Cohomology of the discrete de Rham complex on domains of general 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 (Found Comput Math 23:85–164, 2023, https://doi.org/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 a fully computable arbitrary-order complex built from a general polyhedral mesh.