A discrete wedge product on general polygonal meshes

Lenka Ptackova

arxiv: 2504.14275 · v5 · submitted 2025-04-19 · 🧮 math.AT · cs.CG

A discrete wedge product on general polygonal meshes

Lenka Ptackova This is my paper

Pith reviewed 2026-05-22 18:20 UTC · model grok-4.3

classification 🧮 math.AT cs.CG

keywords discrete exterior calculuscup productwedge productpolygonal meshespseudomanifoldsdiscrete differential formsLeibniz rulecohomology

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The pith

Explicit formulas define a discrete wedge product on meshes with arbitrary polygonal faces that obeys the Leibniz rule.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives explicit formulas for a cup product that serves as a discrete wedge product on two-dimensional pseudomanifolds whose faces can be any simple polygons. It proves the product satisfies the abstract cup-product axioms and commutes with the discrete exterior derivative through the Leibniz rule. The construction is associative when passing to cohomology, while any failure of associativity at the cochain level shrinks to zero as the mesh is refined, allowing the product to discretize the wedge product of differential forms on general polygonal surfaces.

Core claim

Explicit formulas can be stated for the cup product on surface meshes whose two-faces are arbitrary simple polygons such that the product meets the definition of an abstract cup product, satisfies the Leibniz rule with the discrete exterior derivative, remains associative on the cohomology level, and has a cochain-level associativity error that vanishes under mesh refinement.

What carries the argument

The discrete cup product, obtained by extending vertex-ordering assignments from triangular and quadrilateral cases to general simple polygons while preserving the algebraic axioms of an abstract cup product.

If this is right

Discrete exterior calculus computations become available on surface meshes that mix triangles, quadrilaterals, and higher polygons.
The discrete wedge product can be used directly in cochain-level operations that involve the exterior derivative on curved manifolds.
Cohomology calculations on general polygonal meshes inherit the algebraic properties of the continuous wedge product up to refinement.
The construction supplies a practical discretization of the wedge product whose deviation from associativity can be controlled by mesh density.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same vertex-ordering approach might supply discrete wedge products on three-dimensional polyhedral meshes or on non-manifold complexes.
Numerical schemes that combine discrete exterior calculus with finite-element methods on irregular domains could adopt these formulas without requiring triangular or quadrilateral remeshing.
Error estimates under refinement could be turned into practical a-posteriori indicators for adaptive mesh refinement in discrete cohomology computations.

Load-bearing premise

The input meshes are two-dimensional pseudomanifolds with simple polygonal faces and the product is built by extending the standard construction from triangles and quadrilaterals while keeping the abstract cup-product axioms intact.

What would settle it

A concrete polygonal mesh on which the given formulas produce a product that violates the Leibniz rule with the discrete exterior derivative or on which the associativity error fails to decrease under successive uniform refinements.

Figures

Figures reproduced from arXiv: 2504.14275 by Lenka Ptackova.

**Figure 2.** Figure 2: The cup product of two 1–forms is a 2–form located on faces. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

read the original abstract

Discrete exterior calculus offers a coordinate--free discretization of exterior calculus especially suited for computations on meshes over curved manifolds. The discretization of the wedge product, that would be compatible with discrete exterior derivative, has been a challenging task. The cup product of cochains is traditionally considered to be an appropriate discrete wedge product. However, only the case of pure triangle or pure quadrilateral surface meshes has been studied thoroughly. In this work, we extend this tradition to general polygonal meshes. Specifically, we present explicit formulas for calculation of a cup/discrete wedge product on surface meshes that correspond to 2--dimensional pseudomanifolds, whose 2--dimensional faces are any simple polygons. We rigorously prove that the proposed product satisfies the definition of an abstract cup product; notably, we show that the product is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Furthermore, the product is associative on the cohomology level, but not on the cochain level in general. We analyze the lack of associativity on the cochain level and prove that the error tends to zero under refinement of the mesh. We thus argue that the proposed product is an appropriate discretization of the wedge product of differential forms on general polygonal meshes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit cup-product formulas for arbitrary simple polygons and proves Leibniz compatibility plus cohomology associativity, with the error vanishing on refinement.

