Fast numerical conformal mapping of bounded multiply connected regions via integral equations

Abstract

This study presents a fast numerical conformal mapping of bounded multiply connected region onto a disk with circular slits, an annulus with circular slits, circular slits, parallel slits and radial slits regions and their inverses using integral equations with Neumann type kernel and adjoint generalized Neumann kernel. A graphical user interface is created to illustrate the effectiveness of the approach for computing the conformal maps of bounded multiply connected regions and image transformations via conformal mappings. Some image transformation results are shown via graphical user interface. This study also presents a fast numerical conformal mapping of bounded multiply connected region onto second, third and fourth categories of Koebe’s canonical slits regions using integral equations with adjoint generalized Neumann kernel. The integral equations are discretized using Nystr¨om method with trapezoidal rule. For regions with corners, the integral equations are discretized using Kress’s graded mesh quadrature. All the linear systems that arised are solved using generalized minimal residual method (GMRES) or least square iterative method powered by fast multipole method (FMM). The interior values of the mapping functions and their inverses are determined by using Cauchy integral formula. Some numerical examples are presented to illustrate the effectiveness for computing the conformal maps of bounded multiply connected regions. This study also discussed a fast numerical conformal mapping of bounded multiply connected regions onto fifth category of Koebe’s canonical regions using integral equations with the generalized Neumann kernel. An application of fast numerical conformal mapping to some coastal domains with many obstacles is also shown.