Lui, S. H,

Numerical Analysis of Partial Differential Equations

• Gebundenes Buch

Produktbeschreibung

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.

Produktdetails

• Wiley Series in Pure and Applied Mathematics
• Verlag: Wiley & Sons
• 1. Auflage
• Seitenzahl: 512
• Erscheinungstermin: 30. August 2011
• Englisch
• Abmessung: 262mm x 184mm x 35mm
• Gewicht: 36096g
• ISBN-13: 9780470647288
• ISBN-10: 0470647280
• Artikelnr.: 33379898

Autorenporträt

S. H. Lui, PhD, is Associate Professor of Mathematics in the Department of Mathematics at the University of Manitoba, Canada.

Inhaltsangabe

Preface. Acknowledgments. 1. Finite Difference. 1.1 Second-Order
Approximation for ". 1.2 Fourth-Order Approximation for ". 1.3 Neumann
Boundary Condition. 1.4 Polar Coordinates. 1.5 Curved Boundary. 1.6
Difference Approximation for "². 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.

Rezensionen

"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)