Numerical Treatment and Analysis of Time-Fractional Evolution Equations 9783031210495, 9783031210501

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

Volume 214

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.

Bangti Jin · Zhi Zhou

Numerical Treatment and Analysis of Time-Fractional Evolution Equations

Bangti Jin Department of Mathematics The Chinese University of Hong Kong Shatin, Hong Kong

Zhi Zhou Department of Applied Mathematics Hong Kong Polytechnic University Hung Hom, Hong Kong

ISSN 0066-5452 ISSN 2196-968X (electronic) Applied Mathematical Sciences ISBN 978-3-031-21049-5 ISBN 978-3-031-21050-1 (eBook) https://doi.org/10.1007/978-3-031-21050-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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

Preface

The purpose of this book is to present a self-contained and up-to-date survey of numerical treatment for the so-called time-fractional diffusion model and their mathematical analysis. The time-fractional diffusion model involves a fractionalorder derivative (in the sense of Djrbashian–Caputo or Riemann–Liouville) of order α ∈ (0, 1) in time, and is often employed to describe anomalously slow diffusion processes arising in many applications in physics, engineering, biology, and finance (hence also known as subdiffusion in the literature). The nonlocality of the fractional-order time derivative can faithfully capture the hereditary nature of these physical processes. The model can be viewed as a time-fractional analogue of classical parabolic problems. Meanwhile, the presence of the nonlocal derivative also completely changes the analytical and numerical treatment of the mathematical model, and often poses big mathematical and computational challenges. Nonetheless, due to its broad range of applications, the numerical treatment has witnessed significant progress in the last two decades, and many powerful numerical methods have been developed. Many of these methods are based on the Galerkin finite element method/finite difference method in space, and then suitable time-stepping schemes for approximating the fractional derivative in time. The goal of the book is to provide a comprehensive and in-depth discussion over these developments. It is needless to say that the selection of topics is not meant to be exhaustive, but rather reflects the authors’ involvement in the field over the past ten years and thus it is strongly biased by our own experiences and knowledge. The goal has been mainly pedagogical, with an emphasis on collecting powerful ideas and methods of analysis in simple model situations, rather than on pursuing each approach to its limits. The book summarizes recent important developments on numerical schemes, efficient implementation and rigorous numerical analysis, as the overview [JLZ19a], and the interested reader is often referred to the literature for more complete results on a given topic. The following is an outline of the contents of the book. In the introductory Chap. 1, we describe basic results for time-fractional diffusion, including fundamental concepts in fractional calculus, e.g., Riemann–Liouville fractional integral and Riemann–Liouville and Djrbashian–Caputo fractional derivatives and basic mapping properties. The fractional derivatives are basic building blocks of v

vi

Preface

the mathematical model. Then we introduce the two-parameter Mittag-Leffler function E α,β (z), one useful tool for representing the solutions, and several important properties of the function, e.g., asymptotic decay and complete monotonicity. Last we recall the fundamental regularity theory for solutions to the subdiffusion problem with general problem data. These results are obtained from the solution representation, which is itself derived using Laplace transform and separation of variables separately. These regularity results will be used extensively in the error analysis throughout the book. In Chap. 2, we consider the simplest Galerkin finite element method for approximating initial boundary value problems for the subdiffusion equation with a zero Dirichlet boundary conditions on a bounded convex polyhedral domain. The semidiscrete problem resulting from discretization in the space variables only is based on the standard weak formulation of the associated second-order elliptic problem and employing piecewise linear approximating functions vanishing on the boundary of the domain. For this model problem, we demonstrate optimal (sometimes up to a logarithmic factor) error estimates in the H 1 () and L 2 () norms for the approximation. We present two separate analysis techniques, one based on the separation of variables technique (more precisely, eigenpansion and Mittag-Leffler functions), and the other based on Laplace transform and resolvent estimate. In addition, we present the lumped mass method, which results in a diagonal mass matrix and facilitates the computation, and provide nearly optimal error estimates. In the next four chapters we consider two popular classes of time-stepping schemes and show error estimates in a variety of norms, mostly in the pointwise-in-time L 2 () norm. First, in Chap. 3, we consider a first class of time-stepping schemes based on convolution quadrature generated by backward differentiation formula (on uniform temporal meshes). By construction, this class of schemes inherits the excellent stability of the underlying schemes for classical odes. First, we present a complete analysis of the schemes for smooth functions (under suitable compatibility conditions). Second, we develop an initial correction scheme, which modifies only the first few steps of the standard discretization in order to restore high-order convergence rate for the subdiffusion model with general problem data. Third, we develop and analyze a fractional version of the classical Crank–Nicolson scheme, and its corrected versions. The analysis in this chapter employs the discrete Laplace transform/generating function and the resolvent estimate. Fourth and last, we provide effective strategies for memory-efficient implementation, e.g., fast convolution and parallel-in-time, to overcome the outstanding computational challenge arising from the memory effect of the fractional time derivative. Second, in Chaps. 4–6, we investigate a second class of time-stepping schemes of finite difference nature based on piecewise polynomial interpolation. This class of schemes is straightforward to construct and implement on an arbitrary temporal mesh, but its mathematical analysis is often challenging. First, in Chap. 4, we describe the construction of several representative discrete schemes, including L1 scheme, L2 scheme, L1-2 scheme and L2-1σ scheme (i.e., Alikhanov’s scheme). We also derive the local truncation errors for the schemes under the assumption that the

Preface

vii

underlying function is sufficiently smooth. These bounds provide benchmarks for optimal convergence rates. The efficient implementation of the schemes based on the sum of exponential approximations technique is also described, where the key idea is to approximate the weakly singular kernel t −β , β ∈ (0, 2), with a sum of exponentials over an interval bounded away from the singularity t = 0, which allows capturing effectively the history part of the fractional time derivative with constant computational complexity per time stepping and greatly reduced memory footprint. We describe two different ways to construct the sum of exponentials, based on different quadrature rules. Then in Chap. 5, we present an analysis of the L1 scheme on uniform grid. This scheme is arguably one of the most popular schemes in the literature. We prove a first-order convergence of the L1 scheme for general problem data, which is lower than the optimal rate expected from the local truncation error in Chap. 4. This loss of accuracy is common to nearly all time-stepping schemes, and it originates from the limited solution regularity. The proof is again based on the discrete Laplace transform, as for convolution quadrature in Chap. 3. In addition, we describe a simple correction scheme for restoring the optimal convergence rate, following the strategy of initial correction for convolution quadrature in Chap. 3. Next in (relatively long) Chap. 6, we discuss the error analysis of time-stepping schemes of finite difference type on graded meshes. Graded meshes have recently been established as a powerful way to handle the weak solution singularity at time t = 0. Depending on the specific grading strategy, the analysis of these schemes can be very technical. We describe two prominent approaches for the error analysis of nonuniform grids, i.e., discrete Gronwall’s inequality and discrete barrier functions. The former imposes suitable conditions on the discrete kernels for the time-stepping scheme and local mesh grading parameter. The latter applies to general quasi-graded meshes and is applicable provided that the time-stepping scheme satisfies a certain discrete maximum principle. In this chapter, we illustrate these approaches on the L1 scheme and Alikhanov’s scheme. Chapters 7 and 8 are concerned with qualitative properties of time-stepping schemes, i.e., maximum principle and discrete maximal L p regularity (i.e., maximal  p regularity). In Chap. 7, we discuss the discrete maximum principle for spatially semidiscrete schemes and fully discrete schemes based on backward Euler convolution quadrature, i.e., whether the discrete solution preserves the nonnegativity of the continuous solution, which represents an important structural property in many physical problems. It turns out that nonnegativity preservation relies heavily on the geometrical property of the underlying finite element mesh and the time step size. We also discuss briefly the maximum norm contraction property. In Chap. 8, we investigate maximal  p regularity of time-stepping schemes. Maximal L p regularity represents an extremely powerful tool in the analysis of nonlinear evolution equations, and its discrete analogue is important for the numerical analysis of nonlinear problems. In this chapter, we establish maximal  p regularity for several classes of time-stepping schemes, including convolution quadrature generated by backward differentiation formula, L1 scheme, explicit Euler discretization, and fractional Crank–Nicolson scheme, described in Chaps. 3 and 4. We also discuss the maximal  p regularity for the homogeneous problem. The main tool in the analysis is a discrete version of Weis’

viii

Preface

Fourier multiplier theorem in UMD spaces. These results will be used extensively in the analysis of time-variable coefficients in Chap. 9, nonlinear problems in Chap. 10, and inverse problems and optimal control problems in Chaps. 14and 15. In Chaps. 9 and 10, we extend the discussions in Chaps. 2 and 3 to more complex problems, i.e., subdiffusion with time-dependent elliptic operators or nonlinearity, respectively, for which the powerful (discrete) Laplace transform technique no longer applies directly. In Chap. 9, we develop both spatially semidiscrete and fully discrete schemes (with the latter based on backward Euler convolution quadrature and secondorder backward differentiation formula with initial correction) for subdiffusion problems involving a time-dependent diffusion coefficient, and present sharp error estimates. The analysis strategy is based on a perturbation argument, i.e., viewing the problem as a perturbed problem with the time-independent coefficient and a source term depending on the unknown solution. In Chap. 10, we discuss semilinear subdiffusion with a globally Lipschitz nonlinearity. First we prove sharp regularity estimates for smooth initial data. Then we develop a linearized scheme for the problem and provide sharp error estimates. This is achieved using suitable fractional Gronwall’s inequalities proved in this chapter and the regularity results. The development of corrected high-order time-stepping schemes to semilinear problems with additional compatibility conditions on the initial data is also discussed. By suitably splitting the semillinear term into an irregular linear part and a smoother nonlinear part, we finally derive an improved convergence rate. In Chaps. 11 and 12, we develop space-time variational formulations for the subdiffusion model with zero initial condition. This idea is very natural in view of the nonlocality of the fractional derivative in time. In Chap. 11, we lay the foundation of a space-time formulation of Petrov–Galerkin type, which involves different trial and test spaces. Using properties of the Riemann–Liouville fractional derivative operator, we prove the well-posedness of the formulation by establishing the inf-sup condition. Further, we develop a fully discrete scheme based on the Petrov–Galerkin formulation, using the fractionalized piecewise polynomials for time discretization and finite element for spatial discretization. We prove the well-posedness of the discrete formulation by establishing the discrete inf-sup condition and provide optimal error estimates. In Chap. 12, we present a spectral method in time based on a space-time Galerkin formulation posed on suitable fractional-order Sobolev-Bochner spaces and log-orthonormal functions in time. This is motivated by the observation that log-orthonormal functions can approximate Mittag-Leffler functions exponentially fast. Unlike time-stepping schemes in Chaps. 3 and 4, the spectral approach can achieve spectral convergence for weakly singular solutions. The analysis is achieved by developing suitable weighted regularity estimates of the solution. In Chap. 13, we discuss the numerical solution of the linear systems arising from the fully discrete schemes, by iteratively solving the linear systems approximately, leading to the so-called incomplete iterative schemes. We provide an analysis of the fully discrete scheme based on backward Euler convolution quadrature for the homogeneous problem and establish that if the number of iterations at each time level is chosen properly (which in turn depends crucially on the regularity of the initial data), then incomplete iterative solutions satisfy error estimates nearly identical with

Preface

ix

that obtained by exact linear solve. The analysis employs crucially maximal  p regularity in Chap. 8 and certain nonstandard nonsmooth data estimates in weighted norms. In the last two chapters, Chaps. 14 and 15, we present two canonical applications of nonsmooth data error estimates, i.e., optimal control and inverse problems for subdiffusion. In these settings, the involved problem data often have low regularity. In Chap. 14, we study a distributed optimal control problem subject to a subdiffusion constraint, discretized using the piecewise linear finite element method in space and backward Euler convolution quadrature/L1 scheme in time for both state and adjoint variables, and variational discretization of the control variable. We prove nearly optimal error estimates for a fully discrete scheme, by proving sharp regularity of the control variable, suitably adapting the error estimates for weak data and applying maximal  p regularity. In Chap. 15, we study the numerical solution of backward subdiffusion of recovering the initial condition from the noisy terminal measurement. This is a model inverse problem for the subdiffusion model which enjoys an excellent stability estimate. For the stable numerical solution, we first regularize the inverse problem using the quasi-boundary value method, and then discretize the regularized problem using backward Euler convolution quadrature in time, and Galerkin finite element method in space. We provide a full error analysis of the spatially semidiscrete and fully discrete approximations. The final error estimates provide useful guidelines to properly balance important algorithmic parameters, e.g., the mesh size, time step size, the regularization parameter, with the noise level. Throughout, further references to the existing literature where the reader may find more complete treatments of the different topics, and some (historical) comments as well as other related topics (not covered in the book), are given at the end of each chapter. A desirable mathematical background for reading the text includes basic theory of partial differential equations and functional analysis, including Sobolev spaces; for the convenience of the reader, we give a short appendix collecting several basic facts (Gamma function, polylogarithmic function, Laplace and Fourier transforms) and also give further references to the literature concerning such matters. The book has largely developed from the research we have conducted over the last ten years. The writing has benefited enormously from the collaborations with several researchers on the topic and stimulating discussions with them in the past few years, which is gratefully acknowledged. The materials in the book have been used at a summer school at Tianyuan Mathematical Center at Northeast China, Jilin, in March 2022. Hong Kong September 2022

Bangti Jin Zhi Zhou

Contents

1

Existence, Uniqueness, and Regularity of Solutions . . . . . . . . . . . . . . . 1.1 Basics of Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Mittag–Leffler Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Existence, Uniqueness, and Sobolev Regularity . . . . . . . . . . . . . . .

1 2 6 7

2

Spatially Semidiscrete Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Galerkin Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Error Analysis via Mittag–Leffler Functions . . . . . . . . . . . . . . . . . . 2.3 Error Analysis via Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . 2.4 Lumped Mass FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 25 34 40

3

Convolution Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Convolution Quadrature Generated by BDF . . . . . . . . . . . . . . . . . . 3.2 BDFk CQ with Initial Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Fractional Crank–Nicolson Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Parallel in Time Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Fast Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 50 57 69 79 86

4

Finite Difference Methods: Construction and Implementation . . . . . 95 4.1 Construction of Time-Stepping Schemes . . . . . . . . . . . . . . . . . . . . . 96 4.2 Sum of Exponential Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 109

5

Finite Difference Methods on Uniform Meshes . . . . . . . . . . . . . . . . . . . 125 5.1 Error Analysis of L1 Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2 Corrected L1 Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6

Finite Difference Methods on Graded Meshes . . . . . . . . . . . . . . . . . . . . 6.1 Error Analysis via Nonuniform Gronwall’s Inequality . . . . . . . . . 6.2 Error Analysis of the L1 Scheme via Barrier Functions . . . . . . . . 6.3 Error Analysis of Alikhanov’s Scheme via Barrier Functions . . .

149 150 165 175

xi

xii

Contents

7

Nonnegativity Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Nonnegativity Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Spatially Semidiscrete Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Fully Discrete Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Maximum-Norm Contractivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 193 197 203

8

Discrete Maximal Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 R-Boundedness, UMD Spaces, and Fourier Multiplier Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Convolution Quadrature Generated by BDF . . . . . . . . . . . . . . . . . . 8.3 L1 Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Explicit Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Fractional Crank–Nicolson Method . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Inhomogeneous Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 208 212 215 218 221 222

Subdiffusion with Time-Dependent Coefficients . . . . . . . . . . . . . . . . . . 9.1 Regularity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Semidiscrete Galerkin FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Time Discretization by Backward Euler CQ . . . . . . . . . . . . . . . . . . 9.4 Time Discretization by Corrected BDF2 CQ . . . . . . . . . . . . . . . . .

227 228 230 238 247

10 Semilinear Subdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Discrete Gronwall’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Error Estimates for the Linearized Scheme . . . . . . . . . . . . . . . . . . . 10.3 High-Order Time-Stepping Schemes . . . . . . . . . . . . . . . . . . . . . . . .

259 260 268 275

11 Time-Space Finite Element Approximation . . . . . . . . . . . . . . . . . . . . . . 11.1 Time-Space Petrov–Galerkin Formulation . . . . . . . . . . . . . . . . . . . 11.2 Petrov–Galerkin FEM on Tensor-Product Meshes . . . . . . . . . . . . . 11.3 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 294 299 305

12 Spectral Galerkin Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Time-Space Galerkin Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Log Orthogonal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Spectral Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Fully Discrete Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Fast Linear Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 316 318 323 326 329

13 Incomplete Iterative Solution at Time Levels . . . . . . . . . . . . . . . . . . . . . 13.1 Incomplete Iterative Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Error Analysis for Smooth Initial Data . . . . . . . . . . . . . . . . . . . . . . 13.3 Error Analysis for Nonsmooth Initial Data . . . . . . . . . . . . . . . . . . .

333 334 345 349

14 Optimal Control with Subdiffusion Constraint . . . . . . . . . . . . . . . . . . . 14.1 Regularity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Numerical Approximation of the Forward Problem . . . . . . . . . . . . 14.3 Numerical Approximation of the Optimal Control Problem . . . . .

359 360 362 368

9

Contents

15 Backward Subdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Stability and Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Spatially Semidiscrete Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Fully Discrete Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

385 387 390 394

Appendix A: Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Chapter 1

Existence, Uniqueness, and Regularity of Solutions

In this chapter, we describe the basic mathematical theory of the standard timefractional diffusion model, whose numerical treatment and analysis is the main topic of this book. Let  be an open bounded Lipschitz domain with a boundary ∂. Consider the following initial boundary value problem for the function u: ⎧ α ⎪ ⎨ ∂t u(x, t) = Lu(x, t) + f (x, t), u(x, t) = 0, ⎪ ⎩ u(x, 0) = u 0 (x),

in  × (0, T ], on ∂ × (0, T ], in .

(1.1)

In the model, α ∈ (0, 1) is the order of the derivative, ∂tα u denotes the left-sided Djrbashian–Caputo fractional derivative of the function u of order α ∈ (0, 1) with respect to time t (based at zero): ∂tα u(x, t)

1 = (1 − α)



t

(t − s)−α ∂s u(x, s) ds,

0

with (z) denotes Gamma function, defined by  (z) =

∞

t z−1 e−t dt, (z) > 0,

0

(see Appendix A.1 for properties of the function (z)), and L denotes a timeindependent second-order elliptic differential operator, defined by Lu(x) =

d 

∂xi (ai j (x)∂x j u) − q(x)u(x), x ∈ ,

(1.2)

i, j=1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Jin and Z. Zhou, Numerical Treatment and Analysis of Time-Fractional Evolution Equations, Applied Mathematical Sciences 214, https://doi.org/10.1007/978-3-031-21050-1_1

1

2

1 Existence, Uniqueness, and Regularity of Solutions

where q(x) ≥ 0 is smooth, and the symmetric matrix-valued function a = [ai j (x)] :  → Rd×d (i.e., ai j = a ji , i, j = 1, . . . , d) is smooth and uniformly elliptic, i.e., λ|ξ |2 ≤ ξ · a(x)ξ ≤ λ−1 |ξ |2 , ∀ξ ∈ Rd , x ∈ , for some λ ∈ (0, 1), where · and | · | denote the Euclidean inner product and norm on Rd , respectively. The precise regularity condition on the source f and initial data u 0 will be specified later. We shall derive explicit solution representations using the separation of variables technique and Laplace transform separately, and then establish the existence, uniqueness, and regularity theory in Sobolev spaces. Throughout the book, for a function f (x, t) defined on  × (0, T ), we denote by f (t) the function f (·, t), and in the model (1.1), we only consider the case α ∈ (0, 1), which corresponds to subdiffusion. The notation c, with or without a subscript/superscript, denotes a generic constant, which may change at each occurrence, but it is always independent of other parameters of interest, e.g., mesh size h and time step size τ . The rest of the chapter is organized as follows. We first recall basic facts in fractional calculus and Mittag–Leffler functions in Sects. 1.1 and 1.2, respectively, and then discuss the well-posedness of problem (1.1) in Sect. 1.3.

1.1 Basics of Fractional Calculus First, we review basic concepts and results in fractional calculus, including Riemann– Liouville fractional integral, Riemann–Liouville/Djrbashian–Caputo fractional derivatives. These tools form the basic building blocks of relevant mathematical models and will be used extensively below. Throughout, for real numbers a < b, we denote D = (a, b) the interval. The interval (a, b) is often taken to be (0, T ), with T being the final time. Definition 1.1 For any v ∈ L 1 (D), the left-sided Riemann–Liouville fractional integral of order α > 0 based at t = a, denoted by a Itα v, is defined by (a Itα v)(t) =

1 (α)



t

(t − s)α−1 v(s) ds,

(1.3)

a

and the right-sided Riemann–Liouville fractional integral of order α > 0 based at t = b, denoted by t Ibα v, is defined by (t Ibα v)(t) =

1 (α)



b

(s − t)α−1 v(s) ds.

(1.4)

t

The Riemann–Liouville integrals satisfy a semigroup property, which is a direct consequence of Fubini’s theorem.

1.1 Basics of Fractional Calculus

3

Theorem 1.1 For v ∈ L 1 (D), α, β ≥ 0, there hold α β a It a It v

β

α+β

= a It a Itα v = a It

α β t Ib t Ib v

v and

β

α+β

= t Ib t Ibα v = t Ib

v.

The fractional integral operator a Itα satisfies several useful mapping properties on L (D). The second assertion in (ii) is known as the Hardy–Littlewood inequality. p

Theorem 1.2 The following mapping properties hold for a Itα with α > 0. (i) a Itα is bounded on L p (D) for any 1 ≤ p ≤ ∞. (ii) For 1 ≤ p < α −1 , a Itα is a bounded operator from L p (D) into L r (D) for 1 ≤ p r < (1 − αp)−1 . If 1 < p < α −1 , then a Itα is bounded from L p (D) to L 1−αp (D). (iii) For p > 1 and p −1 < α < 1 + p −1 or p = 1 and 1 ≤ α < 2, a Itα is bounded 1 from L p (D) into C 0,α− p (D), and moreover, a Itα f (0) = 0 for f ∈ L p (D). The fractional integral operators a Itα and t Ibα are adjoint to each other in L 2 (D). Lemma 1.1 For any v ∈ L p (D), w ∈ L q (D), p, q ≥ 1 with p −1 + q −1 ≤ 1 + α ( p, q = 1, when p −1 + q −1 = 1 + α), the following identity holds: 

b

a

w(t)(a Itα v)(t) dt

 = a

b

v(t)(t Ibα w)(t) dt.

Next ,we define Riemann–Liouville fractional derivatives. Definition 1.2 For v ∈ L 1 (D) and n − 1 < α ≤ n, n ∈ N, its left-sided Riemann– Liouville fractional derivative of order α (based at t = a), denoted by RaDtα v, is defined by

R α a Dt v(t)

:=

dn dn 1 (a Itn−α v)(t) = n dt (n − α) dt n



t

(r − s)n−α−1 v(s) ds,

a

if the integral on the right-hand side exists. Its right-sided Riemann–Liouville fractional derivative of order α (based at t = b), denoted by RtDbα v, is defined by R α t Db v(t)

:= (−1)n

dn (−1)n dn (t Ibn−α v)(t) = n dt (n − α) dt n



b

(s − t)n−α−1 v(s) ds,

t

if the integral on the right-hand side exists. The next result gives the fundamental theorem of calculus for the Riemann– Liouville fractional derivative: RaDtα is the left inverse of a Itα , but not the right inverse. The notation AC n (D) denotes the space of functions whose nth derivative belongs to L 1 (D).

4

1 Existence, Uniqueness, and Regularity of Solutions

Theorem 1.3 Let α > 0, n − 1 < α ≤ n, n ∈ N. Then the following statements hold. (i) For any v ∈ L 1 (D), RaDtα (a Itα v) = v. (ii) If a Itn−α v ∈ AC n (D), then αR α a It a Dt v

=v−

n−1 

(t R α− j−1 v(a + ) a Dt

j=0

− a)α− j−1 . (α − j)

Next, we introduce the definition of the Djrbashian–Caputo fractional derivative (more commonly known as Caputo fractional derivative), which is frequently used in the subdiffusion model, since it allows specifying the initial condition as in the classical diffusion model. This classical definition requires the existence of the weak derivative v (n) , and thus it is more restrictive than the Riemann–Liouville one. Under suitable regularity condition, Ct Dbv (t) does recover the classical nth derivative v (n) as α → n−. Definition 1.3 For v ∈ L 1 (D) and n − 1 < α ≤ n, n ∈ N, its left-sided Djrbashian– Caputo fractional derivative Ca Dtα v of order α based at t = a is defined by C α a Dt v(t)

:=

(a Itn−α v (n) )(t)

1 = (n − α)



t

(t − s)n−α−1 v (n) (s) ds,

a

if the integral on the right-hand side exists. Likewise, its right-sided Djrbashian– Caputo fractional derivative Ct Dbα v of order α based at t = b is defined by C α t Db v(t)

:= (−1)

n

(t Ibn−α v (n) )(t)

(−1)n = (n − α)



b

(s − t)n−α−1 v (n) (s) ds,

t

if the integral on the right-hand side exists. The Riemann–Liouville and Djrbashian–Caputo fractional derivatives are closely connected with each other. The following result is often used to define the Djrbashian– Caputo fractional derivative in terms of the Riemann–Liouville one as ( Ca Dtα v)(t)

:=

R α a Dt



n−1 ( j)  v (a)(t − a) j v− (t), j! j=0

so as to relax the regularity requirement on the function v. Theorem 1.4 Let n − 1 < α ≤ n, n ∈ N, and v ∈ AC n (D). Then, for the left-sided fractional derivatives, there holds (RaDtα v)(t) = ( Ca Dtα v)(t) +

n−1  (t − a) j−α ( j) v (a), ( j − α + 1) j=0

(1.5)

1.1 Basics of Fractional Calculus

5

and, likewise, for the right-sided fractional derivatives, there holds (RtDbα v)(t)

=

( Ct Dbα v)(t)

+

n−1  (−1) j (b − t) j−α j=0

( j − α + 1)

v ( j) (b).

Similar to the Riemann–Liouville case, the Djrbashian–Caputo fractional derivative Ca Dtα v satisfies the following fundamental theorem of calculus. Theorem 1.5 Let α > 0, n − 1 < α < n, n ∈ N. Then the following statements hold. (i) If v ∈ L 1 (D) with a Itα−n+1 v ∈ AC(D) and a Itα−n+1 v(a) = 0, then C α α a Dt a It v

= v, a.e. D.

(ii) If v ∈ AC n (D), then αC α a It a Dt v = v −

n−1 ( j)  v (a)(t − a) j , a.e. D. j! j=0

The Djrbashian–Caputo fractional derivative C0 Dtα v has a Laplace transform relation similar to integer-order derivatives. See Appendix A.3 for the definition of Laplace transform. Note that the Laplace transform relation involves only integerorder derivatives at t = 0, indicating that one can specify initial conditions as usual. This property represents one major difference between the Djrbashian–Caputo and Riemann–Liouville fractional derivatives. Lemma 1.2 Let α > 0, n − 1 < α ≤ n, n ∈ N, such that v ∈ C n (R+ ), |v(t)| ≤ ce p0 t for large t, v (z) and for some p0 ∈ R, and v ∈ AC n [0, b] for any b, the Laplace transforms

(k)  (n) v (z) exist, and limt→∞ v (t) = 0, k = 0, 1, . . . , n − 1. Then the following relation holds for (z) > p0 : v (z) − L[C0 Dtα v](z) = z α

n−1  k=0

z α−k−1 v (k) (0).

6

1 Existence, Uniqueness, and Regularity of Solutions

1.2 Mittag–Leffler Function We shall use extensively the following two-parameter Mittag–Leffler function E α,β (z) (where (α) > 0) defined by E α,β (z) =

∞  k=0

zk , z ∈ C, (kα + β)

where (z) denotes the Gamma function; see Appendix A.1 for its basic properties. The Mittag–Leffler function E α,β (z) is a two-parameter family of entire functions in z of order α −1 and type 1 [Jin21, Proposition 3.1]. It generalizes the exponential function e z in the sense that E 1,1 (z) = e z . In the limit α = 0, the series reduces to the standard geometric series in z: E 0,β (z) = (β)−1 (1 − z)−1 . Two important members of this family are E α,1 (z) and E α,α (z), which occur in the solution operators for the subdiffusion model (1.1). There are several important properties of the Mittag–Leffler function E α,β (z) that will play a very important role in the mathematical and numerical analysis of the subdiffusion model (1.1). The notation Z− = {−1, −2, −3, . . .} denotes the set of negative integers. Lemma 1.3 Let 0 < α < 2 and β ∈ R be arbitrary, and απ < μ < min(π, απ ). 2 Then there exists a constant c = c(α, β, μ) > 0 such that for μ ≤ |arg(z)| ≤ π  |E α,β (z)| ≤

c(1 + |z|2 )−1 , β − α ∈ Z− ∪ {0}, c(1 + |z|)−1 , otherwise.

(1.6)

Moreover, for λ > 0, α > 0, and t > 0, we have C α α 0 Dt E α,1 (−λt )

= −λE α,1 (−λt α ),

d E α,1 (−λt α ) = −λt α−1 E α,α (−λt α ). dt

(1.7) (1.8)

Proof The estimate (1.6) can be found in [Pod99, Theorem 1.4] or [Jin21, Theorem 3.2] and the fact 1/ (0) = 0 (i.e., 0 is a pole of the Gamma function (z)). The identities (1.7) and (1.8) follow from termwise differentiation, since the function  E α,1 (−λt α ) is entire. Since the function E α,β (z) is entire, by termwise differentiation, we obtain the following result. Lemma 1.4 For any λ > 0, α > 0 and β ∈ R and k ∈ N, we have dk β−1 (z E α,β (−λz α )) = z β−k−1 E α,β−k (−λz α ). dz k

1.3 Existence, Uniqueness, and Sobolev Regularity

7

In particular, for β = 1 and β = α, there holds dk E α,1 (−λz α ) = −λz α−k E α,α−k+1 (−λz α ), dz k dk α−1 (z E α,α (−λz α )) = z α−k−1 E α,α−k (−λz α ). dz k One remarkable property of the function E α,1 (−t) is the complete monotonicity [Pol48]. A function f : R+ := (0, ∞) → R is said to be completely monotone if n (−1)n dtd n f (t) ≥ 0 for all t > 0 and n ∈ N. Lemma 1.5 For any α ∈ (0, 1], the function E α,1 (−t) is completely monotone on [0, ∞). The next lemma provides useful upper and lower bounds of the Mittag–Leffler function E α,1 (−t), t ≥ 0. See [Jin21, Theorem 3.6] for an elementary proof. Lemma 1.6 For any α ∈ (0, 1), there holds 1 1 ≤ E α,1 (−t) ≤ , ∀t ≥ 0. 1 + (1 − α)t 1 + (1 + α)−1 t

1.3 Existence, Uniqueness, and Sobolev Regularity In this section, we discuss the well-posedness of problem (1.1). We shall derive useful solution representations to problem (1.1) using the separation of variables technique and Laplace transform separately, and establish Sobolev regularity of the solutions. First, we describe the functional analytic setting. Let  ⊂ Rd be an open bounded smooth domain with a boundary ∂. Let A : H01 () ∩ H 2 () → L 2 () be the realization of the symmetric uniformly coercive elliptic operator −L defined in (1.2) (with smooth coefficients) in the space L 2 (), i.e., (Au)(x) = (−Lu)(x) for x ∈ , with its domain D(A) = {v ∈ H01 () : Av ∈ L 2 ()}. Then it is unbounded, closed, and self-adjoint, and by standard elliptic regularity theory [Gri85] and Sobolev embedding theorem, its inverse A−1 : L 2 () → L 2 () is compact. Thus, by the standard spectral theory for compact operators [Yos80], the spectrum of A is discrete, positive, and accumulates only at infinity. We repeat each eigenvalue of the operator A according to its (finite) multiplicity: 0 < λ1 < λ2 ≤ · · · ≤ λ j ≤ · · · → ∞, as j → ∞, and denote by ϕ j ∈ H 2 () ∩ H01 () an eigenfunction corresponding to λj:  Aϕ j = λ j ϕ j , in , ϕ j = 0, on ∂. The eigenfunctions (ϕ j )∞ j=1 can be taken to form an orthonormal basis of the space 2 L (). This choice is always employed below.

8

1 Existence, Uniqueness, and Regularity of Solutions

We use the spaces H˙ s () frequently. For s ≥ 0, we denote by H˙ s () ⊂ L 2 () the Hilbert space ∞

 H˙ s () = v ∈ L 2 () : λsj (v, ϕ j )2 < ∞ ,

(1.9)

j=1

equipped with the norm:

v 2H˙ s () =

∞ 

λsj (v, ϕ j )2 ,

j=1

with (·, ·) being the L 2 () inner product. By definition, we have

v 2H˙ s ()

∞ ∞ ∞    s s s s 2 2 2 2 = (v, λ j ϕ j ) = (v, A ϕ j ) = (A 2 v, ϕ j )2 = A 2 v 2L 2 () . j=1

j=1

j=1

For any s ≥ 0, we define the fractional power As by As v(x) =

∞ 

λsj (v, ϕ j )ϕ j ,

j=1

with its domain Dom(As ) = H 2s (). In particular, for a smooth domain , H˙ 0 () = L 2 (),

H˙ 1 () = H01 () and H˙ 2 () = H01 () ∩ H 2 (). 1

Then v H˙ 0 () = v L 2 () = (v, v) 2 is the standard L 2 () norm. Further, v H˙ 1 () is an equivalent norm in H01 () and v H˙ 2 () = Lv L 2 () is equivalent to the norm in H 2 () ∩ H01 () [Tho06, Sect. 3.1]. Since H˙ s () ⊂ L 2 (), by identifying the dual (L 2 ()) of L 2 () with itself, we have H˙ s () ⊂ L 2 () ⊂ ( H˙ s ()) . We set H˙ −s () = ( H˙ s ()) , the dual space of the space H˙ s (), which consists of all bounded linear functionals on H˙ s (). It can be characterized by the space of distributions that can be written as v=

∞  j=1

v j ϕ j (x), with

∞ 

2 λ−s j v j < ∞.

j=1

Next, we specify more explicitly the space H˙ s (). If the domain  is of class C , then H˙ s () = H s () for 0 < s < 21 , and for j = 1, 2, . . ., such that 2 j − 23 < ∞

1.3 Existence, Uniqueness, and Sobolev Regularity

s < 2j +

9

1 2

  H˙ s () = v ∈ H s () : v = Av = . . . = A j−1 v = 0 on ∂ . When s = 2 j − 23 , the condition A j−1 v = 0 on ∂ should be replaced by A j−1 v ∈ 1

H002 (), the so-called Lions–Magenes space. These results can be proved using elliptic regularity theory and interpolation [Tho06, p. 34]. If the domain  is not C ∞ , then one must restrict the range of s accordingly: if  is Lipschitz then the above relations hold for s ≤ 1, and if  is convex or C 1,1 , then we can allow s ≤ 2. Now we derive a solution representation using the separation of variables technique. By multiplying both sides of the governing equation of problem (1.1) by the eigenfunction ϕ j , integrating over the domain , and applying integration by parts twice, we obtain ∂tα (u(t), ϕ j ) = (Lu(t), ϕ j ) + ( f (t), ϕ j ) =(u(t), Lϕ j ) + ( f (t), ϕ j ) = −λ j (u(t), ϕ j ) + ( f (t), ϕ j ). Let u j (t) = (u(t), ϕ j ), f j (t) = ( f (t), ϕ j ) and u 0 j = (u 0 , ϕ j ). Then we arrive at the following system of fractional odes: ∂tα u j (t) = −λ j u j (t) + f j (t), t > 0, with u j (0) = u 0 j , for j = 1, 2, . . .. It remains to find the scalar functions u j (t), j = 1, 2, . . .. To this end, consider the following fractional ode for λ > 0: ∂tα u λ (t) = −λu λ (t) + f (t), t > 0, with u λ (0) = c0 .

(1.10)

By means of Laplace transform, the unique solution u λ (t) to the ode (1.10) is given by [Jin21, Chap. 4] α



u λ (t) = c0 E α,1 (−λt ) +

t

(t − s)α−1 E α,α (−λ(t − s)α ) f (s) ds.

