Symmetry and Invariant Bases in Finite Element Exterior Calculus

TL;DR

This paper investigates the symmetry properties of bases in finite element exterior calculus, identifying conditions for invariant bases under vertex permutation symmetries using representation theory.

Contribution

It introduces conditions for the existence of invariant bases in finite element spaces and constructs explicit examples in 2D and 3D.

Findings

Conditions for invariant bases are established and conjectured to be necessary.

Explicit invariant bases are constructed for 2D and 3D finite element differential forms.

New symmetries of geometric decompositions are demonstrated.

Abstract

We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element spaces by the pullback action. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djokovi\'c and Malzan's classification of monomial irreducible representations of the symmetric group, and show new symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Explicit invariant bases with complex coefficients are constructed in dimensions two and three for different spaces of finite element differential forms.