Discrete Poincaré and Bogovskiĭ operators on cochains and Whitney forms

Abstract: Smooth Poincaré operators are a tool used to show the vanishing of smooth de Rham cohomology on contractible manifolds and have found use in the analysis of finite element methods based on the Finite Element Exterior Calculus (FEEC). We construct analagous discrete Poincaré operators acting on cochains and Whitney forms. We provide explicit, constructive realizations of these operators under various assumptions on the underlying domain or simplicial complex. In particular, we provide simple constructions for the discrete Poincaré operators on simplicial complexes which are collapsible and those with underlying domain being star-shaped with respect to a point. We then provide more abstract constructions on simplicial complexes which are discrete contractible and domains which are Lipschitz contractible. We also modify the discrete Poincaré operator on star-shaped domains to construct a discrete Bogovskiĭ operator which satisfies the requisite homotopy identity while preserving homogeneous boundary conditions. Applications arise in the construction of discrete scalar and vector potentials and in the discrete wedge product of Discrete Exterior Calculus (DEC).

• The paper constructs discrete Poincaré operators by dualizing simplicial and singular cone homotopies, yielding exact homotopy identities on cochains and Whitney forms.
• The framework provides explicit constructions for collapsible, star-shaped, discretely contractible, and Lipschitz-contractible complexes, with numerical residuals at machine precision across tested meshes.
• The paper develops a Whitney-form Bogovskiĭ operator that preserves homogeneous boundary conditions, while identifying open questions about boundedness, mesh-independent estimates, and top-degree geometric assumptions.

This paper constructs discrete analogues of the smooth Poincaré operator and Bogovskiĭ operator on simplicial complexes, working both at the level of cochains and of Whitney forms. The central objects are "generalized cone operators" — chain-level homotopies contracting a complex to a vertex or point — whose duals satisfy the homotopy identity $(dP + Pd)\alpha = \alpha$ and thereby certify vanishing of discrete cohomology and enable explicit computation of discrete scalar and vector potentials.

Framework: cone operators and their duals

The authors formalize two parallel notions. A simplicial cone operator $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$ satisfies $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$ for $k\ge 1$, with $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$ for vertices, where $a$ is a contraction vertex. A singular cone operator $\mathrm{Co}^{\Ss}_k:\C_k(X)\to\Ss^{\sm}_{k+1}(X)$ is the analogous object into Lipschitz singular chains, with contraction point $a\in|X|$. Dualizing via the pairing between chains and cochains yields a combinatorial discrete Poincaré operator $P:\C^k\to\C^{k-1}$; dualizing via integration over simplices yields an operator $\mathcal{P}:V_h^k\to V_h^{k-1}$ on Whitney forms, defined by prescribing degrees of freedom as

$\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$0

Both constructions are shown to satisfy the full homotopy identity, including the $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$1 case $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$2 for an appropriate constant projection $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$3. A Whitney-form-based operator transfers to cochains through $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$4, using the commuting properties of the de Rham and Whitney maps. The paper notes that the complex property $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$5 is not verified directly but can be obtained by the standard modification $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$6 following Čap–Hu.

Constructions under specific hypotheses

The paper provides a hierarchy of constructive realizations:

• Collapsible complexes: if $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$7 admits a collapse sequence to a vertex, reversing it gives elementary expansions; the cone operator is defined inductively by setting $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$8 and $\mathrm{Co}_k:\C_k(X)\to\C_{k+1}(X)$9 on each newly added pair $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$0. This recovers operators known independently from Desbrun–Leok–Marsden (whose "generalized star-shaped" condition is equivalent to strong collapsibility, strictly stronger than collapsibility) and from Pitassi–Ghiloni–Specogna's curl-inversion algorithm, whose back-substitution is algebraically identical when the matching induced by the collapse sequence is used. An appendix shows this operator follows naturally from Whitehead's simple-homotopy chain contractions.
• Star-shaped domains: the singular cone $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$1 is a linear singular simplex; the resulting Whitney-form Poincaré operator assigns $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$2 (zero when $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$3 is coplanar with $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$4). By a result in Whitney's Geometric Integration Theory, this operator coincides with the canonical projection of the continuous straight-line Poincaré operator onto Whitney forms. Implementation requires quadrature over regions cutting through mesh elements.
• Discrete contractibility: if there exists a simplicial map $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$5 on an admissible product complex $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$6 with $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$7 and $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$8, then $$\mathrm{Co}_{k-1}\partial\sigma + \partial\mathrm{Co}_k\sigma = \sigma$$9, where $k\ge 1$0 is an explicit extrusion (prism) operator built from staircase triangulations of $k\ge 1$1, is a simplicial cone operator.
• Lipschitz contractible domains: analogously, $k\ge 1$2 using a Lipschitz contraction $k\ge 1$3 is a singular cone operator; for star-shaped domains with the straight-line contraction this reduces exactly to the linear cone above, since all degenerate terms integrate to zero.