0

Hence, the solution u(t) to problem (1.1) can be formally represented by u(x, t) =

∞ 

E α,1 (−λ j t α )(u 0 , ϕ j )ϕ j (x)

j=1

+

∞   j=1

t

(t − s)α−1 E α,α (−λ j (t − s)α )( f (·, s), ϕ j ) dsϕ j (x).

0

This representation can be succinctly written as  u(t) = F(t)u 0 + 0

t

E(t − s) f (s) ds,

(1.11)

10

1 Existence, Uniqueness, and Regularity of Solutions

where the solution operators F and E are defined by F(t)v =

∞ 

E α,1 (−λ j t α )(v, ϕ j )ϕ j ,

(1.12)

t α−1 E α,α (−λ j t α )(v, ϕ j )ϕ j ,

(1.13)

j=1

E(t)v =

∞  j=1

respectively. The operators F and E denote the solution operators for problem (1.1) with f ≡ 0 and u 0 ≡ 0, respectively. Note that as α → 1− , the two operators F(t) and E(t) coincide and recover that for the standard parabolic case, i.e., E(t)v = F(t)v =

∞ 

e−λ j t (v, ϕ j )ϕ j .

j=1

Next, we derive an alternative solution representation by means of (vector-valued) Laplace transform; see Appendix A.3 for the details about the transform. We begin with the necessary functional analytic framework. Recall that we have defined the operator A in L 2 () by (Au)(x) = (−Lu)(x) for x ∈ , with its domain D(A) = H 2 () ∩ H01 (). Then the operator A satisfies the following resolvent estimate (cf. [ABHN11, Example 3.7.5 and Theorem 3.7.11] and [Tho06, Theorem 6.4]):

(z + A)−1 ≤ cθ |z|−1 , ∀z ∈ θ , ∀θ ∈ (0, π ),

(1.14)

where · denotes the operator norm on L 2 (), with the sector θ ⊂ C defined by

θ := {z ∈ C \ {0} : |arg(z)| ≤ θ }. u (z) = We by

u the Laplace transform of a function u : R+ → R, i.e.,

 ∞ denote −zt e u(t) dt. We extend f from the interval (0, T ) to (0, ∞) by zero. By Lemma 0 ∂tα u(z) = z α

u (z) − z α−1 u(0). 1.2, the Laplace transform  ∂tα u(z) of ∂tα u is given by  Thus, applying Laplace transform to problem (1.1) leads to u (z) + A

u (z) =

f (z) + z α−1 u 0 . z α

The Laplace transform

u of the solution u is given by f (z) + z α−1 u 0 ).

u (z) = (z α + A)−1 (

By inverse Laplace transform and the convolution rule for Laplace transform, we have

1.3 Existence, Uniqueness, and Sobolev Regularity

 1 e zt z α−1 (z α + A)−1 u 0 dz + 2π i C  1 e zt z α−1 (z α + A)−1 u 0 dz + = 2π i C

11

 1 e zt (z α + A)−1

f (z) dz 2π i C  t   1 e zs (z α + A)−1 dz f (t − s) ds, 2π i 0 C

u(t) =

where C ⊂ C is a Hankel contour, oriented with an increasing imaginary part. Therefore, we obtain the following representation of the solution u(t):  u(t) = F(t)u 0 +

t

E(t − s) f (s) ds,

(1.15)

0

where the solution operators F(t) and E(t) are, respectively, defined by  1 e zt z α−1 (z α + A)−1 dz, 2π i θ,σ  1 E(t) := e zt (z α + A)−1 dz, 2π i θ,σ F(t) :=

(1.16) (1.17)

with the integrals over a contour θ,σ ⊂ C (oriented with an increasing imaginary part), deformed from the contour C: θ,σ = {z = ρe±iθ : ρ ≥ σ } ∪ {z = σ eiϕ : |ϕ| ≤ θ }.

(1.18)

Throughout, we fix θ ∈ ( π2 , π ) so that z α ∈ αθ for all z ∈ θ . One may deform the contour θ,σ to obtain explicit representations of F(t) and E(t) in terms of the ∞ eigenexpansion (λ j )∞ j=1 and (ϕ j ) j=1 of the operator A, and then recover the expressions (1.12) and (1.13). In this book, we use both representations interchangeably and very frequently. Now, we give several useful results. The first connects the operators E and F, where I denotes the identity operator. This result is direct from the definition of the operators. Lemma 1.7 The following identities hold AE(t) = −

d F(t) and dt

lim I − F(t) = 0.

t→0+

The next theorem summarizes smoothing properties of F(t) and E(t). The notak tion F (k) (t) = dtd k F(t) denotes the kth derivative of F(t) in t, etc. (ii) indicates that E can absorb A2 , which agrees with the asymptotic of E α,α (z) in Lemma 1.3. Theorem 1.6 For any k ∈ N ∪ {0}, the operators F and E defined in (1.16)–(1.17) satisfy that, for any t ∈ (0, T ],

12

1 Existence, Uniqueness, and Regularity of Solutions

(i) t −α A−1 (I − F(t)) + t 1−α A−1 F  (t) ≤ c; (ii) t k+1−α E (k) (t) + t k+1 AE (k) (t) + t k+1+α A2 E (k) (t) ≤ c; (iii) t k F (k) (t) + t k+α AF (k) (t) ≤ c. Proof Obviously, the following identity holds: A(z α + A)−1 = I − z α (z α + A)−1 , and thus by the resolvent estimate (1.14),

A(z α + A)−1 ≤ c, ∀z ∈ θ,σ .

(1.19)

In part (i), by Lemma 1.7 and choosing σ = t −1 in the contour θ,σ and letting zˆ = t z (with |dz| being the arc length element of θ,σ ):

A

−1

 1 F (t) = E(t) ≤ e(z)t (z α + A)−1 |dz| 2π θ,σ  α−1 ≤ct e(ˆz ) |ˆz |−α | dˆz | θ,1  α−1 ecos(θ)|ˆz | (1 + |ˆz |−1 ) |dˆz | ≤ ct α−1 . ≤ct 

θ,1

Now, for any k ∈ N0 and m = 0, 1, by choosing σ = t −1 in θ,σ and changing variables z = ρ cos ϕ + iρ sin ϕ, we have

Am F (k) (t)      zt k+α−1 m α −1  ≤ c e z A (z + A) dz  e(z)t |z|k−1+mα |dz| ≤c  θ,σ θ,σ  ∞  θ ρt cos θ k−1+mα ≤c e ρ dρ + c ecos ϕ σ k+mα dϕ ≤ ct −mα−k . σ

This implies

−θ

t k F (k) (t) + t k+α AF (k) (t) ≤ c, ∀t > 0,

thereby showing (iii). Similarly, for m = 0, 1, . . . , we can show (ii) by     1  zt k m α −1  e z A (z + A) dz

Am E (k) (t) =   2π i  θ,σ  ≤c e(z)t |z|k+(m−1)α |dz| ≤ ct (1−m)α−k−1 . θ,σ

Next, direct computation gives

1.3 Existence, Uniqueness, and Sobolev Regularity

1 2π i

AE(t) = −

 θ,σ

13

e zt z α (z α + A)−1 dz.

It follows from this identity and (1.19) that

A E 2

(k)

1 (t) ≤ 2π

 θ,σ

e(z)t |z|k+α |dz| ≤ ct −k−1−α .

Last, in view of the identity F  (t) = −AE(t) from Lemma 1.7, we have A−1 F  (t) = −E(t). Thus, (ii) implies the estimate t 1−α A−1 F  (t) ≤ c in (i). Now the bound on A−1 (I − F(t)) follows from Lemma 1.7, 

t

I − F(t) = 0

d (I − F(s)) ds = ds



t

AE(s) ds, 0

from which and (ii) it follows directly that 

−1

A (I − F(t)) ≤

t



E(s) ds ≤ c

0

t

s α−1 ds = ct α .

0

This completes the proof of the theorem.



Now, we can show the existence, uniqueness, and regularity of a solution to problem (1.1), using the solution representation (1.15). First, we introduce the concept of weak solutions for problem (1.1). Definition 1.4 For some q ∈ [0, 1], we call u a weak solution to problem (1.1) if the equation in (1.1) holds in H˙ −q () and u(t) ∈ H01 () for almost all t ∈ (0, T ] and u ∈ C([0, T ]; H˙ −q ()), with lim u(t) − u 0 H˙ −q () = 0.

t→0+

The following existence and uniqueness result holds for problem (1.1) with f ≡ 0. The inhomogeneous case can be analyzed similarly. Alternatively, one may employ the standard Galerkin approximation and energy estimates [Jin21, Sect. 6.1]. Theorem 1.7 If u 0 ∈ L 2 () and f ≡ 0, then there exists a unique weak solution u ∈ C([0, T ]; L 2 ()) ∩ C((0, T ]; H˙ 2 ()) in the sense of Definition 1.4. Proof First, we show that the function u(t) = F(t)u 0 gives a weak solution to problem (1.1). By Theorem 1.6(iii), we have

u(t) L 2 () = F(t)u 0 L 2 () ≤ c u 0 L 2 () ,

(1.20)

and, by Theorem 1.6(iii), for any t > 0,

u(t) H˙ 2 () = F(t)u 0 H˙ 2 () ≤ ct −α u 0 L 2 () .

(1.21)

14

1 Existence, Uniqueness, and Regularity of Solutions

Further, by Theorem 1.6, u ∈ C([0, T ]; L 2 ()) ∩ C((0, T ]; H˙ 2 ()). Now using the governing equation ∂tα u = Au in (1.1), ∂tα u ∈ C((0, T ]; L 2 ()), and thus the equation is satisfied in L 2 () almost everywhere. Meanwhile, by Lemma 1.7, u(t) = F(t)u 0 satisfies lim u(t) − u 0 L 2 () ≤ lim+ F(t) − I

u 0 L 2 () = 0.

t→0+

t→0

(1.22)

Thus, the function u(t) = F(t)u 0 is indeed a solution to problem (1.1) in the sense of Definition 1.4. Next, we show the uniqueness of the weak solution. It suffices to show that problem (1.1) with u 0 ≡ 0 and f ≡ 0 has only a trivial solution. By taking inner product with ϕ j and setting u j (t) = (u(t), ϕ j ), we obtain ∂tα u j (t) = −λ j u j (t), ∀t ∈ (0, T ]. Since u(t) ∈ L 2 () for t ∈ (0, T ], it follows from (1.22) that u j (0) = 0. Due to the existence and uniqueness of solutions to fractional odes [Jin21, Chap. 4], we deduce u j (t) = 0,

j = 1, 2, . . . .

2 Since the set (ϕ j )∞ j=1 is an orthonormal basis of L (), we have u ≡ 0 in  × (0, T ). This shows the uniqueness and completes the proof. 

Now, we derive the H˙ p () regularity for f ≡ 0. The regularity result in Theorem 1.8 contrasts sharply with that for homogeneous parabolic problems, which is formally recovered by setting α = 1 in the model (1.1). Note that, in the standard parabolic case, smooth and compatible data imply smooth solutions, and when f ≡ 0, the solution u is smooth for t > 0, irrespective of the smoothness of the initial data. However, in the fractional case, the solution u typically exhibits weak singularity at t = 0, and can have at most two more derivatives in space, reflecting the inherent limited smoothing property of the solution operators in the fractional case. This is intimately connected with the behavior of the Mittag–Leffler functions E α,1 (−λt α ) and t α−1 E α,α (−λt α ), especially the slow asymptotic decay in Lemma 1.3, when compared with the exponential function e−λt . The weak solution singularity at t = 0 represents the major hurdle in the analysis of the numerical methods for solving the model (1.1). Theorem 1.8 If u 0 ∈ H˙ q (), with 0 ≤ q ≤ 2, and f ≡ 0, then the solution u to problem (1.1) belongs to C([0, T ]; H˙ q ()) ∩ C((0, T ]; H˙ p ()) for q ≤ p ≤ q + 2, and ∂tα u ∈ C((0, T ]; H˙ p ()) and satisfies for any k ∈ N,

u (k) (t) H˙ p () ≤ ct −

p−q 2

∂tα u(t) H˙ p () ≤ ct −

p−q+2 α 2

α−k

u 0 H˙ q () , q ≤ p ≤ q + 2,

(1.23)

u 0 H˙ q () , q − 2 ≤ p ≤ q.

(1.24)

1.3 Existence, Uniqueness, and Sobolev Regularity

15

Proof By (1.15), the solution u is given by u(t) = F(t)u 0 . Then the bound (1.23) follows from Theorem 1.6(iii), and the bound (1.24) from the governing equation ∂tα u = Au. The continuity of u(t) in H˙ q () up to t = 0 is given in Lemma 1.7.  Next, we turn to problem (1.1) with u 0 ≡ 0. In this case, Bochner–Sobolev spaces H s, p (0, T ; X ), with X being a Banach space, are useful. We define L r (0, T ; X ) = {v(t) ∈ X for a.e. t ∈ (0, T ) and v L r (0,T ;X ) < ∞}, for any r ≥ 1, and the norm · L r (0,T ;X ) is defined by

v L r (0,T ;X ) =

⎧  ⎪ ⎨ ⎪ ⎩

T

v(t) rX

0

r1 dt

, r ∈ [1, ∞),

esssupt∈(0,T ) v(t) X , r = ∞.

For any s ≥ 0 and 1 < p < ∞, we denote by H s, p (0, T ; X ) the space of functions v : (0, T ) → X , with the norm defined by complex interpolation between H s, p (0, T ; X ) and H s, p (0, T ; X ) (where · and · denote the floor and ceiling operators, respectively). Equivalently, the space is equipped with the quotient norm v H s, p (R;X ) := inf F −1 [(1 + |ξ |2 ) 2 F ( v )(ξ )] L p (R;X ) ,

v H s, p (0,T ;X ) := inf  s

 v

 v

where the infimum is taken over all possible  v that extend v from (0, T ) to R, and F denotes the Fourier transform, cf. Appendix A.3. The space H s, p (0, T ; X ) is closely related to Fourier multiplier theorems, and will be extensively used in Chaps. 8 and 14. For any 0 < s < 1, the Bochner–Sobolev–Slobodeckiˇı seminorm | · |W s, p (0,T ;X ) is defined by |v|W s, p (0,T ;X ) :=

  T 0

0

T

 1p

v(t) − v(ξ ) X dt dξ , |t − ξ |1+ ps p

and the full norm · W s, p (0,T ;X ) by 1  p p

v W s, p (0,T ;X ) = v L p (0,T ;X ) + |v|W s, p (0,T ;X ) p . Likewise, the full norm · W k+s, p (0,T ;X ) , with k ≥ 0 and k ∈ N ∪ {0}, is defined by

v W k+s, p (0,T ;X ) =

k 

v ( j) L p (0,T ;X ) + |v (k) |W s, p (0,T ;X ) p

p

 1p

.

j=0

The space W k+s, p (0, T ; X ) can be obtained via real interpolation between W k, p (0, T ; X ) = H k, p (0, T ; X ) and W k+1, p (0, T ; X ) = H k+1, p (0, T ; X ), i.e.,

16

1 Existence, Uniqueness, and Regularity of Solutions

W k+s, p (0, T ; X ) = [W k, p (0, T ; X ), W k+1, p (0, T ; X )]s, p . Note that W s, p (0, T ) is equivalent to the Besov space B sp, p (0, T ). In general, H s, p (0, T ; X ) and W s, p (0, T ; X ) are not equivalent (unless p = 2). For s1 > s > 0, H s, p (0, T ; X ) ⊂ W s, p (0, T ; X ) ⊂ H s1 , p (0, T ; X ) if p ∈ [2, ∞), whereas W s, p (0, T ; X ) ⊂ H s, p (0, T ; X ) ⊂ W s1 , p (0, T ; X ) if p ∈ (1, 2]. See more details in [Ste70, Chap. V, Theorem 5]. Now, we turn to the L p estimate in time. Given a Banach space X and a closed linear operator A with domain D(A) ⊂ X , the abstract time-fractional evolution equation  ∂tα u(t) + Au(t) = f (t), t ∈ (0, T ], (1.25) u(0) = 0, is said to have the property of maximal L p regularity, if for each f ∈ L p (0, T ; X ), problem (1.25) possesses a unique solution u in the space H α, p (0, T ; X ) ∩ L p (0, T ; D(A)). That is, both terms on the left-hand side belong to L p (0, T ; X ). The following very useful maximal L p -regularity holds. It recovers the classical maximal regularity estimates for standard parabolic problems as α → 1− . It can be found in [Baj01, Chaps. 4 and 5]. The proof below is taken from [JLZ20a, Theorem 2.2] and is a straightforward application of the operator-valued Fourier multiplier theorem [Wei01, Theorem 3.4] (see Theorem 8.1 for the complete statement). The result holds for more general X being a umd space. Theorem 1.9 If u 0 ≡ 0 and f ∈ L p (0, T ; L 2 ()) with 1 < p < ∞, then problem (1.1) has a unique solution u ∈ L p (0, T ; H˙ 2 ()) such that ∂tα u ∈ L p (0, T ; L 2 ()) and

∂tα u L p (0,T ;L 2 ()) + Au L p (0,T ;L 2 ()) ≤ c f L p (0,T ;L 2 ()) , where the constant c does not depend on f and T . Proof For f ∈ L p (0, T ; L 2 ()), extending f to be zero on  × [(−∞, 0) ∪ (T, ∞)] yields f ∈ L p (R; L 2 ()) and

f L p (R;L 2 ()) = f L p (0,T ;L 2 ()) .

(1.26)

Further, we have R α ∂tα f (t) = −∞ ∂t f (t), ∀t ∈ [0, T ] and

 R α −∞∂t f

= (iξ )α  f (ξ )

where  denotes taking Fourier transform in t (i.e.,  f ≡ F [ f ] the Fourier transform of f , cf. Appendix A.3), cf. [Jin21, Theorem 2.8]. Then, the function f (ξ )] u(t) = F −1 [((iξ )α + A)−1 

1.3 Existence, Uniqueness, and Sobolev Regularity

17

is a solution of problem (1.1) and u (ξ ) = (iξ )α ((iξ )α + A)−1  (iξ )α f (ξ ). The self-adjoint operator A : D(A) → L 2 () is invertible from L 2 () to D(A) and generates a bounded analytic semigroup. Thus, the operator (iξ )α ((iξ )α + A)−1

(1.27)

is bounded from L 2 () to D(A) in a small neighborhood N of ξ = 0. Further, in N, the operator ξ

d [(iξ )α ((iξ )α + A)−1 ] =α(iξ )α ((iξ )α + A)−1 − α(iξ )2α ((iξ )α + A)−2 (1.28) dξ

is also bounded. By the resolvent estimate (3.20), for ξ away from zero, the following inequality (iξ )α ((iξ )α + A)−1 ≤ c implies the boundedness of (1.27) and (1.28). Since boundedness of operators is equivalent to R-boundedness of operators in L 2 () (see Chap. 8 or [KW04, p. 75] for the concept of R-boundedness), the boundedness of (1.27) and (1.28) implies that (1.27) is an operator-valued Fourier multiplier (cf. Theorem 8.2 or [Wei01, Theorem 3.4]), and thus u (ξ )] L p (R;L 2 ())

∂tα u L p (R;L 2 ()) ≤ F −1 [(iξ )α −1 α = F [(iξ ) ((iξ )α + A)−1  f (ξ )] L p (R;L 2 ()) ≤ c F −1 [  f (ξ )] L p (R;L 2 ()) ≤ c f L p (R;L 2 ()) . This and (1.26) imply the bound on ∂tα u L p (0,T ;L 2 ()) . The bound on

Au L p (0,T ;L 2 ()) follows similarly by replacing (iξ )α ((iξ )α + A)−1 with A((iξ )α + A)−1 in the proof. If αp > 1, by Sobolev embedding theorem, u ∈ C([0, ∞); L 2 ()), and hence limt→0+ u(t) L 2 = 0. That is, the initial condition u(0) = 0 holds in a classical sense. If pα ≤ 1, the zero initial condition needs to be interpreted in a weak sense [KRY20, Theorem 4.1].  Next, we derive pointwise in time regularity. Theorem 1.10 Let u be the solutionto problem (1.1) with u 0 ≡ 0, and k ∈ N. If f ∈ t C k−1 ([0, T ]; H˙ q ()), q ≥ −1, and 0 (t − s)α−1 f (k) (s) H˙ q () ds < ∞, for any t ∈ (0, T ], then, for any q ≤ p < q + 2 and k ≥ 0, there holds  (k)  u (t) ˙ p

H ()

≤c

k−1  j=0 t

 +

0

t (1−

p−q 2

)α− j−1

(t − s)(1−

p−q 2

f (k− j−1) (0) H˙ q ()

)α−1

f (k) (s) H˙ q () ds.

18

1 Existence, Uniqueness, and Regularity of Solutions

t Similarly, if f ∈ C k ([0, T ]; H˙ q ()), q ≥ −1, and 0 f (k+1) (s) H˙ q () ds < ∞ for any t ∈ (0, T ], then, for any p = q + 2 and k ≥ 0, there holds k   (k)  u (t) ˙ q+2 ≤ c t − j f (k− j) (0) H˙ q () + () H



t 0

j=0

f (k+1) (s) H˙ q () ds.

Proof In view of the solution representation (1.15), the solution u(t) is given by  u(t) =

t



t

E(t − s) f (s)ds =

0

E(s) f (t − s) ds.

0

Differentiating the representation k times in t yields (with the convention summation with lower index greater than upper index being identified with zero) u (k) (t) =

k−1 

E ( j) (t) f (k− j−1) (0) +



t

E(s) f (k) (t − s) ds,

0

j=0

and thus, for q ≤ p < q + 2, by Theorem 1.6(ii), we obtain  (k)  u (t) ˙ p

H ()

≤

k−1 

E ( j) (t) f (k− j−1) (0) H˙ p () +

j=0

≤

k−1 

A

p−q 2

E

( j)

(t)

f

(k− j−1)

 0

t

E(s) f (k) (t − s) H˙ p () ds 

(0) H˙ q () +

≤c

t

(1− p−q 2 )α− j−1

f

(k− j−1)

A

p−q 2

E(s)

f (k) (t − s) H˙ q () ds

0

j=0 k−1 

t

 (0) H˙ q () + c

j=0

t

s (1−

p−q 2

0

)α−1

f (k) (t − s) H˙ q () ds.

This shows the first estimate. In the event p = q + 2, by the identity AE(t) from Lemma 1.7, and integration by parts, we obtain Au (k) (t) =

k−1 

AE ( j) (t) f (k− j−1) (0) +



=

AE ( j) (t) f (k− j−1) (0) +

=

j=0

 0

j=0 k−1 

AE

( j)

(t) f

− F(t)) =

AE(s) f (k) (t − s) ds

0

j=0 k−1 

t

d (I dt

(k− j−1)

t

d (I − F(s)) f (k) (t − s) ds ds

(0) + (I − F(t)) f

(k)

 (0) + 0

t

(I − F(s)) f (k+1) (t − s) ds,

1.3 Existence, Uniqueness, and Sobolev Regularity

19

since I − F(0) = 0, cf. Lemma 1.7. Then, by Theorem 1.6 and repeating the preceding argument, we obtain the second assertion.  In Theorem 1.10, setting k = 0 and q = 0 gives 

u(t) H˙ 2 () ≤ c f (0) L 2 () +

0

t

f  (s) L 2 () ds.

Thus, with f ∈ L ∞ (0, T ; L 2 ()) only, the solution u generally does not belong to L ∞ (0, T ; H˙ 2 ()), indicating again the limited smoothing property of the solution operator E(t). Indeed, if u 0 = 0, and f ∈ L ∞ (0, T ; H˙ q ()), −1 ≤ q ≤ 1, then the solution u ∈ L ∞ (0, T ; H˙ q+2− ()) for any 0 <  < 1, and 

u(t) H˙ q+2− () ≤ c −1 t 2 α f L ∞ (0,t; H˙ q ()) . Actually, by (1.15) and Theorem 1.6(ii),  t  t   

u(t) H˙ q+2− () =  E(t − s) f (s) ds  q+2− ≤

E(t − s) f (s) H˙ q+2− () ds () H˙ 0 0  t   (t − s) 2 α−1 f (s) H˙ q () ds ≤ c −1 t 2 α f L ∞ (0,t; H˙ q ()) , ≤c 0

which shows the desired estimate. The  factor in the estimate reflects the limited smoothing property of the solution operator E(t) of the subdiffusion model (1.1).

Notes Fractional calculus has a long and glorious history, dating back to a letter from Gottfried Wilhelm Leibniz responding to Guillaume de l’Hôpital, dated on September 30, 1695, discussing the meaning of derivative of half an order. There are several excellent textbooks on fractional calculus, e.g., [OS74, MR93, Pod99, KST06, Die10] and the encyclopedic monograph [SKM93]. Our description in Sect. 1.1 can be found in any of these textbooks, and follows largely the book [Jin21, Chap. 1]. The Mittag–Leffler function E α,β (z) plays a central role in the study of fractional differential equations, and the role is so prominent that it has earned the worthy name “the queen function of fractional calculus,” as pointed out by Mainardi and Gorenflo [MG07]. Many important properties of the function were derived by Mkhitar Djrbashian [Djr66] (in Russian) (see also the book [Djr93] for an account of some of these results). A fairly comprehensive treatment of many important properties of the function and its broad range of applications can be found in the recent monograph [GKMR20]. The material in Sect. 1.2 can be found in the book [Jin21, Sect. 3.1]. The solution theory of time-fractional evolution equations is of relatively recent origin. One early pioneering contribution in this direction is Eidelman and Kuchubei

20

1 Existence, Uniqueness, and Regularity of Solutions

[EK04], which contains a detailed study of fundamental solutions of time-fractional diffusion in Rd , using the H-Fox function. The current presentation in Sect. 1.3 largely follows the influential work of Sakamoto and Yamamoto [SY11]: besides several illuminating applications in inverse problems, it gives a first rigorous treatment of the solution theory in the Hilbert space H˙ s (), regularity estimates and covering both cases of subdiffusion and diffusion wave problems (i.e., α ∈ (1, 2)). The analysis uses the standard separation of variables technique and decay property of E α,β (z) in Lemma 1.3 and the complete monotonicity of E α,1 (−t) in Lemma 1.5. See also [Zac09, McL10] for other early works on subdiffusion-type models. Since then, the mathematical analysis of the subdiffusion model (1.1) has flourished, and many fundamental results have been obtained. For example, an operator theoretic treatment of the Djrbashian–Caputo fractional derivative can be found in the papers [GLY15, Yam22a, Yam22b] and the monograph [KRY20], which contain in-depth discussions on the solution theory when the Djrbashian–Caputo fractional derivatives are interpreted within Sobolev spaces. This theory is more involved, but the related results are expected to be fundamental in the setting of low solution regularity, as is commonly encountered in numerical analysis, optimal control, and inverse problems. See also the recent work [Kia21] for a detailed discussion on different concepts of solutions and their equivalence relations. The discussion in Sect. 1.3 is taken from [Jin21, Sect. 6.2].

Chapter 2

Spatially Semidiscrete Discretization

In this chapter, we consider the spatial semidiscretization of the standard subdiffusion model for u(x, t): ⎧ α ⎪ ⎨ ∂t u − u = f, in  × (0, T ], u = 0, on ∂ × (0, T ], (2.1) ⎪ ⎩ u(0) = u 0 , in , where  is a bounded convex polygonal domain in Rd (d = 1, 2, 3) with a boundary ∂, u 0 , and f are, respectively, given initial data and source data, and T > 0 is a fixed terminal time. The spatial discretization methods include the standard Galerkin finite element (fem) and its variant, the lumped mass method. These methods are well suited for the subdiffusion model with general problem data, for which the solution u has only limited spatial regularity, cf. Sect. 1.3. In this chapter, we aim at deriving (nearly) optimal error estimates with respect to the regularity of problem data. Note that spatially semidiscrete schemes are still not implementable in practice. Nonetheless, the study on spatially semidiscrete schemes provides valuable insights into illuminating the role of problem data regularity in the error analysis and also plays an important role in the analysis of fully discrete schemes. In this chapter, we first recall preliminary materials of Galerkin fems for standard second-order elliptic pdes, and introduce a spatially semidiscrete Galerkin scheme for problem (2.1). Then we derive error bounds for the approximate solutions using two different approaches, one based on the standard separation of variables technique and Mittag–Leffler functions, and the other based on Laplace transform (and suitable resolvent estimate). These approaches parallel the regularity analysis in Sect. 1.3. Last, we discuss a variant of the Galerkin fem scheme, i.e., the lumped mass fem.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Jin and Z. Zhou, Numerical Treatment and Analysis of Time-Fractional Evolution Equations, Applied Mathematical Sciences 214, https://doi.org/10.1007/978-3-031-21050-1_2

21

22

2 Spatially Semidiscrete Discretization

2.1 Galerkin Finite Element Method Before discussing the subdiffusion problem (2.1), we briefly review some basic materials about Galerkin fems for the following second-order elliptic problem: 

−u = f, in , u = 0, on ∂,

(2.2)

where f ∈ H −1 () = (H01 ()) or its subspace. We first rewrite the problem into a weak form. By multiplying the elliptic equation by a test function v ∈ H01 (), integrating over the domain , and applying integration by parts, we obtain the following weak formulation : (∇u, ∇v) = ( f, v), ∀v ∈ H01 (),

(2.3)

where we have used the standard L 2 () inner products:  (u, v) =



uvdx and (∇u, ∇v) =

  d ∂u ∂v dx.  j=1 ∂ x j ∂ x j

Note that for a convex polygonal domain , the following elliptic regularity pickup holds: u H p+2 () ≤ c f  H p () , ∀ p ∈ [−1, 0]. (2.4) This estimate with p = 0 does not hold for nonconvex polygonal domains. Now, we describe the finite element discretization of problem (2.2). Let {Th }0 0 by Fh (t)vh =

Nh 

E α,1 (−λhj t α )(vh , ϕ hj )ϕ hj ,

(2.17)

t α−1 E α,α (−λhj t α ) (vh , ϕ hj ) ϕ hj .

(2.18)

j=1

E h (t)vh =

Nh  j=1

26

2 Spatially Semidiscrete Discretization

Repeating the argument in Sect. 1.3 indicates that the solution u h (t) ∈ X h of the discrete problem (2.16) can be expressed by  u h (t) = Fh (t)u 0h +

t

E h (t − s) f h (s) ds.

(2.19)

0

Also, on the finite element space X h , we introduce the discrete norm ||| · ||| H˙ p () for any p ∈ R defined by |||vh |||2H˙ p () =

Nh  (λhj ) p (vh , ϕ hj )2 , ∀vh ∈ X h .

(2.20)

j=1

Clearly, the norm ||| · ||| H˙ p () is well defined for all real p. By the definition of the discrete Laplacian −h , we have |||vh ||| H˙ 1 () = vh  H˙ 1 () and also |||vh ||| H˙ 0 () = vh  L 2 () for any vh ∈ X h . So, there will be no confusion in using vh  H˙ p () instead of |||vh ||| H˙ p () for p = 0, 1 and vh ∈ X h . Analogously, we define the associated spaces ||| · ||| L r (0,T ; H˙ p ()) , r ∈ [1, ∞], on the space X h . Then the following inverse inequality holds. Lemma 2.2 For any p ≥ q, there exists a constant c independent of h such that |||vh ||| H˙ p () ≤ ch q− p |||vh ||| H˙ q () , ∀vh ∈ X h . Proof For quasi-uniform triangulations Th , the following inverse inequality holds: ∇vh  L 2 () ≤ ch −1 vh  L 2 () , ∀vh ∈ X h . By the definition of −h and Courant–Fischer theorem, the above inequality implies max λhj ≤ ch −2 .

j=1,...,Nh

Thus, for the norm ||| · ||| H˙ p () defined in (2.20), there holds for any real p ≥ q |||vh |||2H˙ p ()

≤

max(λhj ) p−q j

Nh  (λhj )q (vh , ϕ hj )2 ≤ ch 2(q− p) |||vh |||2H˙ q () . j=1

This completes the proof of the lemma.



Next, we discuss smoothing properties of the operators Fh and E h . For E h , while p = 0, 1, the parameter q can be arbitrary as long as p − 4 ≤ q ≤ p. This flexibility in the choice of q is essential for deriving error estimates for low-regularity problem data. Lemma 2.3 Let the operators Fh and E h be defined by (2.17) and (2.18), respectively. Then, for any vh ∈ X h , the following estimates for all t > 0.

2.2 Error Analysis via Mittag–Leffler Functions

27

(i) For = 0, q ≤ p ≤ q + 2 and for = 1, p ≤ q ≤ p + 2, the operator Fh satisfies p−q |||(∂tα ) Fh (t)vh ||| H˙ p () ≤ ct −( + 2 )α |||vh ||| H˙ q () . (ii) The operator E h satisfies  |||E h (t)vh ||| H˙ p () ≤

ct −1+(1+

q− p 2 )α

|||vh ||| H˙ q () ,

ct −1+α |||vh ||| H˙ q () ,

p − 4 ≤ q ≤ p, p < q.

Proof (i) By the definition of Fh , for the case = 0, by Lemma 1.3, we get for q ≤ p |||Fh vh (t)|||2H˙ p () =

Nh  (λhj ) p |E α,1 (−λhj t α )|2 |(vh , ϕ hj )|2 j=1

≤ ct −( p−q)α

Nh 

(λhj t α ) p−q (1 + λhj t α )−2 (λhj )q (vh , ϕ hj )2

j=1

≤ ct −( p−q)α

Nh  (λhj )q |(vh , ϕ hj )|2 = ct −( p−q)α |||vh |||2H˙ q () , j=1

since for q ≤ p and p ≤ q ≤ p + 2, by Young’s inequality, there holds max (λhj t α ) p−q (1 + λhj t α )−2 ≤ c.

j=1,...,Nh

The estimate for = 1 is obtained analogously. Indeed, by the identity (1.7), |||∂tα Fh vh (t)|||2H˙ p () =

Nh 

(λhj ) p |(∂tα Fh (t)vh , ϕ hj )|2

j=1

=

Nh 

(λhj )2+ p |E α,1 (−λhj t α )|2 |(vh , ϕ hj )|2 .

j=1

Now, appealing to Lemma 1.3 and Young’s inequality again, we obtain |||∂tα Fh vh (t)|||2H˙ p ()

≤ct

−(2+ p−q)α

Nh  (λhj t α )2+ p−q (1 + λhj t α )−2 (λhj )q |(vh , ϕ hj )|2 j=1

≤ct −(2+ p−q)α

Nh  (λhj )q |(vh , ϕ hj )|2 = ct −(2+ p−q)α |||vh |||2H˙ q () . j=1

The desired estimate follows from this immediately.

28

2 Spatially Semidiscrete Discretization

(ii) By the definition of E h and using Lemma 1.3, we have for any p ∈ R and q≤p |||E h (t)vh |||2H˙ p ()

=t

−2+2α

Nh 

2 E α,α (−λhj t α )(λhj ) p (vh , ϕ hj )2

j=1

≤ ct −2+(2+q− p)α max (λhj t α ) p−q (1 + (λhj t α )2 )−2 j=1,...,Nh

≤ ct

−2+(2+q− p)α

Nh 

(λhj )q (vh , ϕ hj )2

j=1

|||vh |||2H˙ q () ,

where the last line follows from 0 ≤ p − q ≤ 4. The assertion for p < q follows h are bounded away from zero independent of the mesh from the fact that (λhj ) Nj=1 size h.  Now we analyze the semidiscrete Galerkin scheme (2.11), first for u 0 ≡ 0 and f ≡ 0, and then for u 0 ≡ 0 and f ≡ 0. Theorem 2.2 Let u and u h be the solutions of problems (2.1) and (2.11) with f ≡ 0, respectively, and let eh = u h − u. (i) If u 0 ∈ H˙ 2 () and u 0h = Rh u 0 , then eh (t) L 2 () + h∇eh (t) L 2 () ≤ ch 2 u 0  H˙ 2 () . (ii) If u 0 ∈ H˙ q (), q = 0, 1 and u 0h = Ph u 0 , then with h = | log h| q

eh (t) L 2 () + h∇eh (t) L 2 () ≤ ch 2 h t −(1− 2 )α u 0  H˙ q () .

