Numerical Analysis of Partial Differential Equations E-Book

S. H, Lui

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Pure and Applied Mathematics
• Sprache: Englisch

Seitenzahl: 794

Veröffentlichungsjahr: 2012

Beschreibung

A balanced guide to the essential techniques for solving elliptic partial differential equations Numerical Analysis of Partial Differential Equations provides a comprehensive, self-contained treatment of the quantitative methods used to solve elliptic partial differential equations (PDEs), with a focus on the efficiency as well as the error of the presented methods. The author utilizes coverage of theoretical PDEs, along with the nu merical solution of linear systems and various examples and exercises, to supply readers with an introduction to the essential concepts in the numerical analysis of PDEs. The book presents the three main discretization methods of elliptic PDEs: finite difference, finite elements, and spectral methods. Each topic has its own devoted chapters and is discussed alongside additional key topics, including: * The mathematical theory of elliptic PDEs * Numerical linear algebra * Time-dependent PDEs * Multigrid and domain decomposition * PDEs posed on infinite domains The book concludes with a discussion of the methods for nonlinear problems, such as Newton's method, and addresses the importance of hands-on work to facilitate learning. Each chapter concludes with a set of exercises, including theoretical and programming problems, that allows readers to test their understanding of the presented theories and techniques. In addition, the book discusses important nonlinear problems in many fields of science and engineering, providing information as to how they can serve as computing projects across various disciplines. Requiring only a preliminary understanding of analysis, Numerical Analysis of Partial Differential Equations is suitable for courses on numerical PDEs at the upper-undergraduate and graduate levels. The book is also appropriate for students majoring in the mathematical sciences and engineering.

