An Efficient Implementation of a Conformal Mapping Method Based on the Szegö Kernel

Abstract

A fast implementation of a method to compute the Riemann mapping function is presented. The method has been recently introduced by N. Kerzman and the author; it expresses the Szegö kernel as the solution of an integral equation of the second kind. The complexity of this new algorithm is $O(n^2 )$, where n is the number of collocation points on the boundary of the region. Previous algorithms for mapping from the problem domain to the disk require $O(n^2 \log n)$ operations. It is shown how to treat symmetric regions. The algorithm is tested on several examples. The numerical results show that the method is competitive with respect to accuracy, stability, and efficiency.