w in the initial time step and use the velocity to construct the basis function for saturation Sw . For solving single-phase flow problem, we can separate the computation process into two parts. Since Equations (15.25) and (15.26) are independent of saturation and the given source term is independent of time, we know that the velocity is independent of time. Therefore, we can decouple system of equation into two parts which are Equations (15.25)-(15.26) and Equation (15.24). The first part of the computation is computing the velocity and pressure by solving Equations (15.25) and (15.26) by Mixed Generalized Multiscale Finite Element Methods(MGMsFEM) [108]. The second part of the computation is computing the numerical solution for the saturation by solving Equation (15.24). Given a numerical velocity u ms , Equation (15.24) is a standard linear transport equation. We can use the method discussed in Section 15.4.1 to construct the multiscale basis function and then compute the numerical solution for saturation. Next, we will discuss the computation of the saturation. We will consider a multicontinuum version of the method. Assume that an approximation of the velocity u ms has been computed by the MGMsFEM. We derive the upscaled system following the ideas in Section 15.2.1. For each coarse element K i and a continuum j within K i , we solve the following system in an oversampled region K i+ : j ∇ · (u ms ψi ) + μ = 0 (15.27) j

to obtain a basis function ψi . The above system is equipped with the constraints  j

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ψi = δm j δil .

(15.28)

We remark that μ is a piecewise constant function, and plays the role of Lagrange multiplier. We notice that the upscaled system has the form (15.9). Now, we present some numerical examples. We take Ω = [0, 1]2 , and the permeability field κ is shown in Figure 15.6. The coarse grid size H = 1/20. We will use a dual continuum model to solve this problem. For each coarse element K j , we define (2) 3 3 two continua by K (1) j = {x ∈ K j |κ(x) > 10 } and K j = {x ∈ K j |κ(x) ≤ 10 }. In Table 15.2, we present a error comparison at times T = 5 and T = 10 between

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our method and the finite volume method for the saturation equation (15.26) with piecewise constant test functions in each continuum. From the table, we observe the performance of our scheme for various choices of oversampling layers. For all these cases, we observe that our scheme performs better than the usual (dual continuum) finite volume scheme whose error is 13.10% and 14.04% at the observation times T = 5 and T = 10 respectively. We notice that the relative error in the velocity u ms is 5.51%. In Figures 15.7 and 15.8, we present the comparisons between the averaged fine-grid solution and the solution obtained in our proposed method. From these figures, we observe that our proposed method can capture the solution features accurately. 10000

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Table 15.2 L 2 relative errors for our upscaling method. The relative error for dual continuum finite volume scheme on the same grid is 13.10% and 14.04% for T = 5 and T = 10 respectively. In this case, the relative error in velocity u ms is 5.51%.

15.4.3 Two-phase flow In this section, we apply our upscaling method to two-phase flow problem which is described as follows:

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∂t Sw + ∇ · (u w ) = qw , −∇ · (κλt (Sw )∇ Pw ) = q, u w = −κλw (Sw )∇ Pw , λt = λw (Sw ) + λo (Sw ).

(15.29) (15.30) (15.31) (15.32)

In the process of basis construction, we will use a similar approach as solving singlephase flow problem. We notice that the exact velocity depends on the saturation and hence depends on time. To reduce the computational burden, we use the initial total mobility λt (Sw (0)) as the reference mobility to construct the multiscale basis functions. That is, we will construct the multiscale basis functions for approximating the exact velocity field based on the following equations, − ∇ · (u t ) = q,

(15.33)

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(15.34)

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ut Sw ) = qw . λt (Sw (0, ·))

(15.35)

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(MGMsFEM). The multiscale basis functions constructed during the processing are used to span the finite element space Vms , which is used to approximate the exact velocity. The second part is constructing the saturation basis functions by the method discussed in Section 15.3 with a fixed velocity u (0) t,ms . This part is also similar to the single-phase flow. To compute the coarse-grid solution, we will use IMplicit Pressure Explicit Sat(n) and the total velocity u (n) uration (IMPES) scheme. Given the saturation Sms t,ms at the previous time step, we will compute the total velocity at the next time step u (n+1) t,ms by solving   ∇ · u (n+1) qp, ∀ p ∈ Q H t,ms p = Ω Ω  (n+1) (n) (n+1) κ−1 λ−1 Pms , ∇ · v, ∀v ∈ Vms , t (Sw (, ·))u t,ms · v = Ω

Ω

where Q H is the space for pressure. Then, we will use the velocity u (n+1) t,ms to compute the saturation at the next time step as before (similar to (15.27) and (15.28)). We next present some numerical examples. We take Ω = [0, 1]2 . The permeability field κ is shown in Figure 15.6, and we use λw =



 1 − Sw − Sor 2 Sw − Swc 2 −1 μw , λo = μ−1 o (1 − Swc − Sor ) 1 − Swc − Sor

with μw = 1, μo = 1, Swc = 0.2 and Sor = 0.2. The coarse grid size H = 1/20. We are using the dual continuum model to solve this problem. For each coarse (2) 3 element K j , we define two continua by K (1) j = {x ∈ K j |κ(x) > 10 } and K j = {x ∈ K j |κ(x) ≤ 103 }. In Table 15.3, we present a error comparison between our method and the finite volume method for the saturation equation with piecewise constant test functions in each continuum. For this experiment, we observed that the upscaling method works in general better than the finite volume method as before. #layer \ time T = 10 T = 20 4 6 8 ∞

15.82% 7.03% 4.11% 3.52%

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Table 15.3 L 2 relative errors for our upscaling method. The relative error for dual continuum finite volume scheme on the same grid is 10.01% and 14.13% for T = 10 and T = 20 respectively.

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15 Nonlinear non-local multicontinua upscaling

Next we consider another test case with the medium κ shown in Figure 15.9, which is a SPE benchmark test case. The coarse grid size is chosen as H = 1/24. We are again using the dual continuum model to solve this problem. For each (2) coarse element K j , we define K (1) j = {x ∈ K j | log10 (κ(x)) > 0.8} and K j = {x ∈ K j | log10 (κ(x)) ≤ 0.8}. In Figure 15.10, we present the snapshots of the averaged fine solution, finite volume solution and our upscaled solution at the observation time T = 10. We observe that our method is able to capture the dynamics of the solution, while the finite volume method does not produce a good solution. In Table 15.4, we present the errors of our approximation and observe again good accuracy.

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#layer \ time T = 5 T = 10 6 7.19% 11.12% ∞ 6.70% 10.16% Table 15.4 L 2 relative errors for our upscaling method. The relative error for dual continuum finite volume scheme on the same grid is 14.75% and 19.74% for T = 5 and T = 10 respectively.

15.5 Nonlinear approach In this section, we will present a numerical test to show the performance of the nonlinear approach (c.f. Section 15.3). To do, we consider the following single-phase flow problem

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∂t Sw + ∇ · (u w λ(Sw )) = qw , −∇ · (κ∇ Pw ) = q, u w = −κ∇ Pw , where

λ(S) = S β .

We assume q is a function independent of time such that we can compute the velocity in the initial time step and use the velocity to construct the basis function for saturation. The derivation of the upscaled system follows the ideas in Section 15.3.2. Assume n, j that an approximation of the velocity u ms has been computed. Let {Si } be a set of upscaled values for the continuum j in the coarse region K i at the time tn . For each coarse element K i , we solve the following system in an oversampled region K i+ : ∇ · (u ms λ(Ni )) + μ = 0

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to obtain a local downscale function Ni . The above system is equipped with the constraints  1 n, j Ni = Si . (15.37) ( j) ( j) |K i | K i We remark that μ is a piecewise constant function, and plays the role of Lagrange multiplier. Then we define a global downscale field by Shn =



+

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i K i+

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15 Nonlinear non-local multicontinua upscaling

We next present some numerical results. We take Ω = [0, 1]2 , and the permeability field κ is shown in Figure 15.6. The coarse grid size H = 1/10. We are using a dual continuum model to solve this problem. For each K j , we define (2) 3 3 K (1) j = {x ∈ K j |κ(x) > 10 } and K j = {x ∈ K j |κ(x) ≤ 10 }. In Table 15.5, we present a error comparison between our method and the finite volume method for the saturation equation with piecewise constant test functions in each continua. We consider three choices of β, and present the relative errors at times T = 5 and T = 10. From the results, we observe that our method is able to compute more accurate numerical solutions. β \ time T = 5 T = 10 β \ time T = 5 T = 10 2 3 5

9.91% 11.85% 7.54% 12.60% 10.07% 14.89%

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Table 15.5 L 2 norm relative errors. Left: Nonlinear upscaling. Right: Finite volume method.

