A Fast Algorithm for the Numerical Evaluation of Conformal Mappings

An algorithm is presented for the construction of conformal mappings from arbitrary simply connected regions in the complex plane onto the unit disk. The algorithm is based on a combination of the Kerzman–Stein integral equation (see [Math. Anal, 236 (1978), pp. 85–93]) and the Fast Multipole Method for the evaluation of Cauchy-type integrals (see [V. Rokhlin, J. Comput. Phys., 60 (1985), pp. 187–207], [L. Greengard and V. Rokhlin, J. Comput. Phys., 73 (1987), pp. 325–348], [J. Carrier, L. Greengard, and V. Rokhlin, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 669–686], [L. F. Greengard, Ph.D. thesis, Department of Computer Science, Yale University, New Haven, CT, 1987]). Previously published methods for the construction of conformal mappings via the Kerzman–Stein equation have an asymptotic CPU time estimate of the order $O(n^2 )$, where n is the number of nodes in the discretization of the boundary of the region being mapped. The method presented here has an estimate of the order $O(n)$, making it an approach of choice in many situations. The performance of the algorithm is illustrated by several numerical examples.