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 [Formula omitted] and [Formula omitted] families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between [Formula omitted] and [Formula omitted] and between [Formula omitted] and [Formula omitted]. 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 [Formula omitted], [Formula omitted], [Formula omitted] and [Formula omitted] spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between [Formula omitted] and [Formula omitted] and between [Formula omitted] and [Formula omitted]. These isomorphisms can be implemented via a simple computation, which we illustrate with examples.
Author(s): Yakov Berchenko-Kogan [sup.1] Author Affiliations: (1) Honolulu, USA Introduction The finite element method is a tool for solving partial differential equations numerically that approximates solutions to the PDE by [...]