Complexes of Discrete Distributional Differential Forms and their Homology Theory
Abstract
Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus we generalize a notion of Braess and Schöberl, originally studied for a posteriori error estimation. We construct isomorphisms between the simplicial homology groups of the triangulation, the discrete harmonic forms of the finite element complex, and the harmonic forms of the distributional finite element complexes. As an application, we prove that the complexes of finite element exterior calculus have cohomology groups isomorphic to the de Rham cohomology, including the case of partial boundary conditions. PoincaréFriedrichstype inequalities will be studied in a subsequent contribution.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 DOI:
 10.48550/arXiv.1410.6354
 arXiv:
 arXiv:1410.6354
 Bibcode:
 2014arXiv1410.6354L
 Keywords:

 Mathematics  Numerical Analysis;
 65N30;
 58A12
 EPrint:
 revised preprint, 26 pages