The connection between the two viewpoints is transparent: both reduce to integrals of $k\ge 1$4 over the respective cones, and they agree whenever the simplicial contraction $k\ge 1$5 equals the smooth contraction $k\ge 1$6 (e.g., on the closed star of a vertex).

The discrete Bogovskiĭ operator

On star-shaped domains, the plain cone operator does not preserve homogeneous boundary conditions. The authors correct this by subtracting an integral over an infinite cone $k\ge 1$7 (a map from the nonnegative orthant), defining

$k\ge 1$8

Because $k\ge 1$9 vanishes outside $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$0, the infinite-cone term contributes nothing inside $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$1 but cancels boundary degrees of freedom, so $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$2 maps $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$3 and satisfies $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$4 for $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$5. For the top degree $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$6, an additional assumption is required: the contraction point $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$7 must not lie on any facet of the complex. Under this assumption, a geometric lemma shows that $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$8, which vanishes after subtracting the mean of $\partial_1\mathrm{Co}_0(\sigma) = \sigma - a$9, restoring the homotopy identity. This assumption is mild — a local perturbation of the mesh restores it — but it is a genuine restriction that the paper states plainly. This is, to my knowledge, the first construction of a discrete Bogovskiĭ-type right inverse acting directly on Whitney forms while preserving homogeneous boundary conditions at the discrete level.

Numerical experiments

Experiments on a structured triangulation of the unit square (star-shaped), a U-shaped non-star-shaped domain, and the Bogovskiĭ operator confirm the theory: for 100 random $a$0-cochains per degree $a$1, the $a$2 residual of the homotopy identities is at machine precision in every case, for all three operator families (Whitney-based, collapse-based, and discrete-contraction-based). Practical integration details are given: Sutherland–Hodgman polygon clipping for cut-cell regions, quadrature along rays as proxies for infinite cones, and explicit piecewise-linear contractions for the U-shaped domain. The experiments also demonstrate computation of discrete vector potentials ($a$3 projected onto piecewise constants) and scalar potentials (from a projected Nédélec field), including boundary-preserving potentials via the Bogovskiĭ operator with $a$4 chosen off-facet.

Limitations and open questions

Several limitations are acknowledged explicitly. Boundedness of the discrete operators is not investigated, unlike the regularized Poincaré/Bogovskiĭ operators of Costabel–McIntosh; establishing discrete Poincaré inequalities with mesh-independent constants for these operators remains open. The star-shaped and Lipschitz-contractible constructions require numerical integration over cut or curved regions, which can be delicate, though prior work on conservative interpolation addresses parts of this. Finding a discrete contraction for an arbitrary discretely contractible complex may be difficult; the explicit construction given here works only for strongly collapsible complexes, a strictly smaller class than collapsible ones. Finally, whether the top-degree Bogovskiĭ construction can be modified to remove the off-facet assumption on $a$5 without remeshing is not addressed.

Conclusion

The paper supplies a unified cone-operator framework from which discrete Poincaré and Bogovskiĭ operators on cochains and Whitney forms follow systematically, with explicit constructions spanning collapsible complexes, strongly collapsible/discretely contractible complexes, star-shaped domains, and general Lipschitz-contractible domains. The homotopy identities hold exactly (machine precision numerically), yielding direct algorithms for discrete potentials relevant to magnetostatics, eddy-current simulation, discrete de Rham consistency liftings, and the $a$6-structure construction on DEC cochains.