Local behaviour of the error in the Bergman kernel method for numerical conformal mapping (Q1802158)

Let \(\Omega\) be the interior of a smooth Jordan curve \(\Gamma\), let \(K(z,\zeta)\) be the Bergman kernel function of \(\Omega\) with respect to \(\zeta\in\Omega\), let \(f_ \zeta\) be the conformal map of \(\Omega\) onto \(\{w:| w|<r\}\) normalized by \(f_ \zeta(\zeta)=0\), \(f_ \zeta'(\zeta)=1\), and let \(K_ n\) be the polynomial approximation of \(K\) and \(\pi_ n\) the Bieberbach polynomial of degree \(n\), with \(\pi_ n(\zeta)=0\), \(\pi_ n'(\zeta)=1\). The authors are interested in the comparison of the local error \(\| K(\cdot,\zeta)-K_ n(\cdot,\zeta)\|_{L^ 2(B)}\) with the global error \(\|\cdot\|_{L^ 2(\Omega)}\), and in the comparison of \(\| f_ \zeta'-\pi_ n'\|_{L^ 2(B)}\) with \(\|\cdot\|_{L^ 2(\Omega)}\), where \(B\) is a subdomain of \(\Omega\). Their main result is that if \(\Gamma\in C(1,\alpha)\) for \(0<\alpha\leq 1\), and if \(\partial B\) contains a subarc of \(\Gamma\), then \[ \sum^ \infty_{n=1}[\| K(\cdot,\zeta)-K_ n(\cdot,\zeta)\|_{L^ 2(B)}/\| K(\cdot,\zeta)-K_ n(\cdot,\zeta)\|_{L^ 2(\Omega)}]=\infty, \] from which it follows that the rates of convergence of the local error are not essentially different from the global errors. Numerical experiments confirm the results.