Duality in finite element exterior calculus

TL;DR

This paper presents a new basis-free proof of the duality between two families of polynomial differential form spaces in finite element exterior calculus, enhancing the theoretical understanding of these spaces.

Contribution

It introduces an alternative construction of the polynomial differential form spaces and provides a new basis-free proof of their duality using a modified Hodge star operator.

Findings

New basis-free proof of duality between polynomial form spaces

Alternative construction of finite element differential form spaces

Enhanced theoretical foundation for finite element exterior calculus

Abstract

In order to generalize finite element methods to differential forms, Arnold, Falk, and Winther constructed two families of spaces of polynomial differential forms on a simplex $T$, the $\mathcal P_r\Lambda^k(T)$ spaces and the $\mathcal P_r^-\Lambda^k(T)$ spaces, where $k$ is the degree of the form and $r$ is the degree of its coefficients. The geometric decomposition for these finite element spaces hinges on a duality relationship between the $\mathcal P$ and $\mathcal P^-$ spaces proved by Arnold, Falk, and Winther. In this article, we give a natural alternate construction of the $\mathcal P_r\Lambda^k(T)$ and $\mathcal P_r^-\Lambda^k(T)$ spaces, leading to a new basis-free proof of this duality relationship using a modified Hodge star operator.