Conformal Mapping for the Efficient Solution of Poisson Problems with the Kansa-RBF Method

Abstract

We consider the solution of Poisson Dirichlet problems in simply-connected irregular domains. These domains are conformally mapped onto the unit disk and the resulting Poisson Dirichlet problems are solved efficiently using a Kansa-radial basis function (RBF) method with a matrix decomposition algorithm (MDA). In a similar way, we treat Poisson Dirichlet and Poisson Dirichlet---Neumann problems in doubly-connected domains. These domains are mapped onto annular domains by a conformal mapping and the resulting Poisson Dirichlet and Poisson Dirichlet---Neumann problems are solved efficiently using a Kansa-RBF MDA. Several examples demonstrating the applicability of the proposed technique are presented.