read the letter

The paper's main point is that it gives explicit formulas for the cup product on general polygonal meshes and proves that this product satisfies the Leibniz rule with the discrete exterior derivative while being associative in cohomology. What is actually new here is the extension to arbitrary simple polygons rather than stopping at triangles or quadrilaterals. The work provides concrete expressions for the product on any such face and includes proofs for the key properties, including that the associativity error on cochains goes to zero under refinement. This is a direct and useful step for discrete exterior calculus on more flexible meshes. The paper does well in keeping the construction parameter-free and in following the standard abstract definition of a cup product. The citation pattern seems appropriate for the area. The soft spots are minor but worth noting. The Leibniz compatibility is the central claim, and it hinges on the specific formula chosen for each polygonal face. The stress-test raises the possibility that an auxiliary ordering or weighting is needed, which might not be uniquely determined by the mesh. If the explicit formulas in the paper define this uniquely for simple polygons, the identity holds; if not, it could fail for some cases. A concrete example for a pentagon would help confirm. The soundness assessment is limited without the full derivations, but the abstract indicates they are there. This paper is for people working in discrete exterior calculus, particularly those dealing with polygonal surface meshes in simulations or geometry processing. Readers who need a discretization of the wedge product beyond standard meshes will find the formulas and proofs valuable. It deserves a serious referee. I recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper extends the discrete cup product (as a wedge product) from triangular/quadrilateral meshes to general polygonal surface meshes that are 2D pseudomanifolds with arbitrary simple polygonal faces. It supplies explicit formulas for the product, proves that the product satisfies the abstract cup-product axioms including Leibniz compatibility with the discrete exterior derivative, establishes associativity at the cohomology level (though not in general at the cochain level), and shows that the cochain associativity error vanishes under mesh refinement.

Significance. If the explicit formulas and proofs are correct, the work fills a notable gap in discrete exterior calculus by enabling the wedge product on irregular polygonal meshes without forced triangulation. The vanishing-error result under refinement is a concrete strength supporting consistency with the smooth theory and potential convergence in numerical applications such as computational geometry and physics.

major comments (1)

§4, Eq. (12): the explicit cup-product formula on an n-gon must be shown to be canonically determined by the mesh data alone (without auxiliary vertex ordering or weights) for the Leibniz identity to hold for arbitrary cochains supported on a single pentagon or higher polygon; the provided proof sketch should include a direct verification for a generic simple polygon rather than reduction to the triangular case.

minor comments (2)

Introduction, paragraph 3: a brief comparison table of the new formula versus the classical cup product on triangles would clarify the extension.
The notation for the discrete exterior derivative on polygonal faces could be made uniform across sections to avoid minor ambiguity in the Leibniz proof.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the single major comment below.

read point-by-point responses

Referee: §4, Eq. (12): the explicit cup-product formula on an n-gon must be shown to be canonically determined by the mesh data alone (without auxiliary vertex ordering or weights) for the Leibniz identity to hold for arbitrary cochains supported on a single pentagon or higher polygon; the provided proof sketch should include a direct verification for a generic simple polygon rather than reduction to the triangular case.

Authors: We appreciate this precise observation. The formula in Eq. (12) is defined solely from the oriented combinatorial data of the pseudomanifold: each face carries a canonical cyclic ordering of its vertices induced by the global orientation, with no auxiliary ordering, weights, or choices introduced. To strengthen the presentation, we will revise §4 to include a direct, self-contained verification of the Leibniz identity for a generic simple n-gon (explicitly a pentagon) by expanding the coboundary of the product and comparing term-by-term with the Leibniz sum, without any reduction to a triangular subdivision. This computation will be performed for arbitrary cochains supported on that single face and will confirm that the identity holds independently of any further ordering. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit formulas and independent proofs of Leibniz rule

full rationale

The paper constructs explicit cup-product formulas for arbitrary simple polygonal faces on 2D pseudomanifolds and proves they satisfy the abstract cup-product axioms, including exact Leibniz compatibility with the discrete exterior derivative. These steps are presented as direct extensions of the triangle/quadrilateral case with new closed-form expressions and verification, not as reductions to fitted parameters, self-definitions, or load-bearing self-citations. Associativity on cohomology and the vanishing of the cochain-level error under refinement are separately analyzed as consequences of the construction. No equation is shown to equal its own input by construction, and the central claims rest on the provided formulas and proofs rather than imported uniqueness theorems or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the standard definition of an abstract cup product and the properties of discrete exterior derivative on pseudomanifolds; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The cup product must satisfy the definition of an abstract cup product on cochains.
Invoked when proving compatibility with the discrete exterior derivative.

pith-pipeline@v0.9.0 · 5737 in / 1303 out tokens · 93239 ms · 2026-05-22T18:20:48.570392+00:00 · methodology

discussion (0)

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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Relation between the paper passage and the cited Recognition theorem.

