Numerical methods for Riemann-Hilbert problems in multiply connected circle domains
Abstract
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.
Description
Thesis (Ph.D.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics and Physics