15.6 RVE-based non-local multicontinua approaches Previously, we discussed approaches that explore Representative Volume Element (RVE) computations and use them to compute effective properties. These ideas can be used in our nonlinear approaches. We will discuss it next. To demonstrate the main idea of these approaches in nonlinear problems, we consider −divκ(x, ∇u) = f, where κ(x, ·) has a scale separation. The computational domain is divided into coarse blocks and at each coarse block (see Figure 15.11), effective property is computed by solving local problems in each RVE. These local problems are typically formulated as −divκ(x, ∇N ) = 0 subject to N = ξ · x, for all ξ. The effective flux is computed as the average κ∗ (ξ) =

κ(x, ∇N ), where the average is taken over RVE. The use multiple macroscopic variables are critical in multiscale simulations. However, our known approaches use entire coarse blocks to compute macroscale variables, which are not feasible for many applications. For example, in shale gas applications, gas dynamics is described by molecular dynamics of multi-component gas particles. The local simulations are only possible in small RVEs, however, one needs to perform large-scale simulations for predicting flow in reservoirs. In this section, we couple RVE simulations and non-local multicontina approaches to develop efficient numerical simulations where we can explore partly the scale separation ideas. These problems can occur in many applications, where intermediate scales are used to get very large systems and then, explore RVE concepts. Next, we briefly describe these ideas.

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We assume that there are three macroscopic scales. The first scale is denoted by h and can be regarded as a scale, where we apply non-local multicontinua approach ( j) and write down intermediate macroscale equations for Ui , ( j)

G h (Ui ) = 0. These equations are non-local and very expensive to solve. In the next step, we use ( j) RVE ideas to connect these variables to macroscale variables U i defined on H -size grid, where we perform computations. We introduce an intermediate coarse-mesh scale, RVE scale, denoted by H RV E , and assume h H RV E H . In the next step, ( j) ( j) we use RVE computations to connect Ui to U i , ( j)

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This is done via solving RVE problems subject to some constraints that use U . Once we define the map, we perform quadrature of macroscale equations using RVE cells. This assumes some type of periodicity. In the proposed approaches, we make two major assumptions. Though the equations on h-scale are rigorous, their connections to H -scale equations require (1) periodicity (2) identifying macroscale variables. As for periodicity assumptions, our approaches are similar to existing methods, such as HMM, equation free, and so on. However, defining macroscale variables on h-scale and using similar variables on H -scale is one of the main advantages, which allow introducing macroscale variables in a rigorous fashion.

Fig. 15.11 Schematic description of the grids.

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15.6.1 NLNLMC on RVE-scale We will first follow [109] and present nonlinear NLMC on RVE scale using the following model nonlinear problem MUt + ∇ · G(x, t, U ) = g,

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where G is a nonlinear operator that has a multiscale dependence with respect to space (and time, in general) and M is a linear operator. In the above equation, U is the solution and g is a given source term. Next, we briefly recall the NLNLMC method. • The choice of continua The continua serve as our macroscopic variables in each coarse element. Our approach uses a set of test functions to define the continua. To be more specific, we ( j) consider a coarse element K i . We will choose a set of test functions {ψi (x, t)} to define our continua, where j denotes the j-th continuum. Using these test functions, we can define our macroscopic variables as ( j)

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The above equation (15.40) is solved subjected to constraints defined by the following functionals ( j) Iφ (ψi (x, t)). ( j)

This constraint fixes some averages of φ with respect to ψi (x, t). We remark that the function μ serves as the Lagrange multiplier for the above constraints. This local solution builds a downscaling map ( j)

Fims : Iφ (ψi (x, t)) → φ. • The construction of coarse scale model ( j) We will construct the coarse scale model using the test functions {ψi (x, t)} and ms the local downscaling map. Our upscaling solution U is defined as a combination of the local downscaling maps. To compute U ms , we use the following variational formulation ( j)

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15.6.2 RVE-based NLNLMC We denote by H the coarse-mesh size, where the final computations are performed. We denote by h a scale, where we write non-local multicontinua equations; however, they are very large to solve and will be reduced to H -scale. We denote H RV E , the scale of RVE and it is assumed H  H RV E  h. There is also very fine grid, which is subgrid of h. ( j)

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( j)

where U ms depends on Ui , which are defined on h-scale. This equation is very large and we will only use RVE-based solution. ( j) • Our second goal is to use RVE concept and reduce the dimension of Ui . We H introduce a coarse-grid homogenized solution and denote it by U (defined on H -scale) and write its finite element expansion  ( j) j H U = U i Φi , i, j j

where Φ i are standard basis functions, for example, piecewise linear on H -scale. We seek a reduced map ( j) Ui = R H (U ). This map can be local or non-local, in general. Our construction is based on homogenization ideas and uses local maps, which we introduce next. Using these local maps, the quadrature can be approximated on RVEs. In a linear case, R is a matrix of the sizes corresponding to H −d and h −d . H In the simplest approach, we will use R H to be L 2 projection of U (i.e., the H averages of U on h-scale mesh). • There are various ways to use homogenization ideas to construct reduced map. Here, we consider some of them, which differ in a way we impose constraints. Approach 1. In this approach, we solve RVE-based local problem with constraints H given by U on each RVE cell to define a map RhK , which is local for each coarse block. Approach 2. In this approach, we solve local problems on RVE (H RV E -scale) subject to boundary conditions U H (to be determined) Mφt + ∇ · G(x, t, φ) = μ, in K iRV E , subject to boundary condition and initial condition at t = tn H

φ = R H (U ),

(15.43)

438

15 Nonlinear non-local multicontinua upscaling H

where φ = Rh (R H (U )). This equation gives a map between the local solution H and R H (U . Next, we define ( j)

φ, ψi , and the map

( j)

( j)

φ, ψi  = R H (U i ).

The map Rh can be regarded as a reduced dimensional map mentioned in the previous step. • In the last step, we discuss the approximation of the integrals defined in coarse-grid system and the coarse-grid system. We seek U such that ( j)

( j)

MRh (R H (U ))t + ∇ · G(x, t, Rh (R H (U )), ψi ) RV E =

g, ψi  RV E , ( j)

where ψi

are standard (e.g., linear test functions).

15.6.3 Examples Example 1. We consider

−divκ(x, ∇u) = f,

where κ(x, ξ) is monotone with respect to ξ (see e.g., [107]). The algorithm is the  H ( j) j following. We seek U = i, j U i Φ i , such that   j j “ κ(x, ∇RhK (R H (U ))) · Φ i d x  = “ f Φ i d x  , Ω

Ω

where R H (U ) is L 2 projection of U onto h-size mesh in RVE, and Rh (R H (U )) is the local RVE solution defined on the fine grid with constraints given by R H (U ) (which is defined on h-size mesh),    j j κ(x, ∇Rh (R H (U ))) · Φ i d x ≈ ωK κ(x, ∇RhK (R H (U ))) · Φ i d x Ω





K

Ω

j f Φi d x

≈

 K

K RV E

j

ωK K RV E

f Φ i d x.

We again note that RhK is local map, RhK : Vh (K RV E ) → V (K RV E ), and R H : VH → ⊕ K Vh (K RV E ) couples different coarse regions. Example 2. We consider a simpler example −div(κ(x, u)∇u) = f. In this case, we can consider a linearization (Picard) as −div(κ(x, u n )∇u n+1 ) = f. n+1

Then, the algorithm is a special case of Example 1. We seek U , which solves the linearized equations. In this case, we can also define effective permeabilities as we will do in our numerical examples.

Chapter 16

Global-local multiscale model reduction using GMsFEM

16.1 Introduction Our previous studies focused on local model reduction techniques. In these approaches, we develop local (space and time) reduced-order models to approximate the solution of PDEs. The global model reduction techniques have been commonly used to approximate input-output maps. Proper Orthogonal Decomposition (POD) is one of the best-known global model reduction methods. The main purpose of this technique is to reduce the dimension of the dynamical system by projecting the highdimensional system into a lower-dimensional manifold using a set of orthonormal basis functions (POD modes) constructed from a sequence of snapshots [11, 12, 407, 408]. In addition to order reduction, this technique constitutes a powerful mode decomposition technique for extracting the most energetic structures from a linear or nonlinear dynamical process [11, 12, 38, 51, 142, 242, 243, 254, 379, 407, 408]. There are many important differences between global and local model reduction techniques. Global model reduction approaches, though are powerful in reducing the degrees of freedom, they lack local adaptivity and numerical discretization properties (e.g., conservations of local mass and energy,...) that local approaches enjoy. Many successful macroscopic laws (e.g., Darcy’s law, and so on) are possible because the solution space admits a large compression locally. For this reason, it is important to construct local multiscale model reduction techniques that can identify local degrees of freedom and be consistent with homogenization when there is scale separation. In this chapter, we discuss the combined global and local model reduction techniques. In particular, we use local model reduction techniques in approximating the offline computations of the global model reduction techniques. This can be particularly useful if one deals with multiscale problems, where the offline computations can be very expensive. Coarse-grid computational models are often preferred because of the computational cost of solving the systems arising in the approximation of the nonlinear flow equation on the fine grid. In this chapter, we apply the enriched

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_16

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16 Global-local multiscale model reduction using GMsFEM

coarse space construction from the Generalized Multiscale Finite Element Method (GMsFEM). The combining of the aforementioned local and global model reduction schemes has been used for linear problem [170, 225]. A significant reduction in the computational complexity when solving linear parabolic PDE in [225] has been achieved by combining GMsFEM and POD and/or Dynamic Mode Decomposition (DMD). In [170], balanced truncation is used to perform global model reduction and is efficiently combined with the local model reduction tools introduced in [192]. More recently, local and global multiscale methods are combined to derive reduced-order models for nonlinear flows in high-contrast porous media. In [70], the proposed multiscale empirical interpolation method for solving nonlinear multiscale PDEs uses GMsFEM to represent the coarse-scale solution. To avoid performing fine-grid computations, the discrete empirical interpolation method introduced in [83] was used to approximate the nonlinear functions at selected points in each coarse region and then a multiscale proper orthogonal decomposition technique is used to find an appropriate interpolation vector. Although the numerical results presented in [70] proved the applicability of the presented method, the reduction is limited by the full cost of the evaluation of the projected nonlinear function. When dealing with reduced-order models of nonlinear systems obtained by projecting the governing equations onto a subspace spanned by the POD modes, the evaluation of the projected nonlinear term is costly since it depends on the full dimension of the original system. In this chapter, our main contribution is to circumvent this issue by employing DEIM to approximate the nonlinear functions locally (at selected points in each coarse region) at the offline stage and globally (at selected points in the domain) at the online stage. For this reason, we refer to our method as global-local nonlinear approach. The numerical results presented here show that the proposed method enables significant reduction in the computational cost associated with constructing projection-based reduced-order models. In addition to the model reduction, the proposed approach allows us to improve the reduced-order solutions in different ways. For instance, increasing the number of local and global points used at the offline and online stages, respectively, leads to better approximation (see Example 16.4.2). Also, using several offline parameter inputs (Example 16.4.3) improves the reduced-order solutions.

