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Abstract
In this article, a recently derived harmonic sine approximation method is used to obtain approximate solutions to two-dimensional steady-state heat conduction problems with singularities and semi-infinite domains and Dirichlet boundary conditions. The first problem is conduction in a square geometry, and the second one involves a semi-infinite medium 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 computational domain, especially at points close to the singularity at the upper left and right corners of the square. The results from the harmonic sine approximation method for the semi-infinite domain problem with a very shallow rectangular cavity agree well with the analytical solution for a semi-infinite domain without the cavity. The results obtained from the harmonic sine approximation also agree 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.