1. 3 Aims

Tools regarding the discrete specification of scientific problems as well as an abstract treatment of interfaces to both, discretization schemes as well as algebraic solution methods (solvers), are required. In special cases all of these tasks can be implemented in an appropriate manner, however, the flexibility of the implemented code is rather low. Even though a high flexibility can be obtained with single solvers and the handling of discrete topological complexes, an appropriate framework for interfacing these features is still missing.

The aim of this work is to provide a mathematical framework for the discretization, linearization, and matrix interfacing, which closes the gap between flexible and highly performant algebra tools and topological tools for the handling of discrete structures with which discretization schemes can be realized.

For the discretization or the formulation of discrete problems, a functional calculus is provided, with the main aim to preserve the topological opportunities which are offered by the topological structure implemented in the GSSE library. Furthermore, one of the design goals is that with the aid of this calculus it has to be possible to implement different kinds of discretization schemes. As a consequence different methods can be compared, while the implementation effort is kept to a minimum.

By storing the expressions for different discretized differential operators in a library it is easy to use pre-formulated expressions in order to solve one's own differential equations. Within such a library different methods regarding modeling, discretization, and algebraic solution of the respective problem can be tested and optimized.

The next aim is to ease the effort for the specification of derivatives with respect to single solution variables for the assembly of a discrete algebraic equation which is, especially for the implementation of finite volumes, most cumbersome and error prone. Additionally, the formulation of the respective formulae shall remain as simple as possible and no separate framework for the specification shall be required. Furthermore, the assembly of matrices should be treated in a manner that the solver used and its respective matrix format can be used in combination of the other methods.

The overall goal is an arbitrary combination of methods explained in the above listed aims. If combined with an abstract solver interface, a large variety of different problems can be approached using such a framework.