Error analysis of the Bergman kernel method with singular basis functions

TL;DR

This paper provides theoretical estimates and numerical evidence for the Bergman kernel method with singular basis functions, enhancing the understanding of its accuracy in conformal mapping of complex domains.

Contribution

It offers the first complete theoretical justification for using algebraic and pole singular basis functions in the Bergman kernel method for conformal mapping.

Findings

Theoretical error estimates are established.

Numerical experiments support the theoretical results.

The method effectively approximates conformal maps for piecewise analytic boundaries.

Abstract

Let G be a bounded Jordan domain in the complex plane with piecewise analytic boundary. We present theoretical estimates and numerical evidence for certain phenomena, regarding the application of the Bergman kernel method with algebraic and pole singular basis functions, for approximating the conformal mapping of G onto the normalized disk. In this way, we complete the task of providing full theoretical justification of this method.