Approximating the conformal map of elongated quadrilaterals by domain decomposition

Abstract

Let $Q:=\{ \Omega;z_1,z_2,z_3,z_4\}$ be a quadrilateral consisting of a Jordan domain $\Omega$ and four points $z_1$, $z_2$, $z_3$, $z_4$ in counterclockwise order on $\partial \Omega$ and let $m(Q)$ be the conformal module of $Q$. Then, $Q$ is conformally equivalent to the rectangular quadrilateral $\{R_{m(Q)};0,1,1+im(Q),im(Q)\}, $, where $ R_{m(Q)}:=\{(\xi,\eta):0<\xi<1, \ 0 <\eta<m(Q)\},$ in the sense that there exists a unique conformal map $f: \Omega \rightarrow R_{m(Q)}$ that takes the four points $z_1$, $z_2$, $z_3$, $z_4$, respectively onto the four vertices $0$, $1$, $1+im(Q)$, $im(Q)$ of $R_{m(Q)}$. In this paper we consider the use of a domain decomposition method (DDM) for computing approximations to the conformal map $f$, in cases where the quadrilateral $Q$ is "long". The method has been studied already but, mainly, in connection with the computation of $m(Q)$. Here we consider certain recent results of Laugesen, for the DDM approximation of the conformal map $f: \Omega \rightarrow R_{m(Q)}$ associated with a special class of quadrilaterals (viz. quadrilaterals whose two non-adjacent boundary segments $(z_2,z_3)$ and $(z_4,z_1)$ are parallel straight lines) and seek to extend these results to more general quadrilaterals. By making use of the available DDM theory for conformal modules, we show that the corresponding theory for $f$ can, indeed, be extended to a much wider class of quadrilaterals than those considered by Laugesen.