Inmost singularities of S.I.Es influencing their numerical solution in the BEM

When solving numerically the singular integral equations (S.I.Es) using the boundary element method (BEM) or other similar methods, special attention is given to the Cauchy singularity of the singular integrals and to the end point singularities of the unknown function. But in many cases there exist other, inmost, singularities either in the unknown function or in the regular kernel. In fact the unknown function can have essential singularities (poles of order one or two), weak singularities and nearby singularities at isolated points. Usually these singularities are provoked from singularities of the right hand side (r.h.s.) function, whereas the regular kernel can have essential and weak singularities at isolated points and nearby singularities. Neglecting these singularities in the numerical process, we obtain solutions largely diverging from the exact ones. So far these problems are not confronted in the BEM numerical process. In this paper, we classify all these singularities and we give two numerical examples illustrating the important influence of nearby singularities of the regular kernel. Some suggestions concerning appropriate numerical methods for these problems are also given.