(2.21)

Proof (i) For u 0 ∈ H˙ 2 (), q = 1, 2, in a customary way, we split the error eh into eh = (u h − Rh u) + (Rh u − u) := ϑ + . By Lemma 2.1 and Theorem 1.8, we have for any t > 0 and q = 1, 2, (t) L 2 () + h∇(t) L 2 () ≤ ch 2 t −(1− 2 )α u 0  H˙ q () , u 0 ∈ H˙ q (), q

so for (i), it suffices to bound ϑ(t) by ϑ(t) L 2 () + h∇ϑ(t) L 2 () ≤ ch 2 u 0  H˙ 2 () . From the identity (2.15), it follows that ϑ satisfies ∂tα ϑ(t) − h ϑ(t) = −Ph ∂tα (t), 0 < t ≤ T.

(2.22)

2.2 Error Analysis via Mittag–Leffler Functions

29

For u 0 ∈ H˙ q (), q = 1, 2, the Ritz projection Rh u 0 is well defined, so that ϑ(0) = 0 and hence,  t

ϑ(t) = − 0

E h (t − s)Ph ∂sα (s) ds.

By Lemma 2.3(ii) with p = 1 and q = 0, the stability of Ph in Lemma 2.1, and Theorem 1.8, we find that for q = 1, 2, α

∇ E h (t − s)Ph ∂sα (s) L 2 () ≤ c(t − s) 2 −1 ∂sα (s) L 2 () α

α

q

≤ch(t − s) 2 −1 ∂sα u(s) H˙ 1 () ≤ ch(t − s) 2 −1 s (− 2 + 2 )α u 0  H˙ q () . 3

Substituting this inequality gives that, for q = 1, 2,  ∇ϑ(t) L 2 () ≤ ch

t 0

α

q

(t − s) 2 −1 s (− 2 + 2 )α ds u 0  H˙ q () 3

q

≤ cht −(1− 2 )α u 0  H˙ q () ,

(2.23)

since, for α < 1, with B(·, ·) being the Beta function (cf. Appendix A.1), 

t

(t − s)

0

α 2 −1

s

(− 23 + q2 )α

ds = t

−(1− q2 )α



1

α

q

(1 − s) 2 −1 s (− 2 + 2 )α ds 3

0 q

= B( α2 , (− 23 + q2 )α + 1)t −(1− 2 )α . α + 1 > 0, the value B(α, (− 32 + q2 )α + 1) is Since for q = 1, 2, α2 > 0 and −3+q 2 finite. Taking q = 2 yields the desired estimate for ∇ϑ. Next, by Lemma 2.3(ii) with p = q = 0 and Theorem 1.8, we get  t E h (t − s)Ph ∂sα (s) L 2 () ds ϑ(t) L 2 () ≤ 0  t  t (t − s)α−1 ∂sα u(s) H˙ 2 () ds ≤c (t − s)α−1 ∂sα (s) L 2 () ds ≤ ch 2 0 0  t (t − s)α−1 s −α dsu 0  H˙ 2 () = cB(α, 1 − α)h 2 u 0  H˙ 2 () . (2.24) ≤ch 2 0

This shows the estimate in assertion (i). (ii) Since the Ritz projection Rh u 0 is not defined for u 0 ∈ L 2 (), we use instead the L 2 ()-projection Ph u 0 , and split the error eh into ϑ + . eh = (u h − Ph u) + (Ph u − u) :=  By Lemma 2.1 and Theorem 1.8, for q = 0, 1, we have q

(t) L 2 () ≤ ch 2 u(t) H˙ 2 () ≤ ch 2 t −(1− 2 )α u 0  H˙ q () .  (t) L 2 () + h∇

30

2 Spatially Semidiscrete Discretization

Thus, we only need to estimate the term  ϑ: ϑ (t) L 2 () ≤ ch 2 h t −(1− 2 )α u 0  H˙ q () .  ϑ (t) L 2 () + h∇  q

Since Ph ∂tα  = ∂tα Ph (Ph u − u) = 0 and by the identity h Rh = Ph , cf. (2.15),  ϑ satisfies ϑ (t) − h  ϑ (t) = −h (Rh u − Ph u)(t), 0 < t ≤ T, with  ϑ (0) = 0. ∂tα  By the formula (2.19),  ϑ (t) = −



t

E h (t − s)h (Rh u − Ph u)(s) ds.

0

Then, by Lemma 2.3(ii) with p = 0, 1 and q = p − 2 + , for any > 0, we have  ϑ (t) H˙ p () ≤



t

0



E h (t − s)h (Rh u − Ph u)(s) H˙ p () ds t

≤c 0



t

≤c 0



(t − s) 2 α−1 |||h (Rh u − Ph u)(s)||| H˙ p−2+ () ds

(t − s) 2 α−1 |||(Rh u − Ph u)(s)||| H˙ p+ () ds := I.

Further, we apply the inverse inequality in Lemma 2.2 to Rh u − Ph u, the bounds in Lemma 2.1 and Theorem 1.8, and obtain |||(Rh u − Ph u)(s)||| H˙ p+ () ≤ ch − (Rh u − Ph u)(s) H˙ p () q

≤ch 2− p− u(s) H˙ 2 () ≤ ch 2− p− s −(1− 2 )α u 0  H˙ q () . Consequently, we get  t q (t − s) 2 α−1 s −(1− 2 )α ds u 0  H˙ q () I ≤ ch 2− p− 0

q = cB 2 α, 1 − α + q2 α h 2− p− t −(1− 2 − 2 )α u 0  H˙ q () q

≤ c −1 h 2− p− t −(1− 2 )α u 0  H˙ q () . The last inequality follows from the identity B( 2 α, 1

−α+

q α) 2

=

( 2 α) (1 − α + q2 α)

(1 − α +

q+ α) 2

2.2 Error Analysis via Mittag–Leffler Functions

31

(cf. (A.11) in the appendix) and the asymptotic ( 2 α) ∼ The desired assertion in (ii) follows by choosing = −1 h .

2 α

as → 0+ , cf. (A.9). 

The estimates (2.22) and (2.23) imply the following bound on the error eh for u 0 ∈ H˙ 1 () and u 0h = Rh u 0 : α

∇eh (t) L 2 () ≤ cht − 2 u 0  H˙ 1 () .

(2.25)

By Lemma 2.3(ii), we can improve the estimate of ϑ(t) for q = 2 to O(h 2 ) at the α expense of a factor O(t − 2 ): α

∇ E h (t − s)Ph ∂sα (s) L 2 () ≤ ch 2 (t − s) 2 −1 ∂sα u(s) H˙ 2 () α

≤ ch 2 (t − s) 2 −1 s −α u 0  H˙ 2 () , α

which yields ∇ϑ L 2 () ≤ ch 2 t − 2 u 0  H˙ 2 () . Next, we analyze the scheme (2.11) for the inhomogeneous case, i.e., f ≡ 0 and u 0 ≡ 0. First, we give a stability result for the scheme (2.11). Lemma 2.4 Let u h be the solution of the scheme (2.11) with u 0 ≡ 0 and f ≡ 0. Then, for any p > −1 and 0 < < 1, |||∂tα u h |||2L 2 (0,T ; H˙ p ()) + |||u h |||2L 2 (0,T ; H˙ p+2 ()) ≤ ||| f h |||2L 2 (0,T ; H˙ p ()) ,

|||u h (t)||| H˙ p+2− () ≤ c −1 t 2 α ||| f h ||| L ∞ (0,t; H˙ p ()) . Proof The solution u h of (2.11) is represented by (2.19), and hence |||u h (t)|||2H˙ p+2 ()

=

Nh 



 (λhj ) p+2 

j=1

=

Nh 

0

t

2  (t − s)α−1 E α,α (−λhj (t − s)α )( f h (s), ϕ hj ) ds 

 t 2   (λhj ) p  λhj (t − s)α−1 E α,α (−λhj (t − s)α )( f h (s), ϕ hj ) ds  . 0

j=1

Then by Young’s inequality for convolution, we deduce |||u h |||2L 2 (0,T ; H˙ p+2 ())

 T 2  T Nh  h p h 2 h α−1 h α ≤ (λ j ) |( f h (t), ϕ j )| dt |λ j t E α,α (−λ j t )| dt 0

j=1

0

≤ ||| f h (t)|||2L 2 (0,T ; H˙ p ()) , where the last line follows from the nonnegativity of the function E α,α (−λhj t α ), t > 0, cf. Lemma 1.5 so that  T  T |λhj t α−1 E α,α (−λhj t α )| dt = λhj t α−1 E α,α (−λhj t α ) dt = 1 − E α,1 (−λhj T α ) ≤ 1. 0

0

32

2 Spatially Semidiscrete Discretization

Now the first estimate follows from this and the triangle inequality. The second estimate follows from Lemma 2.3(ii) with 0 < < 1 

t

|||E h (t − s) f h (s)||| H˙ p+2− () ds |||u h (·, t)||| H˙ p+2− () ≤ 0  t ≤ c (t − s) 2 α−1 ds||| f h ||| L ∞ (0,t; H˙ p ()) ≤ c −1 t 2 α ||| f h ||| L ∞ (0,t;H p ()) . 0



This completes the proof of the lemma.

Now we can give an error bound for f ∈ L (0, T ; H˙ q ()), −1 < q ≤ 0. Theorem 2.3 Let f ∈ L ∞ (0, T ; H˙ q ()), −1 < q ≤ 0, and u and u h be the solutions of problems (2.1) and (2.11) with u 0 ≡ 0 and f h = Ph f , respectively, and let eh = u h − u. Then, with h = | log h|, there hold ∞

eh  L 2 (0,T ;L 2 ()) + h∇eh  L 2 (0,T ;L 2 ()) ≤ ch 2+q  f  L 2 (0,T ; H˙ q ()) , eh (t) L 2 () + h∇eh (t) L 2 () ≤ ch 2+q 2h  f  L ∞ (0,t; H˙ q ()) . Proof We employ the L 2 ()-projection Ph u, and split the error eh into ϑ + . eh = (u h − Ph u) + (Ph u − u) :=  By Lemma 2.1 and Theorem 1.9, we have  L 2 (0,T ;L 2 ())   L 2 (0,T ;L 2 ()) + h∇ ≤ch 2+q u L 2 (0,T ; H˙ 2+q ()) ≤ ch 2+q  f  L 2 (0,T ; H˙ q ()) . Meanwhile, we have Ph ∂tα  = ∂tα Ph (Ph u − u) = 0, and by the identity (2.15),  ϑ satisfies ϑ (t) − h  ϑ (t) = −h (Rh u − Ph u)(t), 0 < t ≤ T, with  ϑ (0) = 0. ∂tα  In view of the solution representation (2.19),  ϑ (t) can be represented by  ϑ (t) = −



t

E h (t − s)h (Rh u − Ph u)(s) ds.

0

Then by Lemmas 2.4 and 2.1, and Theorem 1.9, we have for p = 0, 1:  ϑ (t)2L 2 (0,T ; H˙ p ())  T  ≤c |||h (Rh u − Ph u)(t)|||2H˙ p−2 () dt ≤ c ≤ch

0 4+2q−2 p

u2L 2 (0,T ; H˙ 2+q ()) ≤ ch 4+2q−2 p 

T

|||(Rh u − Ph u)(t)|||2H˙ p () 0 f 2L 2 (0,T ; H˙ q ()) .

dt

2.2 Error Analysis via Mittag–Leffler Functions

33

Combining the last two estimates yields the first assertion. Likewise, by Lemma 2.1 and Theorem 1.10, the following estimate holds: (t) L 2 () ≤ ch 2+q− u(t) H˙ 2+q− ()  (t) L 2 () + h∇ ≤ c −1 h 2+q−  f  L ∞ (0,t; H˙ q ()) . Now, the choice h = | log h| and = −1 h yields (t) L 2 () ≤ c h h 2+q  f  L ∞ (0,t; H˙ q ()) .  (t) L 2 () + h∇

(2.26)

By Lemma 2.3(ii), we deduce that for p = 0, 1  ϑ (t) H˙ p () ≤



t 0



E h (t − s)h (Rh u − Ph u)(s) H˙ p () ds t

≤c 

0 t

≤c 0



(t − s) 2 α−1 |||h (Rh u − Ph u)(s)||| H˙ p−2+ () ds

(t − s) 2 α−1 |||Rh u(s) − Ph u(s)||| H˙ p+ () ds := I.

Further, we apply the inverse estimate from Lemma 2.2 to Rh u − Ph u and the bounds in Lemma 2.1, to Ph u − u and Rh u − u, respectively, and deduce I ≤ ch −



t



(t − s) 2 α−1 Rh u(s) − Ph u(s) H˙ p () ds 0  t 2+q− p−2 (t − s) 2 α−1 u(s) H˙ 2+q− () ds. ≤ ch 0

Further, by applying Theorem 1.10 and choosing = −1 h , we get I ≤ c

−1 2+q− p−2

h

  f  L ∞ (0,t; H˙ q ())

t





(t − s) 2 α−1 s 2 α ds 0

≤ ch 2+q− p 2h  f  L ∞ (0,t; H˙ q ()) . Consequently, ϑ (t) L 2 () ≤ ch 2+q 2h  f  L ∞ (0,t; H˙ q ()) .  ϑ (t) L 2 () + h∇  This and (2.26) show the second assertion, completing the proof of the theorem.  The proof of Theorem 2.3 indicates that for 0 < q ≤ 1, one can get rid of one factor h , and hence the error estimate can be improved to eh (t) L 2 () + h∇eh (t) L 2 () ≤ ch 2 h  f  L ∞ (0,t; H˙ q ()) .

34

2 Spatially Semidiscrete Discretization

Compared with the L 2 (0, T ; H˙ p ()) estimate of the error eh , the L ∞ (0, T ; H˙ p ()) estimate suffers from a factor 2h . This is due to Lemma 2.3(ii) and the regularity estimate in Theorem 1.10. Note that some of the log factor in Theorem 2.2 is actually due to the limitation of the proof technique, and can be removed using alternative proof strategies, e.g., Laplace transform.

2.3 Error Analysis via Laplace Transform In this section, we revisit the error analysis of the semidiscrete Galerkin scheme (2.11) using Laplace transform, cf. Appendix A.3 for the definition and its basic properties. By Laplace transform (and extending f by zero), the scheme (2.11) can be written into z α uˆ h (z) + Ah uˆ h (z) = z α−1 u 0h + fˆh (z), with uˆ h (z) = L[u h ](z) being the Laplace transform of u h and Ah = −h . Thus, u h (z) = (z α + Ah )−1 z α−1 u 0h + (z α + Ah )−1 fˆh (z). Then, by inverse Laplace transform and convolution rule for Laplace transform, we obtain an alternative representation of the semidiscrete solution u h given by  u h (t) = Fh (t)u 0h +

t

E h (t − s) f h (s) ds,

0

with the operators Fh (t) and E h (t) given, respectively, by  1 e zt z α−1 (z α + Ah )−1 dz, Fh (t) = 2π i θ,σ  1 E h (t) = e zt (z α + Ah )−1 dz, 2π i θ,σ where the contour θ,σ ⊂ C, with θ ∈ ( π2 , π ), is defined by

θ,σ = {z = ρe±iθ : ρ ≥ σ } ∪ {z = σ eiϕ : |ϕ| ≤ θ }. These representations coincide with that in (2.17) and (2.18), and by properly deforming the contour θ,σ , one can show they are indeed identical. Now, we derive error estimates for the scheme (2.11) using an operator trick. The following lemma plays a key role in error analysis. For any θ ∈ (0, π ), the sector θ ⊂ C is defined by θ = {z ∈ C \ {0} : | arg(z)| ≤ θ } .

2.3 Error Analysis via Laplace Transform

35

Lemma 2.5 For any v ∈ H01 () and z ∈ θ with θ ∈ ( π2 , π ), there holds   |z α |v2L 2 () + ∇v2L 2 () ≤ cz α v2L 2 () + ∇v2L 2 () .

(2.27)

Proof By [FS91, Lemma 7.1], we have that for any z ∈ θ   |z|v2L 2 () + ∇v2L 2 () ≤ czv2L 2 () + ∇v2L 2 () . Alternatively, let γ = v2L 2 () and β = ∇v2L 2 () and arg(z) = ϕ, we have |zγ + β|2 ≥ (|z|γ cos ϕ + β)2 + (|z|γ sin ϕ)2 . Therefore, we derive |zγ + β| ≥ |z|γ sin ϕ and |zγ + β|2 ≥ (β cos ϕ + |z|γ )2 + β 2 sin2 ϕ ≥ β 2 sin2 ϕ.

Then, for ϕ ∈ [π − θ, θ ], we have 2|zγ + β| ≥ (|z|γ + β) sin ϕ ≥ (|z|γ + β) sin θ. Meanwhile, for ϕ ∈ [0, π − θ ], we have cos ϕ ≥ cos(π − θ ) > 0. |zγ + β| ≥ |z|γ cos ϕ + β ≥ |z|γ cos(π − θ ) + β ≥ (|z|γ + β) cos(π − θ ). This completes the proof of the lemma.



The next lemma shows an error estimate between the function (z α + A)−1 v, with A = − with a zero Dirichlet boundary condition, and its fem approximation (z α + Ah )−1 Ph v. Lemma 2.6 Let v ∈ L 2 (), z ∈ θ , w = (z α + A)−1 v, and wh = (z α + Ah )−1 Ph v. Then (2.28) wh − w L 2 () + h∇(wh − w) L 2 () ≤ ch 2 v L 2 () . Proof Let e = w − wh . By the definition, w and wh , respectively, satisfy z α (w, vh ) + (∇w, ∇vh ) = (v, vh ), ∀vh ∈ H01 (), z (wh , vh ) + (∇wh , ∇vh ) = (v, vh ), ∀vh ∈ X h . α

Subtracting these two identities yields the following Galerkin orthogonality relation: z α (e, vh ) + (∇e, ∇vh ) = 0, ∀vh ∈ X h . This and Lemma 2.5 imply that, for any vh ∈ X h ,

(2.29)

36

2 Spatially Semidiscrete Discretization

  |z α |e2L 2 () + ∇e2L 2 () ≤ cz α e2L 2 () + ∇e2L 2 ()  = c |z α (e, w − vh ) + (∇e, ∇(w − vh ))| . By taking vh = Ph w and using the Cauchy–Schwarz inequality, we arrive at |z α |e2L 2 () + ∇e2L 2 ()

≤ c |z α |he L 2 () ∇w L 2 () + h∇e L 2 () w H˙ 2 () .

(2.30)

Appealing again to Lemma 2.5, we obtain |z α |w2L 2 () + ∇w2L 2 () ≤ c|((z α + A)w, w)| ≤ cv L 2 () w L 2 () . Consequently, w L 2 () ≤ c|z α |−1 v L 2 () and ∇w L 2 () ≤ c|z α |− 2 v L 2 () . 1

(2.31)

In view of (2.31) and the resolvent estimate (1.14), we can bound w H˙ 2 () by w H˙ 2 () = Aw L 2 () = c(−z α + z α + A)(z α + A)−1 v L 2 () ≤ c(v L 2 () + |z α |w L 2 () ) ≤ cv L 2 () . It follows from this and (2.30) that |z α |e2L 2 () + ∇e2L 2 () ≤ chv L 2 () (|z α | 2 e L 2 () + ∇e L 2 () ), 1

and this yields

|z α |e2L 2 () + ∇e2L 2 () ≤ ch 2 v2L 2 () .

(2.32)

This gives the desired bound on ∇e L 2 () . Next, we bound e L 2 () using a duality argument. For ϕ ∈ L 2 (), by setting ψ = (z α + A)−1 ϕ and ψh = (z α + Ah )−1 Ph ϕ, we have by duality |(e, ϕ)| |z α (e, ψ) + (∇e, ∇ψ)| = sup . ϕ L 2 () ϕ∈L 2 () ϕ L 2 () ϕ∈L 2 ()

e L 2 () = sup

Then the desired estimate follows from (2.29) and (2.32) by |z α (e, ψ) + (∇e, ∇ψ)| = |z α (e, ψ − ψh ) + (∇e, ∇(ψ − ψh ))| ≤|z α | 2 e L 2 () |z α | 2 ψ − ψh  L 2 () + ∇e L 2 () ∇(ψ − ψh ) L 2 () 1

1

≤ch 2 v L 2 () ϕ L 2 () . This completes proof of the lemma.



2.3 Error Analysis via Laplace Transform

37

Now, we can state error estimates for the homogeneous problem. Theorem 2.4 Let u and u h be the solutions of problems (2.1) and (2.11), respectively, with f ≡ 0, and let eh = u h − u. (i) If u 0 ∈ L 2 () and u 0h = Ph u 0 , then, for t > 0, there holds eh (t) L 2 () + h∇eh (t) L 2 () ≤ ch 2 t −α u 0  L 2 () . (ii) If u 0 ∈ H˙ 2 () and u 0h = Rh u 0 , then for t > 0, there holds eh (t) L 2 () + h∇eh (t) L 2 () ≤ ch 2 u 0  H˙ 2 () . Proof (i) For u 0 ∈ L 2 (), by the solution representations of u and u h , the error eh (t) can be represented as 1 eh (t) = 2π i



θ,σ

e zt z α−1 (wh (z) − w(z)) dz,

with w(z) = (z α + A)−1 u 0 and wh (z) = (z α + Ah )−1 Ph u 0 . By Lemma 2.6, and taking σ = t −1 in the contour θ,σ , we have  ∇eh (t) L 2 () ≤ chu 0  L 2 ()

θ,σ

e(z)t |z|α−1 |dz| ≤ cht −α u 0  L 2 () .

A similar argument also yields the L 2 ()-estimate. (ii) For u 0 ∈ H˙ 2 (), the error eh (t) can be represented by eh (t) =

1 2π i



θ,σ

e zt z α−1 ((z α + Ah )−1 Rh − (z α + A)−1 )u 0 dz.

By the identity z α−1 (z α + A)−1 = z −1 − z −1 (z α + A)−1 A, we can obtain eh (t) =

1 2π i



θ,σ

e zt z −1 (w(z) − wh (z)) dz +



θ,σ

 e zt z −1 (Rh u 0 − u 0 ) dz ,

(2.33) with w(z) = (z α + A)−1 Au 0 and wh (z) = (z α + Ah )−1 Ah Rh u 0 . Then Lemmas 2.1 and 2.6, and the identity Ah Rh = Ph A, cf. (2.15), give w(z) − wh (z) L 2 () + h∇(w(z) − wh (z)) L 2 () ≤ ch 2 Au 0  L 2 () . Now it follows from this and the representation (2.33) that

38

2 Spatially Semidiscrete Discretization

 eh (t) ≤ ch 2 Au 0  L 2 () ≤ ch Au 0  L 2 () 2

e(z)t |z|−1 |dz|

θ,σ  ∞

e

ρt cos θ

t −1

ρ

−1

 dρ +

θ

e

cos ϕ

 dϕ

−θ

≤ ch 2 Au 0  L 2 () = ch 2 u 0  H˙ 2 () . Hence, we obtain the L 2 () estimate. The H 1 () estimate follows analogously.  The next result gives error estimates for the inhomogeneous problem. Theorem 2.5 Let u and u h be the solutions of problems (2.1) and (2.11), respectively, with u 0 ≡ 0, and let eh = u h − u. (i) If f ∈ L ∞ (0, T ; L 2 ()), then for t ∈ (0, T ], there holds eh (t) L 2 () + h∇eh (t) L 2 () ≤ ch 2 h  f  L ∞ (0,T ;L 2 ()) . (ii)

If f ∈ C([0, T ]; L 2 ()) ∩ W 1,1 (0, T ; L 2 ()), then for t ∈ (0, T ], there holds  eh (t) L 2 () + h∇eh (t) L 2 () ≤ ch 2  f (0) L 2 () +

t 0

 f  (s) L 2 () ds.

Proof (i) f ∈ L ∞ (0, T ; L 2 ()). Then eh (t) for t > h α can be written as 2

eh (t) =

 0



2

t−h α

+

t 2

t−h α

 (E h (t − s)Ph f (s) − E(t − s) f (s)) ds := I1 (t) + I2 (t).

With w(z) = (z α + A)−1 f (s) and wh (z) = (z α + Ah )−1 Ph f (s), we have 1 E h (t − s)Ph f (s) − E(t − s) f (s) = 2π i



θ,σ

e z(t−s) (wh (z) − w(z)) dz.

By Lemma 2.6, and taking σ = (t − s)−1 in the contour θ,σ , we have for p = 0, 1 ∇ p (E h (t − s)Ph f (s) − E(t − s) f (s)) L 2 ()  2− p ≤ ch  f (s) L 2 () e(z)(t−s) |dz|

θ,σ

≤ ch

2− p

−1

(t − s)  f (s) L ∞ (0,T ;L 2 ()) . 2

This and the condition t > h α implies that the term I1 can be bounded by

2.3 Error Analysis via Laplace Transform

39

 I1 (t) L 2 () + h∇I1 (t) L 2 () ≤ ch 2  f  L ∞ (0,T ;L 2 ())

2

t−h α

(t − s)−1 ds

0

≤ ch 2 h  f  L ∞ (0,T ;L 2 ()) . Meanwhile, Lemmas 1.6 and 2.3(ii) imply ∇ p (E h (t − s)Ph f (s) − E(t − s) f (s)) L 2 () ≤ c(t − s)

2− p 2 α−1

 f (s) L 2 () .

Consequently, we have the following bound on the term I2 as  ∇ I2 (t) L 2 () ≤ c f  L ∞ (0,T ;L 2 ())

t

p

t−h

2 α

(t − s)

2− p 2 α−1

ds

≤ ch 2− p  f  L ∞ (0,T ;L 2 ()) . 2

Further, if t ≤ h α , then the error bound for eh (t) reduces to the one for I2 (t).  (ii) The identities E(t) = −A−1 F  (t) (cf. Lemma 1.7) and E h (t) = −A−1 h Fh (t) and integration by parts imply 

t

eh (t) =

 E h (s)Ph f (t − s) ds −

t

E(s)P f (t − s) ds  t  t (I − Fh (s)) A−1 P f (t − s) ds − (I − F(s)) A−1 f (t − s) ds = h h 0 0  t s=t  P f (t − s)| + (I − Fh (s))A−1 = (I − Fh (s))A−1 h s=0 h h Ph f (t − s) ds 0  t − (I − F(s))A−1 f  (t − s) ds. − (I − F(s))A−1 f (t − s)|s=t s=0 0

0

0

By the identity F(0) = I , we deduce −1 f (0) eh (t) =(I − Fh (t))A−1 h Ph f (0) − (I − F(t))A  t  −1  [(I − Fh (s))A−1 f (t − s)] ds. + h Ph f (t − s) − (I − F(s))A 0

By Theorem 2.4 (ii), we derive for p = 0, 1 −1 2− p , ∇ p (Fh (t)A−1 h Ph − F(t)A ) ≤ ch

which together with Theorem 2.1 leads to assertion (ii).



40

2 Spatially Semidiscrete Discretization

2.4 Lumped Mass FEM The standard Galerkin fem (2.11) represents only one possible way to discretize problem (2.1) in the spatial direction, and there are many alternatives, e.g., lumped mass fem, finite volume element method, and discontinuous Galerkin fem. In this section, we discuss the more practical lumped mass fem [Tho06, Chap. 15] and derive convergence rates. For simplicity, we consider the two-dimensional case. Let z Kj , j = 1, 2, 3, be the three vertices of a triangle K ∈ Th . Consider the quadrature formula |K |  f (z Kj ) ≈ 3 j=1 3

Q K ,h ( f ) =

 f dx. K

We then define an approximate L 2 () inner product in X h by (wh , vh )h =



Q K ,h (wh vh ).

K ∈Th

Then the lumped mass fem reads: with u¯ h (0) = u 0h , find u¯ h (t) ∈ X h such that (∂tα u¯ h , vh )h + (∇ u¯ h , ∇vh ) = ( f, vh ), ∀vh ∈ X h , ∀0 < t ≤ T.

(2.34)

The lumped mass method leads to a diagonal mass matrix, and thus enhances the computational efficiency. Next, we introduce the corresponding discrete Laplacian ¯ h : X h → X h , corresponding to the inner product (·, ·)h , by − ¯ h wh , vh )h = (∇wh , ∇vh ), ∀wh , vh ∈ X h . − (

(2.35)

Also, we introduce a projection operator P¯h : L 2 () → X h by ( P¯h w, vh )h = (w, vh ), ∀vh ∈ X h . The lumped mass method can then be written with f h = P¯h f in an operator form as ¯ h u¯ h (t) = f h (t), 0 < t ≤ T, with u¯ h (0) = u 0h . ∂tα u¯ h (t) −  Next, we define the discrete solution operators F¯h and E¯ h by F¯h (t)vh =

Nh 

E α,1 (−λ¯ hj t α )(vh , ϕ¯ hj )h ϕ¯ hj ,

(2.36)

t α−1 E α,α (−λ¯ hj t α )(vh , ϕ¯ hj )h ϕ¯ hj .

(2.37)

j=1

E¯ h (t)vh =

Nh  j=1

2.4 Lumped Mass FEM

41

h h where (λ¯ hj ) Nj=1 and (ϕ¯ hj ) Nj=1 are respectively the eigenvalues and the orthonormal ¯ h with respect to the inner product (·, ·)h . eigenfunctions of the discrete Laplacian − Then the solution u¯ h to problem (2.34) can be represented as

u¯ h (t) = F¯h (t)u 0h +



t

E¯ h (t − s) f h (s) ds.

(2.38)

0

Analogous to (2.20), we introduce a discrete norm ||| · ||| H˙ p () on the space X h |||vh |||2H˙ p () =

Nh 

(λ¯ hj ) p (vh , ϕ¯ hj )2h , ∀ p ∈ R.

(2.39)

j=1

Then we have the following norm equivalence result and inverse estimate. Lemma 2.7 For −1 ≤ p ≤ 1, the norm ||| · ||| H˙ p () defined in (2.39) is equivalent to the norm  ·  H˙ p () on X h . Further, for any real p > q, |||vh ||| H˙ p () ≤ ch q− p |||vh ||| H˙ q () , vh ∈ X h .

(2.40)

We have an analogue of Lemma 2.3(ii). The proof is identical, and hence omitted. Lemma 2.8 For the operator E¯ h , there holds for vh ∈ X h and all t > 0,  ||| E¯ h (t)vh ||| H˙ p () ≤

ct −1+α(1+

q− p 2 )

|||vh ||| H˙ q () ,

ct −1+α |||vh ||| H˙ q () ,

p − 4 ≤ q ≤ p, p < q.

We recall also the quadrature error operator Q h : X h → X h defined by (∇ Q h wh , ∇vh ) := (wh , vh )h − (wh , vh ), ∀wh , vh ∈ X h ,

(2.41)

which represents the quadrature error due to mass lumping. It satisfies the following error estimate for p = 0, 1 [CLT12, Lemma 2.4]: ¯ h Q h vh  L 2 () ≤ ch p+1 ∇ p vh  L 2 () , ∀vh ∈ X h . ∇ Q h vh  L 2 () + h

(2.42)

We now establish error estimates for the lumped mass fem for smooth initial data, i.e., u 0 ∈ H˙ 2 () (and f ≡ 0). Theorem 2.6 Let u 0 ∈ H˙ 2 (), u and u¯ h be the solutions of problems (2.1) and (2.34), respectively, with u 0h = Rh u 0 and f ≡ 0, and e¯h = u¯ h − u. Then e¯h (t) L 2 () + h∇ e¯h  L 2 () ≤ ch 2 u 0  H˙ 2 () .

42

2 Spatially Semidiscrete Discretization

Proof We split the error e¯h into e¯h = u h − u + δ with δ = u¯ h − u h and u h being the solution obtained by the standard Galerkin fem (2.11). In view of Theorem 2.2, it suffices to show δ(t) L 2 () + h∇δ(t) L 2 () ≤ ch 2 u 0  H˙ 2 () .

(2.43)

It follows from the definitions of u h (t), u¯ h (t), and Q h that ¯ h δ(t) =  ¯ h Q h ∂tα u h (t), 0 < t ≤ T, with δ(0) = 0 ∂tα δ(t) −  and by the solution representation (2.38), we have  δ(t) = 0

t

¯ h Q h ∂sα u h (s) ds. E¯ h (t − s)

(2.44)

Using Lemmas 2.7 and 2.8, and (2.42), we get for vh ∈ X h : ¯ h Q h vh  L 2 () ≤ ct α2 −1  ¯ h Q h vh  L 2 () ≤ ct α2 −1 h∇vh  L 2 () . ∇ E¯ h (t) Similarly, for vh ∈ X h , ¯ h Q h vh  L 2 () ≤ ct α2 −1 ||| ¯ h Q h vh ||| H˙ −1 ()  E¯ h (t) α

α

≤ct 2 −1 ∇ Q h vh  L 2 () ≤ ct 2 −1 h 2 ∇vh  L 2 () . Then using Lemma 2.3 with p = 1 and q = 2, we get  δ(t) L 2 () + h∇δ(t) L 2 () ≤ ch 2

t

0

 ≤ ch 2

0

t

α

(t − s) 2 −1 |||∂sα u h (s)||| H˙ 1 () ds α

α

(t − s) 2 −1 s − 2 ds |||u h (0)||| H˙ 2 () .

From the identity h Rh = Ph , cf. (2.15), we deduce |||u h (0)||| H˙ 2 () = h Rh u(0) L 2 () = Ph u(0) L 2 () ≤ cu 0  H˙ 2 () , which yields the desired estimate (2.43) and concludes the proof.



Now, we consider the case of nonsmooth initial data u 0 ∈ L 2 () and the intermediate case u 0 ∈ H˙ 1 . Due to the lower regularity, we take u 0h = Ph u 0 . Like before, the idea is to split the error e¯h (t) = u¯ h (t) − u(t) into e¯h (t) = u h (t) − u(t) + δ(t) with δ(t) = u¯ h (t) − u h (t) and u h (t) being the solution of the Galerkin scheme (2.11). Thus, in view of Theorem 2.4, it suffices to establish proper bounds for δ(t).

2.4 Lumped Mass FEM

43

Theorem 2.7 Let u ∈ H˙ q (), q = 0, 1, f ≡ 0, and u and u¯ h be the solutions of problems (2.1) and (2.34), respectively, with u 0h = Ph u 0 , and e¯h = u¯ h − u. Then with h = | log h|, the following estimates hold for t > 0: q

∇ e¯h (t) L 2 () ≤ ch h t −α(1− 2 ) u 0  H˙ q () , q = 0, 1, e¯h (t) L 2 () ≤ ch

q+1 −α(1− q2 )

t

u 0  H˙ q () , q = 0, 1.

(2.45) (2.46)

Furthermore, if the quadrature error operator Q h in (2.41) satisfies Q h vh  L 2 () ≤ ch 2 vh  L 2 () , ∀vh ∈ X h ,

(2.47)

then the following error estimate is valid: e¯h (t) L 2 () ≤ ch 2 h t −α u 0  L 2 () .

(2.48)

Proof Using the solution representation (2.44) and the smoothing property of E¯ h in Lemma 2.8 and the inverse inequality (2.40), we get for vh ∈ X h , > 0, and p = 0, 1 ¯ h Q h vh ||| H˙ p () ||| E¯ h (t) α−1 ¯ h Q h vh ||| H˙ p−2+ () = ct 2 α−1 |||Q h vh ||| H˙ p+ () ≤ ct 2 ||| ≤ ct

2 α−1

h − |||Q h vh ||| H˙ p () ≤ ct

2 α−1

(2.49)

h − Q h vh  H˙ p () .

Then by the estimate (2.42) and Lemma 2.3 and H˙ 1 ()- and L 2 ()-stability of the operator Ph from Lemma 2.1, we deduce for q = 0, 1  ∇δ(t) L 2 () ≤ ch q+1−

t

0





(t − s) 2 α−1 ∂sα u h (s) H˙ q () ds

t



(t − s) 2 α−1 s −α dsu h (0) H˙ q () 0

= ch q+1− t −α(1− 2 ) B 2 α, 1 − α Ph u 0  H˙ q () ≤ ch

q+1−



≤ c −1 h q+1− t −α(1− 2 ) u 0  H˙ q () . Now, the estimate (2.45) follows from the triangle inequality, the preceding bound and the estimate (2.21) with = 1 and = −1 h for the cases q = 1 and 0, respectively. To derive (2.46), note that, for vh ∈ X h , we have ¯ h Q h vh  L 2 () ≤ ct α2 −1 ||| ¯ h Q h vh ||| H˙ −1 () ≤ ct α2 −1 ∇ Q h vh  L 2 () .  E¯ h (t) This estimate and (2.42) give for q = 0, 1,

44

2 Spatially Semidiscrete Discretization

 δ(t) L 2 () ≤ ch q+1  ≤ ch

t 0 t

q+1 0

α

(t − s) 2 −1 ∂tα u h (s) H˙ q () ds α

(t − s) 2 −1 s −α ds u h (0) H˙ q ()

α

α

≤ ch q+1 t − 2 Ph u 0  H˙ q () ≤ ch q+1 t − 2 u 0  H˙ q () , which shows the desired estimate (2.46). Finally, if (2.47) holds, by applying (2.49) with p = 0 and ∈ (0, 21 ), we get δ(t) L 2 () ≤ch −



t

0

 ≤ch

2−

≤ch

2−

 ≤c



(t − s) 2 α−1 Q h ∂tα u h (s) L 2 () ds t

0 t



(t − s) 2 α−1 ∂tα u h (s) L 2 () ds

(t − s) 2 α−1 s −α ds u h (0) L 2 ()

0 −1 2− −α(1− 2 )

h

t

u 0  L 2 () .

