1st Edition
Practical Fourier Analysis for Multigrid Methods
234 Pages
48 B/W Illustrations
by
Chapman & Hall
240 Pages
by
Chapman & Hall
Also available as eBook on:
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... Read more
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
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
