A numerical method for quasiconformal mapping of doubly connected domains onto
annuli is presented. The annulus itself is not known a priori and is determined as
part of the solution procedure. The numerical method requires solving a sequence
of inhomogeneous Beltrami equations, each within a different annulus, in an iterative
mode. The annulus within which the equation is being solved is also updated
during the iterations using an updating procedure based on the bisection method.
This quasiconformal mapping method is based on Daripa's method of quasiconformal
mapping of simply connected domains onto unit disks. The performance of
the quasiconformal mapping algorithm has been demonstrated on several doubly
connected domains with two different complex dilations.
The solution of the Beltrami equation in an annulus requires evaluating two
singular integral operators. Fast algorithms for their accurate evaluation are presented.
These are based on extension of a fast algorithm of Daripa. These algorithms
are based on some recursive relations in Fourier space and the FFT (fast
Fourier transform), and have theoretical computational complexity of order log N
per point.
Author(s): Daoud Sulaiman Mashat
Series: Doctoral Dissertation
Publisher: Texas A&M University
Year: 1997
Language: English
Commentary: 214 Extra Pages Added (Papers & Articles) related to the Work.
Pages: C+xi, 119+214
Cover
FAST ALGORITHMS AND THEIR APPLICATIONS TONUMERICAL QUASICONFORMAL MAPPINGS OF DOUBLY CONNECTED DOMAINS ONTO ANNULI
Approved as to style and content
ABSTRACT
DEDICATION
ACKNOWLEDGMENTS
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF TABLES
CHAPTER I INTRODUCTION
1.1. Motivation
1.2. Problem Definition and Background
1.3. Objectives
CHAPTER II MATHEMATICAL PRELIMINARIES
CHAPTER III RAPID EVALUATION OF SINGULAR OPERATORS T1 AND T2
CHAPTER IV FAST ALGORITHMS FOR COMPUTING T1 AND T2 OPERATORS
4.1 The Algorithm
4.2. The Algorithmic Complexity
CHAPTER V BOUNDARY VALUE PROBLEMS
5.1. Dirichlet Problem for Homogeneous Cauchy-Riemann Equation in an Annulus
5.2. Dirichlet Problem for Nonhomogeneous Beltrami Equation in an Annulus
CHAPTER VI A NUMERICAL METHOD FOR QUASICONFORMAL MAPPINGS
CHAPTER VII NUMERICAL RESULTS
CHAPTER VIII SUMMARY AND CONCLUSION
REFERENCES
APPENDIX
VITA
Papers
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Numerical Algorithms 18 (1998) 159–175
Numerical Algorithms 23 (2000) 71–96
Journal of Computational Physics 169; 151–192 (2001)
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PAMM · Proc. Appl. Math. Mech. 7; 2020065–2020066 (2007)
