Project:

Computational
Cohomology of Finite Element Meshes

Researcher(s) | : | Prof. Dr. R. Hiptmair |

Funding | : | no external funding |

Duration | : | ongoing project |

**Description.** Scalar potentials are a
very convenient tool in
computational electromagnetics when it comes to modelling irrotational
vectorfields.
For instance, if denotes a topologically trivial
bounded domain, we have

This amounts to the Poincaré lemma of the calculus of differential forms cast in the language of vector analysis. The statement (1) remains true, when replacing with a simply connected surface and by the surface rotation .

There is a counterpart of (1) for co-chains on topologically simple simplicial complexes: if a 1-co-chain has zero rotation for all bounding cycles, it will the discrete gradient of 0-co-chain, that is, a function defined on the vertices of the complex. This is relevant for numerics, because co-chains are closely related to discrete differential forms, which provide popular finite element schemes for electromagnetic fields, for instance.

However, (1) and its discrete counterparts break down, when and the simplicial complex have a non-vanishing first Betti number, which ``measures the number of handles of ''. In this case, (1) has to be modified into

where is a

- finding a fundamental set of non-bounding edge cycles for the case of a simplicial mesh on a surface,
- finding a full set of abstract Seifert surfaces (``cuts'')for the case of a 3D triangulation.

It is the goal of the project to investigate whether there are efficient algorithms for solving the discrete cut problem in three dimensions. By efficient we mean, that the computational effort should scale linearly in the size of the triangulation.

**Activities.** Currently, a term project
by J. Kayatz is
experimentally
examining the use of edge contractions [DEGN99]
for mesh simplifications.