A Multipole Method for Schwarz-Christoffel Mapping of Polygons with Thousands of Sides

A method is presented for the computation of Schwarz--Christoffel maps to polygons with tens of thousands of vertices. Previously published algorithms have CPU time estimates of the order O(N3) for the computation of a conformal map of a polygon with N vertices. This has been reduced to O(N log N) by the use of the fast multipole method and Davis's method for solving the parameter problem. The method is illustrated by a number of examples, the largest of which has $N \approx 2 \times 10^5$.

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