A Fast Boundary Integral Equation Method for Conformal Mapping of Multiply Connected Regions

Abstract

This paper presents a fast boundary integral equation method for approximating the conformal mapping from bounded and unbounded multiply connected regions of connectivity $m+1$ onto the canonical region obtained by removing $m$ rectilinear slits from a strip. The method is based on a combination of a uniquely solvable boundary integral equation with generalized Neumann kernel and the fast multipole method. The presented method requires $O((m+1)n\ln n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component. The numerical results of some test calculations illustrate that our method has the ability to handle regions with complex geometry and very high connectivity. Besides the above canonical region, the presented fast method also can be used to compute the numerical conformal mapping from multiply connected regions onto Koebe's 39 canonical slit regions.