
Hybridization and postprocessing in finite element exterior calculus
We hybridize the methods of finite element exterior calculus for the Hod...
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Symmetry and Invariant Bases in Finite Element Exterior Calculus
We study symmetries of bases and spanning sets in finite element exterio...
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The Bubble Transform and the de Rham Complex
The purpose of this paper is to discuss a generalization of the bubble t...
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Unifying the geometric decompositions of full and trimmed polynomial spaces in finite element exterior calculus
Arnold, Falk, Winther, in _Finite element exterior calculus, homolog...
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An hphierarchical framework for the finite element exterior calculus
The problem of solving partial differential equations (PDEs) on manifold...
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Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra
In this work, merging ideas from compatible discretisations and polyhedr...
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Finite element simulation of ionicelectrodiffusion in cellular geometries
Mathematical models for excitable cells are commonly based on cable theo...
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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 P_rΛ^k and P_r^Λ^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 P_rΛ^k and P_r+k+1^Λ^nk and between P_r^Λ^k and P_r+kΛ^nk. In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard nsphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the P_rΛ^k, P_r^Λ^k, P_rΛ^k and P_r^Λ^k spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between P_rΛ^k and P_r+k+1^Λ^nk and between P_r^Λ^k and P_r+kΛ^nk, which we illustrate with examples.
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