Unifying the geometric decompositions of full and trimmed polynomial spaces in finite element exterior calculus

TL;DR

This paper introduces a unified operator that simplifies the geometric decomposition of polynomial spaces in finite element exterior calculus, improving the compatibility and conditioning of basis functions.

Contribution

It presents a single operator that unifies the geometric decompositions and extension operators for full and trimmed polynomial spaces, enhancing their compatibility and conditioning.

Findings

Unified operator $ abla{ m star}_T$ simplifies geometric decompositions.

Single extension operator $ ext{dot}E_{ ext{sigma},T}$ handles all differential forms.

Improved basis conditioning and compatibility in finite element spaces.

Abstract

Arnold, Falk, & Winther, in "Finite element exterior calculus, homological techniques, and applications" (2006), show how to geometrically decompose the full and trimmed polynomial spaces on simplicial elements into direct sums of trace-free subspaces and in "Geometric decompositions and local bases for finite element differential forms" (2009) the same authors give direct constructions of extension operators for the same spaces. The two families -- full and trimmed -- are treated separately, using differently defined isomorphisms between each and the other's trace-free subspaces and mutually incompatible extension operators. This work describes a single operator $\mathring{\star}_T$ that unifies the two isomorphisms and also defines a weighted-$L^2$ norm appropriate for defining well-conditioned basis functions and dual-basis functionals for geometric decomposition. This work also describes a single extension operator $\dot{E}_{\sigma,T}$ that implements geometric decompositions of all differential forms as well as for the full and trimmed polynomial spaces separately.