Many linear boundary value problems can be reformulated as integral equations. These can in turn be rapidly solved using FMM based schemes. We refer to the resulting methods as Fast Integral Equation Methods (FIEM).

The most important advantage of FIEM over competing methods is that it offers the possibility of working with mathematical equations that are themselves very well-conditioned. This is in contrast to methods such as FEM that discretize mathematical operators that are unbounded. As a result, very high order accurate discretizations can be used in FIEM. Codes that produce ten or fifteen accurate digits are common. (When FEM is used, it is typically necessary to use so called "pre-conditioners" to reduce the condition number of the system of linear equations. In contrast, it can be said that when integral equation methods are used, the pre-conditioning is done analytically, which is to say exactly.)

FIEM is particularly useful for solving linear PDEs that have constant coefficients and no body loads. In such cases, the equations solved are defined only on the boundary of the domain. This advantage can lead to dramatic speed-ups over competing methods.

Recent developments of FIEM has opened the possibility of not only solving PDEs, but also: (1) computing an approximation to the inverse of an operator, (2) computing full or partial spectral decompositions of an operator, and (3) multiplying operators, see this page. Here, the term "operator" can refer to a partial differential operator or an integral operator (and more generally, a Calderon-Zygmund operator).

An introduction to FIEM can be found here. This primer gives multiple references to more thorough treatments for different classes of equations.