Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.