# Matrix-Based Multigrid

## Theory and Applications

• Yair Shapira
Textbook

Part of the Numerical Methods and Algorithms book series (NUAL, volume 2)

1. Front Matter
Pages i-xxiii
2. ### Concepts and Preliminaries

1. Front Matter
Pages 1-3
2. Pages 5-21
3. Pages 23-43
3. ### Partial Differential Equations and Their Discretization

1. Front Matter
Pages 45-47
2. Pages 49-66
3. Pages 67-83
4. ### The Numerical Solution of Large Sparse Linear Systems of Algebraic Equations

1. Front Matter
Pages 85-87
2. Pages 89-107
3. Pages 109-129
5. ### Matrix-Based Multigrid for Structured Grids

1. Front Matter
Pages 131-133
2. Pages 135-144
3. Pages 145-153
4. Pages 155-164
5. Pages 165-181
6. Pages 183-198
6. ### Matrix-Based Multigrid for Semistructured Grids

1. Front Matter
Pages 199-201
2. Pages 203-222
3. Pages 223-233
7. ### Matrix-Based Multigrid for Unstructured Grids

1. Front Matter
Pages 235-237

### Introduction

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)

### Keywords

Helmholtz equation computer computer science finite elements image processing linear algebra model numerical analysis

### Editors and affiliations

• Yair Shapira
• 1
1. 1.Department of Computer ScienceTechnion – Israel Institute of TechnologyHaifaIsrael

### Bibliographic information

• Book Title Matrix-Based Multigrid
• Book Subtitle Theory and Applications
• Authors Yair Shapira
• Series Title Numerical Methods and Algorithms
• DOI https://doi.org/10.1007/978-0-387-49765-5
• Copyright Information Springer-Verlag US 2008
• Publisher Name Springer, Boston, MA
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Hardcover ISBN 978-0-387-49764-8
• Softcover ISBN 978-1-4419-4321-7
• eBook ISBN 978-0-387-49765-5
• Series ISSN 1571-5698
• Edition Number 2
• Number of Pages XXIV, 318
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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## Reviews

From the reviews of the second edition:

"Shapira delivers a systematic and unified presentation of the multigrid method that is used for the efficient solution of partial differential equations. … The notations are consistent and the presentation is self-contained. The book is recommended to readers involved in the field of computational science and engineering, from the postgraduate to the expert level. Additionally, the book is suitable for courses in numerical analysis, numerical linear algebra, scientific computing, and numerical solution of partial differential equations." (George A. Gravvanis, ACM Computing Reviews, May, 2009)

“This book provides an introduction into this area. Basically, it presupposes only a sound knowledge of analysis and linear algebra and introduces all other necessary concepts on its own. … Many exercises are included. The presentation is well suited for seminars in this area.” (H. Muthsam, Monatshefte für Mathematik, Vol. 156 (3), March, 2009)