# Boundary element method (BEM) and BETI

The boundary element method (BEM) is an efficient tool for modelling boundary value problems for linear partial differential equations. It relies on an analytical expression of the fundamental solution. BEM can be used, e.g. for reaction-diffusion equations, mechanics, acoustics and electromagnetism in the stationary, time-harmonic, as well as general time-dependent problems. When compared to the frequently used finite element method (FEM), in BEM we only discretize the boundary, which significantly reduces number of unknowns in resulting linear algebraic systems. This can be exploited in areas such as shape optimization. Another advantage of BEM is that the fundamental solution correctly models decay conditions at infinity. The latter is useful when solving problems in unbounded domains, e.g. in electromagnetism. On the other hand, BEM can only treat linear material laws, a remedy to which is a coupling with FEM by means of domain decomposition methods. The popularity of BEM suffers from the fact that it is difficult to implement and that the resulting system matrices are densely populated, thus, more computationally demanding. Therefore, for a solution to engineering problems one has to make use of a parallel implementation and/or the so-called fast BEM.

## Fast BEM

The densely populated system matrices can be sparsified by means of hierarchical matrices completed by adaptive cross approximation (ACA) or fast multipole method (FMM). The idea behind the sparsification relies on a hierarchical decomposition of the related discretization and the approximation of the weakly-interacting blocks of the matrix by a sparse approximation. The method can be implemented in parallel by assigning the blocks to concurrent processors.

## BETI

Boundary Element Tearing and Interconnecting (BETI) is a BEM-counterpart of the FETI method. In both methods local Neumann problems are solved on substructures. In terms of mechanics for a given boundary traction one solves for the displacements. This mapping is the (pseudo)inverse of the so-called Steklov-Poincare operator. In case of FETI, a volume discretization in each subdomain is empoyed. On the other hand, BETI needs to discretize only the boundary. Typically, the Neumann data, i.e. boundary tractions in mechanics, calculated by BETI are more accurate in comparison to FETI.

## Areas of use

- heat conduction, mechanics, acoustics, electromagnetism, some fluid dynamic problems
- modelling of physical fields in domains with a linear material law
- modelling of exterior fields
- shape optimization