Then the estimate (2.48) follows immediately by choosing = −1 h .



By interpolation, the estimate (2.48) holds also for 0 < q < 1. The condition (2.47) on the quadrature error operator Q h is satisfied for symmetric meshes [CLT12, Sect. 5]. In the one-dimensional case, the symmetry condition can be relaxed to almost symmetry [CLT12, Sect. 6]. When (2.47) does not hold, we can only show a suboptimal O(h)-convergence rate for L 2 ()-norm of the error, which is reminiscent of that in the classical parabolic case [CLT12, Theorem 4.4]. Now, we derive an L 2 (0, T ; H˙ p ())-error estimate, p = 0, 1, for the inhomogeneous problem. Note that without condition (2.47), the L 2 (0, T ; L 2 ()) estimate is suboptimal for any q < 1, and an optimal estimate can only be obtained under condition (2.47). Theorem 2.8 Let f ∈ L ∞ (0, T ; H˙ q ()), −1 < q ≤ 1, and u 0 ≡ 0, u and u h be the solutions of (2.1) and (2.34), respectively, and e¯h = u¯ h − u. Then there hold ∇ e¯h  L 2 (0,T ;L 2 ()) ≤ ch 1+min(q,0)  f  L 2 (0,T ; H˙ q ()) , e¯h  L 2 (0,T ;L 2 ()) ≤ ch 1+q  f  L 2 (0,T ; H˙ q ()) . In addition, if the operator Q h satisfies condition (2.47), then e¯h  L 2 (0,T ;L 2 ()) ≤ ch 2+min(q,0)  f  L 2 (0,T ; H˙ q ()) .

2.4 Lumped Mass FEM

45

Proof By the argument of Lemma 2.4, we deduce from Lemma 2.7 and (2.42) that 



T

0



T

≤c 0



T

¯ h Q h ∂tα u h (t)|||2˙ −1 dt ||| H ()  T |||Q h ∂tα u h (t)|||2H˙ 1 () dt ≤ c ∇ Q h ∂tα u h (t)2L 2 () dt

∇δ(t)2L 2 ()

dt ≤ c

0

0

T

≤ch 2 0

∂tα u h (t)2L 2 () dt.

The assertion for the case q ≥ 0 now follows from Lemma 2.4. For −1 < q < 0, we use Lemmas 2.7, 2.4 and 2.1 and get 

T

0

 ∇δ(t)2L 2 () dt 

T

≤ch 2+2q 0

≤ ch

2+2q 0

T

|||∂tα u h (t)|||2H˙ q () dt

||| f h (t)|||2H˙ q () dt ≤ ch 2+2q  f 2L 2 (0,T ; H˙ q ()) .

For the L 2 (0, T ; L 2 ()) estimate, by repeating the preceding arguments, we arrive at  T  T 2 ¯ h Q h ∂tα u h (t)|||2˙ −2 dt δ(t) 2 dt ≤ c ||| L ()

0

 =c 0

≤ch

T



H

0

Q h ∂tα u h (t)2L 2 () dt ≤ c

4 0

T



()

T

∇ Q h ∂tα u h (t)2L 2 () dt  T α 2 2+2q ∂t u h (t) H˙ 1 () dt ≤ ch |||∂tα u h (t)|||2H˙ q () dt, 0

0

where the second line follows from the inequality vh  L 2 () ≤ C∇vh  L 2 () for vh ∈ X h and Lemma 2.7. The rest of the proof is identical, and hence omitted. If condition (2.47) holds, then 

T



T

¯ h Q h ∂tα u h (t)|||2˙ −2 dt ||| H () 0 0  T  T |||Q h ∂tα u h (t)|||2L 2 () dt ≤ ch 4 ∂tα u h (t)2L 2 () dt ≤c 0 0  T 4+2 min(q,0) |||∂tα u h (t)|||2H˙ q () dt. ≤ch δ(t)2L 2 ()

dt ≤ c

0

The rest of the proof is same as before, and this completes the proof.



Last, we derive an L ∞ (0, T ; H˙ p ()) error estimate on u¯ h . Like the L (0, T ; L 2 ())-estimate, the L ∞ (0, T ; L 2 ()) estimate is suboptimal for any q ∈ (−1, 1), and can be improved to an almost optimal one by imposing condition (2.47). 2

46

2 Spatially Semidiscrete Discretization

Theorem 2.9 Let f ∈ L ∞ (0, T ; H˙ q ()), −1 < q ≤ 1, and u 0 ≡ 0, u and u¯ h be the solutions of (2.1) and (2.34), respectively, and e¯h = u¯ h − u. Then for −1 < q ≤ 0, with h = | log h|, the following estimate holds for t > 0: ∇ e¯h (t) L 2 () ≤ ch 1+q 2h  f  L ∞ (0,t; H˙ q ()) . Moreover, for −1 < q ≤ 1, there holds e¯h (t) L 2 () ≤ ch 1+q 2h  f  L ∞ (0,t; H˙ q ()) . In addition, if condition (2.47) holds, then for −1 < q ≤ 1, with h = | log h|, there holds e¯h (t) L 2 () ≤ ch 2+min(q,0) 2h  f  L ∞ (0,t; H˙ q ()) . Proof By Lemma 2.8 and (2.40), we have for vh ∈ X h , > 0, and p = 0, 1, ¯ h Q h vh ||| H˙ p () ≤ ct 2 α−1 ||| ¯ h Q h vh ||| H˙ p−2+ () ||| E¯ h (t)



=ct 2 α−1 |||Q h vh ||| H˙ p+ () ≤ ct 2 α−1 h − |||Q h vh ||| H˙ p () .

(2.50)

Setting vh = ∂tα u h (t) in the estimate and condition (2.42) yields

¯ h Q h ∂tα u h (s)||| H˙ 1 () ≤ c(t − s) 2 α−1 h − Q h ∂sα u h (s) H˙ 1 () ||| E¯ h (t − s)

≤ ch 1− (t − s) 2 α−1 ∂sα u h (s) L 2 () . Then it follows from (2.11) and the triangle and inverse inequalities that ∂sα u h (s) L 2 () ≤ h u h (s) L 2 () +  f h (s) L 2 () ≤ c(h − |||h u h (s)||| H˙ − () +  f h (s) L 2 () ), and consequently,  ∇δ(t)

L 2 ()

≤ ch

t

1− 0



(t − s) 2 α−1 (h − |||h u h (s)||| H˙ − () +  f h (s) L 2 () ) ds.

Further, using Lemmas 2.4 and 2.1, we get for −1 < q ≤ 0 ∂sα u h (s) H˙ q () ≤ c(h − |||h u h (s)||| H˙ q− () + ||| f h (s)||| H˙ q () )

≤ ch − ( −1 s 2 α ||| f h ||| L ∞ (0,s; H˙ q ()) + ||| f h (s)||| H˙ q () ) ≤ c −1 h − ||| f h ||| L ∞ (0,s; H˙ q ()) . Then, with the choice = −1 h ,

2.4 Lumped Mass FEM

47

∇δ(t) L 2 () ≤ c −1 h 1+q−2 ||| f h ||| L ∞ (0,t; H˙ q ())



t



(t − s) 2 α−1 ds

0

≤ c −2 h 1+q−2 ||| f h ||| L ∞ (0,t; H˙ q ()) ≤ ch 1+q 2h  f  L ∞ (0,t; H˙ q ()) . This and Theorem 2.3 show the first assertion. Next, similar to (2.50), we get ¯ h Q h ∂tα u h (s) L 2 () ≤ c(t − s) α2 −1 ∇ Q h ∂sα u h (s) L 2 ()  E¯ h (t − s) α

≤ ch 2 (t − s) 2 −1 ∂sα u h (s) H˙ 1 () . Then by the triangle inequality, Lemmas 2.2 and 2.4, there holds  t α δ(t) L 2 () ≤ ch 2 (t − s) 2 −1 (h u h (s) H˙ 1 () + Ph f (s) H˙ 1 () ) ds 0  t α 1+q −1 (t − s) 2 (h − |||h u h (s)||| H˙ q− () + ||| f h (s)||| H˙ q () ) ds ≤ ch 0  t α (t − s) 2 −1 ( −1 h − s 2 α ||| f h ||| L ∞ (0,s; H˙ q ()) + ||| f h (s)||| H˙ q () ) ds. ≤ ch 1+q 0

The L 2 ()-estimate follows by setting = −1 h and Theorem 2.3. Last, if condition (2.47) holds, then applying (2.50) with p = 0 gives ¯ h Q h ∂tα u h (s) L 2 () ≤ c(t − s) 2 α−1 h − Q h ∂sα u h (s) L 2 ()  E¯ h (t − s)

≤ ch 2− (t − s) 2 α−1 ∂sα u h (s) L 2 () . Then this together with Lemmas 2.2, 2.4, and 2.1 yields ∂sα u h (s) L 2 () ≤ h u h (s) L 2 () +  f h (s) L 2 () ≤ch min(q,0) (h − |||h u h (s)||| H˙ − +q () + ||| f h (s)||| H˙ q () )

≤ch min(q,0) (h − −1 s 2 α ||| f h ||| L ∞ (0,s; H˙ q ()) + ||| f h (s)||| H˙ q () ) ≤c −1 h min(q,0)−  f  L ∞ (0,s; H˙ q ()) . This and the choice = −1 h directly yield δ(t) L 2 () ≤ c −2 h 2+min(q,0)−2  f  L ∞ (0,t; H˙ q ()) ≤ ch 2+min(q,0) 2h  f  L ∞ (0,t; H˙ q ()) . This and Theorem 2.3 conclude the proof of the theorem.



48

2 Spatially Semidiscrete Discretization

Notes The error analysis of both standard Galerkin fem and lumped mass fem for the homogeneous problem is taken from [JLZ13], and that for the inhomogeneous problem from [JLPZ15]. In these works, the error analysis was carried out using the separation of variables and Mittag–Leffler functions. See also [JLPZ13] for error estimates of the standard Galerkin fem with even weaker initial data, i.e., u 0 ∈ H˙ q (), q ∈ (−1, 0). To remove the log factor in the case of nonsmooth data, the Laplace transform approach was firstly introduced for parabolic equations by Fujita and Suzuki [FS91] and then this technique was adapted to time-fractional evolution models in [BJLZ15, JLZ16c]. Formally, the final error estimates in the fractional and standard parabolic cases (see [Tho06]) are largely comparable (except small differences in the singularity factor tn−α for the subdiffusion case versus tn−1 for the standard parabolic case), and unsurprisingly, the overall proof strategy is often similar. Of course the proofs in the fractional case are often more involved due to the limited smoothing property of the solution operators. The proof techniques in this chapter rely heavily on the fact that the elliptic operator A is self-adjoint and time-independent. In the absence of these conditions, these strategies do not work anymore, and require suitable modifications; see Chap. 9 for the case of time-dependent coefficients. Thus, it is of much interest to develop more versatile analysis strategies, of which one natural candidate is the energy argument, which represents one of the most commonly used and most powerful strategies in the standard parabolic case. Unfortunately, the energy type argument is generally more tricky in the fractional case. This is due to the nonlocality of the Djrbashian– Caputo fractional derivative ∂tα u and, consequently, that many powerful pde tools, like integration by parts formula and product rule, are either invalid or require substantial modifications. Some first encouraging theoretical results in this important direction were obtained by Mustapha [Mus18], where optimal error estimates for the homogeneous problem were obtained using an energy argument; see also [LMM16] for the time-fractional Fokker–Planck equation. Karaa [Kar18] gave a unified analysis of different kinds of fems for the homogeneous subdiffusion problem based on an energy argument, which generalizes the corresponding technique for standard parabolic problems in [Tho06, Chap. 3]. Most of these works assume a convex domain, which is needed for ensuring the full elliptic regularity pickup. The case of less regular domains, e.g., nonconvex/curved domains, involves additional technicalities and has been less studied for the subdiffusion model; see [LML17] for a domain with a re-entrant corner. The standard Galerkin fem is predominant in spatial semidiscretization, but other methods have also been investigated, e.g., finite volume element method [KMP17].

Chapter 3

Convolution Quadrature

In this chapter, we aim to develop and analyze a class of time-stepping schemes for approximately solving the subdiffusion problem (1.1), i.e., convolution quadrature (cq) generated by backward differentiation formulas (bdfs). Note that the discretization of the Djrbashian–Caputo fractional derivative ∂tα u will inherit its nonlocality, so that the time-stepping scheme determines the solution at the n-th time level in terms of the solutions at all previous time levels. This contrasts sharply with normal diffusion models, and the nonlocality results in substantial challenges in numerical analysis, storage requirement and efficient implementation, which will also be discussed in this chapter. To make the discussion of time-stepping clearer, we separate it from the spatial discretization in Chap. 2 and consider time stepping in an abstract setting. Let the operator A denote the negative Laplacian A ≡ − : H˙ 2 () ≡ H01 () ∩ H 2 () → L 2 () on a convex polyhedral domain  ⊂ Rd (d = 1, 2, 3) with a zero Dirichlet boundary condition. The operator A satisfies the following resolvent estimate (cf. [ABHN11, Example 3.7.5 and Theorem 3.7.11] and [Tho06]) (z + A)−1  ≤ cθ |z|−1 , ∀z ∈ θ ,

(3.1)

for all θ ∈ ( π2 , π ), with the sector θ := {z ∈ C \ {0} : | arg z| ≤ θ }. The notation  ·  denotes the operator norm from L 2 () to L 2 (). Consider the following abstract initial value problem ∂tα u(t) + Au(t) = f (t), 0 < t ≤ T, with u(0) = u 0

(3.2)

in the Hilbert space L 2 (), with α ∈ (0, 1). This chapter focuses on time-stepping N be a uniform schemes with a uniform temporal mesh. Specifically, let (tn = nτ )n=0 −1 partition of the time interval [0, T ], with a time step size τ = N T , N ∈ N. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Jin and Z. Zhou, Numerical Treatment and Analysis of Time-Fractional Evolution Equations, Applied Mathematical Sciences 214, https://doi.org/10.1007/978-3-031-21050-1_3

49

50

3 Convolution Quadrature

The rest of this chapter is organized as follows. In Sect. 3.1, we introduce the idea of cq generated by bdf. The error analysis in terms of problem data regularity is given in Sect. 3.2. In Sect. 3.3, we develop a fractional version of the classical Crank–Nicolson scheme and its corrected version. Finally, in Sects. 3.4 and 3.5, we describe two strategies for the efficient implementation of time-stepping schemes based on convolution quadrature, i.e., parallel-in-time algorithm and a fast convolution algorithm, and provide relevant theoretical underpinnings.

3.1 Convolution Quadrature Generated by BDF First we introduce convolution quadrature (cq) generated by k-step bdf methods. cq was proposed by Lubich in a series of pioneering works [Lub86, Lub88a, Lub04]. It approximates the Djrbashian–Caputo fractional derivative ∂tα u(tn ) by a discrete convolution: ∂tα u(tn ) ≈ τ −α

n 

¯α ω(α) j (u(tn− j ) − u(0)) = ∂τ u(tn ),

(3.3)

j=0

 ∞ where the weights ω(α) are given by the following power series expansion j j=0 ∞ 1  (α) j ω ζ . δτ (ζ ) = α τ j=0 j α

(3.4)

Then δ(ζ ) = τ δτ (ζ ) is the characteristic polynomial of a linear multistep method (ρ, σ ), where, as usual, ρ and σ denote the generating polynomials of the linear multistep method [HNW10], and δ(ζ ) =

σ (ζ −1 ) . ρ(ζ −1 )

Examples of linear multistep methods include backward Euler method, backward differentiation formulas, trapezoidal rule, Runge–Kutta methods, Newton–Gregory formulas, etc. The most popular one is the bdf of order k (bdfk), k = 1, . . . , 6, for which the generating function δτ (ζ ) is given by 1 δ(ζ ) with δ(ζ ) = (1 − ζ ) j . τ j j=1 k

δτ (ζ ) :=

(3.5)

For α = 1, the approximation in (3.3) reduces to the standard bdfk method, which is known to be A(ϑk )-stable (i.e., the set {μ ∈ C \ {0} : | arg(−μ)| < ϑk } is contained

3.1 Convolution Quadrature Generated by BDF

51

in the stability region of the method) with angle ϑk = 90◦ , 90◦ , 86.03◦ , 73.35◦ , 51.84◦ , 17.84◦ for k = 1, 2, 3, 4, 5, 6, respectively [HW96, p. 251]. The special case k = 1, i.e., the backward Euler cq, is commonly known as Grünwald–Letnikov approximation in fractional calculus [OS74, Pod99] and the corresponding coefficients ω(α) j are given explicitly by the following recursion ω0(α) = 1, ω(α) j =−

α − j + 1 (α) ω j−1 , j

j = 1, 2, . . . .

This approximation is customarily used to define fractional integral/derivatives. Generally, the weights ω(α) j can be evaluated via recursion or discrete Fourier transform [Sou12]. The time-stepping scheme based on cq for problem (3.2) reads: given U 0 = u 0 , N to the exact solution u(tn ) by find an approximation (U n )n=1 ∂¯τα U n + AU n = f (tn ), n = 1, . . . , N .

(3.6)

It can be combined with the spatially semidiscrete schemes in Chap. 2 to arrive at fully discrete schemes, which are implementable on computers. To study the cq approximation ∂¯τα U n , we need the following lemma on the sectorial property and approximation property of the generating function δτ (ζ ). Lemma 3.1 For any , there exists θ ∈ ( π2 , π ) such that for any θ ∈ ( π2 , θ ), there exist positive constants c, c1 , andc2 (independent of τ ) such that c1 |z| ≤ |δτ (e−zτ )| ≤ c2 |z|,

δτ (e−zτ ) ∈ π−ϑk + ,

|δτ (e−zτ ) − z| ≤ cτ k |z|k+1 ,

|δτ (e−zτ )α − z α | ≤ cτ k |z|k+α ,

τ ∀ z ∈ θ,σ ,

τ where σ > 0 and the contour θ,σ ⊂ C is defined by τ θ,σ := {z = ρe±iθ : ρ ≥ σ, |(z)| ≤

π } τ

  ∪ z = σ eiϕ : |ϕ| ≤ θ .

δ(ζ ) Proof Since the function 1−ζ has no zero in a neighborhood N of the unit circle [CM75, proof of Lemma 2] and for θ sufficiently close to π2 , e−zτ lies in the neighborhood N, there are positive constants c1 and c2 such that

c1 ≤

|δτ (e−zτ )| |δ(e−zτ )| τ = ≤ c2 , ∀ z ∈ θ,σ . −zτ |1 − e | |(1 − e−zτ )/τ |

τ Since  c1 |zτ | ≤ |1 − e−zτ | ≤  c2 |zτ | for z ∈ θ,σ , the first estimate follows. When |ζ | ≤ 1 and ζ = 0, we have δτ (ζ ) ∈ π−ϑk for the A(ϑk ) stable bdfk [HW96]. Hence, by expressing e−zτ as e−|z|τ cos(θ) e−i|z|τ sin(θ) , we have

52

3 Convolution Quadrature

|δτ (e−zτ ) − δτ (e−i|z|τ sin(θ) )| = |δτ (e−|z|τ cos(θ) e−i|z|τ sin(θ) ) − δτ (e−i|z|τ sin(θ) )|   ≤ ce−η|z|τ cos(θ) δ  (e−η|z|τ cos(θ) e−i|z|τ sin(θ) )zτ cos(θ ), τ

τ for some η ∈ (0, 1), by the mean value theorem. For θ close to π2 and z ∈ θ,σ , π by Taylor expansion, |z|τ ≤ sin θ and the first estimate, we have τ |δτ (e−η|z|τ cos(θ) e−i|z|τ sin(θ) )| ≤ c and |δτ (e−i|z|τ sin(θ) )| ≥ c|z|. Consequently, we deduce |δτ (e−|z|τ cos(θ) e−i|z|τ sin(θ) ) − δτ (e−i|z|τ sin(θ) )| (3.7) ≤ c| cos(θ )||δτ (e−i|z|τ sin(θ) )| ≤ c|θ − π2 ||δτ (e−i|z|τ sin(θ) )|.

Hence, δτ (e−τ z ) lies in a sector π−ϑk +c|θ− π2 | . If θ > π2 is sufficiently close to π2 , then c|θ − π2 | < . This proves the second assertion. The third estimate is given in [Tho06, (10.6)]. The last estimate follows from   |δτ (e−zτ )α − z α | = α 

δτ (e−zτ )

  ζ α−1 dζ  ≤ max |ζ |α−1 |δτ (e−zτ ) − z|, ζ

z

(3.8)

where ζ lies on the line segment with end points δτ (e−zτ ) and z. Since δ(e−iθ ) > 0 for θ ∈ (0, π ) (see [Gea71, pp. 214–216] or [HW96, p. 246]), it follows from the continuity estimate (3.7) and by choosing θε sufficiently close to π2 that (δτ (e−zτ )) > τ with (z) > 0, from which and the first estimate we deduce 0 for z ∈ θ,σ |ζ |α−1 ≤ max(|z|α−1 , |δτ (e−zτ )|α−1 ) ≤ c|z|α−1 . 

This inequality and (3.8) yield the last estimate. The next lemma provides useful bounds of the convolution weights ωn(α) . Lemma 3.2 The weights ωn(α) satisfy |ωn(α) | ≤ c(n + 1)−α−1

and

n     −α ω(α)  j  ≤ c(n + 1) . j=0

Proof By the definition of the weight ωn(α) and Cauchy’s integral formula, we obtain ωn(α)

τα = 2π i



α −n−1

|ζ |=1

δτ (ζ ) ζ

τ 1+α dζ = 2π i

τ θ,σ

e ztn δτ (e−zτ )α dz,

with θ ∈ ( π2 , π ) and σ ≥ 0. Then with σ = 0, Lemma 3.1 implies for n ≥ 1

3.1 Convolution Quadrature Generated by BDF

|ωn(α) | ≤ cτ 1+α

τ θ,σ

53

|e ztn ||δτ (e−zτ )|α |dz| ≤ cτ 1+α



∞

er tn cos θ r α d r

0

≤ cτ 1+α tn−α−1 ≤ cn −α−1 ≤ c(n + 1)−α−1 . This and the uniform bound of ω0(α) lead to the desired result. The second estimate follows similarly from the identity n  j=0

ω(α) j

τα = 2π i

|ζ |=1

δτ (ζ )α −n−1 τα ζ dζ = 1−ζ 2π i

τ θ,σ

e ztn δτ (e−zτ )α

τ dz. 1 − e−zτ 

This completes the proof of the lemma.

For a sufficiently smooth (and compatible) function u, the discrete convolution ∂¯τα u(tn ) approximates the Djrbashian–Caputo fractional derivative ∂tα u(tn ) with an accuracy O(τ k ). See more discussion in [Lub88a, Theorem 3.1]. Theorem 3.1 If u ∈ C k+1 [0, T ] and u (m) (0) = 0 with m = 1, 2, . . . , k, then |∂tα u(tn ) − ∂¯τα u(tn )| ≤ cτ k . Proof Let w(t) = u(t) − u(0). Upon noting that w(m) = u (m) (0) = 0 with m = k 1, 2, . . . , k, Taylor’s expansion implies w(t) = p(t) ∗ w(k+1) (t) with p(t) = tk! , where ∗ denotes Laplace convolution. First, we claim |∂tα p(tn ) − ∂¯τα p(tn )| ≤ ctn−α τ k . Using Laplace transform, ∂tα p(t) can be written as ∂tα p(t) = where the contour θ,σ ⊂ C is defined by

(3.9) 1 2πi

θ,σ

  θ,σ = {z = ρe±iθ : ρ ≥ σ } ∪ z = σ eiϕ : |ϕ| ≤ θ ,

e zt z α−k−1 dz,

(3.10)

oriented with an increasing imaginary part. Multiplying ζ n with ∂¯τα p(tn ), summing over n from 1 to ∞ and then applying the discrete convolution rule, we obtain ∞  n=1

∂¯τα p(tn )ζ n = τ −α

∞  n=0

ζn

n 

(α) ωn− j p(t j )

j=0

∞ k ∞     tn n τ k γk (ζ ) ζ = δτ (ζ )α , ωn(α) ζ n = τ −α k! k! n=0 n=0

with γk (ζ ) :=

∞ n=0

n k ζ n . Then Cauchy’s integral theorem implies that for ρ > 0

54

3 Convolution Quadrature



1 ∂¯τα p(tn ) = 2π i

|ζ |=ρ



1 = 2π i

ζ −n−1 δτ (ζ )α

τ k+1 γk (e−zτ ) dz, k!

e ztn δτ (e−zτ )α

τ

τ k γk (ζ ) dζ k!

where the contour  τ is given by  τ := {z = − logτ  + iy : y ∈ R and |y| ≤ πτ }. By the analyticity of the generating function and the periodicity of the function e−zτ , τ := {z ∈ θ,σ : |(z)| ≤ Cauchy’s theorem allows deforming the contour  τ to θ,σ π }, oriented with an increasing imaginary part: τ

1 ∂¯τα p(tn ) = 2π i

τ θ,σ

e ztn δτ (e−zτ )α

τ k+1 γk (e−zτ ) dz. k!

Then the error ∂tα p(tn ) − ∂¯τα p(tn ) can be written as ∂tα p(tn )

1 − ∂¯τα p(tn ) = 2π i +



 τ k+1 γk (e−zτ ) dz e ztn z α−k−1 − δτ (e−zτ )α k!

τ θ,σ

1 2π i



τ θ,σ \θ,σ

e ztn z α−k−1 dz = I1 + I2 .

In view of the estimate (3.29) below and Lemma 3.1, we have   k+1 −zτ   δτ (e−zτ )α τ γk (e ) − z α−k−1  ≤ c|z|α−1 τ k , ∀z ∈  τ . θ,σ   k! Upon choosing σ = tn−1 , we arrive at |I1 | ≤ cτ

k



∞

e

−cρtn

ρ

α−1

dρ +

0

θ

−θ

σ α dϕ ≤ cτ k tn−α .

Meanwhile, for the term I2 , we have |I2 | ≤ c

∞ π τ sin θ

e−cρtn ρ α−k−1 dρ ≤ cτ k



∞ π τ sin θ

e−cρtn ρ α−1 dρ ≤ cτ k tn−α .

Thus we complete the proof of the estimate (3.9). A similar argument yields |∂tα tn − ∂¯τα tn | ≤ cτ  tn−α ,  = 1, 2, . . . , k.

(3.11)

Then by the association property of Laplace convolution (noting that y(0) = 0), we deduce ∂tα w(tn ) = ∂tα ( p ∗ w (k+1) )(tn ) = ((∂tα p) ∗ w (k+1) )(tn ).

3.1 Convolution Quadrature Generated by BDF

Let E τ (t) = τ −α

∞ 

55

ωn(α) δtn (t),

n=0

with δtn being the Dirac-delta function at tn (from the left side). Then we have ∂¯τα w(tn ) = (E τ ∗ ( p ∗ w (k+1) ))(tn ) = ((E τ ∗ p) ∗ w (k+1) )(tn ). Next, we prove |(∂tα p − E τ ∗ p)(t)| ≤ cτ k t −α , ∀t ∈ (tn−1 , tn ].

(3.12)

Indeed, applying Taylor expansion at tn gives ∂tα p(t) =∂tα p(tn ) +

tn

+ t

k−1  (−1) (tn − t) α () ∂t p (tn ) ! =1

(−1)k (s − t)k−1 s −α ds. (k − 1)! (1 − α)

This expansion holds also for (E τ ∗ p)(t). Then (3.11) yields k−1    (−1) (tn − t) α ¯ α ()   α ¯α (∂t − ∂τ ) p (tn ) ≤ cτ k tn−α . (∂t − ∂τ ) p(tn ) + ! =1

Meanwhile, a simple computation yields for t ∈ (tn−1 , tn )   

tn t

tn  (s − t)k−1 s −α  k−1 ds  ≤ cτ s −α ds ≤ cτ k t −α . (k − 1)! (1 − α) t

Similarly, from Lemma 3.2, we deduce   

tn t

n       k −α (s − t)k−1 (E τ ∗ p (k) )(s) ds  ≤ cτ k τ −α  ω(α) j  ≤ cτ tn . j=0

The estimate (3.12) follows directly by tn−α ≤ t −α for t ∈ (tn−1 , tn ). Then we arrive at |∂tα u(tn ) − ∂¯τα u(tn )| = |∂tα w(tn ) − ∂¯τα w(tn )| tn (tn − s)−α |w (k+1) (s)| ds ≤ cτ k . ≤cτ k 0

This completes the proof of the theorem.



56

3 Convolution Quadrature

The next result gives the error estimate for the scheme (3.6) when the solution u to problem (3.2) is sufficiently smooth and satisfies suitable compatibility conditions. Theorem 3.2 Let u be the solution to problem (3.2), and U n the solution of the scheme (3.6). If u ∈ C k+1 ([0, T ]; L 2 ()) and u (m) (0) = 0 with m = 1, 2, . . . , k, then U n − u(tn ) L 2 () ≤ c(u)τ k . Proof The error en ≡ U n − u(tn ) satisfies en = 0 and ∂¯τα en + Aen = ∂tα u(tn ) − ∂¯τα u(tn ) = (∂tα − ∂¯τα )w(tn ), with w(t) = u(t) − u 0 . Then Theorem 3.1 implies (∂tα − ∂¯τα )w(tn ) L 2 () ≤ cτ k , n = 1, 2, . . . , N . Now since e0 = 0, the desired estimate follows from the maximal  p regularity in Chap. 8, i.e., for any 1 < p < ∞, N N N  p (L 2 ()) + (Aen )n=1  p (L 2 ()) ≤ c p ((∂tα − ∂¯τα )w(tn ))n=1  p (L 2 ()) , (∂¯τα en )n=1

and the discrete Hardy inequality in Theorem 10.3.



Theorem 3.2 implies that the approximation U n converges at a rate O(τ k ) uniformly in time t, if the exact solution u is smooth enough and has sufficiently many vanishing derivatives at t = 0. This assumption is true if the following compatibility condition holds f (0) + u 0 = 0 and f () (0) = 0,  = 1, . . . , k − 1.

(3.13)

In the absence of the compatibility condition, the time-stepping scheme (3.6) generally only exhibits a first-order accuracy when solving time-fractional evolution equations even for smooth u 0 and f [CLP06, JLZ16c], since the requisite compatibility condition (3.13) is usually not satisfied. This loss of accuracy is one distinct feature for most time-stepping schemes, since they are usually derived under the assumption that the solution u is sufficiently smooth (and compatible), which holds only if the problem data satisfy certain rather restrictive compatibility condition (3.13). In summary, they tend to lack robustness with respect to the regularity of problem data.

3.2 BDFk CQ with Initial Correction

57

3.2 BDFk CQ with Initial Correction To restore the k th -order accuracy of the bdfk cq scheme for general problem data, we correct bdfk at the starting k − 1 steps by (as usual, the summation disappears if the upper index is smaller than the lower one) ⎧ ¯α n ∂τ U + AU n =an(k) (−Au 0 + f (0)) + f (tn ) ⎪ ⎪ ⎪ ⎪ k−2 ⎨  (k)  () + b,n τ f (0), n = 1, . . . , k − 1, ⎪ ⎪ =1 ⎪ ⎪ ⎩ α n ∂¯τ U + AU n = f (tn ), n = k, . . . , N ,

(3.14)

(k) where an(k) and b,n are coefficients to be determined. They are chosen so as to improve the accuracy of the scheme to O(τ k ) for a general u 0 ∈ D(A) and a possibly incompatible source f . The difference between (3.14) and (3.6) lies in the correction terms at the starting k − 1 steps. Hence, the scheme (3.14) is easy to implement. Moreover, the derivative f () (0) may be replaced by its (k −  − 1)-order finite difference approximation, without sacrificing its accuracy. The correction in (3.14) is also minimal in the sense that there is no other correction scheme that modifies only the k − 1 starting steps while retaining the O(τ k ) rate. Of course, this does not rule out corrections with more steps. (k) Now we derive the criteria for choosing the coefficients a (k) j and b, j , cf. (3.25) and (3.26) below, using Laplace transform and its discrete analogue. Like before, we denote by  taking Laplace transform, and for a given sequence ( f n )∞ n=0 , denote by n n  f ζ its generating function. First we split the source f into f (ζ ) := ∞ n=0

f (t) = f (0) +

k−2   t =1

!

f () (0) + Rk (t),

(3.15)

and Rk is the corresponding local truncation error, given by k−2   t

t k−1 t k−1 f (k−1) (0) + ∗ f (k) . ! (k − 1)! (k − 1)! =1 (3.16) Thus the function w(t) := u(t) − u 0 satisfies Rk (t) = f (t) − f (0) −

f () (0) =

∂tα w + Aw = −Au 0 + f (0) +

k−2   t =1

!

f () (0) + Rk ,

(3.17)

with w(0) = 0. Since w(0) = 0, the identity  ∂tα w(z) = z α w (z) holds (cf. Lemma 1.2), and thus by Laplace transform, we obtain

58

3 Convolution Quadrature

zα w (z) + A w (z) = z −1 ( f (0) − Au 0 ) +

k−2 

k (z). z −−1 f () (0) + R

=1

By inverse Laplace transform, the function w(t) can be readily represented by 1 w(t) = 2π i

θ,σ

1 + 2π i

  e zt K (z) f (0) − Au 0 dz



with the kernel function

k−2  1 () k (z) dz, e z K (z) f (0) + R z +1 =1

(3.18)

zt

θ,σ

K (z) = z −1 (z α + A)−1 .

(3.19)

and the contour θ,σ defined by (3.10). Throughout, we choose the angle θ in θ,σ such that π2 < θ < π and hence, z α ∈ θ  with θ  = αθ < π for all z ∈ θ . By the resolvent estimate (3.1), there exists a constant c which depends only on θ and α such that (z α + A)−1  ≤ c|z|−α and K (z) ≤ c|z|−1−α , ∀z ∈ θ .

(3.20)

Next, we give a representation of the discrete solution W n := U n − u 0 .

t Theorem 3.3 Let f ∈ C k−1 ([0, T ]; L 2 ()) and 0 (t − s)α−1  f (k) (s) L 2 () ds < ∞ for any t ∈ (0, T ]. Then the discrete solution W n := U n − u 0 is represented by 1 e ztn μ(e−zτ )K (δτ (e−zτ ))( f (0) − Au 0 ) dz τ 2π i θ,σ  k−2   γ (e−zτ ) k−1  (k) 1 + e ztn δτ (e−zτ )K (δτ (e−zτ )) b, j e−zt j τ +1 f () (0) dz + τ 2π i θ,σ ! j=1 =1 1 k (e−zτ ) dz, + e ztn δτ (e−zτ )K (δτ (e−zτ ))τ R (3.21) τ 2π i θ,σ Wn =

τ with the contour θ,σ := {z ∈ θ,σ : |(z)| ≤ πτ } (oriented with an increasing imaginary part), where the functions μ(ζ ) and γ (ζ ) are respectively defined by



 (k) ζ + μ(ζ ) = δ(ζ ) aj ζ j 1−ζ j=1 k−1





d and γ (ζ ) = ζ dζ



Proof The functions W n , n = 1, . . . , N , satisfy (with W 0 = 0):

1 . 1−ζ

(3.22)

3.2 BDFk CQ with Initial Correction

59

∂¯τα W n + AW n =(1 + an(k) )( f (0) − Au 0 )  k−2    tn (k)  + b,n + τ f () (0) + Rk (tn ), n = 1, . . . , k − 1, ! =1 ∂¯τα W n + AW n = −Au 0 + f (0) +

k−2   t n

=1

!

f () (0) + Rk (tn ), n = k, . . . , N .

