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

12/03/2021

∙

by   Toby Isaac, et al.

∙

0

∙

share

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. The two families – full and trimmed – are treated separately, using differently defined isomorphisms onto their trace-free subspaces. This work describes a single map h_T: Λ^k(T) →Λ^n-k(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.