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

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.