Unifying the geometric decompositions of full and trimmed polynomial spaces in finite element exterior calculus
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 tracefree 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 tracefree 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 wellconditioned basis functions and dualbasis 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.02174
 Bibcode:
 2021arXiv211202174I
 Keywords:

 Mathematics  Numerical Analysis
 EPrint:
 21 pages