{
  "abstract": "The geometrically decomposed bases of the polynomial differential form spaces $\\mathcal{P}_rΛ^1$ and $\\mathcal{P}_r^-Λ^1$ 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 $\\mathcal{P}_r^-Λ^1$ (trimmed) space and introduce a new basis for the $\\mathcal{P}_rΛ^1$ (full) space, for arbitrary polynomial order $r \\geq 1$ and dimension $D \\geq 2$, 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 $π$ 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 $π$. The inverse transformation is obtained by computing the formulas at $π^{-1}$, 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39511",
  "author_total": 3,
  "authors": [
    {
      "name": "Santiago Badia"
    },
    {
      "name": "Jordi Manyer"
    },
    {
      "name": "Antoine Marteau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6h0azqw9f9hz7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39511",
  "fulltext_url": null,
  "id": "6h0azqw9f9hz7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39511",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7215081791",
  "pdf_url": "https://arxiv.org/pdf/2609.39511",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6h0azqw9f9hz7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39511",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39511"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Rotating bases for finite element exterior calculus: closed-form change of basis under vertex permutations",
  "title_text": "Rotating bases for finite element exterior calculus: closed-form change of basis under vertex permutations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/6h0azqw9f9hz7",
  "venue": "arXiv",
  "versions": []
}