A Fornberg-like Method for the Numerical Conformal Mapping of Bounded Multiply Connected Domains

A new Fornberg-like method is presented for computing conformal maps from the interior of the unit disk with m−1 circular holes to the interior of a smooth closed curve with m−1 holes bounded by smooth curves. The method is a Newton-like method for computing the boundary correspondences and the conformal moduli (centers and radii of the circles). The inner linear systems are derived from conditions for analytic extension of functions defined on the circles to the interior domain. These systems are N -point trigonometric discretizations of the identity plus a compact operator and are solved efficiently with the conjugate gradient method at a cost of O(N) per step. Two formulations of the map are given: the first uses a disk with circular holes, and the second uses an annulus with circular holes. Some numerical examples are given.