An iterative method for the conformal mapping of doubly connected regions

摘要

An iterative method is described for the determination of the conformal mapping of a circular annulus Rq ≔ {z:q < |z| < 1} onto a doubly connected region G with smooth boundary curves. The method is analogous to the method of [7] for simply connected regions. It requires in each step the solution of a Riemann-Hilbert problem on Rq. This problem can be solved explicitly in terms of a generalized conjugation operator K. A degeneracy of this problem leads in a natural way to an adjustment of the parameter q in each step. If the boundary curves of G admit parametrizations by functions ην with second derivatives which are Hölder continuous with exponent μ, 0 < μ ⩽ 1, then the method converges locally in the Sobolev space W of 2π-periodic absolutely continuous functions with square-integrable derivatives. The order of convergence is at least 1 + μ. The convergence is quadratic if the ην have Lipschitz-continuous second derivatives. For the numerical implementation of the method the operator K can be approximated by a Wittich type method, which can be performed very effectively using FFT. A calculation with N = 2m grid points on the computer requires storage of the order O(N) and computing time O(mN). The results of some test calculations are reported and compared with results of calculations with the method of Theodorsen-Garrick.