ABSTRACT
There are essentially two distinct categories of boundary integral equation methods: direct and indirect ones. The construction of these two types is based on different approaches. Since the original problem is the same in both cases, whichever representation we choose, it must ultimately produce the same solution. This chapter provides a comparison of these representations, determines the mathematical connections between the various unknown functions, and helps to decide which of the techniques seems to be best suited for computational work. The solutions of the boundary integral equations in any of the indirect methods can parametrize those in all the other methods. If we consider only the interior Dirichlet problem, then the two clear favourites are the refined indirect method and the direct method. However, the picture changes if we widen the discussion to include the exterior Dirichlet problem. It is clear that the refined indirect method offers the best overall computational prospects.
