Duality in finite element exterior calculus and Hodge duality on the sphere

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary polynomial degree $r$, and arbitrary differential form degree $k$. The study of finite element exterior calculus began with the $\mathcal Pr\Lambda^k$ and $\mathcal Pr^-\Lambdak$ families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between $\mathcal Pr\Lambda^k$ and $\mathring{\mathcal P}{r+k+1}^{-\Lambda{n-k}$} and between $\mathcal Pr^-\Lambdak$ and $\mathring{\mathcal P}{r+k}\Lambda^{n-k}$. In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard $n$-sphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the $\mathcal Pr\Lambda^k$, $\mathcal Pr^-\Lambdak$, $\mathring{\mathcal P}r\Lambda^k$ and $\mathring{\mathcal P}r^-\Lambdak$ spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between $\mathcal Pr\Lambda^k$ and $\mathring{\mathcal P}{r+k+1}^{-\Lambda{n-k}$} and between $\mathcal Pr^-\Lambdak$ and $\mathring{\mathcal P}{r+k}\Lambda^{n-k}$, which we illustrate with examples.