Matrix-Based Multigrid

Theory and Applications

  • Yair Shapira

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

Introduction

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.

Keywords

algebra algorithms calculus linear algebra partial differential equation

Authors and affiliations

  • Yair Shapira
    • 1
  1. 1.Computer Science departmentTechnion — Israel Institute of TechnologyHaifaIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-3726-4
  • Copyright Information Springer-Verlag US 2003
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4757-3728-8
  • Online ISBN 978-1-4757-3726-4
  • Series Print ISSN 1571-5698
  • About this book
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