A Choice of Fundamental Solutions

A key ingredient in the formulation of boundary integral equations via the direct method is the so called “fundamental solution”. This is generally obtained as the particular singular solution of an elliptic boundary value problem which corresponds to a “concentrated load” (i.e. the right hand side is a delta function; see, for example, Chapter 4 of Brebbia and Walker, 1980). Now it turns out that in general, the singular part of the fundamental solution is not uniquely determined by this specification, but the indeterminacy is usually eliminated by requiring that the solution be single valued in a neighborhood of the singular point. Indeed, single valuedness is an essential requirement when generating the boundary integral representation for interior points. However, for the boundary integral equations needed to determine the unspecified boundary data the singular point is itself on the boundary of the region and it is therefore not necessary to use the single valued fundamental solution in the representation.