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Analytical properties of polygonal Stokes and BGG Hessian complexes with application to Kirchhoff--Love plates

Daniele A. Di Pietro, Jérôme Droniou, Kaibo Hu, Arax Leroy

arXiv:2609.10150v1math.NAcs.NA

TL;DR

The paper addresses numerical approximation of the Kirchhoff–Love plate problem on general polygonal meshes. It constructs a discrete Hessian complex through BGG and establishes analytical properties of the underlying complexes. The resulting scheme is coercive and converges at order h^(k+1).

  • Problem

    The paper develops an arbitrary-order numerical method for the Kirchhoff–Love plate problem on general polygonal meshes.

    Source: §1 · p. 1

  • Method

    A discrete Hessian complex is derived through BGG from discrete Stokes and de Rham complexes, whose consistency and Poincaré properties are analyzed.

    Source: §1 · pp. 1, 3

  • Results

    The scheme is coercive and converges at order h^(k+1) in the natural energy norm for polynomial degree k ≥ 0.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The discrete Hessian complex provides a polygonal-mesh scheme whose cohomology is isomorphic to that of the continuous complex.

    Source: §1 · p. 2

    The cited analysis assumes u∈C0(Ω)∩H^(k+4)(T_h) and Δu∈H^(k+4)(T_h), with additional elementwise regularity for tensor fields.

    Source: §5.2 · p. 17

Abstract

from arXiv · show

We develop and analyse a new arbitrary-order method for the Kirchhoff--Love plate problem on general polygonal meshes. The construction relies on a discrete Hessian complex obtained through the Bernstein--Gelfand--Gelfand construction, which stacks a Stokes complex on top of a tensorised de Rham complex. A key ingredient of the analysis is a detailed study of the former. We specifically establish primal and adjoint consistency estimates, as well as uniform Poincar\'e inequalities, and use these results to derive the analytical properties of the resulting discrete Hessian complex. The proposed method is then shown to be coercive and to converge with order $k+1$ with respect to the mesh size, where $k\ge 0$ is the polynomial degree of the complex. Numerical experiments on polygonal meshes confirm the theoretical convergence rates.

1 Introduction

The paper develops an arbitrary-order Kirchhoff–Love plate scheme for general polygonal meshes using a discrete Hessian complex derived by BGG. It also studies the analytical properties of the underlying discrete complexes and proves the scheme’s coercivity and convergence.

1 Introduction

  • The BGG construction stacks a Stokes complex with enhanced regularity above a tensorised de Rham complex, yielding the Hessian complex linked to thin-plate models.The paper follows the BGG complex to construct its plate method.Source: §1 · pp. 1–2
  • The paper focuses on clamped boundary conditions, since natural boundary conditions involve technicalities and practical applications often use clamped plates.This requires replacing each space in the complex with its zero-trace subspace.Source: §1 · p. 2
  • For polynomial degree k ≥ 0, the polygonal-mesh scheme is coercive and converges at order h^(k+1) in the natural energy norm.Its discrete Hessian complex has cohomology isomorphic to that of the continuous complex.Source: §1 · p. 2
  • Polygonal alternatives include C1-conforming and non-C1 Virtual Element methods, Hybrid High-Order schemes, and a mixed-formulation method based on the discrete de Rham paradigm.The cited C1-conforming Virtual Element method achieves comparable convergence rates despite using a lower-degree Hessian reconstruction.Source: §1 · p. 3
  • The analysis establishes primal and adjoint consistency estimates and uniform Poincaré-type inequalities for the discrete Stokes and Hessian complexes.These analytical properties support the analysis of the proposed scheme.Source: §1 · p. 3

2 The Hodge Laplacian operator of the Hessian complex

This section formulates the Hodge–Laplace problem for the Hessian complex, including its weak and mixed forms. It relates the unrestricted formulation to natural plate boundary conditions and its homogeneous-boundary version to the clamped problem.

