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

Santiago Badia, Jordi Manyer, Antoine Marteau

arXiv:2609.39511v1math.NAcs.NA

TL;DR

Ordering-dependent basis functions make conformity nontrivial when neighboring simplex cells use different vertex orderings. The paper gives closed-form relabelling formulas for full and trimmed spaces; most pullbacks are single signed basis functions, with bounded signed sums on filter hits.

  • Problem

    Ordering-dependent basis functions can disagree across shared faces when neighboring cells use different vertex orderings, making conformity nontrivial.

    Source: §1 · p. 1

  • Method

    The paper constructs a new geometrically decomposed basis for the full space and gives closed-form combinatorial relabelling formulas for it and the standard trimmed basis.

    Source: §9 · p. 20

  • Results

    Most pullbacks are single signed basis functions; filter hits require at most D full-space or two trimmed terms, while scrambled- and sorted-mesh solutions coincide to 10^-15.

    Source: §8.4 · p. 19, §9 · p. 20

  • Takeaways & Limitations

    The new rotating bases and relabelling logic are cheaper to implement because directional forms yield a degree-independent catalogue of forms and transformations.

    Source: §9 · p. 20

    Extension to K-forms is left for future work; for K ≥ 2, a filter hit may require several resummations at once.

    Source: §9 · p. 20

Abstract

from arXiv · show

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 $r \geq 1$ 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 $\pi$ 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.

1. Introduction

The paper addresses how to maintain conformity when neighboring simplices use different local vertex orderings, and develops closed-form relabelling laws for geometrically decomposed bases.

1. Introduction

  • At higher polynomial orders, neighboring cells’ face-owned basis functions require a non-trivial change of basis, for which no closed form was available.Conformity requires matching form traces across shared faces.Source: §1 · p. 1

1.1. Numerically computed transformation matrices.

  • Existing transformation-matrix approaches compute matrices numerically at setup time and require consumers of reference-element tabulations to apply them.The computation introduces floating-point error; the paper instead proposes closed-form formulas for the transformation and its inverse.Source: §1.1 · pp. 1–2

1.2. Global ordering conventions and orientation-embedded bases.

  • Orientation-embedded bases parameterize face-owned functions by local orientation, whereas this work supplies closed-form relabelling laws for complete cell spaces.The cited orientation-embedded bases are unions of face-owned bases.Source: §1.2 · p. 2

1.3. Mesh reorientation.

  • Mesh reorientation seeks globally compatible local vertex orderings, but compatible orderings need not exist and reorientation can be costly to parallelize and repeat.The paper’s approach accepts arbitrary local orderings without global orientation or preprocessing.Source: §1.3 · p. 2

1.4. Symmetry, invariance, and Whitney-form combinatorics.

  • Berchenko-Kogan [9] and Licht [20] study when relabelling-invariant bases exist, but do not give basis-function expansions when invariance is unavailable.Their results are mostly restricted to dimensions two and three.Source: §1.4 · p. 2

1.5. Contributions and outline.

  • The paper gives uniform-in-degree closed-form combinatorial transformation laws for the standard trimmed basis and a new full-space basis.Most pullbacks are single signed basis functions; filter-hit cases are signed sums of at most D terms for the full space or exactly two for the trimmed space.Source: §1.5 · pp. 2–3
  • The authors implement the bases in Gridap and report numerical validation, including a curl–curl problem on meshes with arbitrary local vertex orderings.The results are presented for 1-forms; extension to higher form degrees is left for future work.Source: §1.5 · p. 3

2. Preliminaries

The preliminaries set up barycentric coordinates and vertex relabellings, describe the polynomial 1-form spaces, and reduce mesh conformity to a relabelling relation on a reference simplex.

2.1. The simplex, barycentric coordinates, and vertex relabellings.

  • A vertex relabelling permutes barycentric coordinates and differentials, so pullbacks of indexed data can be computed by relabelling their indices.The relabelling acts on barycentric monomials by permuting exponent entries.Source: §2.1 · p. 4

2.2. Polynomial differential forms and geometric decomposition.

  • The full and trimmed polynomial 1-form spaces admit geometrically decomposed bases built from scalar barycentric monomials and directional or Whitney 1-forms.The geometric decomposition assigns basis functions to faces, whose traces are used to characterize conformity.Source: §2.2 · p. 5

2. Preliminaries

  • The setup assumes arbitrary local vertex orderings and uses the associated affine maps to transport reference fields to mesh cells.The simplex vertex labels provide the local ordering used for the reference coordinates.Source: §2 · p. 4, §2.1 · p. 4, §2.2 · p. 5

2.3. From mesh conformity to reference relabellings.

  • Conformity across a shared face is equivalent to matching the traces after applying the reference-simplex relabelling associated with the two local face orderings.Pullback commutes with restriction to subsimplices.Source: §2.3 · p. 5

3. The rotating basis for PrΛ1

The section develops geometrically decomposed bases for the full and trimmed spaces and gives closed-form sparse pullback rules under vertex relabellings, including an anchored-pair filter.

