Cohomology of the discrete de Rham complex on domains of general topology

TL;DR

This paper proves that the cohomology spaces of a discrete de Rham complex on general polyhedral domains in  are isomorphic to those of the continuous complex, extending previous results to arbitrary-order complexes and general meshes.

Contribution

It establishes the isomorphism of cohomology spaces between discrete and continuous de Rham complexes on general polyhedral domains, a novel result for arbitrary-order complexes.

Findings

Cohomology spaces of discrete and continuous complexes are isomorphic.

First such result for arbitrary-order complexes on general polyhedral meshes.

Extends the understanding of discrete de Rham complexes in computational topology.

Abstract

In this work we prove that, for a general polyhedral domain of $\mathbb{R}^3$, the cohomology spaces of the discrete de Rham complex of [Di Pietro and Droniou, An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincar\'e inequalities, and consistency, Found. Comput. Math., 2021, DOI: 10.1007/s10208-021-09542-8] are isomorphic to those of the continuous de Rham complex. This is, to the best of our knowledge, the first result of this kind for an arbitrary-order complex built from a general polyhedral mesh.