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.