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Convergence of a variant of the zipper algorithm for conformal mapping. (English) Zbl 1157.30006

In the early 1980s Kühnau and Marshall proposed a fast and accurate algorithm for computing conformal maps. The present paper is aimed to prove its convergence in the sense of uniformly close boundaries. The approximation is a composition of conformal maps onto slit half-planes. Depending on the type of slit the authors obtain different approximation versions, namely, the geodesic, slit and zipper algorithms. Given points \(z_0,\dots,z_n\) in the plane, the algorithms compute an explicit conformal map of the half-plane onto a region bounded by a Jordan curve \(\gamma\) with \(z_0,\dots,z_n\in\gamma\). The two following theorems refer to smooth domains or quasiconformal disks.
Theorem 3.10. Suppose \(\Omega\) is a Jordan region bounded by a \(C^1\) curve \(\partial\Omega\). Then there exists \(\delta_0>0\) depending on \(\partial\Omega\) so that for \(\delta<\delta_0\), \(\partial\Omega\subset\cup_k(D(z_k,z_{k+1})\cup\{z_k\})\), where \(D=\cup D(z_k,z_{k+1})\) is a \(\delta\)-diamond-chain, and so that the boundary \(\partial\Omega_c\) of the region computed by the geodesic algorithm is contained in \(D\cup(\cup_k\{z_k\})\). Moreover, if \(\zeta\in\partial\Omega_c\) and if \(\alpha\in\partial\Omega\) with \(|\zeta-\alpha|<\delta\), then \(|\eta_{\zeta}-\eta_{\alpha}|<6\delta\), where \(\eta_{\zeta}\) and \(\eta_{\alpha}\) are the unit tangent vectors to \(\partial\Omega\) and \(\partial\Omega_c\) at \(\zeta\) and \(\alpha\), respectively.
Theorem 3.11. There is a constant \(K_0>1\) so that if \(\Gamma\) is a \(K\)-quasicircle with \(K=1+\delta<K_0\) and if \(\{z_k\}\) are locally evenly spaced on \(\Gamma\), then the geodesic algorithm finds a conformal map of the upper half-plane onto a region \(\Omega_c\) bounded by a \(C(K)\)-quasicircle containing the data points \(\{z_k\}\), where \(C(K)\) is a constant depending only on \(K\).
Next, the authors show that if \(\partial\Omega\) is contained in a chain of disks of radius less than \(\epsilon\) with the data points being the contact points of the disks, or if \(\partial\Omega\) is a \(K\)-quasicircle with \(K\) close to 1 and the data points are consecutive points on \(\partial\Omega\) of distance comparable to \(\epsilon\), then the Hausdorff distance between \(\partial\Omega\) and the boundary \(\partial\Omega_c\) of the domain computed by the geodesic algorithm is at most \(\epsilon\). Moreover, the conformal maps \(\varphi\), \(\varphi_c\) onto the unit disk satisfy \(\sup_{\Omega\cap\Omega_c}|\varphi-\varphi_c|\leq C\epsilon^p\), where \(p<1/2\) for the disk-chain case, and \(p\) is close to 1 if \(K\) is close to 1. A brief discussion of numerical results ends the article.

MSC:

30C30 Schwarz-Christoffel-type mappings
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
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