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

Abstract: 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 $\mathcal P_r\Lambda<sup>k$ and $\mathcal P_r<sup>-\Lambda<sup>k$ families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between $\mathcal P_r\Lambda<sup>k$ and $$\mathring{\mathcal P}<em>{r+k+1}<sup>-\Lambda<sup>{n-k}$$ and between $\mathcal P_r<sup>-\Lambda<sup>k$ and $\mathring{\mathcal P}</em>{r+k}\Lambda<sup>{n-k}$. 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 $\mathcal P_r\Lambda<sup>k$, $\mathcal P_r<sup>-\Lambda<sup>k$, $\mathring{\mathcal P}<em>r\Lambda<sup>k$ and $\mathring{\mathcal P}_r<sup>-\Lambda<sup>k$ spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between $\mathcal P_r\Lambda<sup>k$ and $$\mathring{\mathcal P}</em>{r+k+1}<sup>-\Lambda<sup>{n-k}$$ and between $\mathcal P_r<sup>-\Lambda<sup>k$ and $\mathring{\mathcal P}_{r+k}\Lambda<sup>{n-k}$, which we illustrate with examples.