6. Summary and Outlook

The main achievements in which the proposed methods differ from state-of-the-art frameworks are, that arbitrary discretization schemes and different kinds of algebraic equations (linear, nonlinear, eigenvalue) can be treated using the same mechanisms.

At the level of discretizing differential equations enormous flexibility can be obtained, because different discretization schemes can be tested at the same time. Furthermore, the formulation of single discretized equations using topological incidence functions for traversing the respective elements makes the formulation independent of the dimension and the archetype of the used elements, even if different mechanisms for the calculation of the geometry related factors have to be used.

At the level of assembly, the use of linearized equations eases filling the matrices and implicitly couples a basis function with a governing equation (which is mainly evaluated in the neighborhood of the local support of the basis function). Furthermore, the use of linearized equations implicitly forms the derivatives required for the use of gradient methods for the solution of nonlinear equation systems. Therefore, gradient based schemes such as the Newton scheme can be implemented straightforwardly.