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 π«rΞ1 and π«r-Ξ1 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 π«r-Ξ1 (trimmed) space and introduce a new basis for the π«rΞ1 (full) space, for arbitrary polynomial order r β₯ 1 and dimension D β₯ 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 Ο in the symmetric group SD+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 {β1, +1} for all r, D and Ο. The inverse transformation is obtained by computing the formulas at Οβ1, 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.
Related
- Finite element exterior calculus for spectra and pseudospectra of advection-diffusion of differential forms
- HDG methods in finite element exterior calculus
- A finite element exterior-to-interior reconstruction in the fractional conductivity inverse problem
- Finite Element Approximation of Rotating MHD-Boussinesq Flows with Nonlinear Power-Law Viscosity
- Discrete Exterior Calculus-based Whittle-MatΓ©rn Fields on Meshes
BibTeX
@misc{badia2026rotating,
title = {{Rotating bases for finite element exterior calculus: closed-form change of basis under vertex permutations}},
author = {Santiago Badia and Jordi Manyer and Antoine Marteau},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39511},
eprint = {2609.39511},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39511},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rotating bases for finite element exterior calculus: closed-form change of basis under vertex permutations
AU - Santiago Badia
AU - Jordi Manyer
AU - Antoine Marteau
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39511
UR - https://arxiv.org/abs/2609.39511
AB - The geometrically decomposed bases of the polynomial differential form spaces π«_rΞ^1 and π«_r^-Ξ^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 π«_r^-Ξ^1 (trimmed) space and introduce a new basis for the π«_rΞ^1 (full) space, for arbitrary polynomial order r β₯ 1 and dimension D β₯ 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 Ο 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 {β1, +1} for all r, D and Ο. The inverse transformation is obtained by computing the formulas at Ο^β1, 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.
ER -