Multiplying both sides by ζ n and summing over n from 1 to ∞ yield ∞ 

ζ n ∂¯τα W n +

n=1

=

 ∞

=

AW n ζ n

n=1

k−1 

ζn +

n=1



∞ 

j=1

ζ + 1−ζ

k−1  j=1

k (ζ ) = where R ∞ 

  k−2   ∞  k−1  tn n  (k)  j (k) k (ζ ) a j ζ j ( f (0) − Au 0 ) + b, j τ ζ f () (0) + R ζ + ! =1

n=1

j=1

  k−2  k−1  γ (ζ )  (k) j  () (k) k (ζ ), a j ζ j ( f (0) − Au 0 ) + b, j ζ τ f (0) + R + ! =1

∞ n=1

j=1

Rk (tn )ζ n and we have used the elementary identities

ζ ζ = 1−ζ n=1 n

  d  1 := γ (ζ ). n ζ = ζ dζ 1−ζ n=1

∞ 

and

 n

(3.23)

Next we simplify the summations. Since W 0 = 0, by the discrete convolution rule, ∞ 

 (ζ ), ζ n ∂¯τα W n = δτ (ζ )α W

n=1

and consequently, we obtain   W (ζ ) = K (δτ (ζ )) τ −1 μ(ζ )( f (0) − Au 0 )   k−1 γ (ζ )  (k) j  ()  + + δτ (ζ ) b, j ζ τ f (0) + δτ (ζ ) Rk (ζ ) , ! j=1 =1 

k−2 

where the kernel K is given by (3.19), and the functions μ(ζ ) and γ (ζ ) are given  (ζ ) is analytic with respect to ζ in the unit disk on the complex by (3.22). Since W plane C, thus Cauchy’s integral formula and the change of variables ζ = e−zτ lead to the following representation for any  ∈ (0, 1) 1 W = 2π i



n

|ζ |=

ζ

−n−1

 (ζ ) dζ = τ W 2π i

τ

 (e−zτ ) dz, e ztn W

(3.24)

60

3 Convolution Quadrature

where  τ is given by  τ := {z = − logτ  + iy : y ∈ R and |y| ≤

π }. τ

Note that

τ (ζ ) (1) η(ζ ) := δ1−ζ is a polynomial without roots in a neighborhood N of the unit circle [CM75]. Thus, η(ζ )α is analytic in N. (2) By choosing θ and  sufficiently close to π2 and 1, and 0 < δ < − logτ  , the function e−τ z lies in N for

τ = {z ∈ θ : |z| ≥ σ, |(z)| ≤ z ∈ θ,σ

τ , π

(z) ≤ − logτ  };

τ (3) (1 − e−τ z )α is analytic for z ∈ C\(−∞, 0] ⊃ θ,σ . τ Hence, δτ (e−τ z )α = τ −α (1 − e−τ z )α η(e−τ z )α is analytic for z ∈ θ,σ . By choosing τ .  small enough, Lemma 3.1 implies 0 = δτ (e−τ z )α ∈ α(ϑk +) ⊂ π− for z ∈ θ,σ τ −τ z −τ z −1 −τ z α −1 Thus K (δτ (e )) = δτ (e ) (δτ (e ) + A) is analytic for z ∈ θ,σ , which τ τ and the two lines ± := R ± i πτ (oriented from is a region enclosed by  τ , θ,σ ztn  −zτ τ left to right). Since the values of e W (e ) on ± coincide, Cauchy’s theorem τ in the integral (3.24) to obtain the desired allows deforming the contour  τ to θ,σ representation. 

In view of (3.18) and (3.21), in order to have O(τ k ) accuracy, the following three τ : conditions should be satisfied for z ∈ θ,σ |μ(e−zτ ) − 1| ≤ c|z|k τ k , |δτ (e−zτ ) − z| ≤ c|z|k+1 τ k ,    k−1  γ (e−zτ )  1  (k) −zt j +1  ≤ c|z|k−−1 τ k . τ + b e − , j  +1  ! z j=1 Note that for bdfk, the estimate |δτ (e−zτ ) − z| ≤ c|z|k+1 τ k holds automatically (cf. Lemma 3.1). Thus, it suffices to impose the following two criteria (by changing e−zτ (k) k−1 k−1 to ζ and zτ to 1 − ζ ): for bdfk, choose the coefficients (a (k) j ) j=1 and (b, j ) j=1 such that |μ(ζ ) − 1| ≤ c|1 − ζ |k ,   k−1  γ (ζ )  1  (k) j  ≤ c|1 − ζ |k−−1 ,  = 1, . . . , k − 2, + b ζ − , j  ! +1  δ(ζ ) j=1

(3.25) (3.26)

with μ(ζ ) and γ (ζ ) defined in (3.22). It can be verified that for bdfk, k = 3, . . . , 6, the leading singularities on the left hand side of (3.26) do cancel out, and thus the criterion (3.26) can be satisfied.

k−1 (k) j Next we compute the coefficients a (k) j . To this end, we rewrite j=1 a j ζ as

3.2 BDFk CQ with Initial Correction k−1 

61

j a (k) j ζ =ζ

j=1

k−2 

c j (1 − ζ ) j .

(3.27)

j=0

Thus, by writing ζ = 1 − (1 − ζ ), expanding the summation and collecting terms, we obtain (with the convention c−2 = c−1 = 0)

μ(ζ ) =

  k k−2   1 ζ c j (1 − ζ ) j (1 − ζ ) j +ζ j 1−ζ j=1

=

k−1  j=0

=

  1 c j−2 (1 − ζ ) j + c j−1 (1 − ζ ) j (1 − ζ ) j 1 − (1 − ζ ) − j +1 k

k−1

j=0

k−1  j=0

+

j=0



1 (1 − ζ ) j − j +1

j k−1    =0

j=1

=1+

k−1  

k−1   j=1

j=1

1 (1 − ζ ) j − j

j=0

j k−1    j=2

=0

 1 c−2 (1 − ζ ) j j −+1

   c−1 (1 − ζ ) j + O (1 − ζ )k

 j j    1 1 1 1  − − c−2 + c−1 (1 − ζ ) j + O (1 − ζ )k j +1 j j −+1 j −+1 =0

j=1

=1+

1 j −+1

k 



−

1 − j ( j + 1)

j−1  =1

=0

1 c−1 + j −

j−1  =0

   1 c (1 − ζ ) j + O (1 − ζ )k . j −

Thus by choosing c ,  = 0, . . . , k − 2, such that j−1  =0

 1 1 1 c = + c−1 , j − j ( j + 1) =1 j −  j−1

j = 1, . . . , k − 1,

(3.28)

Criterion (3.25) follows. The coefficients a (k) j can be computed recursively from (3.28) and (3.27), and are given in Table 3.1. (k) γ (ζ ) 1 Next we compute the coefficients b, j . First we expand ! − δ(ζ )+1 in 1 − ζ as k−−2  (k) 1 γ (ζ ) − = g, j (1 − ζ ) j + O(|1 − ζ |k−−1 ), ! δ(ζ )+1 j=0

(3.29)

(k) and then choose the coefficients b, j , j = 1, . . . , k − 1 to satisfy (3.26). To this end,

k−1 (k) j we rewrite j=1 b, j ζ into the following form:

62

3 Convolution Quadrature (k)

(k)

Table 3.1 The coefficients an and b,n a1(k)

a2(k)

5 − 12

k=4

1 2 11 12 31 24

− 76

3 8

k=5

1181 720

− 177 80

341 240

k k=2 k=3

a3(k)

a4(k)

a5(k)



(k) b,1

(k) b,2

=1

1 12 1 6

0

=1 251 − 720

− 2543 720

17 5

− 1201 720

95 288

=2 0

0

0

=1

29 − 120 1 − 240

19 240

0

0

0

0

0

7 − 15

73 240 1 160

3 − 40

0

0

0

=2

59 240 1 240 1 720 77 240 1 96

=3

1 − 360

=1

=4 0

k−1  j=1

(k) j b, jζ

=ζ

k−2  j=0

d,(k)j (1

− ζ) = j

k−2 

(k) b,5

0

=3 2837 1440

(k) b,4

1 − 12

=2 k=6

(k) b,3

d,(k)j (1

− ζ) − j

j=0

0

1 − 60 1 720

0

0

0

0

0

0

0

k−1 

d,(k)j−1 (1 − ζ ) j . (3.30)

j=1

Then it suffices to choose ⎧ (k) (k) ⎪ d,0 = −g,0 , ⎪ ⎪ ⎨ (k) (k) (k) d, j = d, j−1 − g, j = 1, . . . , k −  − 2, j, ⎪ ⎪ ⎪ (k) ⎩ d, j = 0, j = k −  − 1, . . . , k − 2.

(3.31)

(k) Now the coefficients b, j can be computed recursively using (3.29), (3.31) and (3.30), and the results are given in Table 3.1. Note that for k = 4 and 6, the coefficients (k) bk−2, j , j = 1, 2 . . . , k − 1 vanish identically. The coefficients are independent of the order α. Next, we investigate the error estimates of the time-stepping scheme (3.14). The error analysis relies on the following splitting

u(tn ) − U n = w(tn ) − W n and the representations (3.18) and (3.21), and then bounding each term using the resolvent estimate (3.20). The details are given below. First, we give some useful estimates. τ Lemma 3.3 Let Criteria (3.25) and (3.26) hold. Then for z ∈ θ,σ , there hold

3.2 BDFk CQ with Initial Correction

63

μ(e−zτ )K (δτ (e−zτ )) − K (z) ≤ cτ k |z|k−1−α ,     k−1    (k) − j zτ +1 − (δτ (e−zτ )α + A)−1 1 γ (e−zτ ) +  ≤ cτ k |z|k−−1−α . τ b e − z K (z) , j   ! j=1

τ Proof Since |1 − e−zτ | ≤ cτ |z| for z ∈ θ,σ , by Criterion (3.25), there holds

|μ(e−zτ ) − 1| ≤ c|1 − e−zτ |k ≤ cτ k |z|k . Meanwhile, by the triangle inequality, we have K (δτ (e−zτ )) − K (z) = δτ (e−zτ )−1 (δτ (e−zτ )α + A)−1 − z −1 (z α + A)−1  ≤ |δτ (e−zτ )−1 − z −1 |(δτ (e−zτ )α + A)−1  + |z|−1 (δτ (e−zτ )α + A)−1 − (z α + A)−1 . The identity (δτ (e−zτ )α + A)−1 − (z α + A)−1 =(z α − δτ (e−zτ )α )(δτ (e−zτ )α + A)−1 (z α + A)−1 , Lemma 3.1 and the resolvent estimate (3.20) imply directly K (δτ (e−zτ )) − K (z) ≤ c|τ |k |z|k−1−α . Thus, we obtain the first estimate by μ(e−zτ )K (δτ (e−zτ )) − K (z) ≤ |μ(e−zτ ) − 1|K (δτ (e−zτ )) τ + K (δτ (e−zτ )) − K (z) ≤ cτ k |z|k−1−α , ∀z ∈ θ,σ .

Next we show the second estimate. By Lemma 3.1, there holds τ . |δτ (e−zτ )+1 − z +1 | ≤ c|δτ (e−zτ ) − z||z| ≤ cτ k |z|k++1 , ∀z ∈ θ,σ

By Criterion (3.26), there holds   k−1  γ (e−zτ )   1 (k) − j zτ   ≤ cτ k−−1 |z|k−−1 , ∀z ∈  τ . + b e − θ,σ , j  !  −zτ +1 δ(e ) j=1 τ Hence, for any z ∈ θ,σ , we have

64

3 Convolution Quadrature

    k−1    (k) − j zτ +1 − (δτ (e−zτ )α + A)−1 1 γ (e−zτ ) +  b e − z K (z) τ , j   ! j=1

    k−1    1 (k) − j zτ −zτ α −1 −zτ +1 −zτ −−1  (δ ≤ γ (e ) + A) (e ) + b e − δ (e ) τ  τ , j  τ  ! j=1

+ δτ (e

−zτ −

) K (δτ (e

−zτ

)) − z

−

K (z) ≤ cτ k |z|k−−1−α . 

This completes the proof of the lemma. Now we can state the following error estimate. Theorem 3.4 Let Criteria (3.25) and (3.26) hold, and f ∈C

k−1

([0, T ]; L ()) and

t

2

0

(t − s)α−1  f (k) (s) L 2 () ds < ∞.

(3.32)

Then for the solution U n to (3.14), the following error estimate holds for any tn > 0  k−1  U n − u(tn ) L 2 () ≤cτ k tnα−k  f (0) − Au 0  L 2 () + tnα+−k  f () (0) L 2 () =1



tn

+ 0

 (tn − s)α−1  f (k) (s) L 2 () ds .

Proof By (3.18) and (3.21), we employ the splitting U n − u(tn ) = W n − w(tn ) = I1 +

k−2 

I2, − I3 + I4 ,

=1

where the terms I1 , . . . , I4 are given by I1 = I2,

1 2π i

1 = 2π i

1 I3 = 2π i

τ θ,σ

 e ztn μ(e−zτ )K (δτ (e−zτ )) − K (z) ( f (0) − Au 0 ) dz,

   k−1 γ (e−zτ )  (k) −zτ j +1 −zτ τ K (δτ (e−zτ )) + e δτ (e ) b, j e τ ! θ,σ j=1  − z − K (z) f () (0) dz,

ztn

τ θ,σ \θ,σ

k−2    e ztn K (z) f (0) − Au 0 + z − f () (0) dz, =1

3.2 BDFk CQ with Initial Correction

I4 =

1 2π i −



k (e−zτ ) dz e ztn (δτ (e−zτ )α + A)−1 τ R

τ θ,σ

1 2π i

65



k (z) dz. e ztn (z α + A)−1 R

θ,σ

It suffices to bound them separately. By Lemma 3.3, and choosing σ = tn−1 in the τ , we can bound the first term I1 by contour θ,σ I1  L 2 () ≤ cτ k  f (0) − Au 0  L 2 () ≤ cτ

k

(tnα−k

+σ

k−α



∞ σ

eρtn cos θ ρ k−1−α dρ +

) f (0) − Au 0  L 2 () ≤



θ

eσ tn | cos ϕ| σ k−α dϕ

−θ k α−k cτ tn  f (0) −



Au 0  L 2 () .

τ Similarly, by Lemma 3.3 and choosing σ = tn−1 in θ,σ , we bound the terms I2, by

I2, 

L 2 ()

≤ cτ  f k

()

 (0)

L 2 ()

∞

e σ

ρtn cos θ

ρ

k−−1−α

dρ +

θ

−θ

 σ

k−−α

dϕ

≤ cτ k tnα+−k  f () (0) L 2 () ,  = 1, 2..., k − 1. Direct computation yields the following estimate on the term I3 :   k−2  I3  L 2 () ≤ cτ k tnα−k  f (0) − Au 0  L 2 () + tnα+−k  f () (0) L 2 () . =1

The term I4 is the error of the numerical solution with a compatible right-hand side Rk . Upon recalling the definition of Rk in (3.16), we use the splitting Rk =

t k−1 t k−1 f (k−1) (0) + ∗ f (k) (t) =: Rk1 + Rk2 . (k − 1)! (k − 1)!

Then we have I4 = I41 + I42 with I4i =

1 2π i

−



1 2π i

τ θ,σ

ki (e−zτ ) dz e ztn (δτ (e−zτ )α + A)−1 τ R



θ,σ

ki (z) dz. e ztn (z α + A)−1 R

By repeating the preceding argument and (3.29), we have the following estimate for I41 : (3.33) I41  L 2 () ≤ cτ k tnα−1  f (k−1) (0) L 2 () . It remains to bound I42 . We define an operator E(t) by

66

3 Convolution Quadrature

E(t) =

1 2π i

θ,σ

e zt (z α + A)−1 dz.

(This is actually the solution operator for the inhomogeneous problem, cf. (1.17).) Then we define the auxiliary function y(tn ) = (E ∗ Rk2 )(tn ) = E ∗

 t k−1  t k−1 ∗ f (k) (tn ). ∗ f (k) (tn ) = E ∗ (k − 1)! (k − 1)!

Similarly, we define Yn = −

1 2π i

θ,σ

k2 (z) dz, e ztn (z α + A)−1 R

which can be represented by Yn = τ

n 

E τn− j Rk2 (t j ) with E τn =

j=1

1 2π i

τ θ,σ

e ztn (δτ (e−zτ )α + A)−1 dz.

By the resolvent estimate (3.1) and Lemma 3.1, the following estimate holds E τn  ≤ ctnα−1 . Let E τ (t) = τ

∞ 

(3.34)

E τn δtn (t),

n=0

with δtn being the Dirac-delta function at tn (from the left side). Then we have Y n = (E τ ∗ Rk2 )(tn ) = E τ ∗

 t k−1  t k−1 ∗ f (k) (tn ) = E τ ∗ ∗ f (k) (tn ). (k − 1)! (k − 1)!

Then (3.33) implies    (E τ − E) ∗

t k−1   (tn ) ≤ cτ k tnα−1 . (k − 1)!

Next, we aim to show that for t > 0    (E τ − E) ∗

t k−1   (t) ≤ cτ k t α−1 , ∀t ∈ (tn−1 , tn ). (k − 1)!

To see the claim, we recall the Taylor expansion of E(t) at t = tn

(3.35)

3.2 BDFk CQ with Initial Correction



67

 (t − tn ) j  t k−1 (t) = E ∗ t k−1− j (tn ) + (k − 1)! (k − 1 − j)! j=0 k−1

E∗



t

(t − s)k−1 E(s) ds.

tn

This expansion holds also for (E τ ∗ t)(t). Then the preceding argument yields ((E τ − E) ∗ t k−1− j )(tn ) ≤ cτ j+1 tnα−1 . and hence for any t ∈ (tn−1 , tn ], k−1    j=0

 (t − tn ) j   (E τ − E) ∗ t k−1− j (tn ) ≤ cτ k tnα−1 . (k − 1 − j)!

Meanwhile, by the resolvent estimate (3.1) (or from Theorem 1.6), we have E(t) ≤ ct α−1 , and consequently, there holds  t     (t − s)k−1 E(s) ds  ≤ c   tn

tn

(s − t)k−1 s α−1 ds ≤ cτ k t α−1 .

t

Similarly, appealing to (3.34), we deduce  t     (t − s)k−1 E τ (s) ds  ≤ τ k E n  ≤ cτ k t α−1 . τ n   tn

Then (3.35) follows directly by tnα−1 ≤ t α−1 for t ∈ (tn−1 , tn ) and α ∈ (0, 1), concluding the proof of the theorem.  The error estimate in Theorem 3.4 requires Au 0 ∈ L 2 (), i.e., u 0 is reasonably smooth. For u 0 ∈ L 2 (), the term Au 0 should be interpreted in a distributional sense. Upon minor modifications of the proof, one can derive a similar error estimate for u 0 ∈ L 2 ():  k−1  tnα+−k  f () (0) L 2 () U n − u(tn ) L 2 () ≤cτ k tn−k u 0  L 2 () + + 0

=0

tn

 (tn − s)α−1  f (k) (s) L 2 () ds .

(3.36)

So far we have focused on the temporal semidiscretization. Nonetheless, the argument extends to fully discrete schemes, which we outline briefly next. Let {Th }0 0, if u 0 ∈ H01 () ∩ H 2 () and f satisfies (3.32). Similarly, if u 0 ∈ L 2 () and f satisfies (3.32), the fully discrete solution Uhn (with u 0h = Ph u 0 ) satisfies for any tn > 0,   Uhn − u(tn ) L 2 () ≤ c h 2 tn−α + τ k tn−k .

3.3 Fractional Crank–Nicolson Scheme

69

3.3 Fractional Crank–Nicolson Scheme In this section, we present and analyze a second-order fractional Crank–Nicolson scheme for approximating the model (3.2): given U 0 = u 0 , with f n = f (tn ), find N ⊂ H01 () such that (U n )n=1     ∂¯τα U n + 1 − α2 AU n + α2 AU n−1 = 1 − α2 f n +

α 2

f n−1 , n = 1, . . . , N . (3.39) In (3.39), ∂¯τα ϕ n denotes the backward Euler cq approximation defined by ∂¯τα ϕ n := τ −α

n 

(α) j 0 ωn− j (ϕ − ϕ ), with

∞ 

j=0

j α ω(α) j ξ := (1 − ξ ) ,

(3.40)

j=0

where the weights ω(α) j are available in closed form: j ω(α) j = (−1)

(α + 1) . ( j + 1)(α − j + 1)

Clearly, for α = 1, the scheme (3.39) recovers the classical Crank–Nicolson method, i.e., ∂t ϕ(tn− 21 ) ≈ τ −1 (ϕ n − ϕ n−1 ), which is second-order accurate for sufficiently smooth ϕ. For α ∈ (0, 1), the scheme (3.39) hybridizes the backward Euler cq with the θ -type method with a weight θ = α2 . This choice is taken so that it yields a local truncation error O(τ 2 ) under certain regularity and compatibility conditions. In order to achieve an O(τ 2 ) accuracy, we derive the weight θ by Fourier transform. We denote by Ft the Fourier transform in t and by Fξ−1 the inverse Fourier transform in ξ . Assuming that the function ϕ is smooth over R and ϕ = 0 for t ≤ 0, then the function ∞  ∂¯τα ϕ(t) := τ −α ω(α) j ϕ(t − jτ ) j=0

coincides with the scheme (3.40) at t = tn and satisfies Ft [∂¯τα ϕ(t)](ξ ) =



=τ

R

∂¯τα ϕ(t)e−itξ dt = τ −α

−α

∞  j=0

(1 − e

−iτ ξ α

R

−itξ ω(α) dt j ϕ(t − jτ )e

) F [ϕ](ξ ) = (iξ )α (1 −

iα τξ 2

+ O(τ 2 ξ 2 ))F [ϕ](ξ ).

In view of the identity   Ft [ R∂tα ϕ(t − s)](ξ ) = (iξ )α e−isξ F [ϕ](ξ ) = (iξ )α 1 − isξ + O(s 2 ξ 2 ) F [ϕ](ξ ),

70

3 Convolution Quadrature

and by the choice s = α2 τ , formally we derive ∂¯τα ϕ(t) = R∂tα ϕ(t − α2 τ ) + Fξ−1 [O(τ 2 ξ 2 )(iξ )α F ϕ(ξ )] = R∂tα ϕ(t − α2 τ ) + O(τ 2 ) = (1 − α2 ) R∂tα ϕ(t) +

αR α ∂t ϕ(t 2

− τ ) + O(τ 2 ).

By choosing t = tn in the expression, we obtain the scheme (3.39). The preceding discussion also indicates that if ϕ(0) = 0, uncorrected high-order cqs usually can achieve only an O(τ ) rate, which is also the case for (3.39). To derive the correction, we define f˜(t) := f (t) − f (0) and rewrite (3.2) into ∂tα (u(t) − u 0 ) = −A(u(t) − u 0 ) − Au 0 + f˜(t) + f (0) = −A(u(t) − u 0 ) − ∂t ∂t−1 Au 0 + f˜(t) + ∂t ∂t−1 f (0). Next we apply (3.39) and approximate ∂t ∂t−1 by ∂˜τ ∂t−1 , where ∂˜τ denotes bdf2, i.e., ∂¯τα U n = (1 − α2 )(−A(U n − u 0 ) + f˜n ) + α2 (−A(U n−1 − u 0 ) + f˜hn−1 ) + (1 − α )∂˜τ ∂t−1 (−Au 0 + f (0))n + α ∂˜τ ∂t−1 (−Au 0 + f (0))n−1 . 2

2

The purpose of keeping ∂t−1 intact in the discretization and using the approximation ∂˜τ ∂t−1 instead of ∂¯τ ∂t−1 is to maintain the desired O(τ 2 ) accuracy. Then direct computation gives (∂˜τ t)0 = 0, (∂˜τ t)1 = τ −1

3  3   −1 3 ˜ 2 ˜ n 2 t1 = 2 , (∂τ t) = τ 2 t2 − 2t1 = 1, (∂τ t) = 1, ∀n ≥ 3.

Therefore, ∂˜τ ∂t−1 1 = 1τ on the grid points tn , with 1τ = (0, 23 , 1, 1, . . .). Then the corrected fractional Crank–Nicolson scheme is given by ∂¯τα U 1 + (1 − α2 )AU 1 + ( 21 − α4 )Au 0 = (1 − α2 )( f 1 +

∂¯τα U 2 + (1 − α2 )AU 2 + α2 AU 1 + α4 Au 0 = (1 − α2 ) f 2 + ∂¯τα U n + (1 − α2 )AU n + α2 AU n−1 = (1 − α2 ) f n +

1 0 f ), 2 α 1 f + α4 2 α n−1 f , 2

f 0, n = 3, . . . , N . (3.41)

Note that the correction only changes the first two steps. Now we analyze the corrected scheme (3.41), and discuss homogeneous and inhomogeneous problems separately. The convergence analysis relies crucially on the integral representations of the solution w(t) := u(t) − u 0 in (3.18) (with k = 2), and time-stepping solution W n := U n − u 0 in the next lemma.

3.3 Fractional Crank–Nicolson Scheme

71

Lemma 3.4 Let K (z) be given by (3.19) and g n := f n − f 0 . Then there exists an 0 ∈ (0, π2 ) (independent of τ ) such that for σ ∈ (0, 0 ] and θ ∈ ( π2 , π2 + 0 ], the time-stepping solution W n := U n − u 0 by the scheme (3.41) can be represented by Wn =

1 2π i



 e ztn μ(e−zτ )K (βτ (e−zτ ))( f 0 − Au 0 )

τ θ,σ

f (e−zτ )τ dz, − δτ (e−zτ )K (βτ (e−zτ )) 

τ τ with the contour θ,σ (oriented with an increasing imaginary part) defined by θ,σ := π {z ∈ θ,σ : |(z)| ≤ τ }, and the functions βτ (ζ ) and μ(ζ ), respectively, given by

βτ (ζ ) =

1−ζ τ (1 −

α 2

+ α2 ζ )

and

1 α

μ(ζ ) =

3ζ − ζ 2 α 2

2(1 −

+ α2 ζ ) α 1

.

(3.42)

Proof It follows from the scheme (3.41) that the function W n satisfies ∂¯τα W 1 + (1 − α2 )AW 1 − ( 23 − 43 α)Au 0 = (1 − ∂¯τα W 2 + (1 − α2 )AW 2 − α2 AW 1 − (1 + α4 )Au 0 = (1 −

α )( f 1 + 1 f 0 ), 2 2 α ) f 2 + α f 1 + α f 0 , (3.43) 2 2 4 ∂¯τα W n + (1 − α2 )AW n − α2 AW n−1 − Au 0 = (1 − α2 ) f n + α2 f n−1 , 3 ≤ n ≤ N .

with W 0 = 0. By multiplying both sides by ζ n and summing the results for n = 1, 2, . . . , we obtain ∞ 

ζ n ∂¯τα W n +

n=1

=

∞  

∞ 

((1 − α2 )AW n +

α 2

AW n−1 )ζ n + Au 0

 ∞

n=1

(1 − α2 ) f n +

ζ n + ( 21 −

3α 4 )ζ

+ α4 ζ 2



n=1 α 2





f n−1 ζ n + ( 21 −

3α 4 )ζ



+ α4 ζ 2 f 0 .

(3.44)

n=1

Next we simplify the summations. Since W 0 = 0, by the discrete convolution rule, we have ∞ 

ζ n ∂¯τα W n =

n=1 ∞ 

∞ 

h (ζ ), ζ n ∂¯τα W n = τ −α (1 − ζ )α W

n=0

 (ζ ). ((1 − α2 )AW n + α2 AWnn−1 )ζ n = −((1 − α2 ) + α2 ζ )A W

n=1

Meanwhile, by a simple computation, we have ∞  n=1

ζn +

1 2

−

3α 4

 ζ + α4 ζ 2 =

3ζ −ζ 2 2(1−ζ )

 1−

α 2

 + α2 ζ .

72

3 Convolution Quadrature

Consequently, with g(t) = f (t) − f (0), we derive ∞   (1 − α2 ) f n +

α 2

  f n−1 ζ n + ( 21 −

3α )ζ 4

 + α4 ζ 2 f 0

n=1

∞ ∞     n n α α n−1 (1 − 2 )g + 2 g ζ + = ζ n + ( 21 − n=1

 3α )ζ 4

+

α 2 ζ 4

f0

n=1

=(1 −

α 2

+

α ζ ) g (ζ ) 2

+

3ζ −ζ 2 (1 2(1−ζ )

−

α 2

+ α2 ζ ) f 0 .

Substituting these identities into (3.44) yields  (ζ ) = −κ(ζ )Au 0 + κ(ζ ) f 0 +  g (ζ ), ((βτ (ζ ))α + A) W 3ζ −ζ . Since |ζ | ≤ 1, βτ (ζ )α ∈ θ  for some θ  ∈ ( π2 , π ) (see Lemma with κ(ζ ) = 2(1−ζ ) 3.5 below), by the resolvent estimate (3.1), we have 2

 (ζ ) = ((βτ (ζ ))α + A)−1 (κ(ζ )( f 0 − Au 0 ) +  g (ζ )). W

(3.45)

We can assume f n = f n (so g n = 0) for n > N . Otherwise we redefine Fhn := Fh0 for n > N , and this modification does not affect of the value of W n for n =  (ξ ) defined in (3.45) is analytic with respect to 1, . . . , N . Clearly, the function W ξ in a neighborhood of the origin. Then, like in the proof of Theorem 3.3, the assertion follows from Cauchy’s integral formula, change of variables ζ = e−zτ and then deforming the contour.  The next result shows an important mapping property: βτ (eiθ )α ∈  α2 π . Lemma 3.5 For θ ∈ (0, 2π ), τ −α δ(eiθ )α ∈  απ2 . Proof In fact, for θ ∈ (0, 2π ), we have 2α [sin( θ2 )]α i(− α π+ α θ−ψ(θ)) δ(eiθ )α e 2 2 = , τα τ α ρ(θ ) where the functions ρ(θ ) and ψ(θ ) are defined respectively by  2 (1 − α2 )2 + α4 + α(1 − α2 ) cos θ ,  α α α ψ(θ ) := arg 1 − + cos θ + i sin θ = arctan 2 2 2 1− ρ(θ ) :=

α 2 α 2

sin θ . + α2 cos θ

It is straightforward to compute α (1 − α)(1 − α2 )(1 − cos θ ) α − ψ  (θ ) = 2 α α ≥ 0. 2 2 (1 − 2 + 2 cos θ )2 + α4 sin2 θ

3.3 Fractional Crank–Nicolson Scheme

73

Thus α2 θ − ψ(θ ) is an increasing function of θ , taking values from 0 to απ as θ  changes from 0 to 2π . Thus τ −α δ(eiθ )α ∈  απ2 . The next lemma gives a crucial sector mapping property of the function βτ (e−zτ )α . The proof relies on the observation that βτ (e−zτ )α is very close to βτ (e−is )α for τ z ∈ θ,σ (if θ ∈ ( π2 , π ) is sufficiently close to π2 ) and Lemma 3.5. Lemma 3.6 For α ∈ (0, 1), let φ ∈ ( α2 π, π ) be fixed. Then there exists an 0 > 0 (independent of τ ) such that for σ ∈ (0, 0 ] and θ ∈ ( π2 , π2 + 0 ], we have τ βτ (e−zτ )α ∈ φ , ∀ z ∈ θ,σ ∪  π2 \{0}.

(3.46)

Moreover, the operator (βτ (e−zτ )α + A)−1 is analytic with respect to z in the region τ τ and ± := R ± i πτ , and satisfies enclosed by the curves  τ , θ,σ τ (βτ (e−zτ )α + A)−1  ≤ c|βτ (e−zτ )|−α , ∀z ∈ θ,σ ,

(3.47)

where the constant c is independent of τ (but may depend on φ). τ into two parts, i.e., Proof For the proof, we split the contour θ,σ τ τ θ,σ := σ ∪ θ,± := {z ∈ C : |z| = σ, | arg z| ≤ θ }  ∪ z ∈ C : z = r e±iθ , σ ≤ r ≤

π τ | sin(θ)|



.

(3.48)

τ To prove (3.46), we consider the following three cases z ∈ σ , z ∈ θ,± and z ∈ π  2 \{0}, separately. First, for z ∈ σ ⊂ θ , by choosing σ > 0 sufficiently small and using Taylor’s expansion, we have

βτ (e−zτ )α =

(1 − e−zτ )α = |z|α eiα arg(z) (1 + O(zτ )) ∈ αθ+εσ , τ α (1 − α2 + α2 e−zτ )

for some εσ > 0 with limσ →0+ εσ = 0, showing the relation (3.46) for z ∈ σ . Secτ , we have e−zτ = e−s cot(θ) e−is , s = |z|τ sin(θ ) ∈ (0, π ). ond, for z = |z|eiθ ∈ θ,+ Let γτ (ξ ) := βτ (ξ )α . There exists some c(s) ∈ (0, 1) such that |βτ (e−zτ )α − βτ (e−is )α | = |γτ (e−zτ ) − γτ (e−is )| ≤ cs cot(θ )|γτ (e−c(s)s cot(θ) e−is )|. By choosing θ ∈ ( π2 , π ) sufficiently close to π2 , we have e−c(s)s cot(θ) ≈ 1 and so   |γτ (e−c(s)s cot(θ) e−is )| = 1 − α −

   α −c(s)s cot(θ) −is  ≤ c. + 2e e 1

1−

α 2

The last two inequalities imply |βτ (e−zτ )α − βτ (e−is )α | ≤ cs cot(θ ) ≤ c|θ − π2 |s.

74

3 Convolution Quadrature α

Now if s ∈ (0, π ) is small, then Taylor’s expansion yields βτ (e−is )α ≈ s α ei 2 π asymptotically, and thus |βτ (e−zτ )α − βτ (e−is )α | ≤ c|θ − π2 |s ≤ c|θ − π2 |s 1−α |βτ (e−is )α |. Since βτ (e−is )α ∈  απ2 , cf. Lemma 3.5, it follows that βτ (e−zτ )α ∈ φ when s is sufficiently small. Meanwhile, if s ∈ (0, π ) is away from 0, then |βτ (e−is )α | ≥ c and so |βτ (e−zτ )α − βτ (e−is )α | ≤ c|θ − π2 | ≤ c|θ − π2 ||βτ (e−is )α |. By choosing θ ∈ ( π2 , π ) sufficiently close to π2 , we again have βτ (e−zτ )α ∈ φ . The τ τ is similar as the case θ,+ and thus omitted. Third and last, proof for z = |z|eiθ ∈ θ,− −zτ π for z ∈  2 \{0}, we have |e | ≤ 1. In this case, Lemma 3.5 implies βτ (e−zτ )α ∈  α2 π ⊂ φ . Next we show the analyticity. Since the spectrum of the operator −A lies on the negative real axis, the result (3.46) (with an arbitrary σ ∈ (0, 0 ] and θ ∈ ( π2 , π2 + 0 ]) implies that (βτ (e−zτ )α + A)−1 is analytic with respect to z on the right side of the curve θτ0 ,0 , with θ0 := π2 + 0 . The resolvent estimate (3.47) follows directly from (3.1) and (3.46).  Next, we derive an error estimate for the corrected Crank–Nicolson scheme (3.41). We begin with the case f ≡ 0. By (3.18) and Proposition 3.4, we have 1 e ztn K (z)Au 0 dz, 2π i θ,σ 1 e ztn μ(e−zτ )K (βτ (e−zτ ))Au 0 dz. Wn = τ 2π i θ,σ

w(tn ) =

Hence, the convergence analysis hinges on properly bounding the approximation τ error of the kernel K (βτ (e−zτ )) to K (z) along the contour θ,σ . We begin with basic 1 α α −zτ α −zτ α estimates on the functions (1 − 2 + 2 e ) and (1 − e ) from Proposition 3.4. Lemma 3.7 Let α ∈ (0, 1). Then there exists an 1 > 0 (independent of τ ) such that τ for σ ∈ (0, 1 ] and θ ∈ ( π2 , π2 + 1 ], there hold for any z ∈ θ,σ c0 ≤ |(1 −

+ α2 e−zτ ) α | ≤ c1 , 1

1 |(1 − + α2 e−zτ ) α − (1 − zτ2 )| e−zτ )α − τ α z α (1 − α2 + α2 e−zτ )|

α 2

|(1 −

α 2

(3.49)

≤ cτ |z| ,

(3.50)

≤ c|z|2+α τ 2+α ,

(3.51)

2

2

where the constants c0 , c1 and c are independent of τ , θ and σ (but may depend on 1 ).

