3. 3. 1 Standard Formulation

In contrast to finite elements or finite volumes, shape functions do not have local support but they are non-zero throughout the simulation domain. Normally, basis functions are defined in a manner that they are defined by functions having boundary facets as local support. These facets are incident with a common vertex. In the two-dimensional case, the facets are (boundary) edges incident with a boundary vertex.

The typical formulation of a boundary element scheme can be written in the following manner (Galerkin formulation)

$\displaystyle R_i := \int_{\partial \mathcal{D}} \int_{\partial \mathcal{D}} \mathcal{M} f_i(s) f_j(s') G(s, s') ds ds' = 0 \; ,$ (3.46)

where $ i$ and $ j$ are indices of vertices. For each vertex an equation $ R_i=0$ is assembled. The fundamental solution [81] function $ G$ denotes a fundamental solution of the respective differential operator $ \mathcal{L}(f)$ :

$\displaystyle \mathcal{L}(G(\mathrm{r}, \mathrm{r}')) = \delta(\mathrm{r} - \mathrm{r}')$ (3.47)

The parameters $ s$ and $ s'$ denote the position at the boundary surface or curve. The linear boundary operator $ \mathcal{M}$ can be written as linear combination in the following manner.

$\displaystyle \mathcal{M}(f) := \alpha f + \beta \partial_n f$ (3.48)

Michael 2008-01-16