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Practical Fourier Analysis for Multigrid Methods

By Roman Wienands, Wolfgang Joppich

Series Editor: Achim Sydow

Chapman and Hall/CRC – 2004 – 240 pages

Series: Numerical Insights

Purchasing Options:

  • Add to CartHardback: $119.95
    978-1-58488-492-7
    October 27th 2004

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.

Reviews

"… 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

Contents

PRACTICAL APPLICATION OF LFA AND xlfa

Introduction

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

THE THEORY BEHIND LFA

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

APPENDIX

Fourier Representation of Relaxation

Two-Dimensional Case

Three-Dimensional Case

Name: Practical Fourier Analysis for Multigrid Methods (Hardback)Chapman and Hall/CRC 
Description: By Roman Wienands, Wolfgang JoppichSeries Editor: Achim Sydow. 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, ...
Categories: Computational Numerical Analysis, Mathematics & Statistics for Engineers, Fluid Mechanics