3.3 Fractional Crank–Nicolson Scheme

75

Proof Let g(z) = (1 −

α 2

+ α2 e−zτ ) α . 1

π τ , and write z = r eiθ , r ∈ (δ, τ sin ]. Then with s = First we consider z ∈ θ,+ θ 2 r τ sin θ ∈ (0, π ) and γ = − cot θ > 0, η = α − 1 > 1, there holds

|g(z)|α = α2 (η2 + e2γ s + 2ηeγ s cos s) 2 ≥ α2 (η2 + e2γ s − 2eγ s ) 2 ≥ α2 (η − eγ π ). 1

1

Since α ∈ (0, 1), we have η − eγ π > 0, for θ ∈ ( π2 , π ) close to π2 . Next we consider z ∈ σ , with z = σ eiϕ , 0 ≤ ϕ ≤ θ and small σ . Then by letting ρ = τ σ ∈ (0, 1) and 1 s = ρ cos ϕ ∈ [−, ρ], for small  > 0, and h(s) = (ρ 2 − s 2 ) 2 ≤ ρ ≤ 1, we have cos h(s) ≥ 0 and thus |g(z)|α = α2 (η2 + 2ηe−s cos h + e−2s ) 2 ≥ α2 (η2 + e−2s ) 2 ≥ α2 η. 1

1

This shows the lower bound on |g(z)| in (3.49). The upper bound on |g(z)| follows by τ , |g(z)|α ≤ 1 − α2 + α2 e−π cot θ ≤ c, ∀z ∈ θ,+ and a similar bound for z ∈ σ . For the estimate (3.50), it suffices to show (|z|2 τ 2 )−1 |g(z) − (1 −

zτ )| 2

τ ≤ c, ∀z ∈ θ,σ .

(3.52)

If |z|τ ≤ , where  ∈ (0, 1) is to be determined, then by Taylor expansion, we deduce ∞  C( α1 , k)(− α2 + α2 e−zτ )k + O(|z|2 τ 2 ), g(z) − (1 − zτ2 ) = k=2 (γ +1) . (k+1)(γ −k+1)

with C(γ , k) =

Meanwhile, we have

  ∞    α α e − 1 |zτ |k  − + α e−zτ  ≤ α |zτ | ≤ |z|τ .  2  2 2 (k + 1)! 2  k=0 

Since f () = e −1 is increasing in  for  ∈ (0, ∞) and lim→0+ f () = 1, for any  α ∈ (0, 1), there exists an  ∈ (0, 1) such that α(e2−1) < 1. By ratio test, ∞     C( α1 , k)(− α2 + α2 e−zτ )k  ≤ c|z|2 τ 2 ,  k=2

and thus (3.52) holds. Meanwhile, for |z|τ > , there exists an 1 > 0 (independent of τ ) such that for σ ∈ (0, 1 ] and θ ∈ ( π2 , π2 + 1 ], |g(z)| ≤ c. Since |z|τ ≤ sinπ θ τ for z ∈ θ,σ , this again yields (3.52), showing the estimate (3.50). Last we turn to

76

3 Convolution Quadrature

τ the third estimate (3.51). Since |z|τ ≤ c for z ∈ θ,σ , like before, it suffices to show (3.51) for |z|τ ≤ 1. For |z|τ ≤ 1, by Taylor expansion, we deduce

1 − e−zτ = zτ

∞  (−zτ ) j−1

j!

j=1

= zτ + zτ

∞  (−zτ ) j−1 j=2

j!

.

In view of the identity ∞  (−zτ ) j−1

j!

j=2

∞

=

 (−zτ ) j−2 −zτ + (−zτ )2 , 2 j! j=3

we have ∞ ∞ ∞   (−zτ ) j−2   1 (−zτ ) j−1    ≤ e and  ≤  ≤ |z|τ ( 21 + (e − 2 − 21 )) < |z|τ.  j! j! j! j=3

j=3

j=2

Thus we have   ∞  (−zτ ) j−1 α (1 − e−zτ )α = z α τ α 1 + j! j=2 = z α τ α + αz α τ α

∞  (−zτ ) j−1 j=2

j!

+ zα τ α

∞  k=2

C(α, k)

∞  (−zτ ) j−1 k j=2

j!

= z α τ α − α2 z α+1 τ α+1 + O(|z|α+2 τ α+2 ), and

τ α z α (1 −

α 2

+ α2 e−zτ ) = τ α z α − α2 τ α+1 z α+1 + O(|z|α+2 τ α+2 ).

Combining the last two estimates completes the proof of the lemma.



The next lemma provides the crucial estimate on the functions μ and βτ . Lemma 3.8 Let μ(ξ ), βτ (ξ ) be defined as (3.42) and the constant 1 be given in τ Lemma 3.7. Then for σ ∈ (0, 1 ] and θ ∈ ( π2 , π2 + 1 ], we have for any z ∈ θ,σ |μ(e−zτ ) − 1| ≤ cτ 2 |z|2 , |βτ (e−zτ ) − z| ≤ cτ 2 |z|3 , |βτ (e−zτ )α − z α | ≤ cτ 2 |z|2+α ,

and

c0 |z| ≤ |βτ (e−zτ )| ≤ c1 |z|.

The constants c0 , c1 and c are independent of τ , θ and σ (but may depend on 1 ). Proof The first three estimates are direct from Lemma 3.7. The upper bound on βτ (e−zτ ) follows from these estimates and the triangle equality

3.3 Fractional Crank–Nicolson Scheme

77

|βτ (e−zτ )| ≤ (|βτ (e−zτ ) − z| + |z|) ≤ (1 + c2 τ 2 |z|2 )|z|  z ∈ σ , (1 + c2 τ 2 δ 2 )|z|, ≤ (1 + c2 ( sinπ θ )2 )|z|, z ∈ θ . −zτ

Since c0 |z| ≤ | 1−eτ | ≤ c1 |z| (see Lemma 3.1), the lower bound on βτ (e−zτ ) follows from the fact that |1 − α2 + α2 e−zτ | is uniformly bounded from below in τ for all τ , cf. Lemma 3.7.  z ∈ θ,σ Using Lemmas 3.6 and 3.8, we have the following error estimate for K (βτ (e−zτ )). Lemma 3.9 Let 0 and 1 be defined in Lemma 3.4 and Lemma 3.7, respectively. Then by choosing  = min(0 , 1 ) and θ = π2 + , we have τ , μ(e−zτ )K (βτ (e−zτ )) − K (z) ≤ cτ 2 |z|1−α , ∀ z ∈ θ,σ

where the constant c is independent of τ . Proof By the triangle inequality, we obtain μ(e−zτ )K (βτ (e−zτ )) − K (z) ≤ |μ(e−zτ ) − 1|K (z) + |μ(e−zτ )|K (βτ (e−zτ )) − K (z) =: I + II. The bound on the term I follows from (3.1) and Lemma 3.8. Appealing to Lemma 3.8 again yields |βτ (e−zτ )−1 − z −1 | = |z − βτ (e−zτ )||βτ (e−zτ )|−1 |z|−1 ≤ cτ 2 |z|. Similarly, using (3.1) and (3.47), and Lemma 3.8, and the identity (βτ (e−zτ )α + A)−1 − (z α + A)−1 = (z α − βτ (e−zτ ))(βτ (e−zτ )α + A)−1 (z α + A)−1 ,

we obtain (βτ (e−zτ )α + A)−1 − (z α + A)−1  ≤|βτ (e−zτ )α − z α |(βτ (e−zτ )α + A)−1 (z α + A)−1  ≤cτ 2 |z|2+α (βτ (e−zτ )α + A)−1 (z α + A)−1  ≤ cτ 2 |z|2−α , and hence, the term II can be bounded by II ≤ c|βτ (e−zτ )−1 − z −1 |(z α + A)−1  + c|z|−1 (βτ (e−zτ )α + A)−1 − (z α + A)−1  ≤ cτ 2 |z|1−α , which completes the proof of the lemma.



78

3 Convolution Quadrature

Now we state an error estimate for the fractional Crank–Nicolson scheme (3.41). Theorem 3.5 Let u and U n be the solutions of problem (3.2) and the scheme (3.41), respectively. Then the following statements hold. (i) If f ≡ 0 and u 0 ∈ D(Aq ) with q ∈ [0, 1], then there holds qα

u(tn ) − U n  L 2 () ≤ cτ 2 tn2

−2

(ii) If u 0 ≡ 0 and f ∈ C 1 ([0, T ]; L 2 ()) and t ∈ [0, T ], then

Aq u 0  L 2 () , n ≥ 1.

t

0 (t

− s)α−1  f  (s) L 2 () ≤ c for all

 u(tn ) − U n  L 2 () ≤ cτ 2 tnα−2  f (0) L 2 () + tnα−1  f  (0) L 2 () tn + (tn − s)α−1  f  (s) L 2 () ds . 0

Proof (i) f = 0 and u 0 ∈ D(A). With the constants 0 and 1 from Lemmas 3.4 and 3.7, respectively, we choose  = min(0 , 1 ) and θ = π2 + . By Lemma 3.4, we split the error into 1 u(tn ) − U = 2π i



n

+

τ θ,σ

  e ztn K (z) − μ(e−zt )K (βτ (e−zτ )) Au 0 dz



1 2π i

τ θ,σ \θ,σ

e ztn K (z)Au 0 dz =: I + II.

By Lemma 3.9 and choosing σ = tn−1 , we bound the first term I by  I L 2 () ≤ cτ 2 Au 0  L 2 () ≤

c(tnα−2

+σ

2−α

π τ sin θ

σ

eρtn cos θ ρ 1−α dρ +

)τ Au 0  L 2 () ≤ 2



θ

eσ tn | cos ϕ| σ 2−α dϕ



−θ 2 α−2 cτ tn Au 0  L 2 () .

For the second term II, by the estimate (3.1) and the change of variables s = ρtn , we obtain ∞ eρtn cos θ ρ −α−1 dρ II L 2 () ≤ cAu 0  L 2 () π τ sin θ



≤ cτ Au 0  2

L 2 ()

≤ cτ 2 tnα−2 Au 0 

∞

0∞ 0

eρtn cos θ ρ 1−α dρ eρ cos θ ρ 1−α dρ ≤ cτ 2 tnα−2 Au 0  L 2 () ,

3.4 Parallel in Time Algorithm

79

where the last inequality follows since for θ ∈ ( π2 , π ) and α ∈ (0, 1),

∞ ρ cos θ 1−α ρ dρ < c. Then we confirm the assertion (i) for u 0 ∈ D(A). Then proof 0 e for the case u 0 ∈ L 2 () is similar, and the intermediate case follows by interpolation. (ii) The argument for the case u 0 = 0 and f = 0 follows from the expansion f = f (0) + t f  (0) + t ∗ f  = t f  (0) + t ∗ f  , and the argument in the proof of Theorem 3.4.  The initial correction in the fractional Crank–Nicolson scheme (3.41) is crucial to achieve the desired O(τ 2 ) convergence. Otherwise, we can only derive an O(τ ) rate u(tn ) − U n  L 2 () ≤ cτ tnα−1 Au 0  L 2 () , even if the initial data u 0 is smooth. This was numerically verified in [JLZ18a, Table 2]. The key of correction is to choose a proper function μ in (3.42), such that the estimate |μ(e−zτ ) − 1| ≤ cτ 2 |z|2 in Lemma 3.8 holds.

3.4 Parallel in Time Algorithm Now we develop a parallel-in-time (PinT) algorithm for the bdfk cq scheme with initial correction. For n ≥ 1, the corrected bdfk cq scheme looks for U n ∈ H01 () such that  ∂¯τα U n + AU n = f¯n , n = 1, 2, · · · , N , (3.53) U n = u 0 , n ≤ 0, with the modified source f¯n given by f¯n = f (tn ) + an(k) ( f (0) − Au 0 ) +

k−2 

(k)  () b,n τ f (0),

(3.54)

=1 (k) where the coefficients an(k) and b,n are given in Table 3.1. Note that f¯n coincides f (tn ) for n ≥ k. To develop a PinT solver, we proceed as follows. For given the n N )n=1 (from the (m − 1)th iteration), we compute the update Umn by iterate (Um−1

⎧ α n n ¯ ¯ n = 1, 2, . . . , N , ⎪ ⎨ ∂τ Um + AUm = f n , N −n −n N −n Um = u 0 + κ(Um − Um−1 ), n = 0, 1, . . . , N − 1, ⎪ ⎩ Umn = u 0 , n ≤ −N ,

(3.55)

with the corrected source f¯n given in (3.54), where κ ∈ (0, 1) is a relaxation paramN to the corrected scheme (3.53) is a fixed eter to be determined. The solution (U n )n=1 point of the iteration (3.55). In the fully discrete case, we may rewrite the iteration (3.55) into

80

3 Convolution Quadrature

τ −α (Bk (κ) ⊗ Ix )Um + (It ⊗ A)Um = Fm−1 , where ⊗ denotes Kronecker product of two matrices, A ∈ R Nh ×Nh the Galerkin fem approximation of the elliptic operator A (with Nh being the degree of freedom of the fem problem), Ix ∈ R Nh ×Nh and It ∈ R N ×N are identity matrices, and Um = (Um1 , Um2 , · · · , UmN ) , Fm−1 = (F1 , F2 , · · · , FN ) with Fn = f¯n + κτ −α

N −1 

N +n− j

ω(α) j Um−1

+ τ −α

j=n

n−1 

ω(α) j u0,

j=0

and the matrix Bk (κ) ∈ R N ×N (with subscript k indicating the order of bdf) given by ⎤ ⎡ (α) (α) (α) ω0 κω(α) · · · κω κω 2 1 N −1 ⎥ ⎢ (α) ω0(α) · · · κω3(α) κω2(α) ⎥ ⎢ ω1 ⎢ .. ⎥ . ⎥ ⎢ Bk (κ) = ⎢ ω2(α) ω1(α) . . . ⎥. ⎥ ⎢ . .. ⎥ ⎢ . ⎣ . . ω0(α) κω(α) N −1 ⎦ (α) (α) ω(α) ω0(α) N −1 ω N −2 · · · ω1 The following diagonalization lemma lays the foundation for developing a PinT algorithm for the corrected bdfk- cq scheme (3.53). N −1

Lemma 3.10 Let (κ) = diag(1, κ − N , · · · , κ − N ). Then the following identity holds Bk (κ) = S(κ)Dk (κ)S(κ)−1 , S(κ) = (κ)V, 1

where Dk (κ) ∈ R N ×N is a diagonal matrix and V ∈ C N ×N is the Fourier matrix # 2(n−1)π 2(n−1)(N −1)π $ N V = [v1 , v2 , . . . , v N ] ∈ C N ×N , with vn = 1, ei N , . . . , ei ∈ CN . Lemma 3.10 allows developing a parallel solver for the scheme (3.55), by inverting the three factors of the matrix Bk (κ) sequentially. The complete procedure is listed in Algorithm 1. Algorithm 1 PinT bdfk scheme for subdiffusion. 1: Solve (S(κ) ⊗ Ix )H = Fm−1 . 2: Solve (Dk (κ) ⊗ Ix + τ α It ⊗ A)Q = τ α H. 3: Solve (S(κ)−1 ⊗ Ix )Um = Q.

Now we comment on the potential speedup. Note that the discrete operator ∂¯τα requires the solutions at all the previous steps. Indeed, the n-th step of a bdf cq scheme, requires solving a Poisson-like problem

3.4 Parallel in Time Algorithm

81

(ω0(α) I + τ α A)U n =

n 

0 n− j ω(α) ) + ω0(α) U 0 + f¯n . j (U − U

j=1

Let Nh be the number of degrees of freedom and M f the total real floating point operations for solving the Poisson-like equations. The computational cost of this step is O(n Nh + M f ), where the cost O(n Nh ) is associated with the nonlocality of the discrete fractional derivative ∂¯τα . Then summing over n from 1 to N , we deduce that the total cost of a direct implementation of a bdf cq scheme is O(Nh N 2 + M f N ). Now consider parallelizing Algorithm 1 with p used processors. The cost of parallel fast Fourier transform (fft) in Step 1 and Step 3 is O( p −1 [Nh N log(N )]). The computational cost of Fm−1 in Step 1 consists of the following three components: 1. The source term f¯n defined in (3.53): The correction is taken at first few steps and hence the computational is O( p −1 Nh N ).

N −1 cost (α) N +n− j 2. The convolution term j=n ω j Um−1 can be rewritten as the n-th entry of ⎡

(α) ω(α) N ω N −1 ⎢ (α) ⎢ 0 ωN ⎢ . .. ⎢ . . ⎢ . ⎢ ⎣ 0 0 0 0

· · · ω2(α) · · · ω3(α) . . .. . . · · · ω(α) N ··· 0

⎤⎡

⎤ 1 Um−1 ⎥ ⎢U 2 ⎥ ⎥ ⎢ m−1 ⎥ ⎥⎢ . ⎥ ⎥ ⎢ . ⎥ := WUm−1 . ⎥⎢ . ⎥ ⎥ ⎢ N −1 ⎥ ω(α) N −1 ⎦ ⎣Um−1 ⎦ N ω(α) Um−1 N ω1(α) ω2(α) .. .

Since the matrix W is not circulant, one cannot apply the fft algorithm directly. Nonetheless, it can be extended to be circulant to facilitate the use of the fft algorithm ⎡

0 0 (α) ⎢ ω 0    ⎢ 1 Z W 0 ⎢ .. .. , with Z := ⎢ . . W Z Um−1 ⎢ (α) (α) ⎣ω ω N −2 N −3 (α) ω(α) N −1 ω N −2

⎤ ··· 0 0 · · · 0 0⎥ ⎥ . . .. .. ⎥ ∈ R N ×N . . . .⎥ ⎥ · · · 0 0⎦ · · · ω1(α) 0

The extended matrix is circulant and can be diagonalized using fft with O(N log N ) operations [GVL12, Sect. 4.7]. Using the bulk synchronous parallel fft algorithm [IB01], the fft of the extended system and [0, Um−1 ]T lead to O( p −1 [N log(N )]) and O( p −1 [Nh N log(N )]) operations, respectively. Using −1 (α) N +n− j ω j Um−1 can be achieved the inverse fft, the cost of evaluating the term Nj=n −1 )]). at O( p [N

h N log(N (α) ω u can be computed with a cost O( p −1 [N log(N ) + Nh N ]). 3. The term n−1 0 j=0 j

82

3 Convolution Quadrature

 f N ]), where M  f denotes the cost for solving For Step 2, the total cost is O( p −1 [ M a Poisson-like problem obtained after diagonalization (which can be different from  f N ]), M f ). The overall cost for the parallel algorithm is O( p −1 [Nh N log N + M which becomes O(Nh log N + % M f ) if p = O(N ). However, the approach does not reduce the storage requirement. Last we give a convergence analysis of the PinT algorithm. We use the following preliminary estimate. Lemma 3.11 For β, γ ≥ 0, there holds n 

(n + 1 − i)−β i −γ

i=1

⎧ max(1−γ ,0)−β , ⎨ cn ≤ cn −β log(1 + n), ⎩ − min(β,γ ) cn ,

0 ≤ β < 1, γ = 1, 0 ≤ β ≤ 1, γ = 1, β > 1, γ > 1.

Proof We denote by [·] the integral part of a real number. Then [2] n n    (n + 1 − i)−β i −γ = (n + 1 − i)−β i −γ + (n + 1 − i)−β i −γ := I + II. n

i=1

i=[ n2 ]+1

i=1

Then, since for 1 ≤ i ≤ [ n2 ], there holds (n + 1 − i)−β ≤ cn −β and for [ n2 ] + 1 ≤ i ≤ n, there holds i −γ ≤ cn −γ , we deduce [2]  n

I ≤ cn

−β

i −γ and II ≤ cn −γ

n 

(n + 1 − i)−β .

i=[ n2 ]+1

i=1

j −1

j −γ ≤ cj max(1−γ ,0) if γ = 1 and ≤ Simple computation gives i=1 i i=1 i c log( j + 1). Combining these estimates yields the desired assertion.  Now we show the convergence of the iteration (3.55) as m → ∞. Lemma 3.12 Let Umn be the solution to the scheme (3.55) with u 0 = 0 and f¯n = 0 for all n = 1, 2, . . . , N . Then there exists a parameter κ = O((log N )−1 ) such that N   N − j+1 Umn  L 2 () ≤ ctnα−1 γ (κ)m τ t −α  L 2 () , j U0 j=1

where γ (κ) ∈ (0, 1) may depend on α, κ, β0 and T , but is independent of τ , n and m. Proof We use the discrete solution operator E τn defined by (with ρ being a small number)  −1 1 E τn = ξ −n−1 δτ (ξ )α + A dξ. 2π τ i |ξ |=ρ

3.4 Parallel in Time Algorithm

83

Then (3.34) implies the following estimate α−1 , n = 1, 2, . . . . , N , E τn  ≤ ctn+1

(3.56)

where c is independent of τ and n. Now the scheme (3.55) can be recast into 

∂¯τα Umn + AUmn = f¯n − κτ −α G nm , n = 1, 2, · · · , N , Umn = u 0 , n ≤ 0,

where the term G nm is given by G nm = τ

N −1 

N− j  (α)  N − j ωn+ − Um−1 . j Um

j=0

Upon noting the condition f¯n = 0, we obtain the following representation Umn = −κτ 1−α

n 

E τn−i G im = −κτ 1−α

i=1

n 

E τn−i

i=1

N −1 

N− j

N− j ω(α) − Um−1 ). j+i (Um

j=0

Now the estimate (3.56) implies Umn  L 2 () ≤ cκτ 1−α

n 

α−1 tn−i+1

i=1

≤ cκ

n 

N −1 

N− j

N− j |ω(α) − Um−1  L 2 () j+i | Um

j=0

(n − i + 1)α−1

i=1

N −1 

N− j

N− j |ω(α) − Um−1  L 2 () . j+i | Um

j=0

Then Lemma 3.2 implies Umn  L 2 () ≤ cκ

n N −1   N− j (n − i + 1)α−1 ( j + i)−α−1 UmN − j − Um−1  L 2 () i=1

= cκ

N −1 

j=0 N− j

UmN − j − Um−1  L 2 ()

j=0

≤ cκτ

α

N −1 

n  (n − i + 1)α−1 ( j + i)−α−1 i=1

N− j t −α j+1 Um

N− j − Um−1  L 2 ()

j=0

n  (n − i + 1)α−1 ( j + i)−1 i=1

N   N − j+1 N − j+1 ≤ cκtnα−1 log(n + 1) τ t −α − Um−1  L 2 () , j Um j=1

(3.57)

84

3 Convolution Quadrature

n where the last step follows from Lemma 3.11: i=1 (n − i + 1)α−1 ( j + i)−1 ≤ −α α−1 log(n + 1). Multiplying τ t N −n+1 on both sides of (3.57) and then summing cn over n give τ

N 

tn−α UmN −n+1  L 2 ()

n=1 N N     N − j+1 N − j+1 ≤ cκ τ t N−α−n+1 tnα−1 log(n + 1) τ t −α − Um−1  L 2 () j Um n=1

≤ cκ log N

j=1

N 

−α α−1

(N − n + 1)

n

N   N −n+1 τ tn−α UmN −n+1 − Um−1  L 2 ()

n=1

n=1

N   N −n+1 ≤ cκ log N τ tn−α UmN −n+1 − Um−1  L 2 () , n=1

where c in the second inequality depends on α, and in the last inequality, we have used N  (N − n + 1)−α n α−1 n=1

=

N  n=1 N

=

n

(N − n + 1)−α n α−1 ds ≤

n−1

(N − s)−α s α−1 ds =

0



N  n=1

1

n

(N − s)−α s α−1 ds

n−1

(1 − s)−α s α−1 ds = B(α, 1 − α),

0

where B(·, ·) denotes the Beta function, cf. Appendix A.1. By the triangle inequality, choosing κ small enough such that cκ log N < 1, we derive τ

N 

cκ log N   −α N −n+1 τ tn Um−1  L 2 () . 1 − cκ log N n=1 N

tn−α UmN −n+1  L 2 () ≤

n=1

Finally, let

γ (κ) = (1 − cκ log N )−1 cκ log N .

(3.58)

(3.59)

By choosing κ such that cκ log N ∈ (0, 21 ), we have γ (κ) ∈ (0, 1). Then, by (3.57),

3.4 Parallel in Time Algorithm

85

N   N − j+1 N − j+1 Umn  L 2 () ≤ cκtnα−1 log N τ t −α − Um−1  L 2 () j Um j=1 N N    N − j+1 N − j+1 2 ≤ cκtnα−1 log N τ t −α U  + τ t −α  L 2 () L () m j j Um−1 j=1

j=1

≤ cκtnα−1 log N (1 + γ (κ))τ

N 

N − j+1

t −α j Um−1

 L 2 ()

j=1 N   N − j+1 2 () . ≤ ctnα−1 γ (κ) τ t −α U  L m−1 j j=1

Then repeating the estimate (3.58) leads to the desired result.



The stability result in Lemma 3.12 leads to the desired convergence. Corollary 3.1 Let Umn be the solution to the iterative scheme (3.55), and U n the solution to (3.53). Then there exists a κ = O((log N )−1 ) such that for any m ≥ 1 N   N − j+1 N − j+1 U n − Umn  L 2 () ≤ ctnα−1 γ (κ)m τ t −α − U0  L 2 () , j U j=1

where γ (κ), given by (3.59), may depend on κ, β0 and T , but is independent of τ , n and m. N Proof Since (U n )n=1 is the unique fixed point of (3.55), the error emn = Umn − U n satisfies

∂¯τα emn + Aemn = −κτ −α K mn , n = 1, 2, · · · , N , em− j = 0, with K mn = 3.12.

N −1 j=0

N− j

(α) ωn+ j (em

j = 0, 1, . . . , k − 1,

N− j

− em−1 ). Then the assertion follows from Lemma 

Combining Corollary 3.1 with Theorem 3.4 gives the following error estimate. Theorem 3.6 Let u 0 ∈ L 2 () and f ∈ W k, α + (0, T ; L 2 ()) for some  > 0. Let Umn be the solution to the scheme (3.55) with U0n = u 0 , n = 0, 1, . . . , N , and u the exact solution. Then for n = 1, . . . , N , there holds 1

Umn − u(tn ) L 2 () ≤ c(γ (κ)m tnα−1 + τ k tn−k ), with κ = O((log N )−1 ), where c and γ (κ) given by (3.59) may depend on k, κ, β0 , T , u 0 and f , but are independent of τ , n, m and u.

86

3 Convolution Quadrature

Proof We split the error Umn − u(tn ) into Umn − u(tn ) = (Umn − U n ) + (U n − u(tn )). The term U n − u(tn ) is already bounded in Theorem 3.4 and the remark thereafter. By Corollary 3.1 and the triangle inequality, the term Umn − U n can be bounded by U n − Umn  L 2 () ≤ ctnα−1 γ (κ)m τ

N 

N − j+1 t −α − u 0  L 2 () j U

j=1

= ctnα−1 γ (κ)m τ

N 

 N − j+1  U t −α  L 2 () + u 0  L 2 () . j

j=1

Theorem 3.4 and the data regularity imply U n  L 2 () ≤ c, n = 1, 2, . . . , N . Thus, we obtain U n − Umn  L 2 () ≤ cT tnα−1 γ (κ)m . Then the desired result follows immediately.  The explicit formula of γ (κ) in (3.59) indicates that the iteration (3.55) converges linearly when cκ log N ∈ (0, 21 ), and the convergence may deteriorate for a large N and a fixed κ. Numerically the iteration (3.55) converges robustly even for a relatively large κ and the step number N seems not to affect much the convergence rate [WZ21].

3.5 Fast Convolution In this section, we develop an efficient numerical algorithm for approximating fractional integrals using bdf cq, and then apply it to solving subdiffusion. We present the algorithm only for backward Euler cq, but higher-order schemes can be impleβ mented similarly. For any β ∈ (0, 1), 0 It denotes the Riemann–Liouville fractional integral of order β, defined by (cf. Definition 1.1) β 0 It v(t)

=

1 (β)



t

(t − s)β−1 v(s) ds, for t ∈ [0, T ].

0

Consider a uniform partition of the interval [0, T ] with a step size τ = N −1 T and grid points tn = nτ , n = 0, 1, . . . , N . Then the backward Euler cq approximation reads (with the shorthand v j = v(t j )) β 0 It v(tn )

≈ τβ

n 

(−β)

ωn− j v j , with

j=0

(−β)

ωj

ζ j = (1 − ζ )−β .

(3.60)

j=0 (−β)

Note that the convolution weights ωn ωn(−β)

∞ 

τ 1−β sin(βπ ) = π

can be expressed as 0

∞

s −β (1 + τ s)−n−1 ds.

(3.61)

3.5 Fast Convolution

87

Indeed, Cauchy’s integral formula implies ωn(−β)

1 = 2π i

|ζ |=1

(1 − ζ )

−β −n−1

ζ

τ 1−β dζ = 2π i

C

z −β (1 − τ z)−n−1 dz,

where C ⊂ C is a Hankel contour. Since the function z −β is analytic in the complex plane C except for the branch cut on the negative real axis, C can be degenerated into the negative real axis (−∞, 0). Thus we obtain ωn(−β) =

τ 1−β 2π i



∞

(eβπi − e−βπi )s −β (1 + τ s)−n−1 ds,

0

which directly implies the representation (3.61). To approximate the integral in (3.61), we first truncate the interval (0, ∞) to a finite one (0, L) (for some L > 0), and then apply suitable quadrature rules on the finite interval (0, L). First, we truncate the interval (0, ∞) to (0, L) ωn(−β)

τ 1−β sin(βπ ) = π



L

s −β (1 + τ s)−n−1 ds + R(L),

(3.62)

0

where R(L) denotes the truncation error. Let L = aτ −1 , for some a > 0. Then the truncation error R(L) is bounded by sin(βπ ) |R(L)| = π



∞

(1 + s)−n−1 s −β ds ≤

a

sin(βπ ) −β a (1 + a)−n . nπ

Given a tolerance  > 0, choosing a> ensures |R(L)| ≤

1  sin(βπ ) n+β

nπ

(3.63)

sin(βπ ) −β−n sin(βπ ) −β a |a + 1|−n ≤ a < . nπ nπ

The estimate becomes uniform in n > n 0 by setting n = n 0 + 1 in this bound. Next, we use numerical quadrature to approximate the finite integral (3.62). First, we consider the integral over an initial interval (0, L 0 ): I0,n =

τ 1−β sin(βπ ) π



L0

s −β (1 + sτ )−n−1 ds.

0

Changing variables to the reference interval [−1, 1] gives I0,n =

τ 1−β sin(βπ ) π



L0 2

1−β

 −n−1 y+1 τ L0 (y + 1)−β 1 + dy. 2 −1 1

88

3 Convolution Quadrature

Then we approximate the integral by weighted Gauss–Jacobi quadrature with nodes y j and weights w j , j = 1, . . . , Q. The quadrature error RGJ,n (Q) is given by RGJ,n (Q) :=

1 −1

f 0 (y)w(y) dy −

Q 

w j f 0 (y j ),

(3.64)

j=1

with τ 1−β sin(βπ ) f 0 (y) = π



L0 2

1−β  −n−1 y+1 1+ and w(y) = (y + 1)−β . τ L0 2

Now we give an error estimate for (weighted) Gauss–Jacobi quadrature. A proof of this result for w ≡ 1 can be found in [Tre13, Chap. 19], and that for the weighted Gauss quadrature in [BLF19, Theorem 3]. The Bernstein ellipse E is given as the −1 image of the circle of radius  > 1 under the map z → z+z2 . The largest imaginary part on Eρ is

−−1 2

and the largest real part is

+−1 . 2

Lemma 3.13 Let f be analytic inside the Bernstein ellipse E with  > 1 and bounded there by M. Then the error of Gauss quadrature with weight w is bounded by Q   1  −2Q+1 1   w f dy − w j f (y j ) ≤ 4M w dy.   − 1 −1 −1 j=1 Lemma 3.13 implies the following bound on the quadrature error RGJ,n (Q) in (3.64). Theorem 3.7 Let max = 1 + b

−1

log

1 , 1−b

and opt =

4Q γ T L0

2b L0τ

+ (( L2b0 τ )2 +

+ (1 +

( γ 4Q )2 ) T L0

1 2

4b 21 ) , with some b L0τ

∈ (0, 1) and γ =

. Then the following estimate holds

⎧   1−β L 1−β sin(βπ )  ⎪ γ T L0 eγ T L 0 2Q ⎪ γb τ 0 ⎪ e , if opt ∈ (1, max ), 1 + ⎪ ⎨ π(1 − β) 4Q 8Q   |RGJ,n (Q)| ≤ 1−β −2Q+1 ⎪ τ 1−β L 0 sin(βπ ) max T ⎪ ⎪ ⎪ eγ b eγ b τ , otherwise. ⎩ π(1 − β) max − 1 1 Proof For a fixed b ∈ (0, 1) and γ = b−1 log 1−b , it is easy to deduce

|(1 − z)−1 | ≤ eγ (z) ≤ eγ b for 0 ≤ (z) ≤ b.

(3.65)

Next we choose a suitable Bernstein ellipse E in order to avoid the singularities of (1 − z)−n−1 in the right-half complex plane (so that there is a restriction on ρ), −1 − 2) L40 = and then apply Lemma 3.13. We choose max such that τ (max + max 2b δmax b, which implies, with max = e , cosh(δmax ) − 1 = L 0 τ , i.e., δmax = cosh−1 (1 + 2b ) giving the stated expression for max . By Lemma 3.13 and the estimate (3.65), L0τ

3.5 Fast Convolution

89

we derive   −2Q+1  L 0 −n−1  max (1 + τ (ζ + 1) 2 )   − 1 ζ ∈Eρ  −2Q+1  1−β L0 τ 1−β L 0 sin(βπ )  −1 ≤ min eγ tn 4 (+ −2) 1 0. Specifically, the function t −β is approximated by an soe on the interval [δ, T ], δ > 0, with an error . That is, there Ne ⊂ R2+ (the pairs of nodes and weights) such that exist positive numbers (s , ω )=1 t −β ≈

Ne 

ω e−s t , t ∈ [δ, T ].

(4.10)

=1

There are several different ways to construct soe approximations. Below we describe two constructions, whose starting point is the following integral representation of the function t −β , β > 0: t −β =



∞

e−ts ωβ (s) ds, t > 0.

(4.11)

0

This identity follows easily from the definition of the Gamma function (z) in (A.1) and changing variables ξ = ts as 

∞

(β) = 0

ξ

β−1 −ξ

e



∞

dξ = 0

(ts)

β−1 −ts

e

t ds = t

β



∞

e−ts s β−1 ds.

0

Thus the function t −β can be represented as the mixture of infinitely many exponentials. Beylkin and Monzón [BM10] constructed an soe approximation by substituting s = e x into (4.11):  ∞ 1 x t −β = e−te +βx dx.

(β) −∞

110

4 Finite Difference Methods: Construction and Implementation

Then applying the infinite trapezoid rule with a step size h > 0 leads to t −β =

∞ 1  ωn e−sn t , with sn = ehn and ωn = he−βnh .

(β) n=−∞

(4.12)

The relative error satisfies a uniform bound for 0 < t < ∞, and if t is restricted on a compact interval [δ, T ] with 0 < δ < T < ∞, then we can similarly bound the relative error of the finite soe. The next remarkable result shows the relative error of the approximation (4.12) in closed form, and also bounds the two tails in terms of the upper incomplete Gamma function (β, x), cf. Sect. A.1 for the definition. Theorem 4.1 The relative error ρ(t) of the approximation (4.12) is represented by ∞ ∞    tβ  ωn e−sn t = −2 r ( 2nπ ) cos ( 2nπ ) log t − φ( 2nπ ) , h h h

(β) n=−∞ n=1 (4.13) ) iφ(ξ ) = r (ξ )e with where r (ξ ) and φ(ξ ) are real-valued functions defined by (β+iξ

(β) r (ξ ) > 0 and φ(0) = 0. Moreover,

ρ(t) := 1 −

r (ξ ) ≤

e−θ|ξ | , for 0 ≤ θ < (cos θ )β

π , −∞ 2

< ξ < ∞.

