As explained previously, the basic idea of the boundary element method comes from Trefftz
who suggested that in contrast to the method of Ritz, only functions satisfying the differential equations exactly should be used to approximate the solution inside the domain. If we use these functions it means, of course, that we only need to approximate the actual boundary conditions. This approach, therefore, has some considerable advantages:
The solutions obtained inside the domain satisfy the differential equations exactly, approximations (or errors) only occur due to the fact that boundary conditions are only satisfied approximately.
Since functions are defined globally, there is no need to subdivide the domain into elements.
The solutions also satisfy conditions at infinity, therefore, there is no problem dealing with infinite domains, where the FEM has to use mesh truncation or approximate infinite elements.