Rapid methods for the conformal mapping of multiply connected regions
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.
scientific article; zbMATH DE number 3939648
scientific article