Leseprobe

CONTENTS

Cover

Half Title page

Title page

Copyright page

Preface

Acknowledgments

Chapter 1 Finite Difference

1.1 Second-Order Approximation for Δ

1.2 Fourth-Order Approximation for Δ

1.3 Neumann Boundary Condition

1.4 Polar Coordinates

1.5 Curved Boundary

1.6 Difference Approximation for Δ2

1.7 A Convection-Diffusion Equation

1.8 Appendix: Analysis of Discrete Operators

1.9 Summary and Exercises

Chapter 2 Mathematical Theory of Elliptic PDEs

2.1 Function Spaces

2.2 Derivatives

2.3 Sobolev Spaces

2.4 Sobolev Embedding Theory

2.5 Traces

2.6 Negative Sobolev Spaces

2.7 Some Inequalities and Identities

2.8 Weak Solutions

2.9 Linear Elliptic PDEs

2.10 Appendix: Some Definitions and Theorems

2.11 Summary and Exercises

Chapter 3 Finite Elements

3.1 Approximate Methods of Solution

3.2 Finite Elements in 1D

3.3 Finite Elements in 2D

3.4 Inverse Estimate

3.5 L2 and Negative-Norm Estimates

3.6 Higher-Order Elements

3.7 A Posteriori Estimate

3.8 Quadrilateral Elements

3.9 Numerical Integration

3.10 Stokes Problem

3.11 Linear Elasticity

3.12 Summary and Exercises

Chapter 4 Numerical Linear Algebra

4.1 Condition Number

4.2 Classical Iterative Methods

4.3 Krylov Subspace Methods

4.4 Direct Methods

4.5 Preconditioning

4.6 Appendix: Chebyshev Polynomials

4.7 Summary and Exercises

Chapter 5 Spectral Methods

5.1 Trigonometric Polynomials

5.2 Fourier Spectral Method

5.3 Orthogonal Polynomials

5.4 Spectral Galerkin and Spectral Tau Methods

5.5 Spectral Collocation

5.6 Polar Coordinates

5.7 Neumann Problems

5.8 Fourth-Order PDEs

5.9 Summary and Exercises

Chapter 6 Evolutionary PDEs

6.1 Finite Difference Schemes for Heat Equation

6.2 Other Time Discretization Schemes

6.3 Convection-Dominated equations

6.4 Finite Element Scheme for Heat Equation

6.5 Spectral Collocation for Heat Equation

6.6 Finite Difference Scheme for Wave Equation

6.7 Dispersion

6.8 Summary and Exercises

Chapter 7 Multigrid

7.1 Introduction

7.2 Two-Grid Method

7.3 Practical Multigrid Algorithms

7.4 Finite Element Multigrid

7.5 Summary and Exercises

Chapter 8 Domain Decomposition

8.1 Overlapping Schwarz Methods

8.2 Orthogonal Projections

8.3 Non-overlapping Schwarz Method

8.4 Substructuring Methods

8.5 Optimal Substructuring Methods

8.6 Summary and Exercises

Chapter 9 Infinite Domains

9.1 Absorbing Boundary Conditions

9.2 Dirichlet–Neumann Map

9.3 Perfectly Matched Layer

9.4 Boundary Integral Methods

9.5 Fast Multipole Method

9.6 Summary and Exercises

Chapter 10 Nonlinear Problems

10.1 Newton’s Method

10.2 Other Methods

10.3 Some Nonlinear Problems

10.4 Software

10.5 Program Verification

10.6 Summary and Exercises

Answers to Selected Exercises

References

Index

Numerical Analysis of Partial Differential Equations

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Library of Congress Cataloging-in-Publication Data:

Lui, S. H. (Shaun H.), 1961-Numerical analysis of partial differential equations / S.H. Lui.p. cm. — (Pure and applied mathematics, a Wiley-Interscience series of texts, monographs, and tracts)Includes bibliographical references and index.ISBN 978-0-470-64728-8 (hardback)1. Differential equations, Partial-Numerical solutions. I. Title.QA377.L84 2011518′.64—dc232011013570

oBook ISBN: 978-1-118-11113-0 ePDF ISBN: 978-1-118-11110-9 ePub ISBN: 978-1-118-11111-6

PREFACE

This is a book on the numerical analysis of partial differential equations (PDEs). This beautiful subject studies the theory behind algorithms used to approximate solutions of PDEs. The ultimate goal is to design methods which are accurate and efficient. It is a relatively young field which draws upon powerful theory from many branches of mathematics, both pure and applied.

The contents in this book are suitable for a two-semester course at the senior undergraduate or beginning graduate level, for students in mathematical sciences and engineering. The emphasis is on elliptic PDEs, with one chapter discussing evolutionary PDEs. A prerequisite for reading this book is a solid undergraduate course in analysis. Some exposure to numerical analysis and PDEs is helpful.

Our aim is to offer an introduction to most of the important concepts in the numerical analysis of PDEs. The two-dimensional Poisson equation is the model problem upon which the various methods are analyzed. Because of the simplicity of this equation, the analysis can almost always be carried out in full. In this sense, this book is entirely self-contained; the student does not need to consult other books for the proof of a theorem. The only exception is the chapter on the mathematical theory of elliptic PDEs, where very few proofs are given. Many students taking this course may have no prior experience in this area and thus this chapter is simply an introduction to the topic by examples. A proper treatment of PDEs requires several semesters, which most students cannot accommodate in their programs. Some may argue that there is too much emphasis on the Poisson equation, however, our response is that for an introductory course, this equation is appropriate because it gives a good picture of the kinds of results expected without the complications involved in more general PDEs. Convection-diffusion equations and nonlinear equations are of course interesting, but they belong to more advanced courses and many topics there are still under active research. Other omitted topics include finite volume methods, discontinuous Galerkin methods, meshless methods, Monte Carlo methods, wavelets, eigenvalue problems, inverse problems, free boundary value problems, etc. Implementation and visualization issues are not discussed at all.