Furthermore, the following bounds hold for the tail summations: tβ

∞ 

ωn e−sn t ≤ (β, te N h ), if te N h ≥ β,

n=N +1

tβ

−M−1 

ωn e−sn t ≤ (β) − (β, te−Mh ), if te−Mh ≤ β.

n=−∞

Proof For each t > 0, the integrand f (x) = e−te +βx belongs to the Schwarz class of rapidly decreasing C ∞ functions. Then by Poisson summation formula (A.19), we have x

h

∞ 

∞ 

f (nh) =

n=−∞

)= f˜( 2nπ h



n=−∞

∞

−∞

f (x) dx +



), f˜( 2nπ h

n=0

where the Fourier transform f˜ of f is given by f˜(ξ ) =



∞

−∞

e−iξ x f (x) dx =



∞

−∞

e−te

x

+(β−iξ )x

dx,

4.2 Sum of Exponential Approximation

111

cf. Appendix A.3 for the definition. Then substituting p = te x gives 

1

f˜(ξ ) =

t β−iξ

∞

e− p p β−iξ −1 d p =

0

(β − iξ ) , t β−iξ

so, with sn and ωn defined by (4.12), ∞ ) i 2nπ 1  1 1  (β − i 2nπ h t h . ωn e−sn t = β + β

(β) n=−∞ t t n=0

(β)

The formula for the relative error ρ(t) follows directly from the identity (β + iξ ) =

(β − iξ ) for all real ξ ; hence, r (−ξ ) = r (ξ ) and φ(−ξ ) = −φ(ξ ). To bound r (ξ ), let y > 0 and define the ray Cθ = {seiθ : 0 < s < ∞}. By Cauchy’s theorem, 

(β + iy) =

Cθ

e− p p β+iy−1 d p =



∞

e−se (seiθ )β+iy s −1 ds, iθ

0

and thus

| (β + iy)| ≤

 ∞ 0

e−s cos θ e−θ y s β−1 ds =

 ∞ e−θ y e−θ y e−s s β−1 ds =

(β), β (cos θ ) 0 (cos θ)β

which implies the desired bound on r (ξ ). Last, we bound the two tail sums. For x each t > 0, f (x) = e−te +βx is decreasing for x > log βt . Hence, if te N h ≥ β, then using the substitution p = te x , tβh

∞ 

f (nh) ≤ t β



Similarly, f (−x) = e−te −M−1 

f (nh) = t β h

n=−∞

 f (x) dx =

−x

−βx

∞  n=M+1

∞ te N h

Nh

n=N +1

tβh

∞

e− p p β−1 d p = (β, te N h ).

is decreasing for x > log βt so if te−Mh ≤ β, then f (−nh) ≤ t β



∞ Mh

 f (−x) dx =

te−Mh

e− p p β−1 d p,

0

by changing variables p = te−x . This completes the proof of the theorem.



Theorem 4.1 leads to a finite soe approximation of the following form (4.10): t −β =

N 1  ωn e−sn t , with sn = ehn and ωn = he−βnh .

(β) n=−M

(4.14)

112

4 Finite Difference Methods: Construction and Implementation

Theorem 4.1 (and asymptotics of the incomplete Gamma function in Appendix A.1) indicates that the relative error N ρM (t) := 1 −

N tβ  ωn e−sn t

(β) n=−M

is bounded by  provided that h −1 ≥ c| log |,

N ≥ ch −1 log(δ −1 | log |), and M ≥ ch −1 log(T  −1 ).

Jiang et al. [JZZZ17] construct an soe approximation with an absolute error  from (4.11) directly using composite quadrature: first truncate the integral to a finite interval, then subdivide the finite interval into a set of dyadic subintervals and approximate the integral on each dyadic interval with proper quadratures. Theorem 4.2 Fix β ∈ (0, 2), 0 < δ ≤ t ≤ T with δ ≤ 1 and T ≥ 1, tolerance  ∈ Ne such that (0, 1). Then there exist positive tuples (s , ω )=1 Ne    −β   ω e−s t  t − =1

C[δ,T ]

≤ ,

(4.15)

where the number Ne of exponentials is Ne = O(| log |(log | log | + log Tδ ) + | log δ|(log | log | + | log δ|)). Proof To obtain an soe approximation, we first truncate the integral to a finite interval, and then divide the finite interval into a set of dyadic intervals, and approximate the integral with proper quadrature. The details are given next. (i) For t ≥ δ > 0, and p > 1, there holds 



∞

e

−ts

ωβ (s) ds ≤

e−δp ωβ ( p)δ −1 , 0 < β ≤ 1, e−δp 2β−1 ( pω p ( p) + δ −β ), 1 < β < 2.

p

Indeed, for 0 < β ≤ 1, we have 

 ∞ e−ts ωβ (s) ds = e−t p e−t x ωβ (x + p) dx p 0  ∞ e−δx dx ≤ e−δp ωβ ( p)δ −1 . ≤e−t p ωβ ( p) ∞

0

Similarly, for 1 < β < 2,

4.2 Sum of Exponential Approximation



113

 ∞ e−ts ωβ (s) ds = e−t p e−t x ωβ (x + p) dx p 0  p   ∞ ωβ (2 p) dx + e−δx ωβ (2x) dx ≤ e−δp 0 p    ∞   −δp β−1 −δx pωβ ( p) + ≤e 2 e ωβ (x) dx ≤ e−δp 2β−1 pωβ ( p) + δ −β . ∞

0

(ii) On each dyadic interval (2 j , 2 j+1 ), we approximate the integral via standard quadrature, for which the error is nearly uniform, since each dyadic interval is separated from the singular point at the origin, with a distance of its own length. Let (s , ω )n=1 be the nodes and weights for n-point Gauss–Legendre quadrature on the dyadic interval [a, b] = [2 j , 2 j+1 ]. Then for β ∈ (0, 2) and n > 1, 

b

e−ts s β−1 ds −

a

n 

5 1 < 2 2 πa β (4−1 e e )2n .

β−1 −sk t

ωk sk

e

k=1

For any interval [a, b], the error estimate for n-point Gauss–Legendre quadrature on [a, b] [OLBC10, p. 80] yields n  b (b − a)2n+1 (n!)4 d2n  β−1 e−ts s β−1 ds − ω s e−s t ≤ max 2n (e−st s β−1 ) . 3 2n + 1 [(2n)!] s∈(a,b) ds a =1

By the two-sided Stirling’s approximation [OLBC10, p. 138] √ √ 1 1 2π n n+ 2 e−n < n! < 2 πn n+ 2 e−n ,

(4.16)

√ 1 we obtain (n!)4 [(2n)!]−3 < 2 π (e8−1 )2n n −2n+ 2 . Note that for any β ∈ (0, 2), |(β − 1) · · · (β − )| ≤ ! for  > 1, and thus for  > 1, noting  ≤ 2n leads to √  1 | dsd  s β−1 | ≤ 2 π (2n)+ 2 e− s β−−1 . Clearly, this estimate holds also for  = 0, 1. 2n− Since dsd 2n− e−st = (−t)2n− e−st , Leibniz’s rule gives      2n− 2n  d2n  d β−1 d 2n −st e s 2n (e−st s β−1 ) =  ds ds 2n− ds  =0  2n   √ √ 2n 2n− ≤2 π 2ns β−1 e−st (2n) e− s − t  =0 2n √ √ β−1 −st  t + 2n(es)−1 . =2 π 2ns e By combining these estimates and noting b − a = a, we obtain

114

4 Finite Difference Methods: Construction and Implementation



b

e−ts s β−1 ds −

a

n 



β−1 −s t

ω s

e

=1

√ ≤2 2π(b − a)s β−1 e−st ((8n)−1 e(b − a)t + (4s)−1 (b − a))2n ≤2 2 πa β e−at (eat(8n)−1 + 4−1 )2n . 5

The desired assertion follows from the identity ex

max e−x

8n

x>0

+

1 1 2n e e 2n = , n ≥ 2. 4 4

(iii) On the interval [0, a], we use Gauss–Jacobi quadrature with the weight s β−1 to approximate the integral. Let (s , ω )n=1 , n ≥ 2, be the nodes and weights for n-point Gauss–Jacobi quadrature with the weight s β−1 on [0, a]. Then for 0 < t < T and β ∈ (0, 2), 

a

e−ts s β−1 ds −

0

n 

√ 3 ω e−s t < 4 πa β n 2 (8−1 e)2n ((n − 1)−1 aT )2n .

=1

The error estimate for n-point Gauss–Jacobi quadrature [OLBC10, p. 80] yields 

a

e−ts s β−1 ds −

d2n a 2n+β (n!)2 [ (n + β)]2 −st ω e−s t ≤ max e . 2n + β (2n)![ (2n + β)]2 s∈(0,a) ds 2n =1

n 

0

For n > 1 and β ∈ (0, 2), we have (n + β) < (n + β) = (n + 1 − 1)!,

(2n + β) > (2n + β) = (2n + β − 1)!, 2n + β > 2n + β. Consequently, 

a

e

−ts β−1

s

ds −

0

≤ max

0≤s≤a

n 

ω e−s t

=1

(n!)2 [(n + β − 1)!]2 2n −st a t e 2n + β (2n)![(2n + β − 1)!]2 2n+β

a β (n!)2 [(n + β − 1)!]2 (aT )2n 2n (2n)![(2n + β − 1)!]2 √ 3 ≤ 4 πa β n 2 (8−1 e)2n (aT (n − 1)−1 )2n , ≤

by the estimate (4.16), and the trivial inequality β − β ≤ 1. Last, we combine these three results, by letting n o = O(| log |), M = O(log T ), and 0 be the nodes and weights for N = O(log | log | + | log δ|). Further, let (so, , ωo, )n=1 s −M be the nodes and the n o -point Gauss–Jacobi quadrature on [0, 2 ], let (s j, , ω j, )n=1 weights for n s -point Gauss–Legendre quadrature on small intervals [2 j , 2 j+1 ], j =  be the nodes and weights −M, · · · , −1, with n s = O(| log |), and let (s j, , ω j, )n=1

4.2 Sum of Exponential Approximation

115

for n  -point Gauss–Legendre quadrature on [2 j , 2 j+1 ], j = 0, · · · , N , with n  = O(| log | + | log δ|). Then for t ∈ [δ, T ] and β ∈ (0, 2), the preceding estimates indicate no ns n −1  N  1

   β−1 β−1 e−so, t ωo, + e−s j, t s j, ω j, + e−s j, t s j, ω j, ≤ . β − t j=−M =1 j=0 =1 =1



This completes the proof of the theorem.

In practice, the number Ne derived from Theorems 4.1 and 4.2 can be large, often with many small exponents sn . Fortunately, they may be aggregated to yield a more concise approximation. This can be achieved by the modified Prony’s method [BM10].We describe Prony’s method below. Consider a general exponential sum L ω e−s t , with ω , s > 0,  = 1, . . . , L. The aim is to approximate g(t) g(t) = =1 K by an soe with fewer terms g(t) ≈ k=1 ω˜ k e−˜sk t , with 2K − 1 < L and ω˜ k , s˜k > 0, k = 1, . . . , K . Let g j = (−1) j g ( j) (0) =

L 

j

ω s ,

j = 0, . . . , 2K − 1,

(4.17)

=1

and then we aim at finding 2K positives ω˜ k and s˜k such that gj =

K 

j

ω˜ k s˜k ,

j = 0, . . . , 2K − 1.

(4.18)

k=1

Then by Taylor expansion at t = 0, we obtain g(t) ≈

2K −1 

gj

j=0

K K 2K −1    (−t) j (−˜sk t) j = ≈ ω˜ k ω˜ k e−˜sk t . j! j! k=1 j=0 k=1

To enforce the condition (4.18), we employ the following polynomial: Q(z) =

K 

(z − s˜k ) =

k=1

K 

qk z k .

k=0

It follows from direct computation that the unknowns qk must satisfy K  m=0

g j+m qm =

K  K  m=0 k=1

j+m ω˜ k s˜k qm

=

K  k=1

j ω˜ k s˜k

K  m=0

qm s˜km

=

K 

j

ω˜ k s˜k Q(˜sk ) = 0,

k=1

for 0 ≤ j ≤ K − 1 (so that j + m ≤ 2K − 1 for 1 ≤ m ≤ K ), with q K = 1. In this way, we obtain the following linear system:

116

4 Finite Difference Methods: Construction and Implementation K −1 

g j+m qm = b j , with b j = −g j+K ,

j = 0, . . . , K − 1.

(4.19)

m=0

This suggests a numerical procedure given in Algorithm 2. However, there are several potential pitfalls in practical implementation: the optimal choice of the number K of the terms is not known a priori; the matrix [g j+k ] ∈ R K ×K is generally badly conditioned (which is a general feature of a Hankel matrix), the roots of the polynomial Q(z) might not be all real and the weights ω˜ k might not be all positive. Nevertheless, the algorithm can be quite effective in practice, and for the soe approximation, upon reduction, a few dozens of exponentials often suffice high accuracy. Algorithm 2 Prony’s algorithm. 1: Input (s1 , . . . , s L , ω1 , . . . ω L , K ) with 2K − 1 ≤ L. −1 2: Compute (g j )2K j=0 using (4.17). −1 3: Find (q j ) Kj=0 by solving (4.19) and set q K = 1.

K K of the polynomial Q(z) = k 4: Find the roots (˜sk )k=1 k=0 qk z . K j 5: Find ω˜ 1 , …, ω˜ K satisfying k=1 s˜k ω˜ k ≈ g j for 0 ≤ j ≤ 2K − 1 6: Output s˜1 , …, s˜K , ω˜ 1 , …, ω˜ K .

Now we describe a few examples using soe to speed up time-stepping schemes based on polynomial interpolation, including two versions of fast L1 scheme, fast Alikhanov’s scheme, and fast L2 scheme. The overall idea of constructing a fast implementation is direct: in the splitting (4.4) or (4.9), on the local part, we proceed as usual, and on the history part, we approximate the singular kernel with soe, directly or indirectly (e.g., after integration by parts). Example 4.1 First we give a fast variant of the L1 scheme. For the local part of the splitting (4.4), we approximate u(s) on [tn−1 , tn ] by 1,n u(s), and obtain 

tn tn−1

ω1−α u  (s) ds ≈ τn−1 δτ u n



tn

tn−1

ω1−α (tn − s) ds = ω2−α (1)τn−α δτ u n .

For the history part defined on the interval (0, tn−1 ), by integration by parts, we obtain  0

tn−1

ω1−α (tn − s)−α u  (s) ds = ω1−α (τn )u n−1 − ω1−α (tn )u 0 +



tn−1 0

ω−α (tn − s)u(s) ds.

Next we approximate the kernel t −1−α by its soe approximation (4.15) (i.e., with β = 1 + α) and the function u by its linear interpolant 1, j u. Thus, we arrive at  tn−1 ω1−α (tn − s)−α u  (s) ds ≈ ω1−α (τn )u n−1 − ω1−α (tn )u 0 + 0

Ne 1  ω H (tn ),

(−α) =1

4.2 Sum of Exponential Approximation

with H (tn ) =

n−1   j=1

117

tj

e−(tn −s)s (1, j u)(s) ds.

t j−1

Note that the history terms H (tn ),  = 1, . . . , Ne , satisfy H (t1 ) = 0 and the following recurrence relation: H (tn ) = e−s τn H (tn−1 ) +



tn−1

e−s (tn −s) (1,n−1 u)(s) ds.

(4.20)

tn−2

t t Using the elementary identities 0 es ds = es − 1 and 0 ses ds = tet + 1 − et , and by changing variables, the last integral in (4.20) can be evaluated analytically as 

=

tn−1

e−s (tn −s) (1,n−1 u)(s) ds

tn−2 −s τn

 −s τn−1  e (e + s τn−1 − 1)u n−1 + (1 − e−s τn−1 − s τn−1 e−s τn−1 )u n−2 . 2 s τn−1

The weights for u n−1 and u n−2 are subject to cancellation errors when s τn−1 is small, which may be computed using Taylor expansion instead then. In view of the recursion (4.20), at time step tn , we only need O(1) work to compute the history terms H (tn ) instead of O(n), since H (tn−1 ) is known, thereby reducing the total work from O(N 2 ) to O(N Ne ), and the memory from O(N ) to O(Ne ). Clearly, for large N , the speedup is significant. Example 4.2 In this example, we develop an alternative fast variant of the L1 scheme. In the splitting (4.4), approximating u by 1,n u in the local part and using soe for the kernel ω1−α (t − s) and 1, j u for u in the history part lead to ∂¯τα u n =



tn

tn−1

ω1−α (tn − s)τn−1 δτ u n ds +

n−1   j=1

=: a0(n) δτ u n +

Ne 

tj

Ne 

ω e−s (tn −s) (1, j u) (s) ds

t j−1 =1

ω H (tn−1 ), n = 1, 2, . . . ,

=1

with the history terms H (tn ) given by H (tn ) =

n−1   j=1

tj

t j−1

e−s (tn −s) (1, j u) (s) ds, with H (t1 ) = 0.

118

4 Finite Difference Methods: Construction and Implementation

To compute the history terms H (tn ) efficiently, we employ the recursion H (tn ) = e

−s τn

 H (tn−1 ) +

= e−s τn H (tn−1 ) + b

tn−1

tn−2 (n,)

e−s (tn −s) (1,n−1 u) (s) ds

δτ u n−1 ,

with the positive coefficients −1 b(n,) = τn−1



tn−1

e−s (tn −s) ds = (s τn−1 )−1 e−s τn (1 − e−s τn−1 ).

tn−2

Thus the overall discrete scheme ∂¯τα u n , n = 1, 2, . . ., can be written as ∂¯τα u n = a0(n) δτ u n +

n−1 

τ j−1



j=1

tj

Ne 

ω e−s (tn −s) dsδτ u j :=

t j−1 =1

n 

j A(n) n− j δτ u .

j=1

The discrete convolutional form facilitates the stability analysis later on. If the tolerance  of the soe (4.15) satisfies  ≤ min( 13 ω1−α (T ), αω2−α (T )), then the weights A(n) n− j satisfy (n) A(n) j−1 > A j > 0,

j = 1, . . . , n − 1,

(n) (n) 2 (n) A(n) 0 = a0 , and An− j ≥ 3 an− j ,

j = 1, . . . , n − 1,

(n) where the coefficients an− j are for the standard L1 scheme defined in (4.3). These two properties are sufficient to ensure the stability of the time-stepping scheme via discrete Gronwall’s inequality, cf. Chap. 6. Indeed, the definition of a0(n) gives

a0(n) − a1(n) > a0(n) − ω1−α (τn ) = ατn−1 ω2−α (τn ) ≥ αω2−α (T ) ≥ . (n) (n) (n) Thus, A(n) 0 = a0 > a1 +  > A1 . By the construction of the soe (4.15), s and ω are positive for all  = 1, . . . , Ne . Then the mean value theorem yields the first (n) ≤ assertion. By Lemma 4.1 and the condition on , we have  ≤ 13 ω−α (tn ) < 13 an−1 (n) (n) (n) 1 2 (n) a , for j = 1, . . . , n − 1. Hence Theorem 4.2 gives An− j ≥ an− j −  ≥ 3 an− j . 3 n− j

Example 4.3 In this example, we construct a fast variant of Alikhanov’s scheme. In the splitting (4.9), we approximate u by 1, j u in the local part, and apply the soe (4.15) of the kernel (tn − s)−α and 2, j u in the history part. Then we obtain the approximation (∂¯τα u)n−θ , with θ = α2 , defined by

4.2 Sum of Exponential Approximation (∂¯τα u)n−θ =



119

tn−θ

ω1−α (tn−θ − s)τn−1 δτ u n ds +

tn−1 (n)

:= a0 δτ u n +

Ne 

n−1  

tj

Ne 

j=1 t j−1 =1

ω e−s (tn−θ −s) (2, j u) (s) ds

ω H (tn ), n = 1, . . . , N ,

=1

with the history terms H (tn ),  = 1, . . . , Ne , given by H (tn ) =

n−1   j=1

tj

e−s (tn−θ −s) (2, j u) (s) ds, with H (t1 ) = 0.

t j−1

To compute H  (tn ) efficiently, we resort to the following recurrence relation: H (tn ) = e

−s (1−θ)τn −s θτn−1

=e

−s (1−θ)τn −s θτn−1

 H (tn−1 ) + H (tn−1 ) +

tn−1

e−s (tn−θ −s) (2,n−1 u) (s) ds

tn−2 a(n) δτ u n−1

+ b(n) (ρn−1 δτ u n − δτ u n−1 ),

where the coefficients a(n) and b(n) are given by a(n)

=

b(n) =

−1 τn−1



tn−1

e−s (tn−θ −s) ds,

tn−2

2 τn−1 (τn−1 + τn )



tn−1

tn−2

(s − tn− 23 )e−s (tn−θ −s) ds.

These integrals can be evaluated in closed form a(n) = (s τn−1 )−1 e−s (1−θ)τn (1 − e−s τn−1 ), b(n) =

 2e−s (1−θ)τn −s τn−1  s τn−1 s τn−1 (e + 1) − es τn−1 + 1 , 2 2 s τn−1 (τn−1 + τn )

t using the identity 0 (s − 2t )es ds = 2t (et + 1) − et + 1. This variant of Alikhanov’s scheme was analyzed in the work [LMZ21], using discrete Gronwall’s inequality in Chap. 6. Example 4.4 In this last example, we derive a fast variant of the L2 scheme. In the splitting (4.4), we approximate u(s) by 2,n−1 u(s) in the local part as the standard L2 scheme:  tn  tn ω1−α (tn − s)u  (s) ds ≈ ω1−α (tn − s)(2,n−1 u) (s) ds. tn−1

tn−1

For the history term, using integration by parts and then approximating the integral using the soe (4.15) of the kernel (tn − s)−1−α and 2, j u for u give

120

4 Finite Difference Methods: Construction and Implementation



tn−1

ω1−α (tn − s)u  (s) ds ≈

0

n−1   j=1

tj

ω1−α (tn − s)(2, j u) (s) ds

t j−1

e 1  − ω1−α (tn )u + ω H (tn ),

(−α) =1

N

≈ ω1−α (τn )u

n−1

0

with the history terms H (tn ),  = 1, . . . , Ne , given by H (tn ) =

n−1  

tj

e−s (tn −s) 2, j u(s) ds, with H (t1 ) = 0.

t j−1

j=1

To efficiently update the history terms H (tn ), we employ the following recursion: H (tn ) = e−s τn H (tn−1 ) +



tn−1

e−s (tn −s) 2,n−1 u(s) ds.

tn−2

Upon substituting the Lagrangian form of the quadratic interpolant 2,n−1 u in (4.5), i.e., 2,n−1 u(s) =

(s − tn−1 )(s − tn ) n−2 (s − tn−2 )(s − tn ) n−1 (s − tn−2 )(s − tn−1 ) n − u + u u , τn−1 (τn−1 + τn ) τn−1 τn τn (τn−1 + τn )

we deduce 

tn−1 tn−2

e−s (tn −s) 2,n−1 u(s) ds = aˆ (n) u n−2 − bˆ(n) u n−1 + cˆ(n) u n ,

where the coefficients aˆ  , bˆ and cˆ are given by aˆ (n) = bˆ(n) = cˆ(n) =



tn−1



tn−2 tn−1



tn−2 tn−1 tn−2

e−s (tn −s)

(s − tn−1 )(s − tn ) ds, τn−1 (τn−1 + τn )

e−s (tn −s)

(s − tn−2 )(s − tn ) ds, τn−1 τn

e−s (tn −s)

(s − tn−2 )(s − tn−1 ) ds. τn (τn−1 + τn )

All the three integrals can be evaluated in closed form. Indeed, direct computation gives that for any t0 , t1 , t2 > 0, there holds  t0 0

s=t0 s=t0 s=t0 es (s − t1 )(s − t2 ) ds = es (s − t1 )(s − t2 ) − es (2s − t1 − t2 ) + 2es , s=0

s=0

s=0

4.2 Sum of Exponential Approximation

121

with the shorthand τ˜n = s τn (suppressing the dependence on ). Consequently, we have 

e−τ˜n −τ˜n−1 − τ˜n−1 (τ˜n−1 + τ˜n ) + τ˜n eτ˜n−1 − (2τ˜n−1 + τ˜n ) + 2(eτ˜n−1 − 1) , s τ˜n−1 (τ˜n−1 + τ˜n )  e−τ˜n −τ˜n−1

− τ˜n−1 τ˜n eτ˜n−1 − (τ˜n−1 − τ˜n )eτ˜n−1 − (τ˜n + τ˜n−1 ) + 2(eτ˜n−1 − 1) , = s τ˜n−1 τ˜n 

e−τ˜n −τ˜n−1 = − τ˜n−1 (eτ˜n−1 + 1) + 2(eτ˜n−1 − 1) . s τ˜n (τ˜n−1 + τ˜n )

aˆ (n) = (n) bˆ

cˆ(n)

The version of the fast L2 scheme was studied in [ZX19] for uniform meshes.

Notes Finite difference schemes represent a very large class of time-stepping schemes for approximating the Djrbashian–Caputo fractional derivative ∂tα u. Undoubtedly, the L1 scheme is the most popular one within the family. This scheme already appeared in the book [OS74], for approximating the Riemann–Liouville fractional derivative. However, it was until Lin and Xu [LX07] gave a first rigorous error analysis of the scheme under the assumption that the solution is sufficiently smooth. Lemma 4.2 is contained in [LX07] (but with a slightly different proof). The authors also developed a fully discrete scheme based on the spectral method in space for a one-dimensional subdiffusion model, and provided some stability result and error estimates. The idea of using piecewise linear approximation was independently studied by Sun and Wu [SW06], who investigated the approximation for both α ∈ (0, 1) and α ∈ (1, 2), with a focus on the latter (also known as diffusion wave in the literature). Since these seminal works, the research on finite difference schemes has flourished, and many results have been obtained, e.g., high-order schemes, nonsmooth data analysis, and graded meshes. Some representative convergence rate results will be described in Chaps. 5 and 6. Besides these direct constructions of time-stepping schemes using local polynomial interpolation, there are several alternative variants. For example, Mustapha [Mus20] introduced an averaged version of the L1 scheme, which takes the form (with N ) 1 u being the piecewise linear interpolant of u with respect to the grid (tn )n=0 1 ∂ˆτα u n− 2 = τn−1

with ∂˜tα u(t) =



t 0



tn

tn−1

∂˜tα u(t) dt,

ω1−α (t − s)(1 u) (s) ds,

122

4 Finite Difference Methods: Construction and Implementation

i.e., evaluating the discrete derivative at tn− 21 = tn − τ2n . He proved that this modified scheme enjoys a certain positive definite property on an arbitrary temporal mesh, which essentially is a form of discrete stability. Numerically the scheme is observed to achieve a convergence rate O(N − min(2,r α) ), with r being the grading parameter, cf. (6.2) below. A complete analysis is complicated by the consistency error, which appears to be highly nontrivial and is still unavailable. Ji et al. [JLGZ20] analyzed an averaged L1 scheme for the modified subdiffusion model: ∂t u − 0∂tα u = f,

(4.21)

where 0∂tα denotes the Riemann–Liouville fractional derivative operator, cf. Defi2α− 2

nition 1.2. On graded meshes, the authors proved an error estimate O(N −2 tn r ), under suitable a priori regularity assumption on the solution u and certain condition on the grading parameter r . Note also that the construction in this chapter focuses exclusively on collocation which adopts exact matching at selected (grid) points. Of course this is not the only option, and one may employ alternative strategies, e.g., weak formulations of Galerkin, Petrov–Galerkin or discontinuous Galerkin types. This line of research has actually been extensively explored by William McLean, Kassem Mustapha, and other researchers; see [MM11, MM12, MM13, MAF14, Mus15, CM15, MM15] for an incomplete list of works in this direction, often on the closely modified subdiffusion model (4.21). Very recently, Yan et al. [YELY21] developed a continuous Galerkin time-stepping scheme, where the approximate solution U N is sought as a continuous piecewise linear function in time t and the test space is based on the discontinuous piecewise constant functions, and proved a convergence rate O(τ 1+α ) for nonsmooth data (without any correction), with the numerical experiments indicating an O(τ 2 ) convergence rate. The scheme is yet another fractional generalization of the classical Crank–Nicolson scheme. Efficient soe approximations of the function t −β (β > 0) have been extensively studied in [BM05, BM10, Li10, JZZZ17, McL18]. Building on their earlier work [BM05], Beylkin and Monzon [BM10] proved that for any β > 0, t −β admits an efficient soe approximation on the interval [δ, 1] with a relative error , with the number Ne = O(| log |(| log | + | log δ|)). Theorem 4.1 is due to [BM10, Sect. 2], and the current presentation follows [McL18, Theorems 1 and 2]. The modified Prony’s method was also suggested in [BM10]. Around the same time, Li [Li10] summarized several alternative approaches for the fast evaluation of a Riemann– t α−1 by Liouville fractional integral of order α ∈ (0, 1), by representing the kernel (α) t α−1 sin(απ ) =

(α) π



∞

e−st s −α ds, t > 0.

0

Then she developed a quadrature approximation

4.2 Sum of Exponential Approximation

 0

∞

e−st s −α ds ≈

123 Ne 

ω j e−s j t s −α j , δ ≤ t < ∞,

j=1

based on adaptive Gaussian quadrature and utilizes asymptotic error formula for Gauss quadrature. This again provides an soe approximation, and the error can be made smaller than  for all t ∈ [δ, ∞), with Ne of order (| log | + | log δ|)2 . The use of the soe approximation for subdiffusion model was recognized by several groups of researchers independently, notably [JZZZ17], [BH17b], and [McL18]. The approach of Jiang et al. [JZZZ17], given in Theorem 4.2, for t ∈ [δ, T ], employs composite Gauss quadrature on dyadic intervals, with Ne of the order O(| log | log(T δ −1 | log |) + | log δ|(| log δ|| log |)). See also the review [McL18] for further discussions on soe approximations (and yet another construction of soes). t α−1 (or other more In other approximations, the Riemann–Liouville kernel k(t) = (α)  ∞ −zt ˆ general kernels) is exploited via its Laplace transform k(z) = 0 e k(t) dt = z α , ˆ so instead of soe it is natural to look for a sum of poles approximation k(z) ≈  L w , for z in a suitable region of the complex plane C [AGH00]. Baffet and =1 z− p Hesthaven [BH17b] followed the approach of pole approximation, and also applied it to the solution of the subdiffusion model [BH17a].

Chapter 5

Finite Difference Methods on Uniform Meshes

In this chapter, we provide an analysis of one representative time-stepping scheme of finite difference type, i.e., L1 scheme, on uniform temporal meshes using discrete Laplace transform, and derive sharp error bounds for nonsmooth and incompatible problem data. We also describe a correction scheme for the L1 scheme, similar to the corrected bdf2 cq, to restore the optimal convergence rate. The overall strategy for convergence analysis is similar to that in Chap. 3. However, for time-stepping schemes based on polynomial interpolation, the discrete Laplace transform of the kernels involves the polylogarithmic function Liα−1 , which makes the analysis far more complex. Throughout, the time step size is denoted by τ = N −1 T , with N being the total number of steps.

5.1 Error Analysis of L1 Scheme In this section, we analyze the standard L1 scheme on uniform meshes, using Laplace transform. Consider the following abstract time-fractional evolution equation ∂tα u + Au = f, 0 < t ≤ T, with u(0) = u 0

(5.1)

with ∂tα u being the Djrbashian–Caputo fractional derivative, u 0 ∈ L 2 (), and f ∈ L 2 (0, T ; L 2 ()). The elliptic operator A : H 2 () ∩ H01 () → L 2 () in problem (5.1) satisfies the following resolvent estimate (cf. [ABHN11, Example 3.7.5 and Theorem 3.7.11] and [Tho06, Theorem 6.4]) (z + A)−1  ≤ cθ |z|−1 , ∀z ∈ θ ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Jin and Z. Zhou, Numerical Treatment and Analysis of Time-Fractional Evolution Equations, Applied Mathematical Sciences 214, https://doi.org/10.1007/978-3-031-21050-1_5

(5.2)

125

126

5 Finite Difference Methods on Uniform Meshes

for all θ ∈ ( π2 , π ), with θ := {z ∈ C\{0} : | arg z| ≤ θ }. Hence, z α ∈ αθ with αθ < θ < π for all z ∈ θ . Now we describe the semidiscrete scheme in time. Recall that the L1 approximation ∂¯τα u n of the Djrbashian–Caputo fractional derivative ∂tα u(tn ) is given by ∂¯τα u n = τ −α

n 

an− j δτ u j ,

(5.3)

j=1

with δτ u j = u j − u j−1 , and the discrete convolutional weights an− j given by aj =

( j + 1)1−α − j 1−α .

(2 − α)

See Chap. 4 for the detailed construction of the scheme and its efficient implementation. By Lemma 4.2, the local truncation error of the approximation ∂¯τα u n to ∂tα u(tn ) is of order O(τ 2−α ) for u ∈ C 2 [0, T ]. However, the regularity u ∈ C 2 [0, T ] is rather restrictive, and generally does not hold for the subdiffusion model (5.1), unless restrictive compatibility conditions on the problem data hold. Thus, there is an imperative need to analyze the L1 scheme (and also other schemes) under more realistic regularity assumptions. In this section, we provide such an analysis for a scheme based on the L1 approximation. The time semidiscrete scheme for the model (5.1) based on the L1 approximation (5.3) reads: with f n = f (tn ), given U 0 = u 0 , find U n ∈ H01 () such that ∂¯τα U n + AU n = f n , n = 1, 2, . . . , N .

(5.4)

To analyze the scheme (5.4), we first derive an integral representation of the solution U n , using the generating function. We denote by  ω(ζ ) =

∞ 

ωjζ j

j=0

the generating function of a sequence (ω j )∞ j=0 . In the analysis below, it is necessary to use the notation a j instead of a (n) , in order to use the generating function j . Indeed, the original L1 scheme approximates ∂tα u(tn ) by of the sequence (a j )∞ j=0  n− j with the weights b(α) τ −α nj=0 b(α) j,n u j,n given by b(α) j,n

=

⎧ ⎪ ⎨ ⎪ ⎩

a0 , j = 0, a j − a j−1 , j = 1, . . . , n − 1, an−1 , j = n.

