A HARMONIC-SINC SOLUTION OF THE LAPLACE EQUATION FOR PROBLEMS WITH SINGULARITIES AND SEMIINFINITE DOMAINS

In this article, a recently derived harmonic sine approximation method is used to obtain approximate solutions to two-dimensional steady-sta te heat conduction problems with singularities and semi-infinite domai ns and Dirichlet boundary conditions. The first problem is conduction in a square geometry, and the second one involves a semi-infinite medi um with a rectangular cavity. In the case of square geometry, results show that the harmonic sine approximation method performs better than the finite-difference and multigrid methods everywhere within the comp utational domain, especially at points close to the singularity at the upper left and right corners of the square. The results from the harm onic sine approximation method for the semi-infinite domain problem wi th a very shallow rectangular cavity agree well with the analytical so lution for a semi-infinite domain without the cavity. The results obta ined from the harmonic sine approximation also agree well with the res ults from the finite-element package ANSYS for the semi-infinite mediu m conduction problem with a rectangular cavity of aspect ratio 1.