First, the frameworks are discussed which deal with discretized differential equations using distinct discretization schemes on a given topological framework.

One of the main inspirations for the development is the GrAL framework [30,31]. In this work the main ideas of topological traversal as well as treating the underlying cell complex with topological methods are given. Implementations exist for many different problems regarding scientific computing.

SGFramework [32] is a mathematical framework especially designed for the solution of partial differential equations using the finite volume method. Various differential equations can be specified whose discretization can be formulated using finite volume schemes as described in Section 2.3. A similar framework which is intended and designed for the solution of drift-diffusion [33] semiconductor equations is Prophet [34].

Roxie [12] is a design environment for the electromagnetic optimization of accelerator magnets. Many optimization features are treated which allow to solve a (quite restricted) engineering problem, namely how to obtain a magnetic field which is possibly identical with that of a given multipole in a large field of an accelerator magnet.

A topological framework for treating triangular and tetrahedral meshes is the wafer-state server (WSS) [35]. This framework offers the possibility of treating quantities which are related to vertices as well as a very generalized approach towards segmentation and the use of subsets of the given cell complex. The topology treatment is reduced to cells and vertices and the only incidence relation available is to obtain all vertices which are incident with a common cell. For these reasons the application to methods which do not use cell based element matrix assembly is complicated and requires various workarounds.

The smart analysis programs (SAP) [36] are implemented which provide a high performance solution of a small number of equations such as the Laplace equation or diffusion equations.

There are several special purpose environments for the solution of very special kinds of differential equations, especially the Navier Stokes equations for fluid mechanics [37,38]. These environments typically do not offer many different configuration features and are specially optimized for this single purpose. The topological cell complex implementation is often directly integrated into the assembly process which makes these tools apt for this special purpose. For general studies on discretization schemes such environments are not intended.

Other tools which also involve geometrical features of the respective cell complex are mesh generators and computer geometry algorithms [39]. In contrast to the generic scientific simulation environment (GSSE) [40], which is designed for flexible topological operations and quantity treatment, these environments offer flexible methods for changing the geometry as well as re-meshing given domains.

The simulation environment `dealII` [41,42,43] is intended as a rapid prototyping tool for finite element simulation and offers methods for mesh refinement and error estimation. The specification of discretization schemes is usually done by writing the local element matrices for finite elements of arbitrary order. `dealII` provides own mesh refinement strategies, different topological elements, and shape functions of different order.

Especially for the use of graphs the Boost Graph library [44,45] is developed. This library contains various algorithms for graphs and implements STL concepts of accessing data which are associated with vertices and edges. Functional structures can be defined in order to formulate local expressions on single nodes.

Michael 2008-01-16