Numerical analysis of partial differential equations / S.H. Lui.

Format:

Book

Author/Creator:

Lui, S. H. (Shaun H.), 1961-

Series:

Pure and applied mathematics, a Wiley-Interscience series of texts, monographs, and tracts.

Pure and applied mathematics, a Wiley-Interscience series of texts, monographs, and tracts

Language:

English

Subjects (All):

Differential equations, Partial--Numerical solutions.

Differential equations, Partial.

Physical Description:

pages ; cm.

Place of Publication:

Hoboken, N.J. : Wiley, [2011]

Summary:

"This book provides a comprehensive and self-contained treatment of the numerical methods used to solve partial differential equations (PDEs), as well as both the error and efficiency of the presented methods. Featuring a large selection of theoretical examples and exercises, the book presents the main discretization techniques for PDEs, introduces advanced solution techniques, and discusses important nonlinear problems in many fields of science and engineering. It is designed as an applied mathematics text for advanced undergraduate and/or first-year graduate level courses on numerical PDEs"-- Provided by publisher.

Contents:

1 Finite Difference 1

1.1 Second-Order Approximation for Δ 1

1.2 Fourth-Order Approximation for Δ 15

1.3 Neumann Boundary Condition 19

1.4 Polar Coordinates 24

1.5 Curved Boundary 26

1.6 Difference Approximation for Δ² 30

1.7 A Convection-Diffusion Equation 32

1.8 Appendix: Analysis of Discrete Operators 35

1.9 Summary and Exercises 37

2 Mathematical Theory of Elliptic POEs 45

2.1 Function Spaces 45

2.2 Derivatives 48

2.3 Sobolev Spaces 52

2.4 Sobolev Embedding Theory 56

2.5 Traces 59

2.6 Negative Sobolev Spaces 62

2.7 Some Inequalities and Identities 64

2.8 Weak Solutions 67

2.9 Linear Elliptic PDEs 74

2.10 Appendix: Some Definitions and Theorems 82

2.11 Summary and Exercises 88

3 Finite Elements 95

3.1 Approximate Methods of Solution 95

3.2 Finite Elements in 1D 101

3.3 Finite Elements in 2D 109

3.4 Inverse Estimate 119

3.5 L² and Negative-Norm Estimates 122

3.6 Higher-Order Elements 125

3.7 A Posteriori Estimate 133

3.8 Quadrilateral Elements 156

3.9 Numerical Integration 160

3.10 Stokes Problem 144

3.11 Linear Elasticity 156

3.12 Summary and Exercises 160

4 Numerical Linear Algebra 169

4.1 Condition Number 169

4.2 Classical Iterative Methods 173

4.3 Krylov Subspace Methods 178

4.4 Direct Methods 190

4.5 Preconditioning 196

4.6 Appendix: Chebyshev Polynomials 208

4.7 Summary and Exercises 210

5 Spectral Methods 221

5.1 Trigonometric Polynomials 221

5.2 Fourier Spectral Method 232

5.3 Orthogonal Polynomials 242

5.4 Spectral Galerkin and Spectral Tau Methods 262

5.5 Spectral Collocation 264

5.6 Polar Coordinates 278

5.7 Neumann Problems 280

5.8 Fourth-Order PDEs 281

5.9 Summary and Exercises 282

6 Evolutionary PDEs 291

6.1 Finite Difference Schemes for Heat Equation 292

6.2 Other Time Discretization Schemes 331

6.3 Convection-Dominated equations 315

6.4 Finite Element Scheme for Heat Equation 317

6.5 Spectral Collocation for Heat Equation 321

6.6 Finite Difference Scheme for Wave Equation 322

6.7 Dispersion 327

6.8 Summary and Exercises 332

7 Multigrid 345

7.1 Introduction 346

7.2 Two-Grid Method 349

7.3 Practical Multigrid Algorithms 351

7.4 Finite Element Multigrid 354

7.5 Summary and Exercises 363

8 Domain Decomposition 369

8.1 Overlapping Schwarz Methods 370

8.2 Orthogonal Projections 374

8.3 Non-overlapping Schwarz Method 382

8.4 Substructuring Methods 387

8.5 Optimal Substructuring Methods 395

8.6 Summary and Exercises 410

9 Infinite Domains 419

9.1 Absorbing Boundary Conditions 420

9.2 Dirichlet-Neumann Map 424

9.3 Perfectly Matched Layer 427

9.4 Boundary Integral Methods 430

9.5 Fast Multipole Method 433

9.6 Summary and Exercises 436

10 Nonlinear Problems 441

10.1 Newton's Method 441

10.2 Other Methods 446

10.3 Some Nonlinear Problems 449

10.4 Software 467

10.5 Program Verification 468

10.6 Summary and Exercises 469

Answers to Selected Exercises 471.

Notes:

Includes bibliographical references and index.

Machine generated contents note: Preface.Acknowledgments.1. Finite Difference.1.1 Second-Order Approximation for [delta].1.2 Fourth-Order Approximation for [delta].1.3 Neumann Boundary Condition.1.4 Polar Coordinates.1.5 Curved Boundary.1.6 Difference Approximation for [delta]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 Problems5.8 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. .

ISBN:

9780470647288

0470647280

OCLC:

712125079