There are several popular computational methods for solving problems in potential theory and linear elasticity. The most popular, versatile and most commonly used is the finite element method (FEM). Many hundreds of books already exist on the subject and new books get published frequently on a regular basis. Another popular method is the boundary element method (BEM). Compared to the FEM, we view the BEM as a niche method, in that it is particularly well suited, from the point of view of accuracy as well as computational efficiency, for linear problems.
The principal advantage of the BEM, relative to the FEM, is its dimensionality advantage. The FEM is a domain method that requires discretization of the entire domain of a body while the BEM, for linear problems, only requires discretization of its bounding surface.