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)

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

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

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