The topics covered are the three main discretization methods of elliptic PDEs: finite difference, finite elements and spectral methods. These are presented in Chapters 1, 3 and 5, respectively. In between are discussions on the mathematical theory of elliptic PDEs in Chapter 2 and numerical linear algebra in Chapter 4. Time-dependent PDEs make a brief appearance in Chapter 6. Multigrid and domain decomposition, are covered in Chapters 7 and 8. These are among the most efficient techniques for solving PDEs today. Chapter 9 contains a discussion of PDEs posed on infinite domains. The main issue here is how to pose the boundary condition on the artificial boundary which is necessary on a finite computational domain. Methods for nonlinear problems are briefly described in Chapter 10. Here, we also describe some important nonlinear problems in many fields of science and engineering. These can serve as computing projects for students from different disciplines.

Each chapter can be, and have been, expanded by other authors into a course by itself! Most chapters can be covered in approximately 10 hours. The exceptions are: Chapter 3 (finite elements) and Chapter 5 (spectral methods) which require about 15 hours each to cover all sections; Chapter 6 (multigrid), Chapter 9 (infinite domains) and Chapter 10 (nonlinear problems) need about five hours each.

A few words about the ordering of the chapters are called for. The material on the finite difference method requires few prerequisites and thus is placed in the first chapter. The analysis of the finite element and spectral methods uses the language of Sobolev spaces and their properties, which are conveniently covered in the second chapter. Having seen the structured matrices in the finite difference method and the unstructured matrices in the finite element method, readers are well motivated to appreciate and comprehend the issues in numerical linear algebra in Chapter 4. A possible alternative ordering of the first part of the book is to discuss Sobolev spaces first, followed by the three discretization techniques and finally numerical linear algebra. The problem is that the material on PDE theory appears to many students as abstract, dry and unmotivated. Furthermore, the discussion on numerical PDEs does not begin until the fourth week. This book does not have to be read in the order presented but Chapter 2 should precede all subsequent chapters except 4,9 and 10. Chapters 6 through 9 can be read in any order, but they rely heavily on material from the first four chapters.

No one learns mathematics by reading alone. Exercises are an integral part of this book and students are encouraged to try them. They are essential for reinforcing the material and many extend theories and techniques developed in the text. Both theoretical and programming problems are available, with the former prefixed by E and the latter prefixed by P. Answers to selected written exercises are given. Although this book does not emphasize the implementation of the algorithms, readers willing to invest time on the programming exercises will gain a much better appreciation of the subject. As already mentioned, about half of the chapters can be covered in about four weeks. The time frame can easily extend by one to two weeks per chapter if students do a substantial fraction of the written and programming exercises.

Almost all material in this book are well known to numerical analysts and have been gleaned from various sources listed in the bibliography. References to texts or monographs are generally given in place of the original articles. There are already excellent texts on each of the areas discussed in this book but there does not appear to be one which covers all the topics here. The present book should serve as a good preparation for more advanced work. Readers are encouraged to send their comments and corrections to [email protected]. A webpage http://home.cc.umanitoba.ca/~luish/numpde has been created for this book. It will contain errata as well as some MATLAB programs.

ACKNOWLEDGMENTS

I have learned a great deal from the books and papers of many mathematicians (a partial list appears in the References) and whose ideas I have followed in this book. In fact, I was studying several topics for the first time as I was writing this book-experts in the fields should have no difficulty pointing out my level of ignorance. I sincerely thank my colleagues Susanne Brenner, Raymond Chan, Qiang Du, Martin Gander, Laurence Halpern, Ronald Haynes, Felix Kwok, Jie Shen, Xue-Cheng Tai and Justin Wan for reading parts of this book on short notice and making many insightful suggestions. This book is poorer because I have not the time, energy and/or expertise to implement all their proposed changes. I take this opportunity to thank the many students who have taken courses based on earlier drafts of this book. They pointed out many typos, inaccuracies and made numerous suggestions which have immeasurably improved the presentation. Of course, I am responsible for all remaining mistakes in the book. It has truly been a humbling and rewarding experience having so many wonderful students in my classes.

Michael Doob, Amy Hendrickson and Jonas Lippuner have been extremely helpful with the finer points of LATEX. I also thank my editor Susanne Steitz-Filler and the staff at John Wiley & Sons, Inc. Finally, financial support from NSERC and the University of Manitoba are gratefully acknowledged.