Complexes of Discrete Distributional Differential Forms and their Homology Theory

TL;DR

This paper introduces complexes of discrete distributional differential forms within finite element exterior calculus, establishing isomorphisms with homology groups and proving their cohomology matches de Rham cohomology, even with boundary conditions.

Contribution

It generalizes the notion of distributional forms in finite element calculus and links their homology to classical topological invariants, extending prior work by Braess and Sch"oberl.

Findings

Isomorphisms between homology groups and harmonic forms established

Cohomology groups of finite element complexes shown to be isomorphic to de Rham cohomology

Framework applicable to partial boundary conditions

Abstract

Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus we generalize a notion of Braess and Sch\"oberl, 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. Poincar\'e-Friedrichs-type inequalities will be studied in a subsequent contribution.