Semi-analytic treatment of nearly-singular Galerkin surface integrals

Abstract

On utilizing piecewise linear test and interpolation functions over flat triangles, three semi-analytic techniques for evaluating nearly-singular surface integrals of potential theory are developed and investigated in the context of a three-dimensional Galerkin approximation. In all procedures, the inner surface integral is calculated exactly via recursive expressions defined over the edges of the integration triangle. In addition, the proposed methods respectively employ a Duffy transformation, sinh transformations, and polynomial transformations to weaken or smooth out near singularities, followed by a standard product Gauss–Legendre quadrature rule to compute the outer boundary integral. Numerical experiments and comparisons of the proposed approaches are included to provide more insight into the performance of these techniques. Computational tests have demonstrated that the algorithm based on a Duffy transformation is the most efficient and effective method for dealing with nearly-singular as well as well-separated integrals.