Contribution to boundary integrals by the normal derivative of the kernel

We show that the normal derivatives of kernels of boundary integral equations consist of a Dirac function and a remainder term. In the literature the contribution of this singularity is taken into account by an artificial deformation of the surface S in the near vicinity of the observation point, where the resulting integral equation is generally presented in incorrect form. In contrast to an analytical treatment only, the numerical use of the kernel requires such a separate investigation. This is due to the fact that the Dirac function cannot be handled numerically. In our paper we present the correct writing of the integral equation and furthermore the remainder terms of various Green´s functions for different geometries. Their knowledge is an essential precondition for the correct solution of the boundary integral equation