The Harmonic-Sinc Solution of the Laplace Equation for Problems with Singularities and Semi-Infinite Domains

Many numerical methods have been developed for the solution of Poisson and Laplace equations such as finite-difference methods, finite-element methods, boundary-element methods, etc. Sinc numerical methods developed recently by Stenger and co-workers excel over other methods for problems involving singularities, infinite or semi-infinite domains as well as boundary layer behaviors. In this paper, the harmonic sinc approximation is applied to two dimensional steady state heat conduction problems with singularities and semi-infinite domains and Dirichlet boundary conditions. The first problem consists of conduction in a square geometry and the second one was a semi-infinite medium with a rectangular cavity. In case of the square geometry, results show that the harmonic sinc approximation method performed better than the finite-difference and multigrid methods everywhere within the computational domain, especially at points close to singularities at the upper left and right corners of the square. The results from the harmonic sinc approximation method for the semi-infinite domain problem with a very shallow rectangular cavity agreed well with the analytical solution for a semi-infinite domain without the cavity. The results obtained from the harmonic sinc approximation also agreed well with the results from the finite element package ANSYS for the semi-infinite medium conduction problem with a rectangular cavity of aspect ratio 1.0.