5.1 Error Analysis of L1 Scheme

127

In the analysis below we have modified the coefficient for the term u 0 from an−1 to an − an−1 . The resulting solution representation forms the basis of the error analysis. It involves the polylogarithm function Li p (z), p ∈ R, defined by [Lew81] Li p (z) =

∞  zj . jp j=1

See Appendix A.2 for the definition and properties of Li p (z). Recall also the contour

θ,σ ⊂ C, with θ ∈ ( π2 , π ) and σ > 0, defined by

θ,σ = {z = ρe±iθ : ρ ≥ σ } ∪ {z = σ eiϕ : |ϕ| ≤ θ }, oriented with an increasing imaginary part. N Proposition 5.1 Let (U n )n=1 be the solutions to problem (5.4). Then the function W n = U n − U 0 can be represented by

Wn =

∞

  1 e ztn (βτ (e−zτ )α + A)−1 − μτ (e−zτ )−1 Au 0 + τ f (tn )e−nzτ dz, τ 2π i θ,σ n=1

τ with θ,σ = {z ∈ θ,σ : (z)| ≤ τ −1 }, and

μτ (e−zτ ) =

1 − e−zτ (1 − e−zτ )2 −zτ α Liα−1 (e−zτ ). and β (e ) = τ τ e−zτ e−zτ τ α (2 − α)

Proof Note that W n := U n − U 0 satisfies W 0 = 0 and ∂¯τα W n + AW n = −Au 0 + f n , n = 1, 2, ..., N . Multiplying both sides by ζ n and summing over n from 1 to ∞ yield ∞ 

∞   (ζ ) = − ζ Au 0 + f nζ n. ∂¯τα W n ζ n + A W 1 − ζ n=1 n=1

Next we simplify the term ∞ 

∂¯τα W n ζ n = τ −α

n=1

∞

¯α n n n=1 ∂τ W ζ .

n−1 ∞

   a0 W n + (a j − a j−1 )W n− j ζ n n=1

= τ −α

By the definition of ∂¯τα W n , we have

j=1

∞ n−1   n=1

j=0

∞ n−1     a j W n− j ζ n − τ −α a j−1 W n− j ζ n := I − II. n=1

j=1

128

5 Finite Difference Methods on Uniform Meshes

Since W 0 = 0, by the discrete convolution rule of generating functions, the first term I can be written as n ∞  

I = τ −α

n=1

  (ζ ). a j W n− j ζ n = τ −α a (ζ )W

j=0

Similarly, the term II can be written as II = τ −α

n ∞   n=1

∞  n−1     (ζ ). a j−1 W n− j ζ n = τ −α ζ a j W n−1− j ζ n−1 = τ −α ζ a (ζ )W n=1

j=1

j=0

Hence, we arrive at ∞ 

 (ζ ). a (ζ )W ∂¯τα W n ζ n = τ −α (1 − ζ )

n=1

Next we derive the representation of  a (ζ ): ∞

 a (ζ ) =

 1 (( j + 1)1−α − j 1−α )ζ j

(2 − α) j=0 ∞

=

1 − ζ  1−α j (1 − ζ )Liα−1 (ζ ) . j ζ = ζ (2 − α) j=1 ζ (2 − α)

 (ζ ) is represented by Therefore, W  (ζ ) = W



(1 − ζ )2 Liα−1 (ζ ) + A ζ τ α (2 − α)

−1 

−

∞   ζ Au 0 + f nζ n . 1−ζ n=1

 (ζ ) is analytic at ζ = 0. Hence the Simple calculation shows that the function W Cauchy theorem implies that for  small enough, there holds Wn =

1 2π i

|ζ |=

ζ −n−1



(1 − ζ )2 Liα−1 (ζ ) + A ζ τ α (2 − α)

−1 

  ζ f n ζ n dζ. Au 0 + 1−ζ ∞

−

n=1

Upon changing variable ζ = e−zτ , we obtain

−1 (1 − e−zτ )2 τ −zτ Li e ztn (e ) + A α−1 2π i 0 e−zτ τ α (2 − α) ∞    e−zτ n −nzτ dz, × − Au + f e 0 1 − e−zτ n=1

Wn =

5.1 Error Analysis of L1 Scheme

129

where the contour 0 := {z = − ln() + iy : |y| ≤ πτ } is oriented counterclockwise. τ τ 0 By deforming the contour to θ,σ := {z ∈ θ,σ : | (z)| ≤ πτ } and using the periodicity of the exponential function e z , we obtain the desired representation for W n .  Proposition 5.1 motivates the study of the kernel functions in the representation: in the Laplace domain, the L1 scheme approximates z α with βτ (e−zτ ), whose study is the main focus of the analysis below. Due to the presence of the polylogarithmic function Liα−1 (z), the study is lengthy and tedious. This will be carried out in a series of lemmas. It is convenient to introduce the auxiliary function ψ define by ψ(ξ ) =

eξ − 1 Liα−1 (e−ξ ).

(2 − α)

First we bound the function χτ (z) = τ −1 (1 − e−zτ ). τ Lemma 5.1 Let χτ (z) = τ −1 (1 − e−zτ ). Then for all z ∈ θ,σ , there hold

|χτ (z) − z| ≤ c|z|2 τ and c1 |z| ≤ |χτ (z)| ≤ c2 |z|. Proof Since |z|τ ≤ sion

π sin θ

τ for z ∈ θ,σ , the first assertion follows from Taylor expan-

|χτ (z) − z| ≤ |z|2 τ

∞  |z| j τ j τ ≤ c|z|2 τ, ∀z ∈ θ,σ . ( j + 2)! j=0

For the second claim, the upper bound on χτ (z) is trivial, and it suffices to verify the τ τ into three disjoint parts θ,σ = lower bound. To this end, we split the contour θ,σ + c − + −

τ ∪ τ ∪ τ , with τ and τ being the rays in the upper and lower half planes, respectively, and τc being the circular arc. Let ξ = −zτ with ρ ≡ |ξ | ∈ (0, sinπ θ ). First, for z ∈ τ+ , for which ξ = ρe−i(π−θ) , ρ ∈ (1, sinπ θ ), since | cos(ρ sin θ )| ≤ 1, we obtain      1 − e−zτ   eξ − 1  |e−ρ cos θ cos(ρ sin θ ) − 1 − ie−ρ cos θ sin(ρ sin θ )|  =   zτ   ξ  = ρ e−ρ cos θ − 1 (e−2ρ cos θ + 1 − 2e−ρ cos θ ) 2 = ≥ − cos θ > 0, (5.5) ρ ρ 1

≥

due to positivity and monotonicity of ρ −1 (e−ρ cos θ − 1) in ρ ∈ (0, ∞). The case z ∈ τ− follows analogously. Last, for the case z ∈ τc , by means of Taylor expansion, we have ⎛ ⎞ ∞ j j  z τ ⎠. χτ (z) = z ⎝1 + (−1) j ( j + 1)! j=1 This and the fact that ρ = |zτ | < 1 imply directly that |χτ (z)| ≥ c|z| for z ∈ τc . 

130

5 Finite Difference Methods on Uniform Meshes

The next result gives a useful singular expansion of the function Li p (e−z ) [Fla99, Theorem 1]; see also Appendix A.2. The expansion is stated only for z → 0, but it is valid in the sector θ,δ [Fla99, p. 377, proof of Lemma 2]. Lemma 5.2 For p = 1, 2, . . ., the function Li p (e−z ) satisfies the singular expansion Li p (e−z ) ∼ (1 − p)z p−1 +

∞  zk (−1)k ζ ( p − k) , as z → 0, k! k=0

where ζ is the Riemann zeta function. For |z| ≤ α − 1, the series (5.6) converges absolutely.

π sin θ

(5.6)

with θ ∈ ( π2 , 56 π ), and p =

Proof The singular expansion (5.6) is direct from [Fla99, Theorem 1]. To show the absolute convergence, we use the following well-known functional equation for the Riemann zeta function ζ (z) (see, e.g., [KR01] for a short proof): for z ∈ / Z, ζ (1 − z) =

zπ  2

(z)ζ (z), cos (2π )z 2

we obtain for p = α − 1 ∈ (−1, 0) ζ ( p − k) = ζ (1 − (1 − p + k))

2 (1 − p + k)π

(1 − p + k)ζ (1 − p + k). = cos (2π )1− p+k 2 By Stirling’s formula for the Gamma function (x) [AS92, pp. 257] (cf. also (A.7))

(x + 1) =

√

2π x x+ 2 e−x (1 + O(x −1 )), as x → ∞, 1

and that ζ (1 − p + k) → 1 as k → ∞, we have  lim

k→∞

k

1 π |ζ ( p − k)||z|k ≤ , ∀|z| ≤ . k! 2 sin θ sin θ

Since for θ ∈ ( π2 , 56 π ), 2 sin θ > 1, the series converges absolutely.



Next we state an error estimate for the function βτ (e−zτ ) with respect to z α . Lemma 5.3 For θ ∈ ( π2 , 56 π ), there holds   βτ (e−zτ )α − z α  ≤ c|z|2 τ 2−α , ∀z ∈ τ . θ,σ Proof Since 0 ≤ |zτ | ≤ sinπ θ , using Taylor expansion and the singular expansion  (zτ ) j τ , e zτ − 1 = ∞ (5.6), it follows that for z ∈ θ,σ j=1 j! and

5.1 Error Analysis of L1 Scheme

131

Liα−1 (e−zτ ) = (2 − α)(zτ )α−2 +

∞  (zτ )k . (−1)k ζ (1 − α − k) k! k=0

Hence, the function ψ(z) can be represented by ψ(zτ ) =

j=1

=



∞  (zτ ) j

(zτ )

j!

+

∞  (−1)k ζ (−α − k) (zτ )k

(2 − α)

k=0

∞  (zτ )α+ j−2 j=1

α−2

j!

+



k!

∞ ∞  (zτ ) j  (−1)k ζ (−α − k) (zτ )k j=1

j!

k=0

(2 − α)

k!

,

and consequently, 1 − e−zτ ψ(zτ ) τα ⎡ ⎤ ∞ ∞ ∞ ∞  (zτ )+1 ⎣  (zτ )α+ j−2  (zτ ) j  (−1)k ζ (−α − k) (zτ )k ⎦ −α  + =τ (−1) ( + 1)! j! j!

(2 − α) k! βτ (e−zτ )α =

=0

j=1

j=1

k=0

ζ (α − 1) 2 2−α z τ + O(z 2+α τ 2 ). = zα +

(2 − α)

In view of the choice θ ∈ ( π2 , 56 π ) and Lemma 5.2, the bound is uniform, since the τ . Consequently, series converges uniformly for z ∈ θ,δ

  βτ (e−zτ )α − z α  ≤ |z|2 τ 2−α − ζ (α − 1) + O((zτ )α ) ≤ c|z|2 τ 2−α ,

(2 − α) 

from which the desired assertion follows. τ The next result gives a uniform lower bound on ψ(z) on the contour θ,δ . e −1 Lemma 5.4 Let ψ(z) = (2−α) Liα−1 (e−z ). Then for any θ close to π τ . for any σ < 2τ , |ψ(zτ )| ≥ c > 0, for any z ∈ θ,σ z

τ Proof Since for z ∈ θ,σ , | (z)| ≤

ψ(z) = cα 0

π τ ∞

π , 2

there holds

and z ∈ / (−∞, 0], by Lemma A.2, there holds s α−1 1 − e−s ds, 1 − e−z−s s

τ τ with the constant cα = sin(π(1−α)) . Next we split the contour θ,σ into θ,σ = τ+ ∪ π c − + −

τ ∪ τ , where τ and τ are the rays in the upper and lower half planes, respectively, and τc is the circular arc. First, for z ∈ τ+ , set zτ = ρeiθ = ρ cos θ + iρ sin θ with σ < ρ < sinπ θ . Upon letting r = ρ cos θ and φ = ρ sin θ , we have

132

5 Finite Difference Methods on Uniform Meshes

ψ(zτ ) = cα

∞

0

= cα

∞

0

= cα

0

∞

s α−1 1 − e−s ds 1 − e−(r +iφ)−s s s α−2 (1 − e−s ) ds 1 − e−r −s cos φ + ie−r −s sin φ s α−2 (1 − e−s )(1 − e−r −s cos φ − ie−r −s sin φ) ds. (1 − e−r −s cos φ)2 + e−2r −2s sin2 φ

It suffices to show that the real part

∞

(ψ(zτ )) = cα 0

s α−2 (1 − e−s )(1 − e−r −s cos φ) ds 1 − 2e−r −s cos φ + e−2r −2s

is bounded from below by some positive constant c. First, for φ = ρ sin θ ∈ [ π2 , π ], for which cos φ ≤ 0, we have 0 < 1 − e−r −s cos φ ≤ 1 − 2e−r −s cos φ ≤ 1 − 2e−r −s cos φ + e−2r −2s . Consequently,



∞

(ψ(zτ )) ≥ cα

s α−2 (1 − e−s ) ds = c0 .

0

Next we consider the case φ ∈ (0, π2 ), for which cos φ > 0. We fix θ = π2 , and thus r = ρ cos θ = 0 and e−r cos φ = cos(ρ sin θ ) = cos ρ > 0. Then 1 − e−r −s cos φ > 1 − e−s and 0 ≤ 1 − 2e−r −s cos φ + e−2r −2s ≤ 2, and accordingly, (ψ(zτ )) simplifies to

∞

(ψ(zτ )) = cα cα ≥ 2

0

∞

s α−2 (1 − e−s )(1 − e−s cos ρ) ds 1 − 2e−s cos ρ + e−2s s α−2 (1 − e−s )2 ds ≥ c1 .

0

Then by continuity of (ψ(zτ )), we may choose an angle θ ∈ ( π2 , 5π ) such that for 6 any z ∈ τ+ , there holds (ψ(zτ )) ≥ c2 . Repeating the above argument shows also the assertion for z ∈ τ− . It remains to show the case z ∈ τc . For any fixed ρ ∈ (0, π2 ) and θ ∈ [− π2 , π2 ], cos φ = cos(ρ sin θ ) ≥ 0, r = ρ cos θ ≥ 0. Consequently 1 − e−r −s cos φ ≥ 1 − e−s cos φ ≥ 1 − e−s , 1 − 2e−r −s cos φ + e−2r −2s ≤ 1 + e−2r −2s ≤ 2. These two inequalities directly imply

5.1 Error Analysis of L1 Scheme

133



∞

(ψ(zτ )) = cα cα ≥ 2

0

∞

s α−2 (1 − e−s )(1 − e−r −s cos φ) ds 1 − 2e−r −s cos φ + e−2r −2s s α−2 (1 − e−s )2 ds ≥ c3 .

0

Then by continuity, we may choose a θ > c4 > 0.

π 2

such that for z ∈ τc , R(ψ(zτ )) ≥ 

The next result summarizes the preceding lemmas. Proposition 5.2 For any θ ∈ ( π2 , π ) sufficiently close to π2 , τ , c0 |z| ≤ |μτ (e−zτ )| ≤ c1 |z| and |μτ (e−zτ ) − z| ≤ cτ |z|2 , ∀z ∈ θ,σ

|βτ (e

−zτ α

) | ≥ c|z|τ

1−α

and |βτ (e

−zτ α

α

) − z | ≤ c|z| τ

2 2−α

, ∀z ∈

τ

θ,σ .

(5.7) (5.8)

Proof The estimates are directly from Lemmas 5.1–5.4. It suffices to show |βτ (e−zτ )α | ≥ c|z|τ 1−α . This follows from the definition of βτ (e−zτ )α and Lemmas 5.4: |βτ (e−zτ )α | = |χτ (z)|τ 1−α |ψ(zτ )| ≥ c|z|τ 1−α . This completes the proof of the proposition.



The next result shows a “sector-preserving” property of the mapping βτ (e−zτ )α : τ . This property there exists some θ0 < π , such that βτ (e−zτ )α ∈ θ0 for all z ∈ θ,σ plays a crucial role in the error analysis below. Lemma 5.5 Let θ > π2 be sufficiently close to π2 . Then there exists some θ0 ∈ ( π2 , π ) τ . such that βτ (e−zτ )α ∈ θ0 for z ∈ θ,σ τ Proof Note that any z ∈ θ,σ satisfies (zτ ) ∈ (0, π ] and arg(z) = θ with θ > π2 π −zτ α close to 2 . Since βτ (e ) is continuous in z, it suffices to analyze z with arg(z) = π2 and (zτ ) ∈ (0, π ]. Then since βτ (e−zτ )α depends continuously on z, there exists −zτ 3 ) τ θ0 ∈ ( π2 , π ) such that (1−e Liα−1 (e−zτ ) ∈ θ0 for all z ∈ θ,σ . Meanwhile, for e−zτ π any z with arg(z) = 2 , we have (zτ ) ∈ (0, π ], which implies that ζ = e−zτ = e−iφ , φ ∈ (0, π ]. Hence we only need to show

ζ −1 (1 − ζ )3 Liα−1 (ζ ) ∈ θ0 , ∀ζ = e−iφ , φ ∈ (0, π ]. First, the identity ζ −1 (1 − ζ )2 = ζ −1 + ζ − 2 = e−iφ + eiφ − 2 = 2 cos φ − 2 implies arg(ζ −1 (1 − ζ )2 ) = arg(2 cos φ − 2) = −π, ∀φ ∈ (0, π ]. Next, the identity (1 − ζ ) = (1 − cos φ) + i sin φ implies arg(1 − ζ ) ∈ [0, π2 ) for all α−1 (ζ ) φ ∈ (0, π ]. Last, for Li , by [Woo92, (13.1)] (see also (A.16)),

(2−α)

134

5 Finite Difference Methods on Uniform Meshes





Liα−1 (ζ ) φ α−2 φ α−2 = (−2π i)α−2 k+1− k+ + (2π i)α−2

(2 − α) 2π 2π k=0 k=0 ∞

π

= (2π )α−2 e−(α−2) 2 i = (2π )

α−2

(cos(2 −

∞



∞



φ α−2 φ α−2 π k+ + (2π )α−2 e(α−2) 2 i 2π 2π k=0 ∞

k+1−

k=0 α) π2 )(A(φ)

+ B(φ)) − i sin((2 − α) π2 )(A(φ) − B(φ)),

  φ α−2 φ α−2 with A(φ) = ∞ and B(φ) = ∞ . Both series k=0 (k + 2π ) k=0 (k + 1 − 2π ) φ α−2 φ α−2 > (k + 1 − 2π ) > 0, converge for α ∈ (0, 1). Since for φ ∈ (0, π ], (k + 2π ) −1 there holds (A(φ) + B(φ)) (A(φ) − B(φ)) ∈ (0, 1), and we deduce arg(Liα−1 (ζ )) ∈ [−π, −π + α2 π ), for ζ = e−iφ , φ ∈ (0, π ]. Hence, we get arg(ζ −1 (1 − ζ )2 (1 − ζ )Liα−1 (ζ ) ∈ [−2π + 2π, − 23 π + α2 π + 2π ) = [0, π2 + α2 π ).

Choosing θ0 ∈ [ π2 + απ , π ) yields ζ −1 (1 − ζ )3 Liα−1 (ζ ) ∈ θ0 for ζ = e−iφ , φ ∈ 2 (0, π ]. These estimates together complete the proof of the lemma.  Now we can derive error estimates for the scheme (5.4) for f ≡ 0. Theorem 5.1 Let u and U n be the solutions of problems (5.1) and (5.4) with U 0 = u 0 and f ≡ 0, respectively. Then the following statements hold. (i) If u 0 ∈ L 2 (), then u(tn ) − U n  L 2 () ≤ cτ tn−1 u 0  L 2 () . (ii) If u 0 ∈ H 2 () ∩ H01 (), then u(tn ) − U n  L 2 () ≤ cτ tnα−1 Au 0  L 2 () . Proof Let w = u − u 0 . Then clearly with f ≡ 0, w satisfies ∂tα w(t) + Aw(t) = −Au 0 , 0 < t ≤ T, with w(0) = 0. Taking Laplace transform on both sides leads to w (z) = −(z α + A)−1 z −1 Au 0 . By inverse Laplace transform and then deforming the contour to θ,σ , w is represented by 1 e zt z −1 (z α + A)−1 Au 0 dz. w(t) = − 2π i θ,σ

5.1 Error Analysis of L1 Scheme

135

This and Proposition 5.1 show that the error u(tn ) − U n is given by u(tn ) − U n = w(tn ) − W n = I1 + I2 , with the terms I1 and I2 given, respectively, by 1 e ztn z −1 (z α + A)−1 Au 0 dz, τ 2π i θ,σ \ θ,σ 1 e ztn (μτ (e−zτ )−1 (βτ (e−zτ )α + A)−1 A − z −1 (z α + A)−1 A)u 0 dz. I2 = τ 2π i θ,σ I1 = −

Note that the identity (z α + A)−1 A = I − z α (z α + A)−1 implies (z α + A)−1 A ≤ c, ∀z ∈ θ,σ .

(5.9)

In view of this identity, with the choice σ = tn−1 and (5.2), we arrive at I1  L 2 () ≤ c

∞ π τ sin θ

eρtn cos θ ρ −1 dρu 0  L 2 ()

≤ cτ u 0  L 2 ()

0

∞

eρtn cos θ dρ ≤ cτ tn−1 u 0  L 2 () .

Next we claim that for θ close to π2 , there holds τ . (5.10) z −1 (z α + A)−1 A − μτ (e−zτ )−1 (βτ (e−zτ )α + A)−1 A ≤ cτ, ∀z ∈ θ,σ

The claim and direct calculation yield I2  L 2 () ≤ cτ u 0  L 2 ()



π τ sin θ

tn−1

eρtn cos θ dρ +



θ −θ

 ecos ϕ tn−1 dϕ ≤ ctn−1 τ u 0  L 2 () .

These two estimates yield the bound on u(tn ) − U n  L 2 () . Then by the triangle inequality, z −1 (z α + A)−1 A − μτ (e−zτ )−1 (βτ (e−zτ )α + A)−1 A ≤|z −1 − μτ (e−zτ )−1 |(z α + A)−1 A + |μτ (e−zτ )−1 |(z α + A)−1 A − (βτ (e−zτ )α + A)−1 A. Meanwhile, there holds the identity (βτ (e−zτ )α + A)−1 − (z α + A)−1 = (z α − βτ (e−zτ )α )(βτ (e−zτ )α + A)−1 (z α + A)−1 .

(5.11) Then from the estimate (5.9), the resolvent estimate (5.2) and Lemma 5.5, we deduce

136

5 Finite Difference Methods on Uniform Meshes

z −1 (z α + A)−1 A − μτ (e−zτ )−1 (βτ (e−zτ )α + A)−1 A ≤cτ + c|z|−1 · |z|2 τ 2−α · |z|−1 τ α−1 = cτ, and the desired claim (5.10) follows directly. This proves assertion (i). (ii) Using the error representation in (i), we only need to estimate the corresponding kernels. By the triangle inequality, we have z −1 (z α + A)−1 − μτ (e−zτ )−1 (βτ (e−zτ )α + A)−1  ≤|z −1 − μτ (e−zτ )−1 |(z α + A)−1  + |μτ (e−zτ )−1 |(βτ (e−zτ )α + A)−1 − (z α + A)−1 .

The identity (5.11), Lemma 3.1 and the resolvent estimate (5.2) imply τ . (βτ (e−zτ )α + A)−1 − (z α + A)−1  ≤ cτ |z|−α , ∀z ∈ θ,σ

Thus, we obtain τ . z −1 (z α + A)−1 − μτ (e−zτ )−1 (βτ (e−zτ )α + A)−1  ≤ c|z|−α τ, ∀z ∈ θ,σ (5.12) τ , we derive Upon setting σ = tn−1 and for all z ∈ θ,σ

I2  L 2 () ≤ cτ Au 0  L 2 () ≤



π τ sin θ

tn−1 α−1 ctn τ Au 0  L 2 () .

eρtn cos θ ρ −α dρ +



θ

−θ

ecos ϕ tnα−1 dϕ



Now the resolvent estimate (5.2) implies that for all z ∈ θ,σ , I1  L 2 () ≤ cAu 0  L 2 () ≤ cτ Au 0  L 2 ()

0

∞

∞ π τ sin θ

eρtn cos θ ρ −α−1 dρ

eρtn cos θ ρ −α dρ ≤ cτ tnα−1 Au 0  L 2 () .

The estimate in (ii) follows from the last two estimates.



Assuming that Au 0 ∈ L 2 () rather than u 0 ∈ L 2 () reduces the singular behavior of the error bound near t = 0, but it does not improve the convergence rate. The convergence behavior of the L1 scheme is identical with that for the convolution quadrature generated by the backward Euler method, i.e., at an O(τ ) rate. For smooth initial data u 0 ∈ D(A), the time discretization error by both schemes contains a singularity tnα−1 . This singularity reflects the limited smoothing property of the solution u (cf. Theorem 1.8) ∂tα u(t) L 2 () = Au(t) L 2 () ≤ cAu 0  L 2 () ,

5.1 Error Analysis of L1 Scheme

137

whereas u  (t) is unbounded at t = 0. To illustrate the sharpness of the convergence rate in Theorem 5.1(ii), consider the initial value problem for the fractional ode: ∂tα u + u = 0, ∀t > 0, with u(0) = 1. The exact solution u at t = τ is given by u(τ ) = E α,1 (−τ α ). For small τ , the L1 scheme at the first step t = τ is given by U 1 = (1 + (2 − α)τ α )−1 = 1 +

∞  (−1)n ( (2 − α)τ α )n . n=1

Then the difference between the approximation U 1 and the exact solution u(τ ) = E α,1 (−τ α ) is given by u(τ ) − U 1 = ( (2 − α) − (α + 1)−1 )τ α + cτ τ 2α ,  n −1 − (2 − α)n )τ (n−2)α . Since |cτ | ≤ c0 for small with cτ = ∞ n=2 (−1) ( (nα + 1) −1 τ , and (2 − α) < (α + 1) for α ∈ (0, 1), we deduce |u(τ ) − U 1 | ∼ τ α = t1α−1 τ. This short analysis confirms the sharpness of the convergence rate in Theorem 5.1. Last, we discuss the case u 0 ≡ 0 and f = 0. Theorem 5.2 Let u and U n be the solutions of problems (5.1) and (5.4) with u 0 ≡ 0. Then there holds tn

 (tn − s)α−1  f  (s) L 2 () ds . u(tn ) − U n  L 2 () ≤ cτ  f (0) L 2 () + 0

Proof First, we study time-independent f ≡ f (0). By Proposition 5.1, we have 1 e ztn z −1 (z α + A)−1 f (0) dz w(tn ) − W = 2π i θ,σ 1 e ztn μτ (e−zτ )−1 (βτ (e−zτ )α + A)−1 f (0) dz. − τ 2π i θ,σ n

Then repeating the argument in Theorem 5.1(ii) yields w(tn ) − W n  L 2 () ≤ ctnα−1 τ  f (0) L 2 () .

(5.13)

Next for the case u 0 = f (0) ≡ 0, Taylor’s expansion gives f (t) = f (0) + 1 ∗ f  (t) = 1 ∗ f  (t). The solution w(tn ) can be represented by w(tn ) = (E ∗ f )(tn ) = (E ∗ (1 ∗ f  ))(tn ) = ((E ∗ 1) ∗ f  )(tn ),

(5.14)

138

5 Finite Difference Methods on Uniform Meshes

where the solution operator E(t) is defined in (1.17), i.e., E(t)v = A)−1 v dz. Similarly, we have (βτ (ξ )α + A)−1 = τ

∞ 

1 2πi



θ,σ

e zt (z α +

E τn ξ n

n=0



1 with E τn = 2πi e znτ (βτ (e−zτ )α + A)−1 dz. Hence W n can be represented by τ

θ,σ  n− j n W n = τ j=1 E τ f (t j ), and the second inequality of (5.8) implies

E τn  ≤ ctnα−1 . Let E τ, (t) = τ

∞ 

(5.15)

E τn δtn − (t),

n=0

with δtn − ,  ∈ (0, τ ), being the Dirac–Delta function concentrated at tn − . Then the function W n is represented by W n = lim+ (E τ, ∗ f )(tn ) = lim+ (E τ, ∗ (1 ∗ f  ))(tn ) = ( lim+ (E τ, ∗ 1) ∗ f  )(tn ). →0

→0

→0

This and the representation (5.14) together yield w(tn ) − W n  L 2 () ≤ [ lim+ ((E − E τ, ) ∗ 1) ∗ f  ](tn ) L 2 () . →0

(5.16)

Using Laplace transform and Cauchy’s integral formula, we deduce 1 e ztn z −1 (z α + A)−1 dz ( lim+ (E − E τ, ) ∗ 1)(tn ) = →0 2π i θ,σ 1 e ztn μ(e−zτ )−1 (βτ (e−zτ )α + A)−1 dz. − τ 2π i θ,σ Then by the estimate (5.12), we obtain   ( lim (E − E τ, ) ∗ 1)(tn ) ≤ cτ t α−1 . n + →0

(5.17)

It remains to prove the following extension of the estimate (5.17):   ( lim (E − E τ, ) ∗ 1)(t) ≤ cτ t α−1 , ∀t ∈ (0, T ). + →0

(5.18)

This and (5.16) yield the desired estimate and complete the proof of the theorem. To prove (5.18), we employ the Taylor expansion of (E(t) − E τ, (t)) ∗ 1 at t = tn , i.e.,

5.2 Corrected L1 Scheme

139



tn

((E − E τ, ) ∗ 1)(t) = ((E − E τ, ) ∗ 1)(tn ) −

(E − E τ, )(s) ds.

(5.19)

t

In view of smoothing property of E(t) in Theorem 1.6, i.e., E(t) ≤ ct α−1 , there holds   tn tn     E(s) ds  ≤ c s α−1 ds ≤ cτ t α−1 .  t

t

Similarly, appealing to the estimate (5.15), we have    lim  →0+

t

tn

  n α−1 E τ, (s) ds   = τ E τ  ≤ cτ tn . 

Substituting (5.17) and the last two inequalities into (5.19) yields (5.18).

5.2 Corrected L1 Scheme In this section, we describe a corrected L1 scheme, which aims at improving the accuracy of standard L1 scheme from O(τ ) to O(τ 2−α ). Given f n = f (tn ), with U 0 = u 0 , the corrected L1 scheme reads: find U n ∈ H01 () by 

∂¯τα U n + AU n = f n + 21 (−Au 0 + f (0)), n = 1, ∂¯τα U n + AU n = f n , n = 2, 3, . . . , N .

(5.20)

Note that formally the correction coefficient is identical with that for bdf2 cq in Chap. 3. The corrected scheme modifies only the first step of the standard L1 scheme. We give a brief derivation of the corrected scheme, following Lubich, Sloan, and Thomée [LST96]. Equivalently, we write problem (5.1) into an evolution equation with a positive memory for w(t) = u(t) − u 0 : w(t) + 0 Itα Aw(t) = 0 Itα (−Au 0 + f 0 ) + 0 Itα ( f − f 0 ), t > 0, with w(0) = 0,

(5.21) where 0 Itα denotes the Riemann–Liouville fractional integral of order α, cf. Chap. 1. To obtain a high-order scheme, following [LST96], we may introduce a corrected time-discretization scheme to approximate problem (5.21): W n + 0 I¯τα AW n = −0 I¯τα (−Au 0 + f 0 )n + 0 I¯τα ( f − f 0 )n ,

(5.22)

where 0 I¯τα denotes a corrected quadrature formula approximating the Riemann– Liouville fractional integral 0 Itα , defined by

140

5 Finite Difference Methods on Uniform Meshes

¯α n 0 Iτ W

= τα

n 

bn− j W j + c0 bn−1 W 0 , with c0 = 21 ,

j=1

where the weights jb0 , b1 , . . . are given by the power series expansion of the function   b(ζ ) = ∞ j=0 b j ζ , which is to be determined. Then the following result gives b(ζ ). Lemma 5.6 Let  b(ζ ) = β(ζ )−α . Then the schemes (5.20) and (5.22) are equivalent. Proof Taking the discrete Laplace transform on both sides of (5.20) gives ∞ 

∂¯τα W n ζ n +

n=1

∞ 

AW n ζ n = (−Au 0 + f 0 )

∞



n=1

ζn +

n=1

∞

ζ  n + ( f − f 0 )ζ n . 2 n=1

By the condition W 0 = 0, the argument of Proposition 5.1 yields ∞ 

 (ζ ). ∂¯τα W n ζ n = βτ (ζ )α W

n=1

Consequently,  (ζ ) + A W  (ζ ) = (−Au 0 + f 0 ) τ (ζ )α W β

∞

ζ ζ  n + + ( f − f 0 )ζ n . 1−ζ 2 n=1

Now the assumption on  b(ζ ) implies  (ζ ) + τ α  (ζ ) W b(ζ )A W ∞

ζ  ζ + + τ α ( f n − f 0 )ζ n . b(ζ )(−Au 0 + f 0 ) b(ζ ) =τ α 1−ζ 2 n=1 Thus, we obtain ∞  n=1

W n ζ n +τ α

n ∞   n=1

+ τα

j=1

∞

 n=1

n ∞     bn− j AW j ζ n = τ α bn− j (−Au 0 + f 0 ) ζ n n=1



j=1

c0 bn−1 (−Au 0 + f 0 ) ζ n + τ α

n ∞   n=1

 bn− j ( f j − f 0 ) ζ n ,

j=1

which directly implies (5.22). Hence the two schemes are indeed equivalent.



The next result gives a crucial solution representation to the scheme (5.20). Proposition 5.3 Let U n be the solutions to problem (5.20). Then the function W n = U n − u 0 can be represented by

5.2 Corrected L1 Scheme

Wn =

141

1 2π i

τ

θ,σ

∞   etn z (βτ (e−zτ )α + A)−1 τ f n e−nzτ n=1

 e−zτ f (0) − μ¯ τ (e−zτ )−1 Au 0 dz, + 2 with βτ (e−zτ )α and μ¯ τ (e−zτ ), respectively, given by βτ (e−zτ )α =

(1 − e−zτ )2 2(1 − e−zτ ) Liα−1 (e−zτ ) and μ¯ τ (e−zτ ) = −zτ . − α) τ e (3 − e−zτ )

e−zτ τ α (2

Proof The proof is identical with Proposition 5.1. The function W n satisfies W 0 = 0 and  α n ∂¯τ W + AW n = −Au 0 + f n + 21 (−Au 0 + f (0)), n = 1, ∂¯τα W n + AW n = −Au 0 + f n , n = 2, 3, . . . , N . Multiplying both sides by ζ n and summing from 1 to ∞ yields ∞

ζ  ζ ζ  (ζ ) = − + Au 0 + f n ζ n + f (0). ∂¯τα W n ζ n + A W 1 − ζ 2 2 n=1 n=1

∞ 

Meanwhile, the argument in Proposition 5.1 yields ∞ 

(1 − ζ ) Liα−1 (ζ )  ∂¯τα W n ζ n = τ −α W (ζ ). ζ (2 − α) n=1 2

 (ζ ) is represented by Therefore, the semidiscrete solution W

 (ζ ) = W

(1 − ζ )2 Liα−1 (ζ ) + A ζ τ α (2 − α)

!−1



−

∞

ζ   ζ ζ Au 0 + f n ζ n + f (0) . + 1−ζ 2 2 n=1

Then to complete the proof, we repeat the argument in Proposition 5.1, using Cauchy τ . integral formula, changing variables ζ = e−zτ and deforming the contour to θ,σ The solution representations in Propositions 5.1 and 5.3 indicate that the difference lies in replacing μτ (e−zτ ) for the L1 scheme (5.3) with μ¯ τ (e−zτ ) for the corrected L1 scheme (5.20), and the additional term for f (0). Thus it is crucial to analyze the properties of the function μ¯ τ (e−zτ ); see the next lemma for the requisite results. Lemma 5.7 Let θ > estimates hold

π 2

τ be sufficiently close to π2 . Then for any z ∈ θ,σ , the following

c0 |z| ≤ |μ¯ τ (e−zτ )| ≤ c1 |z| and |μ¯ τ (e−zτ ) − z| ≤ cτ 2 |z|3 .

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5 Finite Difference Methods on Uniform Meshes

Proof With χτ (z) =

1−e−zτ z

μ¯ τ (e−zτ )−1 =

, we have the identity

τ e−zτ (3 − e−zτ ) e−zτ (3 − e−zτ ) = χτ (z)−1 . 2 1 − e−zτ 2

In view of Lemma 5.1, for the first estimate, it suffices to show that −zτ −zτ ) τ | ≤ c1 for z ∈ θ,σ . The upper bound holds trivially true, since c0 ≤ | e (3−e 2 −zτ −zτ ) τ is continuous in z, and |zτ | ≤ c for z ∈ θ,σ . Meanwhile, the function e (3−e 2 π π for θ = 2 , for any z with arg(z) = 2 , we have (zτ ) ∈ (0, π ], which implies e−zτ = e−iφ , φ ∈ (0, π ]. Note that 1 −iφ |e (3 2

− e−iφ )| ≥ 21 (3 − cos φ) ≥ 1.

This and the continuity of the function 21 e−zτ (3 − e−zτ ) in z shows the first assertion. Next, with ζ = zτ , Taylor expansion gives μ¯ τ (e−zτ ) =

2  ζ j    (− jζ )k  2(e zτ − 1) = τ (3 − e−zτ ) 3τ j=1 j! 3 j k! j=0 k=0 ∞

∞

∞

  1   j ζ2 2

2 ζ+ + O(|ζ |3 ) −ζ + O(|ζ | ) . 3τ 2 3! 3j j=0 j=0 ∞

=

Using the identities [GR15, 0.231, p. 9] μ¯ τ (e−zτ ) − z =

∞

1 j=0 3 j

∞

=

3 2

and

∞

j j=0 3 j

= 34 , we deduce

 3 3  ζ2 2

ζ+ + O(|ζ |3 ) − ζ + O(|ζ |2 ) − z = τ −1 O(|ζ |3 ). 3τ 2 2 4 

This completes the proof of the lemma. The next result gives a slightly more refined estimate on the kernel βτ (e−zτ )α . Corollary 5.1 Let θ be close to π2 , and σ
0 such that c|zτ |2−α