Duality in finite element exterior calculus

2018/06/30 by Yakov Berchenko-Kogan, Berchenko-Kogan, Yakov
Engineering · Physics and Astronomy · #58A10 #65N30 #Advanced Numerical Methods in Computational Mathematics #Differential Geometry (math.DG) #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1807.01161

openalex publication_date 2018/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In order to generalize finite element methods to differential forms, Arnold, Falk, and Winther constructed two families of spaces of polynomial differential forms on a simplex T, the \mathcal P_rΛ^k(T) spaces and the \mathcal P_r^-Λ^k(T) spaces, where k is the degree of the form and r is the degree of its coefficients. The geometric decomposition for these finite element spaces hinges on a duality relationship between the \mathcal P and \mathcal P^- spaces proved by Arnold, Falk, and Winther. In this article, we give a natural alternate construction of the \mathcal P_rΛ^k(T) and \mathcal P_r^-Λ^k(T) spaces, leading to a new basis-free proof of this duality relationship using a modified Hodge star operator.

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