16.2 Preliminaries 16.2.1 Model problem We will demonstrate the main concept on a model problem, which consists of timedependent nonlinear flow

16.2 Preliminaries

441

∂u − ∇ · (κ(x; u, μ)∇u) = h(x) in Ω, ∂t

(16.1)

with some boundary conditions. The variable u = u(t, x; μ) denotes the pressure, Ω is a bounded domain, h is a forcing term, and in our case the permeability field represented by κ(x; u, μ) is a nonlinear function. Here, μ represents a parameter.

16.2.2 Discrete empirical interpolation method (DEIM) We approximate with the Discrete Empirical Interpolation Method (DEIM) [83] local and global nonlinear functions. DEIM is based on approximating a nonlinear function by means of an interpolatory projection of a few selected snapshots of the function. The idea is to represent a function over the domain while using empirical snapshots and information in some locations (or components). We briefly review DEIM as presented in [83]. Let f (τ ) ∈ Rn denotes a nonlinear function, where τ ∈ Rn s . Here, in general, n s can be different from n. In a reducedorder modeling, τ has a reduced representation τ=

l 

αi ζi ,

i=1

where l  n s . This leads us to look for an approximation of f (τ ) at a reduced cost. To perform a reduced-order approximation of f (τ ), we first define a reduced dimensional space for f (τ ). That is, we would like to find m basis vectors (where m is much smaller than n), ψ1 ,..., ψm , such that we can write f (τ ) ≈ Ψ d(τ ),

(16.2)

where Ψ = (ψ1 , · · · , ψm ) ∈ Rn×m . The goal of DEIM is to find d(τ ) using only a few rows of (16.2). In general, one can define d(τ )’s using m rows of (16.2) and invert a reduced system to compute d(τ ). This can be formalized using the matrix P P = [e℘1 , · · · , e℘m ] ∈ Rn×m , where e℘i = [0, · · · , 0, 1, 0, · · · , 0]T ∈ Rn is the ℘ith column of the identity matrix In ∈ Rn×n for i = 1, · · · , m. Multiplying Equation (16.2) by PT and assuming that the matrix PT Ψ is nonsingular, we obtain f (τ ) ≈ f˜(τ ) = Ψ d(τ ) = Ψ (PT Ψ )−1 PT f (τ ).

(16.3)

To summarize, approximating the nonlinear function f (τ ), as given by Equation (16.3), requires the following: • Computing the projection basis Ψ = (ψ1 , · · · , ψm );

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16 Global-local multiscale model reduction using GMsFEM

• Identifying the indices {℘1 , · · · , ℘m }. To determine the projection basis Ψ = (ψ1 , · · · , ψm ), we collect function evaluations in an n × n s matrix F = [ f (τ1 ), · · · , f (τn s )] and employ POD to select the most energetic modes. This selection uses the eigenvalue decomposition of the square matrix FT F (left singular values) and form the important modes using the dominant eigenvalues. These modes are used as the projection basis in the approximation given by Equation (16.2). In Equation (16.3), the term Ψ (PT Ψ )−1 ∈ Rn×m is computed once and stored. The d(τ ) is computed using the values of the function f (τ ) at m points with the indices ℘1 , · · · , ℘m identified using the following DEIM algorithm. DEIM Input:

Output:

Algorithm [83]: The projection basis matrix Ψ = (ψ1 , · · · , ψm ) obtained by applying POD on a sequence of n s function evaluations. → The interpolation indices − ℘ = (℘1 , · · · , ℘m )T 1: Set [|ρ|, ℘1 ] = max{|ψ1 |} → 2: Set Ψ = [ψ1 ], P = [e℘1 ], and − ℘ = (℘1 ) 3: for k = 2, ..., m do - Solve (P T Ψ )w = P T ψk for some w. - Compute r = ψk − Ψ w - Compute [|ρ|, ℘k ] = max{|r |}  − → ℘ → ℘ = - Set Ψ = [Ψ ψk ], P = [P e℘k ], and − ℘k end for

The computational saving is due to the resulting fewer evaluations of f (τ ). This shows the advantage of using DEIM algorithm in our proposed reduction method. However, applying the DEIM algorithm to reduce the computational cost of the nonlinear function requires additional computations in the offline stage, which will be discussed in Section 16.3.2. Note that these algorithms are successful if the nonlinear functions admit low dimensional approximations.

16.2.3 Generalized multiscale finite element method (GMsFEM) In this section, we very briefly review the GMsFEM (see Chapter 4). As before, in ωi . We denote each the offline computation, we first construct a snapshot space Vsnap snapshot vector (listing the solution at each node in the domain) using a single index and create the following matrix   snap snap Φsnap = ψ1 , . . . , ψ Msnap , snap

where ψ j denotes the snapshots and Msnap denotes the total number of functions to keep in the local snapshot matrix construction.

16.2 Preliminaries

443

In order to construct an offline space VH,off , we reduce the dimension of the snapshot space using a spectral decomposition. At the offline stage, the bilinear forms are chosen to be parameter-independent (through nonlinearity), such that there is no need to reconstruct the offline space for each ν value, where ν is assumed to be a parameter that represents u and μκ in κ(x, u, μκ ). To construct the offline space, we use the average of the parameters over the coarse region ωi in κ(x, ν) while keeping the spatial variations. That is, ν represents both the average of u and μ. We consider the following eigenvalue problem in the space of snapshots (see Chapter 4), off off Aoff Φkoff = λoff k S Φk ,

where

(16.4)

 off Aoff = [amn ]=

 S

off

=

off [smn ]

=

ωi

ωi

T κ(x, ν)∇ψmsnap · ∇ψnsnap = Φsnap AΦsnap , T  κ(x, ν)ψmsnap ψnsnap = Φsnap SΦsnap .

κ(x, ν)areparameter-averagedcoefficients(see[190]). Thecoefficientsκ(x, ν)and The A denotes a fine-scale matrix. To generate the offline space, we then choose the corresponding eigensmallest Moff eigenvalues from Equation (16.4) and form the off snap (for k = vectors in the respective space of snapshots by setting φoff j Φk j φ j k = 1, . . . , Moff ),whereΦkoffj arethecoordinatesofthevectorΦkoff .Wethencreatetheoffline matrices 

off Φoff = φoff 1 , . . . , φ Moff to be used in the online space construction. The online coarse space is used within the finite element framework to solve the original global problem, where continuous Galerkin multiscale basis functions are used to compute the global solution. At the online stage, the bilinear forms are chosen to be parameter dependent. The following eigenvalue problems are posed in the reduced offline space: on on Aon (ν)Φkon = λon k S (ν)Φk ,

where

(16.5)

 Aon (ν) = [a on (ν)mn ] =

ωi

off T κ(x, ν)∇φoff m · ∇φn = Φoff A(ν)Φoff ,



S (ν) = [s (ν)mn ] = on

on

ωi

off T  κ(x, ν)φoff m φn = Φoff S(ν)Φoff ,

and κ(x, ν) and  κ(x, ν) are now parameter dependent. To generate the online space, (16.5) and form the corresponding we then choose the smallest Mon eigenvalues from on off = eigenvectors in the offline space by setting φon j Φk j φ j (for k = 1, . . . , Mon ), k on on where Φk j are the coordinates of the vector Φk . If κ(x, u) = k0 (x)b(u), then one can use the parameter-independent case of GMsFEM. From now on, we denote the online space basis functions by φi .

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16 Global-local multiscale model reduction using GMsFEM

16.3 Global-local nonlinear model reduction 16.3.1 Local multiscale model reduction The finite element discretization of (16.1) yields a system of ordinary differential equations given by ˙ + F(U) = H, MU (16.6) where

U = u1 u2 · · · u N f

is the vector collecting the pressure values at all nodes in the local domain and H is the right-hand-side vector obtained by discretization. Using the offline basis functions, we can write (in discrete form) κ(x, u, μ) =

Q 

κq (x)bq (u, μ).

