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Numerical Analysis of Partial Differential Equations

ISBN: 978-0-470-64728-8
512 pages
August 2011
Numerical Analysis of Partial Differential Equations (0470647280) cover image
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.

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Preface.

Acknowledgments.

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.

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.

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 A Posteriori Estimate.

3.7 Higher-Order Elements.

3.8 Quadrilateral Elements.

3.9 Numerical Integration.

  3.10 Stokes Problem.

3.11 Linear Elasticity.

3.12 Summary and Exercises.

4. Numerical Linear Algebra.

4.1 Condition Numbers.

4.2 Classical Iterative Methods.

4.3 Krylov Subspace Methods.

4.4 Preconditioning.

4.5 Direct Methods.

4.6 Appendix: Chebyshev Polynomials.

4.7 Summary and Exercises.

5. Spectral Methods.

5.1 Trigonometric Polynomials.

5.2 Fourier Spectral Method.

5.3 Orthogonal Polynomials.

5.4 Spectral Gakerkin 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.

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 Different Scheme for Wave Equation.

6.7 Dispersion.

6.8 Summary and Exercises.

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.

8. Domain Decomposition.

8.1 Overlapping Schwarz Methods.

8.2 Projections.

8.3 Non-overlapping Schwarz Method.

8.4 Substructuring Methods.

8.5 Optimal Substructuring Methods.

8.6 Summary and Exercises.

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 Multiple Method.

9.6 Summary and Exercises.

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. 

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S. H. Lui, PhD, is Associate Professor of Mathematics in the Department of Mathematics at the University of Manitoba, Canada.
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“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.”  (Zentralblatt MATH, 1 December 2012)

“Recommended.  Upper-division undergraduates, graduate students, and researchers/faculty.”  (Choice, 1 May 2012)

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