Whereas in the previous chapter the physical basics of diffusion and oxidation have been discussed, now, an introduction to the numerical basics is given, that explains the necessary transformations of a partial differential equation to a discrete formulation of the problem, that can finally be solved numerically.
The basic implementation for AMIGOS started with the support of finite element discretization methods. The examples described in Chapter 4 are all calculated with the finite element method based on the theory of Galerkin's weighted residual. The following chapter describes the mathematical basics that are necessary for the formulation of a partial differential equation (PDE) with the analytical input interface of AMIGOS. Although AMIGOS supports several different elements, only triangles and tetrahedrons are taken into account within this introduction, but the extension to other elements is similar to the process shown below.
Since AMIGOS is independent of the discretization the basics of finite box integration is presented, too. The finite box integration appears to lead to more accurate results in case of equations describing conservative laws since the coupling between the elements implicitly uses a flux description and the numerical error can be kept smaller. Once more the outlined theory can be implemented congruently into AMIGOS analytical input interface.