Rotating bases for finite element exterior calculus: closed-form change of basis under vertex permutations

Abstract

The geometrically decomposed bases of the polynomial differential form spaces ${\mathcal{P}}_r{\Lambda }^1$ and ${\mathcal{P}}_r^{-}{\Lambda }^1$ on a simplex are products of a barycentric scalar polynomial and a directional or Whitney 1-form. These bases depend on an ordering of the simplex vertices. On unstructured meshes, neighbouring cells need not agree on this ordering, making it non-trivial to achieve conformity in finite element software. Existing strategies compute degree-of-freedom transformation matrices numerically, or impose a global vertex ordering through face-bubble spaces or by preprocessing the mesh. We consider the standard basis of the ${\mathcal{P}}_r^{-}{\Lambda }^1$ (trimmed) space and introduce a new basis for the ${\mathcal{P}}_r{\Lambda }^1$ (full) space, for arbitrary polynomial order $r\ge 1$ and dimension $D\ge 2$, and prescribe their basis polynomials as shape functions. We show that the pullback of shape functions under the change of coordinates induced by an arbitrary vertex relabelling $\pi$ in the symmetric group $S_{D+1}$ admits a closed-form, combinatorial expression. For most basis functions this pullback is a single basis function of the relabelled ordering, up to sign. On an explicitly characterised "filter-hit" set, this pullback is a signed sum of at most $D$ (full space) or exactly two (trimmed space) relabelled basis functions. All coefficients are in $\{-1,+1\}$ for all $r$, $D$ and $\pi$. The inverse transformation is obtained by computing the formulas at ${\pi }^{-1}$, so no numerical inversion of a change-of-basis matrix is needed. Conforming assembly and evaluation on simplicial meshes with arbitrary vertex orderings reduce to index manipulation. Our results are compared with the study of relabelling-invariant bases of Berchenko-Kogan and Licht, and verified in an open-source Julia implementation in the Gridap.jl library.