An iterative method for the conformal mapping of doubly connected regions
From author's abstract: An iterative method is described for the determination of the conformal mapping of a circular annulus \(R_ q:=\{z:q<| z| <1\}\) onto a doubly connected region G with smooth boundary curves. The method is analogous to the method for simply connected regions. It requires in each step the solution of a Riemann- Hilbert problem on \(R_ q\). This problem can be solved explicitly in terms of a generalized conjugation operator K. If the boundary curves of G admit parametrizations by functions \(\eta_{\nu}\) with second derivatives which are Hölder continuous with exponent \(\mu\), \(0<\mu \leq 1\), then the method converges locally in the Sobolev space W of \(2\pi\)- periodic absolutely continuous functions with square-integrable derivatives. The order of convergence is at least \(1+\mu\). The convergence is quadratic if the \(\eta_{\nu}\) have Lipschitz-continuous second derivatives. The results of some test calculations are reported and compared with results of calculations with the method of Theodorsen- Garrick.
scientific article; zbMATH DE number 3923052