Direct Evaluation of Hypersingular

A direct algorithm for evaluating hypersingular integrals arising in a three-dimensional Galerkin boundary integral analysis is presented. By integrating two of the four dimensions analytically, the coincident integration, deened as a limit to the boundary, is shown to be divergent. However, the divergent terms can be explicitly calculated and shown to cancel with corresponding singularities in the adjacent edge integrals. A single analytic integration is employed for the edge and vertex singular integrals. This is suucient to display the divergent term in the edge-adjacent integral and to demonstrate that the vertex integral is nite. By explicitly identifying the divergent quantities, the hypersingular integral can be computed without recourse to Stokes' Theorem or Hadamard Finite Part. The computation of the hypersingular integrals for the Laplace equation, using linear triangular elements, is described herein. Simple test calculations are used to demonstrate that the algorithms work.