The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations that have been formulated as integral equations. The boundary element method is complementary to the finite element method (FEM), which most of the other PDE interfaces are based on. The boundary element method is sometimes more efficient than other methods, including finite elements, in terms of computational resources for problems where there is a small surface-to-volume ratio. In other cases, the boundary element method is more convenient in setting up a problem. For general information about the boundary element method in addition to this section, see
Ref. 1.
where the diffusion coefficient c has the additional requirement, as compared to FEM, that it has to be a constant within each modeling region.
To keep derivations simpler, for the remainder of this section assume that c = 1 and, unless otherwise specified, that
Ω is a finite region (domain or finite void):