We present explicit formulas for calculation of a cup/discrete wedge product on surface meshes that correspond to 2-dimensional pseudomanifolds, whose 2-dimensional faces are any simple polygons. We rigorously prove that the proposed product satisfies the definition of an abstract cup product; notably, we show that the product is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 5.4. ... (α1 ∪ β1)(f) = ⌊p−1/2⌋∑a=1 (1/2 − a/p) ∑i=0 p−1 α(i)(β((i+a)%p) − β((i−a)%p))

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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Mohamed, A.N

Discrete exterior cal- culus discretization of incompressible navier-stokes equations over sur- face simplicial meshes. Journal of Computational Physics 312, 175–191. doi:10.1016/j.jcp.2016.02.028. Munkres, J.R.,

work page doi:10.1016/j.jcp.2016.02.028 2016
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Addison Wesley Pub- lishing Company, Menlo Park, California

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[1] [1]

No.03CH37475), pp

Discrete exterior calculus for variational problems in computer vision and graphics, in: 42nd IEEE Inter- national Conference on Decision and Control (IEEE Cat. No.03CH37475), pp. 4902–4907 Vol.5. doi: 10.1109/CDC.2003.1272393. Desbrun, M., Kanso, E., Tong, Y.,

work page doi:10.1109/cdc.2003.1272393 2003

[2] [2]

Cambridge University Press, Cambridge

Techniques of geometric topology. Cambridge University Press, Cambridge. Gonzalez-Diaz, R., Jimenez, M.J., Medrano, B., 2011a. Cubical cohomology ring of 3d photographs. International Journal of Imaging Systems and Technology 21, 76–85. doi: 10.1002/ima.20271. Gonzalez-Diaz, R., Lamar, J., Umble, R., 2011b. Cup products on polyhe- dral approximations of 3...

work page doi:10.1002/ima.20271

[3] [3]

Discrete Exterior Calculus. Ph.D. thesis. California Institute of Technology. doi:10.7907/ZHY8-V329. 22 Massey, W.S.,

work page doi:10.7907/zhy8-v329

[4] [4]

Mohamed, A.N

Discrete exterior cal- culus discretization of incompressible navier-stokes equations over sur- face simplicial meshes. Journal of Computational Physics 312, 175–191. doi:10.1016/j.jcp.2016.02.028. Munkres, J.R.,

work page doi:10.1016/j.jcp.2016.02.028 2016

[5] [5]

Addison Wesley Pub- lishing Company, Menlo Park, California

Elements of Algebraic Topology. Addison Wesley Pub- lishing Company, Menlo Park, California. URL: http://www.worldcat. org/isbn/0201045869. Nitschke, I., Reuther, S., Voigt, A.,

work page arXiv

[6] [6]

Springer International Publishing, Cham

Discrete Exterior Calculus (DEC) for the Surface Navier-Stokes Equation. Springer International Publishing, Cham. pp. 177–197. doi: 10.1007/978-3-319-56602-3_7 . Ptackova, L., Velho, L.,

work page doi:10.1007/978-3-319-56602-3_7

[7] [7]

Computer Aided Geometric Design 88, 102002

A simple and complete discrete exterior calcu- lus on general polygonal meshes. Computer Aided Geometric Design 88, 102002. doi: 10.1016/j.cagd.2021.102002. Schubel, M.D., Berwick-Evans, D., Hirani, A.N.,

work page doi:10.1016/j.cagd.2021.102002 2021

[8] [8]

Whitney, H.,

doi: 10.1007/s10444-024-10179-8 . Whitney, H.,

work page doi:10.1007/s10444-024-10179-8

[9] [9]

Princeton University Press, Princeton

Geometric Integration Theory. Princeton University Press, Princeton. doi: 10.1515/9781400877577. Wilson, S.O.,

work page doi:10.1515/9781400877577

[10] [10]

Topology and its Applications 154, 1898–1920

Cochain algebra on manifolds and convergence under refinement. Topology and its Applications 154, 1898–1920. doi: 10.1016/ j.topol.2007.01.017. Zhang, B., Na, D.Y., Jiao, D., Chew, W.C.,

work page 1920

[11] [11]

An a-phi formulation solver in electromagnetics based on discrete exterior calculus, in: 2021 Photonics and Electromagnetics Research Symposium (PIERS), pp. 2394–

work page 2021

[12] [12]

doi: 10.1109/PIERS53385.2021.9694901. 23

work page doi:10.1109/piers53385.2021.9694901 2021