Matrix-Based Multigrid

1. Front Matter

   Pages i-xvi

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2. The Multilevel-Multiscale Approach

   1. The Multilevel-Multiscale Approach

      • Yair Shapira

      Pages 1-20

3. The Problem and Solution Methods

   1. Front Matter

      Pages 21-23

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   2. Partial Differential Equations and their Discretization

      • Yair Shapira

      Pages 25-42

   3. Iterative Linear-System Solvers

      • Yair Shapira

      Pages 43-59

   4. Multigrid Algorithms

      • Yair Shapira

      Pages 61-67

4. Multigrid for Structured Grids

   1. Front Matter

      Pages 69-71

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   2. The Automug Method

      • Yair Shapira

      Pages 73-78

   3. Applications in Image Processing

      • Yair Shapira

      Pages 79-90

   4. The Black-Box Multigrid Method

      • Yair Shapira

      Pages 91-97

   5. Application to the Indefinite Helmholtz Equation

      • Yair Shapira

      Pages 99-113

   6. Matrix-Based Semicoarsening Method

      • Yair Shapira

      Pages 115-128

5. Multigrid for Semi-Structured Grids

   1. Front Matter

      Pages 129-132

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   2. Matrix-Based Multigrid for Locally Refined Meshes

      • Yair Shapira

      Pages 133-166

6. Multigrid for Unstructured Grids

   1. Front Matter

      Pages 167-169

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   2. A Domain Decomposition Approach

      • Yair Shapira

      Pages 171-177

   3. An Algebraic Multilevel Method with Applications

      • Yair Shapira

      Pages 179-200

   4. Conclusions

      • Yair Shapira

      Pages 201-204

7. Back Matter

   Pages 205-221

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Many important problems in applied science and engineering, such as the Navier­ Stokes equations in fluid dynamics, the primitive equations in global climate mod­ eling, the strain-stress equations in mechanics, the neutron diffusion equations in nuclear engineering, and MRIICT medical simulations, involve complicated sys­ tems of nonlinear partial differential equations. When discretized, such problems produce extremely large, nonlinear systems of equations, whose numerical solution is prohibitively costly in terms of time and storage. High-performance (parallel) computers and efficient (parallelizable) algorithms are clearly necessary. Three classical approaches to the solution of such systems are: Newton's method, Preconditioned Conjugate Gradients (and related Krylov-space acceleration tech­ niques), and multigrid methods. The first two approaches require the solution of large sparse linear systems at every iteration, which are themselves often solved by multigrid methods. Developing robust and efficient multigrid algorithms is thus of great importance. The original multigrid algorithm was developed for the Poisson equation in a square, discretized by finite differences on a uniform grid. For this model problem, multigrid exhibits extremely rapid convergence, and actually solves the problem in the minimal possible time. The original algorithm uses rediscretization of the partial differential equation (POE) on each grid in the hierarchy of coarse grids that are used. However, this approach would not work for more complicated problems, such as problems on complicated domains and nonuniform grids, problems with variable coefficients, and non symmetric and indefinite equations. In these cases, matrix-based multi grid methods are in order.

• algebra
• algorithms
• calculus
• linear algebra
• partial differential equation

From the reviews:

"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)

• Computer Science department, Technion — Israel Institute of Technology, Haifa, Israel

  Yair Shapira

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