3.1. Directional 1-forms and spanning set.

  • For every r ≥ 1, products of barycentric monomials and directional 1-forms span P_rΛ1.The spanning family is indexed by face, vertex, and exponent data.Source: §3.1 · p. 6, §3 · p. 6

3.2. Rotating basis.

  • Excluding indices with supp(α) = f and k = min f yields an anchored basis of P_rΛ1.Functions owned by each face form a basis of its bubble space, giving the basis its geometric decomposition.Source: §3.2 · pp. 7–8

3.3. Pullback under vertex relabellings.

  • A basis function relabels to one basis function unless α_k > 0 and π(k) = min π(f), when its pullback is a signed sum of at most D basis functions.The filter-hit summands are anchored members of the relabelled basis.Source: §3.3 · p. 9
  • The transformation T_p(π) is well defined, has entries in {-1, 0, +1}, and its inverse is T_p(π^-1); it depends on r only through its index set B(Δξ, r).Source: §3.3 · p. 10
  • The pullback rules combine a position-dependent Whitney form with a degree r − 1 scalar factor, and use multi-exponent shifts to give a closed-form two-term expression uniformly in r, D, and π.The Whitney-form catalogue itself transforms as a signed permutation, without combinations at the catalogue level.Source: §3.3 · pp. 10–11, §4.3 · p. 13

4.1. Whitney forms and the spanning set.

  • The spanning set uses a barycentric monomial of degree r − 1 times an ordered-pair Whitney 1-form, with pair order tracking signs by alternation.Vertices own no basis functions.Source: §4.1 · p. 11

4.2. Extraction of the basis.

  • The basis filter retains triplets whose ordered pair starts at the minimum label of its face; every excluded triplet decomposes into two retained basis functions.The filter-hit condition is independent of the multi-exponent, and each basis function has zero trace on faces that do not contain its owning face.Source: §4.2 · pp. 11–12

4.3. Pullback under vertex relabellings.

  • Under a relabelling, each trimmed basis function pulls back to one signed basis function or, on a filter hit, a signed sum of exactly two; the inverse follows by applying the formulas at π^-1.The filter-hit condition is independent of the multi-exponent, and the standard basis from [5, Section 9] has this transformation law.Source: §4.3 · p. 13

5. Degrees of freedom

The paper prescribes the rotating basis as shape functions and defines its dual degrees of freedom implicitly. The resulting functionals are unisolvent, while their numerical construction is separate from assembly and evaluation.

5. Degrees of freedom

  • The rotating basis is prescribed as the shape-function basis, with its dual degree-of-freedom functionals defined implicitly.The pair (K, B) is prescribed, with V = span B; the functionals are computed from a prebasis of V* when needed.Source: §5 · p. 14
  • The dual functionals form a basis of the dual space, so the degree-of-freedom set is unisolvent.Linear independence follows by evaluating a linear combination at the basis functions, and the family has the dimension of the space.Source: §5 · p. 14
  • In practice, the functionals are computed by a numerical change of basis from any unisolvent prebasis, with the matrix assembled and inverted once per reference simplex.The paper uses the canonical FEEC degrees of freedom as the prebasis.Source: §5 · p. 14
  • The induced interpolation operator matches every canonical moment and is the canonical FEEC interpolant, including its commuting diagram.The stated result applies to fields in the corresponding polynomial space.Source: §5 · p. 14
  • Assembly, evaluation, and error computation use only shape functions, so prescribed closed-form shape functions leave neither the change-of-basis matrix nor the functionals in those computations.The functionals remain useful mainly for interpolating user-provided fields such as boundary or initial data.Source: §5 · p. 14

6. Conforming assembly on a mesh

Each cell expresses its rotating basis using barycentric coordinates ordered by global vertex index, then identifies basis functions across cells by face and position. The resulting gluing has matching traces and yields an HΛ1-conforming assembled space.

6. Conforming assembly on a mesh

  • The rotated basis is defined by global vertex indices alone, with face labels mapped between cells by increasing bijections.Position maps assign each face's basis labels to consecutive indices, consistently across faces of equal dimension.Source: §6 · p. 15
  • The mesh's global vertex indices determine each cell's local permutation and sorted-coordinate rotating basis, which is obtained from reference shape functions by the relabelling pullback.The cell's physical map supplies the final mapping; the reference relabelling uses signed combinations of at most D full-space or two trimmed-space shape functions, without floating-point computation.Source: §6 · p. 15
  • Across a shared face, basis functions are identified by their owning mesh face and face-wise position; identified functions have the same trace.A function owned by a subface is identified with exactly one function from the neighbouring cell, while functions owned outside the shared face have zero trace there.Source: §6 · pp. 15–16
  • Gluing the rotated functions under this identification produces an HΛ1-conforming space on simplicial meshes with arbitrary local vertex orderings.The proof establishes single-valued traces on every mesh face.Source: §6 · pp. 15–16

7. Implementation in Gridap.jl

The Gridap.jl implementation evaluates the rotating bases and applies vertex relabellings through combinatorial index maps. Cell-local face ordering then adapts the reference basis for conforming assembly.

