| Abstract: | This paper reports on some problems that can arise with the use of regularized derivative boundary integral equations. It concentrates on developing a formulation for the simple Laplace equation using a cubic Hermite interpolation and shows how certain combinations of derivative and conventional boundary integral equations can result in a solution scheme severely lacking in stability. With some simple two- and three-dimensional geometries, the derivative equations on their own do not provide enough information to solve a Dirichlet problem. Even combinations of the conventional and derivative equations fail for some simple geometries. We conclude that the only consistently successful combination is that of the conventional equation with the tangential derivative equation, which showed cubic convergence of results with mesh refinement. Numerical results are presented for this scheme in both two and three dimensions. |