(16.7)

q=1

This results in F(U, μ) =

Q 

q

Aq Λ1 (U, μ)U,

q=1

where we have    q 0 0 0 0 Aq := [ai j ] = κq ∇φi · ∇φ j , M := [mi j ] = φi φ j , H := [hi ] = φi0 h , Ω Ω Ω

q Λ1 (U, μ) = diag bq (u 1 , μ) bq (u 2 , μ) · · · bq (u N f , μ) , and φi0 are piecewise linear basis functions defined on a fine triangulation of Ω. Employing the backward Euler scheme for the time marching process, we obtain Un+1 + Δt M−1 F(Un+1 ) = Un + Δt M−1 H,

(16.8)

where Δt is the time-step size and the superscript n refers to the temporal level of the solution. The residue is defined as R(Un+1 ) = Un+1 − Un + Δt M−1 F(Un+1 ) − Δt M−1 H

(16.9)

with derivative (Jacobian) J(Un+1 ) = DR(Un+1 ) = I + Δt M−1 DF(Un+1 ) =I+

Q  q=1

Δt M−1 Aq Λ1 (Un+1 ) + q

Q 

Δt M−1 Aq Λ2 (Un+1 ), q

q=1

(16.10)

16.3 Global-local nonlinear model reduction

where

445

 q Λ2 (U, μ) = diag u 1 ∂bq (u 1 ,μ) u 2 ∂bq (u 2 ,μ) · · · u N f ∂u ∂u

∂bq (u N f ,μ) ∂u



,

and D is the multi-variate gradient operator defined as [DR(U)]i j = ∂ Ri /∂U j . The scheme involves, at each time step, the following iterations   n+1 n+1 n+1 n −1 −1 J(Un+1 (k) )ΔU(k) = − U(k) − U + Δt M F(U(k) ) − Δt M H n+1 n+1 Un+1 (k+1) = U(k) + ΔU(k) , n where the initial guess is Un+1 (0) = U and k is the iteration counter. The above iterations are repeatedly applied until  ΔUn+1 (k)  is less than a specific tolerance. In our simulations, we use Q = 1 in (16.7) as our focus is on localized multiscale interpolation of nonlinear functionals that arise in discretization of multiscale PDEs. With this choice, we do not need to compute the online multiscale space (i.e., the online space is the same as the offline space). We use the solution expansion (i.e., u = Φz) and employ the multiscale framework to obtain a set of Nc ordinary differential equations that constitute a reducedorder model; that is,

z˙ = −(Φ T MΦ)−1 Φ T F(Φz) + (Φ T MΦ)−1 Φ T H.

(16.11)

Thus, the original problem with N f degrees of freedom is reduced to a dynamical system with Nc dimensions where Nc  N f . The nonlinear term (Φ T MΦ)−1 Φ T F(Φz) in the reduced-order model, given by Equation (16.11), has a computational complexity that depends on the dimension of the full system N f . As such, solving the reduced system still requires extensive computational resources and time. To reduce this computational requirement, we use multiscale DEIM as described in the previous section. To solve the reduced system, we employ the backward Euler scheme; that is,  −1  −1  F(z n+1 ) = z n + Δt M H, z n+1 + Δt M

(16.12)

 = Φ T MΦ,  where M F(z) = Φ T F(Φz), and  H = Φ T H. We let   −1  −1  R(z n+1 ) = z n+1 − z n + Δt M F(z n+1 ) − Δt M H

(16.13)

with derivative  −1 D  R(z n+1 ) = I + Δt M F(z n+1 ) J(z n+1 ) = D =I+

Q  q=1

 −1 Φ T Aq Λq (Φz n+1 )Φ + Δt M 1

Q  q=1

 −1 Φ T Aq Λq (Φz n+1 )Φ. Δt M 2

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16 Global-local multiscale model reduction using GMsFEM

The scheme involves, at each time step, the following iterations   n+1 n+1 n+1 n+1   −1  −1  J(z (k) F(z (k) H )Δz (k) = − z (k) − z n + Δt M ) − Δt M n+1 n+1 n+1 z (k+1) = z (k) + Δz (k) ,

(16.14) (16.15)

n+1 n+1 where the initial guess is z (0) = z n . The above iterations are repeated until  Δz (k)  is less than a specific tolerance. Furthermore, we use multiscale DEIM to approximate the nonlinear functions that appear in the residual  R and the Jacobian  J to reduce the number of function evaluations.

16.3.2 Global-local nonlinear model reduction approach We denote the offline parameters by θoff which include samples of the right-hand side h(x) denoted by h ioff , samples of μ denoted by μioff , and samples of initial conditions off . Similarly, the online parameter set is denoted by θon and includes the denoted by U0,i online source term h on , the online μ (μon ), and the online initial conditions U0on . We follow a global-local nonlinear model reduction approach that includes the following steps: • Offline Stage The offline stage includes the following steps: off – Consider the offline parameters set θoff = {θioff } = {h ioff , μioff , U0,i }. off – Use θi to define the fine-scale stiffness and mass matrices, source terms and multiscale basis functions. – Compute the local snapshots of the nonlinear functions and use DEIM algorithm, as described in the previous section, to set the local DEIM basis functions and ). local DEIM points (L local 0 – Generate snapshots of the coarse-grid solutions using local DEIM. – Record Nt instantaneous solutions (usually referred as snapshots) using coarsegrid approximations from the above step and collect them in a snapshot matrix as

Z Nt = {Z1 , Z2 , Z3 , · · · , Z Nt },

(16.16)

where Nt is the number of snapshots and Nc is the size of the column vectors Zi . – Compute the POD modes and use these modes to approximate the solution field on the coarse grid. As such, we assume an expansion in terms of the modes ψi ; that is, we let Nr  αi (t)ψi (x) (16.17) z(x, t) ≈ z˜ (x, t) = i=1

or in a matrix form

where Ψ = ψ1 · · · ψ Nr .

Zn ≈ Z˜ n = Ψ αn ,

(16.18)

16.3 Global-local nonlinear model reduction

447

• Online Stage The online stage includes the following steps: – Given online θon = {h on , μon , U0on } – Use the solution expansion given by (16.17) and project the governing equation of the coarse-scale problem onto the space formed by the modes to obtain a set of Nr ordinary differential equations that constitute a reduced-order model; that is, T T −1 Ψ T  Φ T F(ΦΨ α) +(Ψ T Φ T MΦΨ )−1 Ψ T Φ T H. α˙ = −(Ψ  Φ MΦΨ)     Nr ×Nr

Nr ×Nc Nc ×N f

N f ×1

(16.19) – Employ Newton’s method to solve the above reduced system. The Newton scheme involves at each time step the following iteration. We need to solve the linear system   n+1 n+1 n+1 n+1   −1  −1  J(α(k) F(α(k) H , (16.20) )Δα(k) = − α(k) − αn + Δt M ) − Δt M where ˜  = Ψ T Φ T MΦΨ = Ψ T MΨ, M

˜  H = Ψ T Φ T H = Ψ T H,

˜  F = Ψ T Φ T F = Ψ T F.

Then   n+1 n+1 n+1 −1 n+1 n+1  −1  −1  F(α(k) H . = α(k) − ( J(α(k) )) α(k) − αn + Δt M ) − Δt M α(k+1) Thus, the original problem with N f degrees of freedom is reduced to a dynamical system with Nr dimensions where Nr  Nc  N f . – Use global DEIM to approximate the nonlinear functions that appear in the residual and Jacobian. To do so, we write the nonlinear function F(ΦΨ α) in Equation (16.19) as (16.21) F(ΦΨ α) ≈ Ψ ∗ d, where Ψ ∗ = [ψ1∗ , ..., ψ ∗ global ] is the matrix of the global DEIM basis functions global

L0 {ψi∗ }i=1

L0

. These functions are constructed using the snapshots of the nonlinear function F(Φz) computed offline and employ the POD technique to select the most energetic modes (see Section 16.2.2). The coefficient vector d is computed global global points. using the values of the function F at L 0 – Use the solution expansion given by (16.17) in terms of POD modes to approximate the coarse-scale solution and then use the operator matrix Φ to downscale the approximate solution and evaluate the flow field on the fine grid.

448

16 Global-local multiscale model reduction using GMsFEM

16.4 Numerical results In this section, we use representative numerical examples to illustrate the applicability of the proposed global-local nonlinear model reduction approach for solving nonlinear multiscale partial differential equations. Before presenting the individual examples, we describe the computational domain used in constructing the GMsFEM basis functions. This computation is performed during the offline stage. We discretize with linear finite elements a nonlinear PDE posed on the computational domain Ω = [0, 1] × [0, 1]. For constructing the coarse grid, we divide [0, 1] × [0, 1] into 10 × 10 squares. Each square is divided further into 10 × 10 squares each of which is divided into two triangles. Thus, the mesh size is 1/100 for the fine mesh and 1/10 for the coarse one. The fine-scale finite element vectors introduced in this section are defined on this fine grid. The fine-grid representation of a coarse-scale vector z is given by Φz, which is a fine-grid vector. In the following numerical examples, we consider (16.1) with specified boundary and initial conditions, where the permeability coefficient and the forcing term are given by κ(x; u, μ) = κq (x)bq (u, μ) and h(x) = 1 + sin(2πx1 ) sin(2πx2 ). Here, κq represents the permeability field with high-conductivity channels as shown in Figure 16.1 and bq (u, μ) is defined later for each example. We use the GMsFEM along with the Newton method to discretize (16.1). Furthermore, we employ the local multiscale DEIM in the offline stage and the global multiscale DEIM in the online stage to approximate the nonlinear functions that arise in the residual and the Jacobian.

