Curvilinear crosscuts of subdivision for a domain decomposition method in numerical conformal mapping

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. We consider a domain decomposition method for computing approximations to the conformal module m(Q) of Q in cases where Q is ‘long’ or, equivalently, m(Q) is ‘large’. This method is based on decomposing the original quadrilateral Q into two or more component quadrilaterals Q1,Q2,… and then approximating m(Q) by the sum ∑j m(Qj) of the modules of the component quadrilaterals. The purpose of this paper is to consider ways for determining appropriate crosscuts of subdivision (so that the sum ∑j m(Qj) does indeed give a good approximation to m(Q)) and, in particular, to show that there are cases where the use of curved crosscuts is much more appropriate than the straight line crosscuts that have been used so far.