Numerical Treatment of Partial Differential Equations

Christian Grossmann Hans-Görg Roos Numerical Treatment of Partial Differential Equations Translated and revised by Martin Stynes

Prof. Dr. Christian Grossmann Prof. Dr. Hans-Görg Roos Institute of Numerical Mathematics Department of Mathematics Technical University of Dresden D-01062 Dresden, Germany e-mail: Christian.Grossmann@tu-dresden.de Hans-Goerg.Roos@tu-dresden.de Prof. Dr. Martin Stynes School of Mathematical Sciences Aras na Laoi University College Cork Cork, Ireland e-mail: m.stynes@ucc.ie Mathematics Subject Classification (2000): 65N, 65F Translation and revision of the 3rd edition of Numerische Behandlung Partieller Differentialgleichungen Published by Teubner, 2005. Library of Congress Control Number: 2007931595 ISBN: 978-3-540-71582-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, 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. Cover design: WMXDesign, Heidelberg Typesetting by the authors and SPi using a Springer L A TEX macro package Printed on acid-free paper SPIN: 11830948 46/2244/SPi 5 4 3 2 1 0

Preface Many well-known models in the natural sciences and engineering, and today even in economics, depend on partial differential equations. Thus the efficient numerical solution of such equations plays an ever-increasing role in state-ofthe-art technology. This demand and the computational power available from current computer hardware have together stimulated the rapid development of numerical methods for partial differential equations a development that encompasses convergence analyses and implementational aspects of software packages. In 1988 we started work on the first German edition of our book, which appeared in 1992. Our aim was to give students a textbook that contained the basic concepts and ideas behind most numerical methods for partial differential equations. The success of this first edition and the second edition in 1994 encouraged us, ten years later, to write an almost completely new version, taking into account comments from colleagues and students and drawing on the enormous progress made in the numerical analysis of partial differential equations in recent times. The present English version slightly improves the third German edition of 2005: we have corrected some minor errors and added additional material and references. Our main motivation is to give mathematics students and mathematicallyinclined engineers and scientists a textbook that contains all the basic discretization techniques for the fundamental types of partial differential equations; one in which the reader can find analytical tools, properties of discretization techniques and advice on algorithmic aspects. Nevertheless, we acknowledge that in fewer then 600 pages it is impossible to deal comprehensively with all these topics, so we have made some subjective choices of material. Our book is mainly concerned with finite element methods (Chapters 4 and 5), but we also discuss finite difference methods (Chapter 2) and finite volume techniques. Chapter 8 presents the basic tools needed to solve the discrete problems generated by numerical methods, while Chapter 6 (singularly perturbed problems) and Chapter 7 (variational inequalities and optimal control) are special topics that reflect the research interests of the authors.

VI Preface As well as the above in-depth presentations, there are passing references to spectral methods, meshless discretizations, boundary element methods, higher-order equations and systems, hyperbolic conservation laws, wavelets and applications in fluid mechanics and solid mechanics. Our book sets out not only to introduce the reader to the rich and fascinating world of numerical methods for partial differential equation, but also to include recent research developments. For instance, we present detailed introductions to a posteriori error estimation, the discontinuous Galerkin method and optimal control with partial differential equations; these areas receive a great deal of attention in the current research literature yet are rarely discussed in introductory textbooks. Many relevant references are given to encourage the reader to discover the seminal original sources amidst the torrent of current research papers on the numerical solution of partial differential equations. A large portion of Chapters 1 5 constitutes the material for a two-semester course that has been presented several times to students in the third and fourth year of their undergraduate studies at the Technical University of Dresden. We gratefully acknowledge those colleagues who improved the book by their comments, suggestions and discussions. In particular we thank A. Felgenhauer, S. Franz, T. Linß, B. Mulansky, A. Noack, E. Pfeifer, H. Pfeifer, H.-P. Scheffler and F. Tröltzsch. We are much obliged to our colleague and long standing friend Martin Stynes for his skill and patience in translating and mathematically revising this English edition. Dresden, June 2007