2 The Hodge Laplacian operator of the Hessian complex

  • The section sets the problem in an abstract Hilbert complex with adjoint operators and L2-like inner products on its spaces.The closed, densely defined operators provide the adjoints used in this framework.Source: §2 · p. 3
  • The mixed formulation introduces ω = D* to avoid explicitly defining adjoint-operator domains, which embed boundary conditions.The operators in the complex are closed and densely defined, giving meaning to their adjoints.Source: §2 · pp. 3–4
  • For the first part of the Hessian complex, X_k = H2(Ω), X_(k−1) = {0}, and D_k = ∇2 yield the standard weak Kirchhoff–Love formulation with natural boundary conditions.The load functional is F(q) = (f, q) in L2(Ω).Source: §2 · p. 4
  • Replacing H2(Ω) with H2,0(Ω) gives the homogeneous-boundary Hodge–Laplace problem, corresponding to the Kirchhoff–Love problem with essential, clamped conditions.The homogeneous version uses zero-trace subspaces of H2(Ω) and Hrot(Ω, S).Source: §2 · p. 4

3 Discrete setting

The section builds degree-k discrete spaces and operators on polygonal meshes, then assembles them through a discrete BGG construction into a Hessian complex. Serendipity operators and potential reconstructions provide polynomial consistency, with the richer boundary representation supporting a higher-degree Hessian reconstruction.

3.1 Mesh and inequalities up to a constant

  • The mesh comprises polygonal elements, straight edges, and vertices, under compatibility conditions linking each element boundary to its edges.The mesh size is the maximum element diameter.Source: §3.1 · p. 4

3.3 Discrete BGG diagram

  • The discrete BGG construction combines a Stokes complex with a tensorised de Rham complex to form the discrete Hessian complex.The bottom row uses a serendipity discrete de Rham complex of degree k+1; the top row is the Stokes complex.Source: §3.3 · pp. 5, 7
  • The richer boundary representation permits a Hessian reconstruction one degree higher than the cited C1-conforming VEM, while the construction also starts at k=0.The lower-order schemes are described as less expensive.Source: §3.3 · p. 6

3.4 Serendipity operators

  • Serendipity operators reduce discrete-space dimension while preserving polynomial consistency; the curl reconstruction exactly recovers polynomials of degree at most k.The reconstruction uses the interpolated information in the discrete Hrot space.Source: §3.4 · pp. 8–9

3.5 Potential reconstruction on H^(k+1)

  • The potential reconstructions on H^(k+1) are polynomially consistent at their respective degrees, while the P1 reconstruction is consistent only to degree k.The latter limitation is stated in Lemma 2.Source: §3.5 · p. 9

3.6 Potential reconstruction on H^k

  • The excerpt says this would allow reconstruction of a potential with primal consistency properties up to degree k+3, while its adjoint consistency properties seem limited to degree k+1.The authors detail this in Appendix B.Source: §3.6 · p. 10
  • The potential reconstruction formula extends to all polynomial vector fields through a decomposition into divergence-free and complementary polynomial spaces.The extension follows from the discrete Stokes complex property.Source: §3.6 · p. 10

3.7 L2-like norms and scalar products

  • The local L2-like norms arise from inner products, and their global counterparts are assembled from local contributions.Uniform norm equivalences and boundedness of local DS(k) operators are stated.Source: §3.7 · pp. 10–11

4 Analytical properties of the discrete Stokes complex

The section establishes consistency estimates and Poincaré inequalities for the discrete Stokes complex. These results support the analytical treatment of the discrete Hessian complex, including variants with homogeneous boundary conditions.

