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Numerical conformal mapping methods for exterior and doubly
connected regions

by
Thomas K. DeLillo
Department of Mathematics and Statistics
Wichita State University
Wichita, KS 67260-0033
and
John A. Pfaltzgraff
Department of Mathematics
University of North Carolina
Chapel Hill, NC 27599-3250
Abstract. Methods are presented and analysed for approximating
the conformal map from the exterior of the unit disk to the
exterior of a smooth simple closed curve and from an annulus
to a bounded, doubly-connected region with smooth boundaries.
The methods are Newton-like methods for computing the boundary
correspondences and conformal moduli and are extensions of
Fornberg's original method for the interior of the disk (1980).
Conditions on the Laurent coefficients for a function on the
boundary of the computational region (disk, annulus) to extend
analytically to the region are given. These analyticity conditions
are used to derive a linear Ax = b at each Newton step. In each
case, A is a discretization using N-point trigonometric
interpolation of a linear operator of the form I + R, where R is
a compact operator. It can therefore be shown that the eigenvalues
are well-grouped and the conjugate gradient method converges
superlinearly as in the case of Fornberg's original method.
The matrix-vector multiplications can be performed in O(N log N)
using the fast Fourier transform.
Computational examples and remarks on other methods will be
given. Extensions of these methods to other simply and multiply
connected computational regions, such as ellipses and regions
exterior to non-overlapping circles, may also be indicated.