Spaces of finite element differential forms

TL;DR

This paper develops finite element spaces of differential forms that fit into the finite element exterior calculus framework, ensuring compatibility with the de Rham complex on simplicial and cubical meshes.

Contribution

It introduces two families of finite element differential form spaces for simplicial and cubical meshes using exterior calculus and the Koszul complex, unifying various approaches.

Findings

Spaces form subcomplexes of the de Rham complex

Spaces admit commuting projections

Applicable to a wide range of mesh types

Abstract

We discuss the construction of finite element spaces of differential forms which satisfy the crucial assumptions of the finite element exterior calculus, namely that they can be assembled into subcomplexes of the de Rham complex which admit commuting projections. We present two families of spaces in the case of simplicial meshes, and two other families in the case of cubical meshes. We make use of the exterior calculus and the Koszul complex to define and understand the spaces. These tools allow us to treat a wide variety of situations, which are often treated separately, in a unified fashion.