Fig. 16.1 Permeability field that model high-conductivity channels within a homogeneous domain. The minimum (background) conductivity is taken to be κmin = 1, and the high conductivity (gray regions) with value of κmax = η (η = 106 ).

Using the fine-scale stiffness matrix A that corresponds to (16.1), we introduce the relative energy error as

16.4 Numerical results

449

 EA =

(U −  U)T A(U −  U) . T U AU

(16.22)

Moreover, we define w0 to be the solution of the problem

− ∇ · κq (x)∇w0 = h(x) in Ω,

(16.23)

to use it in the following examples as our initial guess. In the following, we show • In the first example, we compare the approximate solution of the reduced system obtained by applying the global-local approach against the solution of the original system with full dimension (N f ) and show the reduction we achieve in terms of the computational cost. • In the second example, we show the variations of the error as we increase the global , and global DEIM points, L 0 for one number of local DEIM points, L local 0 selection of the parameter μ. • In the third example, we show the effect of using several offline parameters to improve the reduced-order solutions. As such, we use two offline values of the parameter μ and solve an online problem for a different value of μ. • In the fourth example, we use two offline values of μ and show the variations of the errors as we increase the number of local and global points. • Random values of the parameter μ with a probability distribution are used in the fifth example. We demonstrate the applicability of our approach in this setup.

16.4.1 Single offline parameter Example 16.4.1. We consider (16.1) along with the following offline and online parameters ⎧ off ⎪ ⎨h = 1 + sin(2πx1 ) sin(2πx2 ), θoff = μoff = 10, ⎪ ⎩ off U 0 = w0 ,

θon

⎧ on ⎪ ⎨h = 1 + sin(2πx1 ) sin(2πx2 ), = μon = 40, ⎪ ⎩ on U0 = w0 ∗ 0.5.

where the nonlinear function bq is defined as bq (u, μ) = eμu . Here, the source term does not need to be fixed for the method to work as we see below. We employ GMsFEM for the spatial discretization and the backward Euler method for time advancing as described in Section 16.3.1. Furthermore, we follow = 3) per coarse the steps given in Section 16.3.2 using three DEIM points (L local 0

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16 Global-local multiscale model reduction using GMsFEM

region to approximate bq in the offline stage. After generating the snapshots of the coarse-grid solutions using local DEIM, we compute the multiscale POD modes that global = 5 in the online stage to approximate are used in the online problem. We use L 0 bq globally and then use the generated POD modes to approximate the coarse-scale solution. In Figure 16.2, we compare the approximate solution obtained from the global-local nonlinear model reduction approach with the solution of the original system without using the DEIM technique to approximate the nonlinear function. A good approximation is observed in this figure, which demonstrates the capability of global-local nonlinear model reduction to reproduce accurately the fully resolved solution of a nonlinear PDE. We have also considered a permeability field that is obtained by rotating the permeability field κq in Figure 16.1 such that the three long channels are in the vertical direction. Our numerical results show similar accuracy and computational cost compared to the previous case (see Figure 16.1). In general, we expect nonhomogeneous boundary conditions to affect the numerical results. The approximate solution shown in Figure 16.2a is obtained using only two POD modes. As expected, increasing the number of POD modes used in the online stage yields a better approximation. That is, the error decreases as we increase the number of POD modes used as shown in Figure 16.3. The error using two POD modes decreases slightly from 12% (at steady state) to 11.5% when using three POD modes. The decreasing trend is steeper when considering more POD modes. For instance, the use of 5 modes yields an error of 4.5%.

(a) Reference Solution

(b) Approximate Solution

Fig. 16.2 Comparison between reference solution of the fine-scale problem with that obtained from the global-local multiscale approach.

In order to illustrate the computational savings, we compute the time for solving the system of ordinary differential equations given in (16.6) with and without using the proposed method. We denote the time for solving the full system by T f ine and the time for solving the reduced system using global-local nonlinear model reduction by TG L . Then, the percentage of the simulation time is given by

16.4 Numerical results

451

Fig. 16.3 Variations of the solution error with the number of POD modes.

R=

TG L ∗ 100. T f ine

(16.24)

We compute R with respect to different number of DEIM points and POD modes and present the results in Tables 16.1 and 16.2, respectively. In Table 16.1, the first ), the second column represents column shows the number of local DEIM points (L local 0 global ), and the third column illustrates the the number of global DEIM points (L 0 and/or percentage of the simulation time. Here two POD modes are used. As L local 0 global increase, the percentage decreases accordingly. For example, R decreases L0 global from two to three, and to 3.2093% from 3.7832% to 3.3741% by increasing L 0 global local and L 0 from two to three. Decreasing R means that by increasing both L 0 TG L , time for solving the reduced system, decreases as we increase the number of DEIM points. Therefore, increasing the number of local and global DEIM points may speed up the simulation in addition to improving the accuracy as we see in the next example. In Table 16.2, the numbers of POD modes used for the global reduction are listed in the first column and the corresponding values of R are shown in the second column. In this case, we keep the number of local and global DEIM points constant and equal to two and three, respectively. Now, increasing the number of POD modes inversely affects the simulation speed-up. That is, increasing the number of POD modes increases the value of R which means TG L is increasing and hence the speed-up of our simulation is decreasing. For example, R increases from 3.3741% when we use two POD modes to 4.0387% with three POD modes and keeps increasing as we increase the number of POD modes to be 6.1414% with five POD modes. Although, increasing the number of POD modes slows down the simulation, it improves the accuracy of the approximate solution (see Figure 16.3). However, the following examples show the capability in terms of the accuracy of this method when using two POD modes for the global reduction.

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16 Global-local multiscale model reduction using GMsFEM global

L local L0 0 2 2 2 3 3 3

R(%) 3.7832 3.3741 3.2093

Table 16.1 Variation of the percentage of the simulation time corresponding to different number of local and global DEIM points. Here we use two POD modes. POD modes 2 3 4 5

R(%) 3.3741 4.0387 4.9158 6.1414

Table 16.2 Variation of the percentage of the simulation time corresponding to different number global = 2 and L 0 =3 of POD modes. Here we use L local 0

Example 16.4.2. In this example, we use different numbers of local and global global = {1, 2, 3} and L 0 = {1, 2, 3}, to investigate how these DEIM points, L local 0 numbers affect the error. As in Example 16.4.1, we consider bq (u, μ) = eμu and the following offline and online parameters:

θoff

⎧ off ⎪ ⎨h = 1 + sin(2πx1 ) sin(2πx2 ), = μoff = 10, ⎪ ⎩ off U 0 = w0 ,

θon

⎧ on ⎪ ⎨h = 1 + sin(2πx1 ) sin(2πx2 ), = μon = 40, ⎪ ⎩ on U0 = 0.5w0 .

In Figure 16.4a, we plot the transient variations of the error while using different numbers of global DEIM points for a fixed number of local DEIM points equal to one. Increasing the number of global DEIM points from one to three results in a decrease in the error from 13% to 11% (at steady state). Further increases in the number of global DEIM points does not yield any improvement in the total error. This is due to the dominance of the local error. Figure 16.4b shows the decreasing trend of the error as we increase the number of local DEIM points. In Figure 16.4c, we show the variations of the error with increasing the number of both local and global DEIM points. Increasing the number of DEIM points enables smaller error and then improves the solution accuracy. These examples show that the number of local and global DEIM points need to be chosen carefully to balance the local and global errors.

16.4 Numerical results

453

(a) Variations of global DEIM points

(b) Variations of local DEIM points

(c) Variations of both global and local DEIM points Fig. 16.4 Effect of the number of local and global DEIM points on the approximate solution accuracy.

16.4.2 Multiple offline parameters Example 16.4.3. In this example, we define the nonlinear function as bq (u, μ) = off off eμ(0.9+u) and use μ1 = 2 and μ2 = 5, separately, in the offline problem to compute POD modes and DEIM points. We then combine these to use the total number of POD modes in the online problem with a different online value of μ (μon = 3). In this example, we keep the number of local and global DEIM points constant and global = L0 = 3). Furthermore, we use different online initial equal to three (i.e., L local 0 conditions and source term. The following system parameters are considered:

θoff

⎧ off h = ⎪ ⎪ ⎪ ⎨μoff = 1 = ⎪μoff ⎪ 2 = ⎪ ⎩ off U0 =

1 + sin(2πx1 ) sin(2πx2 ), 2, 5, w0 .

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16 Global-local multiscale model reduction using GMsFEM

Fig. 16.5 Transient variations of the error (using different offline values of the parameter μ).

θon

⎧ on ⎪ ⎨h = 1 + sin(4πx1 ) sin(4πx2 ), = μon = 3, ⎪ ⎩ on U0 = 0.