Contents Notation... XI 1 Basics... 1 1.1 Classification and Correctness............................. 1 1.2 Fourier smethod,integraltransforms... 5 1.3 Maximum Principle, Fundamental Solution................. 9 1.3.1 Elliptic Boundary Value Problems................... 9 1.3.2 Parabolic Equations and Initial-Boundary Value Problems... 15 1.3.3 Hyperbolic Initial and Initial-Boundary Value Problems 18 2 Finite Difference Methods... 23 2.1 Basic Concepts.......................................... 23 2.2 Illustrative Examples.................................... 31 2.3 Transportation Problems and Conservation Laws............ 36 2.3.1 The One-Dimensional Linear Case................... 37 2.3.2 Properties of Nonlinear Conservation Laws........... 48 2.3.3 Difference Methods for Nonlinear Conservation Laws... 53 2.4 Elliptic Boundary Value Problems......................... 61 2.4.1 Elliptic Boundary Value Problems................... 61 2.4.2 The Classical Approach to Finite Difference Methods.. 62 2.4.3 Discrete Green s Function.......................... 74 2.4.4 Difference Stencils and Discretization in General Domains......................................... 76 2.4.5 Mixed Derivatives, Fourth Order Operators........... 82 2.4.6 Local Grid Refinements........................... 89 2.5 Finite Volume Methods as Finite Difference Schemes......... 90 2.6 Parabolic Initial-Boundary Value Problems................. 103 2.6.1 Problems in One Space Dimension................... 104 2.6.2 Problems in Higher Space Dimensions................ 109 2.6.3 Semi-Discretization................................ 113

VIII Contents 2.7 Second-Order Hyperbolic Problems........................ 118 3 Weak Solutions...125 3.1 Introduction............................................ 125 3.2 Adapted Function Spaces................................. 128 3.3 Variational Equations and Conforming Approximation....... 142 3.4 Weakening V-ellipticity.................................. 163 3.5 NonlinearProblems...167 4 The Finite Element Method...173 4.1 AFirstExample...173 4.2 Finite-Element-Spaces................................... 178 4.2.1 Local and Global Properties........................ 178 4.2.2 Examples of Finite Element Spaces in R 2 and R 3...189 4.3 Practical Aspects of the Finite Element Method............. 202 4.3.1 Structure of a Finite Element Code.................. 202 4.3.2 Description of the Problem......................... 203 4.3.3 Generation of the Discrete Problem.................. 205 4.3.4 Mesh Generation and Manipulation.................. 210 4.4 Convergence of Conforming Methods....................... 217 4.4.1 Interpolation and Projection Error in Sobolev Spaces.. 217 4.4.2 Hilbert Space Error Estimates...................... 227 4.4.3 Inverse Inequalities and Pointwise Error Estimates..... 232 4.5 NonconformingFiniteElementMethods...238 4.5.1 Introduction...................................... 238 4.5.2 Ansatz Spaces with Low Smoothness................ 239 4.5.3 Numerical Integration.............................. 244 4.5.4 The Finite Volume Method Analysed from a Finite ElementViewpoint...251 4.5.5 Remarks on Curved Boundaries..................... 254 4.6 MixedFiniteElements...258 4.6.1 Mixed Variational Equations and Saddle Points....... 258 4.6.2 Conforming Approximation of Mixed Variational Equations........................................ 265 4.6.3 Weaker Regularity for the Poisson and Biharmonic Equations........................................ 272 4.6.4 Penalty Methods and Modified Lagrange Functions.... 277 4.7 Error Estimators and Adaptive FEM...................... 287 4.7.1 The Residual Error Estimator....................... 288 4.7.2 Averaging and Goal-Oriented Estimators............. 292 4.8 TheDiscontinuousGalerkinMethod...294 4.8.1 The Primal Formulation for a Reaction-Diffusion Problem...295 4.8.2 First-Order Hyperbolic Problems.................... 299 4.8.3 Error Estimates for a Convection-Diffusion Problem... 302

Contents IX 4.9 Further Aspects of the Finite Element Method.............. 306 4.9.1 Conditioning of the Stiffness Matrix................. 306 4.9.2 Eigenvalue Problems............................... 307 4.9.3 Superconvergence................................. 310 4.9.4 p- and hp-versions................................ 314 5 Finite Element Methods for Unsteady Problems...317 5.1 Parabolic Problems...................................... 317 5.1.1 On the Weak Formulation.......................... 317 5.1.2 Semi-Discretization by Finite Elements............... 321 5.1.3 Temporal Discretization by Standard Methods........ 330 5.1.4 Temporal Discretization with Discontinuous Galerkin Methods...337 5.1.5 Rothe s Method................................... 343 5.1.6 Error Control..................................... 347 5.2 Second-Order Hyperbolic Problems........................ 356 5.2.1 Weak Formulation of the Problem................... 356 5.2.2 Semi-Discretization by Finite Elements............... 358 5.2.3 Temporal Discretization............................ 363 5.2.4 Rothe s Method for Hyperbolic Problems............. 368 5.2.5 Remarks on Error Control.......................... 372 6 Singularly Perturbed Boundary Value Problems...375 6.1 Two-Point Boundary Value Problems...................... 376 6.1.1 Analytical Behaviour of the Solution................. 376 6.1.2 Discretization on Standard Meshes.................. 383 6.1.3 Layer-adapted Meshes............................. 394 6.2 Parabolic Problems, One-dimensional in Space.............. 399 6.2.1 The Analytical Behaviour of the Solution............. 399 6.2.2 Discretization..................................... 401 6.3 Convection-Diffusion Problems in Several Dimensions........ 406 6.3.1 Analysis of Elliptic Convection-Diffusion Problems..... 406 6.3.2 Discretization on Standard Meshes.................. 412 6.3.3 Layer-adapted Meshes............................. 427 6.3.4 Parabolic Problems, Higher-Dimensional in Space..... 430 7 Variational Inequalities, Optimal Control...435 7.1 AnalyticProperties...435 7.2 Discretization of Variational Inequalities.................... 447 7.3 PenaltyMethods...457 7.3.1 Basic Concept of Penalty Methods................... 457 7.3.2 Adjustment of Penalty and Discretization Parameters.. 473 7.4 Optimal Control of PDEs................................. 480 7.4.1 Analysis of an Elliptic Model Problem............... 480 7.4.2 Discretization by Finite Element Methods............ 489

