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{lunate} Ω and let K(z, ζ) denote the Bergman kernel function of Ω with respect to ζ. Also, let Kn(z, ζ) denote the nth-degree polynomial approximation to K(z, ζ), given by the classical Bergman kernel method, and let πn 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 {norm of matrix}K(·, ζ)-Kn(·, ζ)|L2(B), |f′ζ - π′n|L2(B), and compare their rates of convergence with those of the corresponding global errors with respect to L2(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. © 1993.