Theory for the Boundary Elements PDE
About the Boundary Element Method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations that have been formulated as integral equations. The boundary element method is complementary to the finite element method (FEM), which most of the other PDE interfaces are based on. The boundary element method is sometimes more efficient than other methods, including finite elements, in terms of computational resources for problems where there is a small surface-to-volume ratio. In other cases, the boundary element method is more convenient in setting up a problem. For general information about the boundary element method in addition to this section, see Ref. 1.
Physics interfaces based on the boundary element method (BEM) differs from those based on the finite element method (FEM) in that they only use mesh elements on the boundaries of the modeled regions (curves in 2D and surfaces in 3D). Physics interfaces based on BEM can be used for modeling in three types of volumetric regions: domains, finite voids, and an infinite void. Physics interfaces based on FEM only supports the domain type.
For modeling with BEM, a geometry model may have multiple domains and multiple finite voids. However, there can be only one infinite void. For more information on domains and voids, see Finite and Infinite Voids.
Unlike FEM, which produces sparse system matrices, BEM leads to filled (dense) matrices when using a direct solver. This means that even though BEM uses less degrees of freedom, as compared to the corresponding FEM discretization of the domain, the memory requirements for using a direct solver grow faster with BEM than with FEM. The problem with handling potentially large filled matrices resulting from BEM is avoided in the physics interfaces based on BEM by using iterative solvers in combination with far-field approximations. These methods avoid explicitly constructing these large matrices. Iterative solvers with far-field approximations is the default setting; however, the options of using a direct solver and no far-field approximations are also available.
The interfaces based on BEM solve the following version of Laplace’s equation:
in
where the diffusion coefficient c has the additional requirement, as compared to FEM, that it has to be a constant within each modeling region.
To keep derivations simpler, for the remainder of this section assume that c = 1 and, unless otherwise specified, that Ω is a finite region (domain or finite void):
in