Local behaviour of the error in the Bergman kernel method for numerical conformal mapping

Abstract

Let Ω be a simply-connected domain in the complex plane, let ζ ϵ Ω and let K( z, ζ) denote the Bergman kernel function of Ω with respect to ζ. Also, let K n ( z, ζ) denote the nth-degree polynomial approximation to K( z, ζ), given by the classical Bergman kernel method, and let π n denote the corresponding nth-degree Bieberbach polynomial approximation to the conformal map f of Ω onto a disc. Finally, let B be any subdomain of Ω. In this paper we investigate the two local errors ‖ K(·, ζ)− K n (·, ζ)| L 2( B) , | f′ ζ − π′ n | L 2( B) , and compare their rates of convergence with those of the corresponding global errors with respect to L 2(Ω). Our results show that if ∂ B contains a subarc of ∂Ω, then the rates of convergence of the local errors are not substantially different from those of the global errors.