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 $\mathcal P_r\Lambda^k$ and $\mathcal P_r^-\Lambda^k$ families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between $\mathcal P_r\Lambda^k$ and $\mathring{\mathcal P}_{r+k+1}^-\Lambda^{n-k}$ and between $\mathcal P_r^-\Lambda^k$ 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 $\mathcal P_r\Lambda^k$, $\mathcal P_r^-\Lambda^k$, $\mathring{\mathcal P}_r\Lambda^k$ and $\mathring{\mathcal P}_r^-\Lambda^k$ spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between $\mathcal P_r\Lambda^k$ and $\mathring{\mathcal P}_{r+k+1}^-\Lambda^{n-k}$ and between $\mathcal P_r^-\Lambda^k$ and $\mathring{\mathcal P}_{r+k}\Lambda^{n-k}$, which we illustrate with examples.
翻译:精度元素外部计算法是指开发不同形式的限定元素方法, 将一些早期的卡路里和矢量场的有限元素空间推广到任意的维度 $, 任意的多元度 $, 任意的差别形式 $k$。 对有限元素外部计算法的研究始于 $mathcal P_r\\Lambda\k$, $mathcal P_r\\\\\\Lambda\kkak$ 。 在这些空间的开发中, Arnold, Falk, 和 Winther 的硬性元素空间空间空间空间空间空间空间, $mr\r\r\Lbdal_macal$, P\r\r\b\macal\ kal$。 Pr\k\macal\k\k\k\ kal$; 硬性硬质的Pr\r\r\\\\\\\ kal_al_al_al_al_al_al_al_maxal_macal_al_al_ maxal_ maxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx