5. Discretization with the Finite Element Method

PARTIAL DIFFERENTIAL EQUATIONS (PDE s) are widely used to describe and model physical phenomena in different engineering fields and so also in microelectronics' fabrication. Only for simple and geometrically well-defined problems analytical solutions can be found, but for the most problems it is impossible. For these problems, also often with several boundary conditions, the solution of the PDE s can only be found with numerical methods.

The most universal numerical method is based on finite elements. This method has a general mathematical fundament and clear structure. Thereby, it can be relative easily applied for all kinds of PDE s with various boundary conditions in nearly the same way. The finite element method (FEM) has its origin in the mechanics and so it is probably the best method for calculating the displacements during oxidation processes [84]. The finite element formulation works on a large number of discretization elements and also on different kinds of meshes within the domain. Furthermore, it also provides good results for a coarse mesh. It can easily handle complicated geometries, variable material characteristics, and different accuracy demands.