Numerical Computation of the Schwarz-Christoffel Transformation for Multiply Connected Domains

We report on recent progress in the computation of Schwarz-Christoffel maps from bounded or unbounded circular domains to conformally equivalent bounded or unbounded multiply connected polygonal domains. The form of the transformation is given in terms of an integral of an infinite product depending on unknown parameters, namely, the prevertices and the centers and radii of the circles. A system of nonlinear equations, which forces the geometry of the given polygonal domain to be correct under the mapping function, is formulated for the unknown parameters and solved by a continuation method. A transformation of the constrained parameters to an unconstrained set of variables is crucial to the effective solution of the system. Several numerical examples are given. The approach here proves to be very robust.

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