Matrix-Based Multigrid: Theory and Applications

Yair Shapira (eds.)0387497641, 9780387497648

Multigrid methods are often used for solving partial differential equations. This book introduces and analyzes the multigrid approach. The approach used here applies to both test problems on rectangular grids and to more realistic applications with complicated grids and domains.

Key Features of this Second Edition:

– Discusses multigrid methods from the domain decomposition viewpoint, thus making the material accessible to beginning undergraduate/graduate students

– Uses the semialgebraic multigrid approach to handle complex topics (such as the solution of systems of PDEs)

– Provides relevant and insightful exercises at the end of each chapter which help reinforce the material

– Uses numerous illustrations and examples to motivate the subject matter

– Covers important applications in physics, engineering and computer science

Matrix-Based Multigrid can serve as a textbook for courses in numerical linear algebra, numerical methods for PDEs, and computational physics at the advanced undergraduate and graduate levels. Since most of the background material is covered, the only prerequisites are elementary linear algebra and calculus.

Excerpts from the reviews of the first edition:

“This book contains a wealth of information about using multilevel methods to solve partial differential equations (PDEs). . . A common matrix-based framework for developing these methods is used throughout the book. This approach allows methods to be developed for problems under three very different conditions. . . This book will be insightful for practitioners in the field. . . students will enjoy studying this book to see how the many puzzle pieces of the multigrid landscape fit together.” (Loyce Adams, SIAM review, Vol. 47(3), 2005)

“The discussion very often includes important applications in physics, engineering, and computer science. The style is clear, the details can be understood without any serious prerequisite. The usage of multigrid method for unstructured grids is exhibited by a well commented C++ program. This way the book is suitable for anyone . . . who needs numerical solution of partial differential equations.” (Peter Hajnal, Acta Scientiarum Mathematicarum, Vol. 70, 2004)

Table of contents :
Front Matter….Pages i-xxiii
Front Matter….Pages 1-3
The Multilevel-Multiscale Approach….Pages 5-21
Preliminaries….Pages 23-43
Front Matter….Pages 45-47
Finite Differences and Volumes….Pages 49-66
Finite Elements….Pages 67-83
Front Matter….Pages 85-87
Iterative Linear System Solvers….Pages 89-107
The Multigrid Iteration….Pages 109-129
Front Matter….Pages 131-133
The Automatic Multigrid Method….Pages 135-144
Applications in Image Processing….Pages 145-153
The Black-Box Multigrid Method….Pages 155-164
The Indefinite Helmholtz Equation….Pages 165-181
Matrix-Based Semicoarsening Method….Pages 183-198
Front Matter….Pages 199-201
Matrix-Based Multigrid for Locally Refined Meshes….Pages 203-222
Application to Semistructured Grids….Pages 223-233
Front Matter….Pages 235-237
The Domain-Decomposition Multigrid Method….Pages 239-248
The Algebraic Multilevel Method….Pages 249-259
Applications….Pages 261-271
Semialgebraic Multilevel Method for Systems of Partial Differential Equations….Pages 273-287
Front Matter….Pages 289-291
Time-Dependent Parabolic PDEs….Pages 293-300
Nonlinear Equations….Pages 301-304
Back Matter….Pages 305-318