|Researcher(s)||:||Prof. Dr. R. Hiptmair|
|Funding||:||no external funding|
Description. Scalar potentials are a
very convenient tool in
computational electromagnetics when it comes to modelling irrotational
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,
``measures the number of handles of ''. In this
case, (1) has to be
where is a fixed low dimensional co-homology space. In the discrete setting, that is, for co-chains, can be determined by
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.