We show in Figure 16.5 that the error decreases when combining two cases that correspond to different values of offline μ. For instance, the error when considering only one offline case is about 16% and it goes down to 13% when combining two cases with two different values of offline μ. Hence, using multiple parameter values in the offline stage improves the method’s accuracy independently of the online parameters. Example 16.4.4. Next, we consider the following parameters ⎧ off h = 1 + sin(2πx1 ) sin(2πx2 ), ⎪ ⎪ ⎪ ⎨μoff = 10, 1 θoff = off ⎪ μ ⎪ 2 = 40, ⎪ ⎩ off U 0 = w0 ,

θon

⎧ on ⎪ ⎨h = 1 + sin(2πx1 ) sin(2πx2 ), = μon = 24, ⎪ ⎩ on U0 = 0,

and the nonlinear function bq (u, μ) = eμu . In this case, we use two offline values of μ while considering different numbers of local and global DEIM points. The effect of the number of local and global DEIM points on the error between the reference and approximate solutions when combining two cases that correspond to two different values of μ is shown in Figure 16.6. Similar trends to those of Example 16.4.2 are

16.4 Numerical results

455

(a) Variations of global DEIM points

(b) Variations of local DEIM points

(c) Variations of both global and local DEIM points Fig. 16.6 Effect of the number of local and global DEIM points on the approximate solution accuracy (using two offline μ).

observed. Increasing both local and global DEIM points improves the approximation to the solution. For instance, the error reduces from about 13% when using a local and a global DEIM point to 2% when using three local and global DEIM points. The error reduction in this case (when we use two offline μ) is bigger than the one we obtained when only using one offline μ value where the error decreased from 13% to 7% (see Figure 16.4c). We conclude that using two offline μ values and increasing number of local and global DEIM points yields a better approximation. Therefore, choosing the number of local and global DEIM points and the offline parameter values are the main factors to achieve high accuracy in the proposed method. Example 16.4.5. In this example, we consider the case with random values of the parameter μ that has a normal distribution with the mean 25 and variance 4. As in Example 16.4.3, we use different values of the offline parameter μoff = {10, 25, 39}, and compute the POD and DEIM modes. Further, we combine these modes to get the global POD and DEIM modes that we use in the online problem. In the online

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16 Global-local multiscale model reduction using GMsFEM

problem, we take uncorrelated random values of μon drawn from the above probability distribution. We rapidly compute the approximate solution and evaluate the relative error corresponding to each value of μon . Comparing the mean solutions of the fully-resolved model and the reduced model demonstrates the capability of the proposed method when random values of the parameter are employed in the nonlinear functional. Furthermore, we observe a good accuracy as shown from the error plotting in Figure 16.7.

Fig. 16.7 Mean error of approximating the solution by using global-local multiscale approach with random values of the online parameter μ.

Chapter 17

Multiscale methods in temporal splitting. Efficient implicit-explicit methods for multiscale problems

17.1 Introduction In this chapter, we discuss how multiscale methods can be used in temporal splitting approaches. Using the multiscale spaces discussed earlier, one can design efficient splitting algorithms. In these algorithms, only some degrees of freedom are handled implicitly, and the rest are handled explicitly. Below, we discuss some details. When the media properties have high contrast, the flow and transport become fast and require small time steps to resolve the dynamics. Implicit discretization can be used to handle fast dynamics; however, this requires solving large-scale nonlinear systems. For nonlinear problems, explicit methods are used when possible to avoid solving nonlinear systems. The main drawback of explicit methods is that they require small time steps that scale as the fine mesh and depend on physical parameters, e.g., the contrast. To alleviate this issue, we have proposed novel splitting algorithms in [116–118, 196]. The main idea of our approaches is to use multiscale methods on a coarse spatial grid such that the time step scales with the coarse mesh size. We follow the works [116–118, 196]. For linear problems, the solution space is divided into two parts, the coarse-grid part and the correction part. The coarse-grid solution is computed using multiscale basis functions with CEM-GMsFEM. The correction part uses special spaces in the complement space (the complement to the coarse space). A careful choice of these spaces guarantees that the method is stable. These splitting algorithms have their origins in earlier works [319, 396]. Earlier approaches include implicit-explicit approaches and other techniques [6, 29, 31, 200, 303]. In many approaches, nonlinear contributions are roughly divided into two parts depending on whether it is easy to implicitly solve discretized system. For easy to solve part, implicit discretization is used, and for the rest, explicit discretization is used. However, in general, one cannot separate these parts for the problems under consideration. Our goal is to use splitting

© Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8_17

457

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17 Multiscale methods in temporal splitting …

concepts and treat implicitly and explicitly some parts of the solution. As a result, we can use larger time steps that scale with the coarse mesh size. In linear problems, for the stability, we formulate a condition that involves the time step, the energy, and the L 2 norm of the solution in the complement space. This is a constraint for the time step. With an appropriate choice of the complement space, this condition guarantees stability for the time steps that scale with the coarse mesh size. We have obtained similar results for nonlinear problems [116]. We remark several important observations. First, we note that the use of additional degrees of freedom (basis functions beyond CEM-GMsFEM basis functions) is needed for dynamic problems, in general, to handle missing information. This is even though CEM-GMsFEM can provide accurate solution for some parabolic equations, the basis functions are computed based on steady-state information and additional degrees of freedom are needed to improve solution adaptively. Secondly, our approaches share some similarities with online methods, where additional basis functions are added and iterations are performed. We note that restrictive time step (e.g., dt = H 2 ) scales as the coarse mesh size and thus much coarser. In this chapter, we present our approach for flow equations. Other applications of our approach can be found in the literature. We consider the following problem. Find u such that u t = ∇ · (κ∇u) + f, in Ω,

(17.1)

where κ ∈ L ∞ (Ω) is a high contrast parameter and f ∈ L 2 (0, T ; L 2 (Ω)) is a source term. We can write the problem in the weak formulation: find u(t, ·) ∈ V such that (u t , v) + a(u, v) = ( f, v), ∀v ∈ V,

(17.2)

where V is an appropriate function space. The energy norm ua is defined as ua = a(u, u)1/2 and the L 2 norm is defined as u = (u, u)1/2 . Cauchy problem consists of finding w(t) in V and 0 < t ≤ T , such that d (w(t), v) + a(w, v) = ( f, v), ∀v ∈ V, 0 < t ≤ T, dt

(17.3)

and initial condition w(0) = w 0 .

(17.4)

Semi-discretization in space is to find u(t) ∈ VH , where VH is a finite-dimensional subspace of V (VH ⊂ V ), such that d (u(t), v) + a(u, v) = ( f, v), ∀v ∈ VH , 0 < t ≤ T, dt

(17.5)

u(0) = u 0 .

(17.6)

17.2 Partially explicit temporal splitting scheme

459

Taking v = du/dt in (17.5) we get u(t)a ≤ u 0 a , 0 < t ≤ T.

(17.7)

For simplicity, we consider a fixed time step, τ and t n = nτ , n = 0, . . . , N , N τ = T , u n = u(t n ). It is known that (e.g., [396]), in the class of two-level schemes, implicit methods (backward Euler) are unconditionally stable, and a forward method (forward Euler) is conditionally stable. n+1 ) (H is the We first consider f = 0. When using implicit method, u n+1 H ≈ u(t coarse mesh size)   n u n+1 H − uH (17.8) , v + a(u n+1 H , v) = 0, ∀v ∈ V H , n = 0, . . . , N − 1. τ n We take in (17.8) v = 2(u n+1 H − u H ) and into account the symmetry in the bilinear form a(·, ·), we have

2 n+1 n+1 2 n 2 n 2 u − u nH 2 + u n+1 H − u H a + u H a − u H a = 0. τ H Thus,

u nH a ≤ u 0H a , n = 1, . . . , N ,

(17.9)

which is a discrete version of (17.7). The estimate (17.9) guarantees the unconditional stability. The stability condition for the explicit scheme   n u n+1 H − uH , v + a(u nH , v) = 0, ∀v ∈ VH , n = 0, . . . , N − 1 (17.10) τ n is carried out by taking into account v = 2(u n+1 H − u H ) in (17.10) and after some manipulations, we have 2 n+1 n+1 2 n 2 n 2 u − u nH 2 − u n+1 H − u H a + u H a − u H a = 0. τ H

Thus, the stability (the estimate (17.9)) will take the place if τ v2 ≥ va2 , ∀v ∈ VH . 2

(17.11)

17.2 Partially explicit temporal splitting scheme In this section, we first introduce a general partial splitting algorithm for u H of problem (17.1) defined as (u H,t , v) + a(u H , v) = ( f, v), ∀v ∈ VH ,

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17 Multiscale methods in temporal splitting …

where VH is a coarse-grid finite element space. We consider VH can be decomposed into two subspaces VH,1 and VH,2 , namely, VH = VH,1 + VH,2 . N ∈ VH,1 and We will use a time discretization scheme: finding {u nH,1 }n=1 N ∈ VH,2 such that {u nH,2 }n=1 n+1 n−1 n n n (u n+1 H,1 − u H,1 , v) + μ(u H,2 − u H,2 , v) + (1 − μ)(u H,2 − u H,2 , v) n n = − τ a(u n+1 H,1 + u H,2 , v)+( f , v), v ∈ V H,1 , n+1 n−1 n n n (u n+1 H,2 − u H,2 , v) + μ(u H,1 − u H,1 , v) + (1 − μ)(u H,1 − u H,1 , v)

(17.12)

n n = − τ a((1 − ω)u nH,1 + ωu n+1 H,1 + u H,2 , v)+( f , v), v ∈ V H,2 .

