Symmetry and Invariant Bases in Finite Element Exterior Calculus
share
We study symmetries of bases and spanning sets in finite element exterior calculus using representation theory. The group of affine symmetries of a simplex is isomorphic to a permutation group and represented on simplicial finite element spaces by the pullback action. We want to know which vector-valued finite element spaces have bases that are invariant under permutation of vertex indices. 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 Djokovic and Malzan's classification of monomial irreducible representations of the symmetric group and use symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Invariant bases are constructed in dimensions two and three for different spaces of finite element differential forms.
READ FULL TEXT