Boundary Integral Equations and the Dirichlet Problem

The standard boundary element method applied to a problem with Dirichlet boundary conditions leads to a Fredholm integral equation of the first kind. A stable second kind equation results from a hypersingular equation formulation of the problem, obtained by differentiating the original boundary integral statement. The evaluation of the hypersingular integral requires that the coefficient function multiplying the hypersingular kernel be differentiable. For two dimensional problems, Overhauser elements are a very convenient basis set for obtaining the necessary smoothness. Test calculations with the Laplace equation demonstrate (a) the stability of the hypersingular approach for the Dirichlet problem and (b) that the hypersingular approach yields results that are more accurate than those obtained with the standard approach