Numerical methods for Riemann-Hilbert problems in multiply connected circle domains

Riemann-Hilbert problems are problems for determining functions analytic in a given domain with speci ed values on the boundary. Since the real and imaginary parts of an analytic function are related by the Cauchy-Riemann equations, both parts cannot be speci ed independently. Riemann-Hilbert problems on multiply connected regions have been studied by several authors in the past. A special kind of Riemann-Hilbert problems on circular regions is necessary for conformal mapping of multiply-connected regions. Wegmann introduced a method of successive conjugation which reduces the general conjugation problem to a sequence of Riemann-Hilbert problems on the circles. Here, we present a new method to solve Riemann-Hilbert problems on the circles. We consider the general conjugation as a Least-Squares problem and use direct and iterative methods to obtain the solution. The resulting linear system has an underlying structure of the form of the identity plus a low rank operator and can be solved e ciently by conjugate gradient-like methods. We present numerical examples and comparisons to the method of Wegmann.

Grantor dc:publisher

Wichita State University

Year dc:date.issued

2020

Author dc:creator

• Balu, Raja

Advisor dc:contributor.advisor

• DeLillo, Thomas K.

Statement dc:rights

• Copyright 2020 by Raja D. Balu All Rights Reserved

Language dc:language.iso

en_US

Dc Identifier Other

d20003

OAI record oai:identifier

https://soar.wichita.edu/oai/request?verb=GetRecord&metadataPrefix=dim&identifier=oai%3Anull%3A10057%2F18800