3. 3 Boundary Element Method

In finite simulation problems the boundary of the simulation domain is treated separately and has to be considered. The actual simulation domain is chosen in a manner that all irregularities such as non-linearities and inhomogeneities are covered. Physical phenomena beyond the boundary are usually not of great importance and are therefore neglected. The boundary conditions which are assumed at the boundaries of the simulation domain are artificially defined by the shape of the considered simulation domain. As a consequence, it may occur, e.g., that for wave equations, waves are reflected at the artificial boundaries so that artefacts occur in the simulation result, which compromise the quality of the results.

Less remarkable but still evident is a behavior that can be observed when applying homogeneous Neumann boundary conditions on the solution of the Laplace equation. If more space is between the boundary and the relevant configurations, the solution can eventually become more precise, when the infinity of the surrounding space is of relevance.

Boundary element methods [56,80] circumvent these difficulties, because the tesselation of the underlying space is only required on the boundaries. The surrounding of the boundary is assumed to be linear, homogeneous, and isotropic. In this case it is not necessary to tessellate the domain far distant from the actual places of interest, but only the boundary has to be tessellated. Therefore, quantities are only stored on topological elements on the boundary.

A feature which makes the application of boundary elements attractive to simulation is that boundary elements and finite elements can be coupled in a simple manner. A practical example of the boundary element method is shown in [12], where a superconductive quadrapole coil is simulated and the effects of the surroundings are explicitly considered.