Direct Integration of the Newton Potential over Cubes

Abstract

In boundary element methods, the evaluation of the weakly singular integrals can be performed either a) numerically, b) symbolically, i.e., by explicit expressions, or c) in a combined manner. The explicit integration is of particular interest, when the integrals contain the singularity or if the singularity is rather close to the integration domain. In this paper we describe the explicit expressions for the sixfold volume integrals arising for the Newton potential, i.e., for a 1/r integrand. The volume elements are axi-parallel bricks. The sixfold integrals are typical for the Galerkin method. However, the threefold integral arising from collocation methods can be derived in the same way.