# From Singular and Nonsingular Boundary Element Methods to Method of Fundamental Solutions

The boundary integral equation method and the method of fundamental solutions (MFS) for solving partial differential equations have different theoretical basis and numerical implementation. However, by alternative ways of implementing these methods, these two methods seem to converge. For example, the indirect boundary integral equation method distributes singularities of fictitious density over the boundary. The discrete distribution density is solved by collocation at the boundary. The first approximation is to distribute the singularity with constant density over linear or flat surface elements covering the boundary. To avoid the treatment of singularity on the boundary, can the elements be moved away from the boundary to enclose the domain? This may no longer be supported by the integral equation theory, but isn’t this similar to the method of fundamental solutions? Method of fundamental solutions is based on distributing singularities on a fictitious contour or surface at a distance away from the boundary. Can we integrate the singularities over a segment or area, similar to the indirect boundary integral equation method? Once integrated, these elements are no longer singular. Can we move them toward the boundary and cover it? These elements can have some regular shapes for easy integration. They do not need to approximate the boundary, or can overlap. This way no fictitious boundary needs to be created. The indirect boundary integral equations distribute source and dipoles, or displacement and tractions over the boundary. The MFS can distribute a wider variety of singularities, such as vortex for potential problems, and displacement potential for elasticity problems, which is of simpler form than displacement and stress. So if we open our mind, the boundary integral equation and MFS can be much more flexible and easier to implement.