Rapid methods for the conformal mapping of multiply connected regions
type: Item
instance of: scholarly article
title: Rapid methods for the conformal mapping of multiply connected regions
zbMATH Open document ID: 0586.30009
DOI: 10.1016/0377-0427(86)90135-4
author: Anita Mayo
published in: Journal of Computational and Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The method which is described applies to simply and multiply connected regions as well and is adopted from a book by \textit{S. G. Mikhlin}, Variation methods in the mathematical physic (1957; Zbl 0082.380). Smooth boundaries are assumed. The general problem under investigation is to map a given region D onto the unit disk in case D is simply connected, onto an annulus with exterior radius one in case D is doubly connected, and onto a disk with concentric circular arcs in case D is at least triply connected. In all cases the problem is reduced to solving Laplace's equation with Dirichlet boundary values and to determine the conjugate function. For solving the Dirichlet problem a Fredholm integral equation of the second kind is used. It is solved by Nyström's method. The solution yields a density function defined on the boundary points of D which is extended into the interior by the application of the Cauchy integral. The author embeds for this purpose the given region into a rectangle in which a uniform mesh is introduced. The purpose is to allow the application of a fast Poisson solver. Some ideas are developed in order to overcome the difficulty that inside and outside of the given region the function defined by the Cauchy integral does not fit together continuously. Several numerical examples including two triply connected regions are presented. There are comments with respect to the accuracy. However, there are no explicit results with respect to rapidity.
Mathematics Subject Classification ID: 30C30
Mathematics Subject Classification ID: 65E05
zbMATH DE Number: 3939648
zbMATH Keywords: fast Poisson solver
zbMATH Keywords: triply connected regions
reviewed by: Gerhard Opfer
MaRDI profile type: Publication
full work available at URL: 0377-0427(86)90135-4
OpenAlex ID: W2070151514
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cites work: The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions
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cites work: Q5822969
cites work: An integral equation method in conformal mapping
cites work: An iterative method for conformal mapping
cites work: An Exterior Poisson Solver Using Fast Direct Methods and Boundary Integral Equations with Applications to Nonlinear Potential Flow
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description: scientific article; zbMATH DE number 3939648