Numerical Analysis of Partial Differential EquationsISBN: 9780470647288
512 pages
August 2011

Numerical Analysis of Partial Differential Equations provides a comprehensive, selfcontained 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

Timedependent 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 handson 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 upperundergraduate and graduate levels. The book is also appropriate for students majoring in the mathematical sciences and engineering.
Acknowledgments.
1. Finite Difference.
1.1 SecondOrder Approximation for Δ.
1.2 FourthOrder Approximation for Δ.
1.3 Neumann Boundary Condition.
1.4 Polar Coordinates.
1.5 Curved Boundary.
1.6 Difference Approximation for Δ^{2}.
1.7 A ConvectionDiffusion 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 L^{2} and NegativeNorm Estimates.
3.6 A Posteriori Estimate.
3.7 HigherOrder 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 FourthOrder 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 ConvectionDominated 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 TwoGrid 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 Nonoverlapping 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 DirichletNeumann 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.
“Requiring only a preliminary understanding of analysis, Numerical Analysis of Partial Differential Equations is suitable for courses on numerical PDEs at the upperundergraduate and graduate levels. The book is also appropriate for students majoring in the mathematical sciences and engineering.” (Zentralblatt MATH, 1 December 2012)
“Recommended. Upperdivision undergraduates, graduate students, and researchers/faculty.” (Choice, 1 May 2012)
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