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 \({\mathcal {P}}_r\varLambda ^k\) and \({\mathcal {P}}_r^-\varLambda ^k\) families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between \({\mathcal {P}}_r\varLambda ^k\) and \(\mathring{{\mathcal {P}}}_{r+k+1}^-\varLambda ^{n-k}\) and between \({\mathcal {P}}_r^-\varLambda ^k\) and \(\mathring{{\mathcal {P}}}_{r+k}\varLambda ^{n-k}\). 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 \({\mathcal {P}}_r\varLambda ^k\), \({\mathcal {P}}_r^-\varLambda ^k\), \(\mathring{{\mathcal {P}}}_r\varLambda ^k\) and \(\mathring{{\mathcal {P}}}_r^-\varLambda ^k\) spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between \({\mathcal {P}}_r \varLambda ^k\) and \(\mathring{{\mathcal {P}}}_{r+k+1}^-\varLambda ^{n-k}\) and between \({\mathcal {P}}_r^-\varLambda ^k\) and \(\mathring{{\mathcal {P}}}_{r+k} \varLambda ^{n-k}\). These isomorphisms can be implemented via a simple computation, which we illustrate with examples.

有限元外部演算是指微分形式的有限元方法的发展，将标量场和矢量场的多个较早的有限元空间推广为任意维n，任意多项式度r和任意微分形式度k。有限元外部演算的研究始于\（{\ mathcal {P}} _ r \ varLambda ^ k \）和\（{\ mathcal {P}} _ r ^-\ varLambda ^ k \）有限元空间族在简单三角剖分上。在开发这些空间时，Arnold，Falk和Winther依赖\（{{mathcal {P}} _ r \ varLambda ^ k \）与\（\ mathing {{\ mathcal {P}}} _ {r + k + 1} ^-\ varLambda ^ {nk} \）和\（{\ mathcal {P}} _ r ^-\ varLambda ^ k \ ）和\（\ mathing {{\ mathcal {P}}} _ {r + k} \ varLambda ^ {nk} \）。在本文中，我们证明了这种对偶关系本质上是标准n球面上微分形式的Hodge对偶性，它通过坐标的变化而掩饰。我们删除了伪装，给出了\（{\ mathcal {P}} _ r \ varLambda ^ k \），\（{\ mathcal {P}} _ r ^-\ varLambda ^ k \），\（\ mathring {{\ mathcal {P}}} _ r \ varLambda ^ k \）和\（\ mathring {{\ mathcal {P}}} _ r ^-\ varLambda ^ k \）空间和球面上微分形式的空间。作为直接推论，我们获得\（{\ mathcal {P}} _ r \ var Lambda ^ k \）与\（\ mathing {{\ mathcal {P}}} _ {r + k + 1}之间的新的点对偶同构^-\ varLambda ^ {nk} \）和\（{\ mathcal {P}} _ r ^-\ varLambda ^ k \）与\（\ mathing {{\ mathcal {P}}} _ {r + k}之间\ varLambda ^ {nk} \）。这些同构可以通过简单的计算来实现，我们将通过示例进行说明。