1st Edition

Practical Fourier Analysis for Multigrid Methods

By Roman Wienands, Wolfgang Joppich Copyright 2004
    234 Pages 48 B/W Illustrations
    by Chapman & Hall

    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.

    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

    Biography

    Roman Wienands, Wolfgang Joppich

    "… 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