4.2 Three-Dimensional Scalar Finite Element Method

Equation (4.11) is used again. The three-dimensional finite element method is very similar to the two-dimensional formulation. Now the analysis is performed in the three-dimensional domain $\mathcal{V}$ with its boundary $\partial\mathcal{V}$ . It leads again to the linear equation system (4.20). The solution is approximated by (4.13). The global matrix $[K]$ and the right hand side vector $\{d\}$ are given analogously to (4.21) by

$$\begin{split}K_{ij} & = \quad \int_{\mathcal{V}}\vec{\nabla}\... ...mathrm{d}A, & \quad \quad i\in[1;n], j\in[1;n]. \end{split}$$ (4.57)

The expression (4.22) for $v(\vec{r})$ is governed by the Dirichlet boundary condition and $D_n$ as in (4.17) complies with the Neumann boundary condition on $\mathcal{A}_N$ . The global functions $\lambda_i$ are constructed similarly as in the two-dimensional case from local ones $\lambda^e$ defined only in a few neighbor elements.

As usual, the finite element procedure starts with the domain discretization. The three-dimensional volume $\mathcal{V}$ is broken into small tetrahedral elements $\mathcal{V}^e$ . As a consequence the boundary $\partial\mathcal{V}$ is subdivided into triangular elements. This kind of elements are very well suited for discretizing of arbitrarily or irregularly shaped regions.