Symmetric bases for finite element exterior calculus spaces
Yakov Berchenko-Kogan
TL;DR
We address which FEEC spaces on a simplex admit a basis that is invariant up to sign under vertex permutations, a property that holds for scalar spaces but depends on the polynomial degree for vector-valued forms. By combining the geometric decomposition of Arnold–Falk–Winther with representation theory (notably induced representations and obstruction theory for $\mathbb Z/3$), we derive complete criteria in dimensions 2 and 3 and provide a framework that can extend to higher dimensions. In 2D, Licht’s list is shown to be complete; in 3D, an expanded, complete list is established, yielding explicit degree constraints (modulo 3) for invariant bases. The approach also connects invariant-basis existence to geometric decomposition and duality mappings, enabling constructive procedures via extension maps and duality.
Abstract
In 2006, Arnold, Falk, and Winther developed finite element exterior calculus, using the language of differential forms to generalize the Lagrange, Raviart--Thomas, Brezzi--Douglas--Marini, and Nédélec finite element spaces for simplicial triangulations. In a recent paper, Licht asks whether, on a single simplex, one can construct bases for these spaces that are invariant with respect to permuting the vertices of the simplex. For scalar fields, standard bases all have this symmetry property, but for vector fields, this question is more complicated: such invariant bases may or may not exist, depending on the polynomial degree of the element. In dimensions two and three, Licht constructs such invariant bases for certain values of the polynomial degree $r$, and he conjectures that his list is complete, that is, that no such basis exists for other values of $r$. In this paper, we show that Licht's conjecture is true in dimension two. However, in dimension three, we show that Licht's ideas can be extended to give invariant bases for many more values of $r$; we then show that this new larger list is complete. Along the way, we develop a more general framework for the geometric decomposition ideas of Arnold, Falk, and Winther.