Approximating the conformal maps of elongated quadrilaterals by domain decomposition

Abstract

Let Q := {Ω

z1, z2, z3, z4} be a quadrilateral consisting of a Jordan domain Ω and four points z1, z2, z3, z4, in counterclockwise order on ∂Ω and let m (Q) be the conformal module of Q. Then Q is conformally equivalent to the rectangular quadrilateral {Rm(Q)

0, 1, 1 + im(Q), im(Q)}, where Rm(Q) := {(ξ, η) : 0 < ξ < 1, 0 < η < m (Q)}, in the sense that there exists a unique conformal map f : Ω → Rm(Q) that takes the four points z1, z2, z3, z4, respectively, onto the four vertices 0, 1, + im(Q), im(Q) of Rm(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 [12], for the DDM approximation of the conformal map f : Ω → Rm(Q) associated with a special class of quadrilaterals (viz., quadrilaterals whose two opposite boundary segments (z2, z3) and (z4, z1) 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.