Here, we will consider some options for μ and ω in [0, 1] and spaces VH,1 and VH,2 . We note that if μ = 0 and ω = 0, the second equation does not require u n+1 H,1 and is totally decoupled. When μ = 0 and ω = 1, the equations can be solved sequentially (the second equation is solved after solving the first equation). When μ = 1, equations need to be solved together at each new time step. As a first case, we briefly consider a case μ = 1 and ω = 1 and VH,1 and VH,2 as two orthogonal spaces such that (v1 , v2 ) = 0, ∀v1 ∈ VH,1 , ∀v2 ∈ VH,2 . The scheme can be simplified as  

n u n+1 H,1 − u H,1

τ n u n+1 H,2 − u H,2

τ

 n n , v1 + a(u n+1 H,1 + u H,2 , v1 ) = ( f , v1 ), ∀v1 ∈ V H,1 ,

 n n , v2 + a(u n+1 H,1 + u H,2 , v2 ) = ( f , v2 ), ∀v2 ∈ V H,2 ,

(17.13)

n = 0, . . . , N − 1. Initial conditions are mapped in corresponding spaces accordingly. In the following, we will discuss the stability of the proposed scheme. To simplify the discussion, we will consider f = 0. Theorem 17.1. The partial explicit scheme (17.13) is stable if v2 2 ≥

τ v2 a2 ∀v2 ∈ VH,2 . 2

(17.14)

Under these conditions, we have u nH a ≤ u 0H a , u nH = u nH,1 + u nH,2 , n = 1, . . . , N .

(17.15)

17.2 Partially explicit temporal splitting scheme

461

Proof. We have the following identity n u n+1 H,1 + u H,2 =

1 n+1 1 1 + u nH ) + (u n+1 − u nH,1 ) − (u n+1 − u nH,2 ). (u 2 H 2 H,1 2 H,2

We take in (17.13) n+1 n n v1 = 2(u n+1 H,1 − u H,1 ), v2 = 2(u H,2 − u H,2 ).

Summing two equations, we have 2 n+1 2 n+1 n n u H,1 − u nH,1 2 + u n+1 − u nH,2 2 + a(u n+1 H + uH, uH − uH) τ τ H,2 n+1 n 2 n 2 + u n+1 H,1 − u H,1 a − u H,2 − u H,2 a = 0.

(17.16)

If (17.14) holds, we get the estimate (17.15). Note that the main finding consists of the constraint on the time step that is due to the explicit part of the scheme. Next, we assume that the spaces VH,1 and VH,2 are not necessarily orthogonal and take μ = 0. We thus obtain the following time discretization scheme: finding N N ∈ VH,1 , {u nH,2 }n=1 ∈ VH,2 {u nH,1 }n=1 n−1 n n (u n+1 H,1 , v) =(u H,1 , v) − (u H,2 − u H,2 , v) n − τ a(u n+1 H,1 + u H,2 , v), v ∈ V H,1

(17.17)

n−1 n n (u n+1 H,2 , v) =(u H,2 , v) − (u H,1 − u H,1 , v) n − τ a((1 − ω)u nH,1 + ωu n+1 H,1 + u H,2 , v), v ∈ V H,2 .

(17.18)

The numerical solution u H ∈ VH is the sum of u H,1 and u H,2 , u H = u H,1 + u H,2 , as before. Now, we will prove stability of the scheme (17.17)-(17.18). To do so, we recall the strengthened Cauchy Schwarz inequality [16]. Let S1 and S2 be finite-dimensional spaces with S1 ∩ S2 = {0}. Then there is a constant 0 < β0 < 1 such that (s1 , s2 ) ≤ β0 s1 s2 , where β0 depends on S1 and S2 . So, there is a constant γ, depending on VH,1 and VH,2 , such that (v1 , v2 ) < 1. (17.19) γ := sup v 1 v2  v1 ∈VH,1 ,v2 ∈VH,2 Theorem 17.2. The partially explicit scheme (17.17)-(17.18) is stable if τ sup

v∈VH,2

1 − γ2 va2 , ≤ v2 (2 − ω)

(17.20)

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17 Multiscale methods in temporal splitting …

where γ is defined in (17.19). Moreover, for n ≥ 1, we have the following stability estimate τ γ2  n τ n 2 γ 2  n+1 2 2 u H,i − u nH,i 2 + u n+1  ≤ u H,i − u n−1 a H H,i  + u H a . 2 i=1,2 2 2 i=1,2 2 The proof can be found in [117].

17.3 Spaces construction In this section, we discuss the spaces satisfying (17.20). We will show that the constrained energy minimization finite element space is a good choice of VH,1 since the CEM basis functions are constructed such that they are almost orthogonal to a space V˜ which can be easily defined. To obtain a VH,2 satisfying the condition (17.20), one of the possible ways is using an eigenvalue problem to construct the local basis function. Before discussing the construction of VH,2 , we will briefly introduce the CEM finite element space VH,1 to keep the chapter selfconsistent.

17.3.1 Construction of VH,1 In this section, we will discuss the CEM method [114] (also Chapter 7) for solving the problem (17.2). We will construct the finite element space by solving a constrained energy minimization problem. We let T H be a coarse-grid partition of Ω with Ne elements. For each coarse element K i ∈ T H , we consider a set of auxiliary basis Li functions {ψ (i) j } j=1 ∈ V (K i ) by solving  Ki

(i) (i) κ∇ψ (i) j · ∇v = λ j si (ψ j , v), ∀v ∈ V (K i )

(17.21)

and collecting the first L i eigenfunctions corresponding to the first L i smallest eigenvalues with  κuv, ˜ (17.22) si (u, v) = Ki

 and κ˜ = κH −2 or κ˜ = κ i |∇χi |2 , where {χi } is a set of partition of unity functions corresponding to an overlapping partition of the domain. In (17.21), we define V (K i ) = H 1 (K i ). (i) ⊂ L 2 (K i ) such that We define a projection operator i : L 2 (K i ) → Vaux (i) si (i u, v) = si (u, v), ∀v ∈ Vaux := span{ψ (i) j : 1 ≤ j ≤ L i }.

We next define a global projection operator by  : L 2 (Ω) → Vaux ⊂ L 2 (Ω)

17.3 Spaces construction

463

s(u, v) = s(u, v), ∀v ∈ Vaux :=

Ne 

(i) Vaux ,

i=1

 Ne where s(u, v) := i=1 si (u| K i , v| K i ). (i) For each auxiliary basis function ψ (i) j , we define a multiscale basis function φ j ∈ V (K i+ ) such that (i) + a(φ(i) j , v) + s(μ j , v) = 0, ∀v ∈ V (K i ), (i) + s(φ(i) j , ν) = s(ψ j , ν), ∀ν ∈ Vaux (K i ),

where K i+ is an oversampling domain of K i obtained by enlarging K i by a few coarse-grid layers. We define the approximation space Vcem by Vcem := span{φ(i) j : 1 ≤ i ≤ Ne , 1 ≤ j ≤ L i }. Then the CEM solution u cem ∈ Vcem is given by (u cem,t , v) + a(u cem , v) = 0, ∀v ∈ Vcem , where u cem,t is the time derivative of u cem . We note that u cem is a multiscale approximation of u, and its convergence is analyzed. A related concept is the global basis functions, which are obtained by solving constraint energy minimization problem in the whole computational domain. The resulting space is called Vglo . We remark that the Vglo is a−orthogonal to a space V˜ := {v ∈ V : (v) = 0}. We also know that Vcem is closed to Vglo and therefore it is almost orthogonal to V˜ . Thus, we can choose Vcem to be VH,1 and construct a space VH,2 in V˜ .

17.3.2 Construction of VH,2 In this section, we present two choices for the space VH,2 which will give an explicit stability condition based on (17.20), and these choices are motivated by reducing errors. Recall that V = H01 (Ω). For any set S, we let V (S) = H 1 (S) and V0 (S) = H01 (S). First choice We will define basis functions for each coarse neighborhood ωi , which is the union of all coarse elements having the i-th coarse-grid node. For each coarse (i) neighborhood ωi , we consider the following eigenvalue problem: find (ξ (i) j , γj ) ∈ (V0 (ωi ) ∩ V˜ ) × R,

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17 Multiscale methods in temporal splitting …

 ωi

κ∇ξ (i) j · ∇v =

γ (i) j



H2

ωi

˜ ξ (i) j v, ∀v ∈ V0 (ωi ) ∩ V .