X Contents 8 Numerical Methods for Discretized Problems...499 8.1 Some Particular Properties of the Problems................. 499 8.2 DirectMethods...502 8.2.1 Gaussian Elimination for Banded Matrices............ 502 8.2.2 Fast Solution of Discrete Poisson Equations, FFT..... 504 8.3 Classical Iterative Methods............................... 510 8.3.1 Basic Structure and Convergence.................... 510 8.3.2 Jacobi and Gauss-Seidel Methods................... 514 8.3.3 Block Iterative Methods............................ 520 8.3.4 Relaxation and Splitting Methods................... 524 8.4 The Conjugate Gradient Method.......................... 530 8.4.1 The Basic Idea, Convergence Properties.............. 530 8.4.2 Preconditioned CG Methods........................ 538 8.5 Multigrid Methods...................................... 548 8.6 Domain Decomposition, Parallel Algorithms................ 560 Bibliography: Textbooks and Monographs...571 Bibliography: Original Papers...577 Index...585

Notation Often used symbols: a(, ) bilinear form D +,D,D 0 difference quotients D α derivative of order α with respect to the multi-index α I identity Ih H,Ih H restriction- and prolongation operator J( ) functional LL differential operator and its adjoint L h difference operator O( ),o( ) Landau symbols P l the set of all polynomials of degree l Q l the set of all polynomial which are the product of polynomials of degree l with respect to every variable R, N real and natural numbers, respectively V, V Banach space and its dual dim V dimension of V V h finite-dimensional finite element space V norm on V (, ) scalar product in V,if V is Hilbert space f(v)or f,v value of the functional f V applied to v V f norm of the linear functional f strong and weak convergence, respectively direct sum L(U, V ) space of continuous linear mappings of U in V L(V ) space of continuous linear mappings of V in V U V continuous embedding of U in V Z orthogonal complement of Z with respect to the scalar product in a Hilbert space V

XII NOTATION Ω given domain in space Ω = Γ boundary of Ω int Ω interior of Ω meas Ω measure of Ω n outer unit normal vector with respect to Ω directional derivative with respect to n n ω h,ω h set of mesh points C l (Ω), C l,α (Ω) space of differentiable and Hölder differentiable functions, respectively L p (Ω) space of functions which are integrable to the power p (1 p ) norm in L (Ω) D(Ω) W l p(ω) infinitely often differentiable functions with compact support in Ω Sobolev space H l (Ω), H l 0(Ω) Sobolev spaces for p =2 H(div; Ω) special Sobolev space TV space of functions with finite total variation l norm in the Sobolev space H l l semi-norm in the Sobolev space H l t, T time with t (0,T) Q = Ω (0,T) given domain for time-depending problems L 2 (0,T; X) quadratically integrable functions with values in the Banach space X W 1 2 (0,T; V,H) special Sobolev space for time-depending problems supp v or grad div h h i,h τ j,τ det(a) cond(a) ρ(a) λ i (A) diag(a i ) span{ϕ i } conv{ϕ i } Π Π h support of a function v gradient divergence Laplacian discrete Laplacian discretization parameters with respect to space discretization parameters with respect to time determinant of the matrix A condition of the matrix A spectral radius of the matrix A eigenvalues of the matrix A diagonal matrix with elements a i linear hull of the elements ϕ i convex hull of the elements ϕ i projection operator interpolation or projection operator which maps onto the finite element space