Numerical conformal mapping of multiply connected regions by Fornberg-like methods
type: Item
instance of: scholarly article
title: Numerical conformal mapping of multiply connected regions by Fornberg-like methods
zbMATH Open document ID: 0931.30004
author: Mark A. Horn
author: John A. Pfaltzgraff
author: T. K. DeLillo
published in: Numerische Mathematik
publication date: 2000-02-20T00:00:00Z
review text: Let \(\Omega\) be a domain outside of \(n\) disjoint Jordan curves \(\Gamma_k\),with \(\infty\in \Omega\). It is well-known (Koebe 1908) that there exists a circular domain \(D\), outside of \(n\) disjoint circles \(C_k\), with \(\infty\in D\), which can be mapped conformally onto \(\Omega\). The authors present a method to compute this mapping \(f\), in particular the positions of the circles \(C_k\) and the boundary correspondence from \(D\) to \(\Omega\) defined by \(f(z_k+ \rho_k e^{i\theta})= \gamma_k (S_k(\theta))\) where \(\gamma_k(S)\) is the representation of \(\Gamma_k\) in terms of arc length. This is done by a Newton like method, that is by linearisation, which is observed (but not proved) to be quadratically convergent. A substantial part of the paper deals with criteria, when a function defined on \(C= \cup C_k\) can be extended analytically into \(D\setminus \{\infty\}\). To approximate \(f\) numerically, the Fourier expansion of \(f(z_k+ \rho_k e^{i\theta})\) has to be truncated. Finally, a large linear system has to be solved. -- The amount of numerical labor seems enormous. Six experiments show good results, but in five of them the \(C_k\) are all analytic Jordan curves. The reviewer believes that no other method can beat Koebe's original method, based on an iteration of mappings of simply connected domains, in implicity and efficiency.
Mathematics Subject Classification ID: 30C30
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 1356170
zbMATH Keywords: numerical conformal mapping
reviewed by: Dieter Gaier
MaRDI profile type: Publication
full work available at URL: s002110050447
OpenAlex ID: W2075779030
DOI: 10.1007/S002110050447
description: scientific article; zbMATH DE number 1356170