1st Edition

Practical Fourier Analysis for Multigrid Methods

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ISBN 9781584884927
Published October 28, 2004 by Chapman and Hall/CRC
240 Pages 48 B/W Illustrations

USD $160.00

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Book Description

Before applying multigrid methods to a project, mathematicians, scientists, and engineers need to answer questions related to the quality of convergence, whether a development will pay out, whether multigrid will work for a particular application, and what the numerical properties are. Practical Fourier Analysis for Multigrid Methods uses a detailed and systematic description of local Fourier k-grid (k=1,2,3) analysis for general systems of partial differential equations to provide a framework that answers these questions.

This volume contains software that confirms written statements about convergence and efficiency of algorithms and is easily adapted to new applications. Providing theoretical background and the linkage between theory and practice, the text and software quickly combine learning by reading and learning by doing. The book enables understanding of basic principles of multigrid and local Fourier analysis, and also describes the theory important to those who need to delve deeper into the details of the subject.

The first chapter delivers an explanation of concepts, including Fourier components and multigrid principles. Chapter 2 highlights the basic elements of local Fourier analysis and the limits to this approach. Chapter 3 examines multigrid methods and components, supported by a user-friendly GUI. Chapter 4 provides case studies for two- and three-dimensional problems. Chapters 5 and 6 detail the mathematics embedded within the software system. Chapter 7 presents recent developments and further applications of local Fourier analysis for multigrid methods.

Table of Contents

Some Notation
Basic Iterative Schemes
A First Discussion of Fourier Components
From Residual Correction to Coarse-Grid Correction
Multigrid Principle and Components
A First Look at the Graphical User Interface

Main Features of Local Fourier Analysis for Multigrid
The Power of Local Fourier Analysis
Basic Ideas
Applicability of the Analysis

Multigrid and Its Components in LFA
Multigrid Cycling
Full Multigrid
xlfa Functionality-An Overview
Implemented Coarse-Grid Correction Components
Implemented Relaxations

Using the Fourier Analysis Software
Case Studies for 2D Scalar Problems
Case Studies for 3D Scalar Problems
Case Studies for 2D SYSTEMS of Equations
Creating New Applications

Fourier One-Grid or Smoothing Analysis
Elements of Local Fourier Analysis
High and Low Fourier Frequencies
Simple Relaxation Methods
Pattern Relaxations
Smoothing Analysis for Systems
Multistage (MS) Relaxations
Further Relaxation Methods
The Measure of h-Ellipticity

Fourier Two- and Three-Grid Analysis
Basic Assumptions
Two-Grid Analysis for 2D Scalar Problems
Two-Grid Analysis for 3D Scalar Problems
Two-Grid Analysis for Systems
Three-Grid Analysis

Further Applications of Local Fourier Analysis
Orders of Transfer Operators
Simplified Fourier k-Grid Analysis
Cell-Centered Multigrid
Fourier Analysis for Multigrid Preconditioned by GMRES

Fourier Representation of Relaxation
Two-Dimensional Case
Three-Dimensional Case

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"… a rather unique distinguished feature is the accompanying LFA software, which is not often found in other multigrid books. … This book presents a thorough and systematic description of [the subject]. Two main features of this book are an extensive selection of problems of different kinds and an accompanying user-friendly software that can perform rather complex local Fourier analysis by just a few mouse clicks. … It was a joy reading this book, and I am happy to have it as a valuable addition to my multigrid book collection."
-Mathematics of Computation, April 2007

"This book enables understanding of basic principles of multigrid and local Fourier analysis, allowing investigation of real multigrid effects."
-Wilhelm Heinrichs, Zentralblatt MATH