This places considerable restrictions on the range and generality of problems to which boundary elements can usefully be applied.
The technique approximates part of the integrand using radial basis functions (local interpolating functions) and converts the volume integral into boundary integral after collocating at selected points distributed throughout the volume domain (including the boundary).
In the dual-reciprocity BEM, although there is no need to discretize the volume into meshes, unknowns at chosen points inside the solution domain are involved in the linear algebraic equations approximating the problem being considered.
The Green's function elements connecting pairs of source and field patches defined by the mesh form a matrix, which is solved numerically.
Galerkin's method is the obvious approach for problems which are symmetrical with respect to exchanging the source and field points.
Nonetheless, the principal source of the computational cost is this double-loop over elements producing a fully populated matrix.
For more general elements, it is possible to design purely numerical schemes that adapt to the singularity, but at great computational cost.
A good example of application of the boundary element method is efficient calculation of natural frequencies of liquid sloshing in tanks.