An exterior calculus framework for polytopal methods

TL;DR

This paper introduces a novel polytopal complexes framework for differential forms, extending discrete de Rham methods to general polytopal meshes, ensuring key mathematical properties and cohomology isomorphisms.

Contribution

It develops the first polytopal complexes of differential forms, combining Discrete De Rham and Virtual Element approaches for general meshes with proven mathematical properties.

Findings

Established commutation properties between interpolators and derivatives.

Proved polynomial consistency of the complexes.

Showed cohomology isomorphism with continuous de Rham complex.

Abstract

We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.