© 2003

Matrix-Based Multigrid

Theory and Applications


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

Table of contents

  1. Front Matter
    Pages i-xvi
  2. The Multilevel-Multiscale Approach

    1. Yair Shapira
      Pages 1-20
  3. The Problem and Solution Methods

    1. Front Matter
      Pages 21-23
    2. Yair Shapira
      Pages 43-59
    3. Yair Shapira
      Pages 61-67
  4. Multigrid for Structured Grids

    1. Front Matter
      Pages 69-71
    2. Yair Shapira
      Pages 73-78
    3. Yair Shapira
      Pages 79-90
    4. Yair Shapira
      Pages 91-97
    5. Yair Shapira
      Pages 115-128
  5. Multigrid for Semi-Structured Grids

    1. Front Matter
      Pages 129-132
  6. Multigrid for Unstructured Grids

    1. Front Matter
      Pages 167-169
    2. Yair Shapira
      Pages 171-177
    3. Yair Shapira
      Pages 201-204
  7. Back Matter
    Pages 205-221

About this book


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

Authors and affiliations

  1. 1.Computer Science departmentTechnion — Israel Institute of TechnologyHaifaIsrael

Bibliographic information

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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)