(17.23)

We arrange the eigenvalues by γ1(i) ≤ γ2(i) ≤ · · · . In order to obtain a reduction in error, we will select the first few Ji dominant eigenfunctions corresponding to smallest eigenvalues of (17.23). We define VH,2 = span{ξ (i) j | ∀ωi , ∀1 ≤ j ≤ Ji }. More discussions on how the condition (17.20) is satisfied can be found in [117]. Second choice The second choice of VH,2 is based on the CEM type finite element space. This choice is more complicated compared to the first choice as it uses CEM type elements, and it can possibly provide faster convergence in some cases. For each coarse element K i , we will solve an eigenvalue problem to obtain the auxiliary basis. More precisely, (i) ˜ we find eigenpairs (ξ (i) j , γ j ) ∈ (V (K i ) ∩ V ) × R by solving  Ki

(i) κ∇ξ (i) j · ∇v = γ j

 Ki

˜ ξ (i) j v, ∀v ∈ V (K i ) ∩ V .

(17.24)

For each K i , we choose the first few Ji eigenfunctions corresponding to the smallest Ji eigenvalues. The resulting space is called Vaux,2 . For each auxiliary basis function (i) (i) (i),2 ξ (i) j , we define the global basis function ζglo, j ∈ V such that μglo, j ∈ Vaux,1 , μglo, j ∈ Vaux,2 and (i) (i),1 (i),2 a(ζglo, j , v) + s(μglo, j , v) + (μglo, j , v) = 0, ∀v ∈ V, (i) s(ζglo, j , ν) (i) (ζglo, j , ν)

(17.25)

= 0, ∀ν ∈ Vaux,1 ,

(17.26)

= (ξ (i) j , ν), ∀ν ∈ Vaux,2 ,

(17.27)

where we use the notation Vaux,1 to denote the space Vaux defined in Section 17.3.1. (i) We define Vglo,2 = span{ζglo, j | j ≤ Ji }. This is our choice of V H,2 , that is, we take VH,2 = Vglo,2 .

17.4 Numerical results In this section, we present representative numerical results that show that proposed approaches can select time step independent of the contrast and predict an accurate approximation for the solution. We consider the following parameters for the mesh sizes, and time steps

17.4 Numerical results

465

H = 1/10, h = 1/100, dt = 10−4 , T = 0.05. Here, H is the coarse mesh size, h is the fine mesh size, dt is the fine time step, and T is the final time step. In all examples, we consider zero Dirichlet boundary conditions and zero initial conditions. The conductivity fields and forcing terms are chosen differently for examples and described in each part. In our numerical example, we choose a smooth source term. Other cases are studied in the literature. In this case, CEM-GMsFEM without additional basis functions provide results similar to those CEM-GMsFEM with additional basis functions that are treated explicitly in our method. We do not dwell on accuracy issues related to the use of additional basis functions in CEM-GMsFEM (that are treated explicitly). These basis functions are needed in many cases to capture dynamics effects, in wave equations, and so on. The medium parameter κ, the reference solution at final time u r e f , and the source term f are shown in Figure 17.1. As we see, the permeability field is heterogeneous with high contrast streaks. The contrast is 106 in all examples. Due to smooth source term, the solution’s features in high conductivity field regions are smeared. In Figure 17.2, we depict the error in L 2 and in energy norm that correspond to three methods. The blue curve denotes the error due to CEM without additional basis functions. Because of smooth source term and problem setup, this method provides an error that is comparable to the error when we consider additional basis functions. The additional degrees of freedom treated both implicitly (red curve) and explicitly (yellow curve). As we see that these two curves coincide. This indicates that the time stepping that is chosen independent of contrast provides as accurate solution as full backward Euler for our proposed partial explicit method. Consequently, this backs up our discussions. In Figure 17.2, we consider VH,2 , which is the first type, and in Figure 17.3, we consider the case with the second type VH,2 . The results are similar, which show that both spaces provide a robust partial explicit discretization. 105 10

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Fig. 17.3 Example 1. Second type of VH,2 (CEM Dof: 300, VH,2 Dof: 300). Left: L 2 error. Right: Energy error. Along x-axis is time, along y-axis is the relative error.

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Index

A Adaptive enrichment, 133, 137 Adaptivity, 78, 82, 133 Anisotropic Gaussian, 379 Auxiliary basis function, 417 Auxiliary space, 179, 182, 198

B Basis function, 13

C Cell problem, 37, 40, 75, 83 Cell resonance, 76 CEM-GMsFEM, 180, 182, 417, 458 Clustering, 350, 352 Complex heterogeneities, 1 Constraint energy minimizing GMsFEM, 16 Continuous Galerkin, 98, 265, 320 Control volume, 392 Convection-dominated diffusion, 251 Cosserat media, 384

D Darcy law, 322 Darcy-Forchheimer, 384 Deep learning, 25 DEIM, 411, 441 DG-GMsFEM, 340 Discontinuous Galerkin, 265, 283 Discontinuous Petrov-Galekin, 251 Discrete stability, 254 Domain decomposition, 80 Downscaled map, 18, 25

Downscaling map, 413 Dynamic mode decomposition, 440

E Effective flux, 414 Elastic wave propagation, 335 Elasticity equations, 230, 265 Elasticity operator, 232 Energy minimizing snapshot, 101, 157 Error indicator, 133, 137

F Fracture network, 417 Fractured geothermal reservoirs, 384 Fractured media, 195, 197, 298, 338

G G-convergence, 80, 394 Generalized finite element method, 13 Geomechanics, 318 Global information, 29, 92 Global learning, 26 Global-local model reduction, 443 GMsDGM, 157, 161 GMsFEM, 14, 95 GMsHDG, 157, 163 Goal oriented adaptivity, 140 Grid, 3, 11

H Harmonic extension, 99, 265, 341 High contrast, 1, 3

© The Editor(s) (if applicable) and Springer Nature Switzerland AG 2023 E. Chung et al., Multiscale Model Reduction, Applied Mathematical Sciences 212, https://doi.org/10.1007/978-3-031-20409-8

489

490 Homogenization, 12, 35, 80, 385 Homogenized coefficient, 386 Homogenized equation, 35, 40, 83, 386 Homogenized flux, 386

I Isotropic Gaussian, 379

K K-mean algorithm, 350 Karhunen-Loéve expansion, 350, 366

L Linear slip model, 336 Local learning, 26 Local residual, 138 Local spectral decomposition, 101

M Macroscopic equation, 13, 15, 23, 413 Macroscopic property, 12 Macroscopic variable, 18, 413 Markov chain Monte Carlo, 374 Marmousi model, 293 MCMC, 370 MCMC estimator, 377 Metropolis-Hastings algorithm, 375 MHD problem, 384 Mixed dimensional flow model, 319 Mixed GMsFEM, 157, 159, 283 MLMC, 370 MLMC-GMsFEM, 370 MLMCMC, 375 Model reduction, 1, 8, 14, 34 Monotone operator, 385 MsFEM, 69 Multicontinuum, 195 Multilevel Metropolis-Hastings MCMC, 378 Multilevel Monte Carlo, 363, 370 Multiple scales, 1 Multiscale Allen-Cahn problem, 384 Multiscale basis function, 15, 16, 183 Multiscale finite element method, 13, 72 Multiscale model reduction, 14 Multiscale stabilization, 251

N NLMC, 196, 209, 320, 413

Index NLNLMC, 436 Nonconforming GMsFEM, 162 Nonlinear diffusion, 9 Nonlinear elliptic equation, 386 Nonlinear fluxes, 25 Nonlinear GMsFEM, 400 Nonlinear homogenization, 17, 413 Nonlinear NLMC, 418 Nonlinear nonlocal multicontinua, 414 Nonlinear spectral problem, 403 Nonlocal multicontinua, 26 Nonlocal multicontinua upscaling, 196 Numerical homogenization, 11, 47, 388, 400

O Offline space, 15, 97, 136, 402 Online adaptive enrichment, 190 Online adaptive procedure, 145 Online basis function, 134, 145, 189, 234, 313 Online space, 105 Orthogonal decomposition, 184 Oversampled region, 11, 16 Oversampled space-time region, 218 Oversampling, 27, 74, 76, 78, 81, 99, 271 Oversampling layer, 187, 188

P P-harmonic extension, 399 P-Laplacian equation, 398 Parallel computation, 29, 81, 82 Parameter realization, 364 Partial explicit scheme, 460 Partial splitting algorithm, 459 Peclet number, 251 Perforated domain, 62, 229 Permafrost, 383 piezoelectric problem, 383 Poroelastic model, 319 Porous media, 349

Q Quantities of interest, 364

R Random boundary condition, 100 Randomized boundary condition, 214 Randomized oversampling, 125 Randomized snapshot, 122, 238, 310 Relaxed CEM-GMsFEM, 188

Index Representative Volume Element, 23, 78, 434 Reservoir simulation, 317, 351 Resonance, 74, 88 RVE, 426, 434

S Scale separation, 1 Shale gas transport, 297 Snapshot space, 15, 19, 97, 229, 402 Space-time GMsFEM, 214, 221 Space-time local domain, 214 Space-time multiscale basis function, 217 Space-time online GMsFEM, 226 SPE benchmark, 432 Spectral convergence, 15 Staggered mesh, 283 Stiffness matrix, 69, 73, 80 Stokes equations, 230, 232 Stokes extension, 240

491 T Temporal splitting, 25, 457 thin domain, 383 Three-level scheme, 209 Two-phase flow, 278, 420

U Uncertainty quantification, 363 Upscaled coefficient, 12 Upscaled equation, 196 Upscaling linear, 77 nonlinear, 17, 413

W Wave equation, 283