Duality in 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.