Fast Algorithms and Their Applications to Numerical Quasiconformal Mappings of Doubly Connected Domains Onto Annuli

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. 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 Numer. Math. 13; 448--457 (1969) Volume 83; Number 6; (1977) JOURNAL OF COMPUTATIONAL PHYSICS 96; 229-236 (1991) SIAM J. ScI. STAT. COMPUT.Vol. 13; No. 6; pp. 1418-1432; 1992 JOURNAL OF COMPUTATIONAL PHYSICS 106; 355-365 (1993) Numerical Algorithms 18 (1998) 133–157 Numerical Algorithms 18 (1998) 159–175 Numerical Algorithms 23 (2000) 71–96 Journal of Computational Physics 169; 151–192 (2001) Numerical Algorithms 30: 199–239; 2002. PAMM · Proc. Appl. Math. Mech. 7; 2020065–2020066 (2007)