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An Introduction to Multigrid Methods

Pieter Wesseling

Introduces the principles, techniques, applications and literature--both current and historical--of multigrid methods.

Bookcover

Format: Hardcover, English, 284 pages
ISBN: 1930217080
Publisher: R.T. Edwards, Inc.
Pub. Date: 2004
Edition: 1
Book Homepage: http://ta.twi.tudelft.nl/nw/users/wesseling/multigridbook.html

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Book Description

Inspired by a series of lectures given in Delft, Bristol, Lyons, Zurich, and Beijing, this book is a corrected reprint of the 1992 classic. Provides a complete introduction to multigrid methods for partial differential equations, without requiring an advanced knowledge of mathematics. Topics such as the basic multigrid principle, smoothing methods and their Fourier analysis, course grid approximation, multigrid cycles and results of multigrid theory are treated. Applications in computational fluid dynamics are discussed extensively.


Author's Comments

Multigrid methods have developed rapidly over the past 25 years and are now a powerful tool for the efficient solution of elliptic and hyperbolic partial differential equations. The impact of multigrid methods on computational fluid dynamics and computational physics is considerable.

Since its first appearance in 1992, this has remained the only book that gives a complete introduction to multigrid methods for partial differential equations, without requiring advanced knowledge of mathematics. Instead, it presupposes only a basic understanding of analysis, partial differential equations and numerical analysis.

The volume begins with an introduction to the literature. Subsequent chapters present an extensive treatment of applications to computational fluid dynamics and discuss topics such as the basic multigrid principle, smoothing methods and their Fourier analysis and results of multigrid convergence theory.

This book will appeal to a wide readership including students and researchers in applied mathematics, engineering and physics, who have to deal with computing-intensive applications.

Publisher Comments

Introduces the principles, techniques, applications and literature--both current and historical--of multigrid methods. Aimed at an audience with non-mathematical but computing-intensive disciplines and basic knowledge of analysis, partial differential equations and numerical mathematics, it is packed with helpful exercises, examples and illustrations.


Table of Contents

1 Introduction  
     
2 The Essential Principle of Multigrid Methods for Partial Differential Equations  
2.1 Introduction  
2.2 The Essential Principle  
2.3 The Two-Grid Algorithm  
2.4 Two-Grid Analysis  
     
3 Finite Difference and Finite Volume Discretization  
3.1 Introduction  
3.2 An Elliptic Equation  
3.3 A One-Dimensional Example  
3.4 Vertex-Centered Discretization  
3.5 Cell-Centered Discretization  
3.6 Upwind Discretization  
3.7 A Hyperbolic System  
     
4 Basic Iterative Methods  
4.1 Introduction  
4.2 Convergenc of Basic Iterative Methods  
4.3 Examples of Basic Iterative Methods: Jacobi and Gauss-Seidel  
4.4 Examples of Basic Iterative Methods: Incomplete Point LU Factorization  
4.5 Examples of Basic Iterative Methods: Incomplete Block LU Factorization  
4.6 Some Methods for Non-M-Matrices  
     
5 Prolongation and Restriction  
5.1 Introduction  
5.2 Stencil Notation  
5.3 Interpolating Transfer Operators  
5.4 Operator-Dependent Transfer Operators  
     
6 Coarse Grid Approximation and Two-Grid Convergence  
6.1 Introduction  
6.2 Computation of the Coarse Grid Matrix with Galerkin Approximation  
6.3 Some Examples of Coarse Grid Operators  
6.4 Singular Equations  
6.5 Two-Grid Analysis; Smoothing and Approximation Properties  
6.6 A Numerical Illustration  
     
7 Smoothing Analysis  
7.1 Introduction  
7.2 The Smoothing Property  
7.3 Elements of Fourier Analysis in Grid-Function Space  
7.4 The Fourier Smoothing Factor  
7.5 Fourier Smoothing Analysis  
7.6 Jacobi Smoothing  
7.7 Gauss-Seidel Smoothing  
7.8 Incomplete Point LU Smoothing  
7.9 Incomplete Block Factorization Smoothing  
7.10 Fourier Analysis of White-Black and Zebra Gauss-Seidel Smoothing  
7.11 Multistage Smoothing Methods  
7.12 Concluding Remarks  
     
8 Multigrid Algorithms  
8.1 Introduction  
8.2 The Basic Two-Grid Algorithm  
8.3 The Basic Multigrid Algorithm  
8.4 Nested Iteration  
8.5 Rate of Convergence of the Multigrid Algorithm  
8.6 Convergence of Nested Iteration  
8.7 Non-Recursive Formulation of the Basic Multigrid Algorithm  
8.8 Remarks on software  
8.9 Comparison with conjugate gradient methods  
     
9 Applications of Multigrid Methods in Computational Fluid Dynamics  
9.1 Introduction  
9.2 The Governing Equations  
9.3 Grid Generation  
9.4 The Full Potential Equation  
9.5 The Euler Equations of Gasdynamics  
9.6 The Compressible Navier-Stokes Equations  
9.7 The Incompressible Navier-Stokes and Boussinesq Equations  
9.8 Final Remarks  

Related Book Categories

Multigrid Methods


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