7.1. Rotating bases implementation.

  • Gridap stores scalar Bernstein data and per-function indices or forms, reusing pointwise Bernstein evaluations to assemble shape functions.The trimmed evaluation multiplies a bare monomial by a pair of barycentric-coordinate-weighted constant 1-forms; its normalization removes the Bernstein multinomial factor.Source: §7.1 · p. 16

7.2. The rotation kernel.

  • The rotation kernel returns signed index expansions, with filter hits producing multiple terms in the full and trimmed spaces.The full-space hit returns at most D terms, while the trimmed-space hit shifts the exponent in two ways.Source: §7.2 · pp. 16–17
  • The inverse pullback reruns the kernel at the inverse permutation, and caching stores rows for encountered permutations rather than assembling the transformation matrix.The hit test uses two integer comparisons in the full space and one in the trimmed space; at most (D+1)! permutations can occur.Source: §7.2 · p. 17

7.3. Global assembly.

  • Cell-local assembly reorders sorted-face blocks to match local faces, then transforms the reference basis and its cell matrix.The rotating reference element is the mesh-independent object; the cell basis uses its local vertex permutation.Source: §7.3 · p. 17

8. Numerical experiments

Experiments validate the relabelling formulas and test conformity, projection, and curl–curl solutions on meshes with arbitrary local vertex orderings. Across the reported tests, scrambled and sorted orderings give matching discrete solutions to machine precision.

8.1. Exactness and the representation property.

  • For D=2,3 and r≤4, transformation entries are exactly -1, 0, or +1, and inverse and composition identities hold in integer arithmetic.Numerical pullbacks agree at 20 interior points up to machine precision.Source: §8 · p. 18

8.2. Sparsity and cost.

  • At r=4 in three dimensions, the mean rotation-map row has at most 1.27 entries; memoised lookup costs about ten nanoseconds.Multi-term rows first appear at r=3 in the full space and r=2 in the trimmed space.Source: Table 2 · p. 18, §8.2 · p. 18

8.3. Conformity on scrambled meshes.

  • With rotation active, trace jumps are at machine precision on both tested two-cell and larger scrambled meshes; without rotation, they are O(1).The tests include non-cyclic shared-face orderings and randomly scrambled simplexified Cartesian grids.Source: §8.3 · p. 18, Table 3 · p. 19

8.4. L2 projection on scrambled meshes.

  • The sorted-mesh and scrambled-mesh L2-projection solutions coincide to 10^-15, with expected rates r+1 for the full space and r for the trimmed space.In 3D, the finest reported rates are 1.97 and 2.75 (full) and 0.91 and 1.92 (trimmed), with pre-asymptotic levels.Source: Table 4 · p. 19, §8.4 · p. 19

8.5. A curl–curl problem on scrambled meshes.

  • In 2D, the curl–curl problem has L2 rates r+1 (full) and r (trimmed), with curl-seminorm rates r for both; sorted and scrambled solutions coincide to 10^-12 or better.In 3D, rates remain pre-asymptotic but approach the same values.Source: §8.5 · p. 19, Table 5 · p. 20

9. Conclusions and future work

The paper presents rotating bases with closed-form relabelling expansions for the full and trimmed 1-form spaces. The analysis leaves higher-form extensions for future work.

9. Conclusions and future work

  • Most basis functions pull back to one signed relabelled basis function; filter hits require at most D terms in the full space or exactly two in the trimmed space.The signed sums are degree-independent, and the transformation matrices are given by closed-form combinatorial formulas.Source: §9 · p. 20
  • The new full-space basis uses directional forms defined through exponent support, yielding a degree-independent catalogue and cheaper relabelling logic.The relabelling theorems also apply to the previously defined geometrically decomposed bases.Source: §9 · p. 20
  • Extension to K-forms is future work; for K≥2, a filter hit may require several resummations at once.The paper expects the structure to carry over through wedge products of catalogue 1-forms.Source: §9 · p. 20

Appendix A. Bernstein normalisation of the trimmed basis

The appendix considers a Bernstein-normalized version of the trimmed basis. Its change-of-basis matrices retain the sparsity structure and inverse rule, while their coefficients can differ from the bare-monomial case.

Appendix A. Bernstein normalisation of the trimmed basis

  • Bernstein normalization scales each trimmed basis function by the multinomial factor of its exponent.The index sets and results independent of the scalar factor remain unchanged; shift identities are affected because a unit of degree moves between exponent entries.Source: Appendix A · p. 20
  • The normalized transformation matrix has the same sparsity and inverse-at-the-inverse-permutation rule as the bare-monomial matrix.The appendix obtains this through a diagonal change of basis.Source: Appendix A · p. 21
  • Normalized matrix rows need not have equal-magnitude entries, and their coefficients may be rational rather than integer.The shift of one degree between exponent entries prevents the multinomial factor from cancelling.Source: Appendix A · pp. 20–21
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