Duality in finite element exterior calculus

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.