Numerical Analysis of Partial Differential Equations

The book presents the three main discretization methods of elliptic PDEs: finite difference, finite elements, and spectral methods.

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.

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.