4.1 Primal and adjoint consistency

  • For every T ∈ T_h and q ∈ H^(k+2)(T), the potential and gradient reconstructions satisfy a consistency estimate bounded by h_T^(k+2)|q|_(H^(k+2)(T)).Source: §4.1 · p. 12
  • The section establishes primal and adjoint consistency estimates for the discrete Stokes operators and products.The listed estimates include the R1 reconstruction, rot product, and G2 adjoint consistency.Source: §4.1 · p. 12

4.2.1 Poincaré–Wirtinger inequalities

  • Poincaré inequalities are established for both operators in the DS(k) complex using an abstract framework and preliminary estimates.The proof transfers inequalities from the DDR(0) complex through extension and reduction maps that are uniformly continuous in h.Source: §4.2.1 · p. 13, §4.2.3 · p. 15

4.2.4 Poincaré inequalities

  • For homogeneous boundary conditions, the Poincaré argument replaces the first two top-row spaces with zero-boundary DS(k) spaces and uses corresponding DDR(0) spaces.The extension and reduction maps remain valid, allowing the abstract framework to yield the stated inequalities.Source: §4.2.4 · pp. 15–16

5 Application to the Kirchhoff–Love plate model

The discrete plate scheme uses a Hessian-complex formulation and is analyzed through Poincaré inequalities and adjoint consistency. Numerical tests report the predicted energy-error rate and benefits from local refinement and higher polynomial degree.

5.1 Discrete problem

  • The discrete scheme uses the Hodge Laplacian associated with the discrete Hessian complex, incorporating boundary conditions through a replacement of the discrete space.The formulation is introduced for the discrete problem in (5.1).Source: §5.1 · p. 16

5.2 Analytical properties of the discrete Hessian complex

  • Poincaré inequalities show that the discrete Hessian operator induces a norm on the stated discrete space.The error-estimate proof also uses adjoint consistency and a Poincaré inequality, under regularity assumptions on u and Δu.Source: §5.2 · pp. 16–18

5.3 Numerical results

  • Locally refined meshes around the Dirac source create hanging nodes, which the polygonal scheme accommodates by splitting physical edges into polygonal cells.The test applies a Dirac force at (0.325, 0.732).Source: §5.3 · pp. 18, 20
  • h^(k+1) is the observed energy-error convergence rate for the trigonometric solution across degrees k=0–3.The reported rate agrees with the expected convergence behavior.Source: §5.3 · p. 18, Figure 2 · p. 19
  • For a target accuracy, locally refined meshes need fewer degrees of freedom and reach the reference energy faster than other mesh families.On hexagonal meshes, higher polynomial degree improves approximation for a fixed number of degrees of freedom, compensating for the lower-order inefficiency.Source: §5.3 · p. 20

A Abstract framework for the transfer of Poincar´e inequalities

The appendix develops a cochain-map framework for transferring Poincaré inequalities between complexes. It extends transfer to the poorer-to-richer direction needed for the degree-0 DDR and degree-k DS complexes, under an additional assumption.

A Abstract framework for the transfer of Poincar´e inequalities

  • The abstract setting considers Hilbert complexes connected by continuous cochain maps and transfers Poincaré inequalities between them.The added assumption controls the discrepancy from mapping to the poorer complex and back.Source: A Abstract framework for the transfer… · p. 21
  • The earlier framework transfers properties only from a richer complex to a poorer one, motivating a more general transfer framework.Its condition requires that no information be lost when returning through the richer complex.Source: A Abstract framework for the transfer… · p. 21
  • The framework enables transfer from the degree-0 DDR complex to the richer degree-k DS complex, with an added assumption compensating for information loss.In practical cases, the added assumption essentially reduces to local Poincaré inequalities for the top sequence.Source: A Abstract framework for the transfer… · p. 21
  • Under Assumption 21 and a Poincaré inequality on the bottom sequence, the top sequence also satisfies a Poincaré inequality.The transfer result depends on the cochain-map framework and the stated assumption.Source: A Abstract framework for the transfer… · p. 22
  • The reconstruction has higher primal consistency, but the corresponding higher adjoint consistency is not established because the needed cancellation does not occur.The missing cancellation prevents completion of the adjoint-consistency proof.Source: A Abstract framework for the transfer… · p. 23

C Notation

The notation is organized around symbols, continuous counterparts, and definitions, with a category for discrete differential operators.

C Notation

  • The notation table is organized by symbol, continuous counterpart, and definition.Source: C Notation · p. 23
  • Discrete differential operators are identified as a notation category.Source: C Notation · p. 23
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