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An arbitrary-order BGG-based discrete scheme for the Reissner--Mindlin plate problem on polygonal meshes

Arax Leroy

arXiv:2609.17122v1math.NAcs.NA

TL;DR

Thickness-uniform stability and accuracy are central challenges for Reissner–Mindlin plate discretisation, particularly on flexible polygonal meshes. The paper constructs an arbitrary-order BGG-based scheme using discrete H2 displacement spaces and proves convergence with a lowest-order locking-free estimate. Numerical comparisons report an enhanced-continuity benefit over DDR–HHO at lowest order.

  • Problem

    Reissner–Mindlin discretisations must remain stable and accurate as plate thickness tends to zero, while polygonal meshes offer flexibility beyond typical triangular or quadrangular mesh restrictions.

    Source: §1 · p. 1

  • Method

    The paper constructs an arbitrary-order polygonal-mesh scheme from a discrete BGG twisted complex, using a discrete H2-based transverse-displacement construction and a modified stabilisation supported by a discrete Korn inequality.

    Source: §1 · p. 2, §2.3 · pp. 5–6

  • Results

    The modified formulation has arbitrary-order consistency and convergence estimates, and its lowest-order error estimate is uniform in plate thickness, establishing locking-free behavior.

    Source: §1 · p. 2

  • Takeaways & Limitations

    At lowest order, DDR–BGG energy error converges at approximately 1.5 on hexagonal and Cartesian meshes, compared with a rate close to one for DDR–HHO.

    Source: §6.3 · p. 20

    The direct Hodge–Laplacian formulation’s required discrete Korn estimate remains an open question, so the fully analysed scheme uses a modified stabilisation.

    Source: §3.1 · pp. 9–10

Abstract

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We design and analyse an arbitrary-order numerical scheme for the Reissner--Mindlin plate problem on general polygonal meshes. The scheme is derived from the Hodge--Laplacian associated with a discrete Bernstein--Gelfand--Gelfand (BGG) twisted complex and exploits a discrete $H_2$-based construction for the transverse displacement. We establish a discrete Korn inequality for the underlying Discrete de Rham method (DDR) spaces and prove the convergence of the method. At the lowest order, the analysis yields an error estimate that is uniform with respect to the plate thickness, showing that the scheme is locking-free. Numerical experiments on several families of polygonal meshes support the theoretical results and illustrate the benefits of the enhanced continuity of the transverse displacement discretisation.

1 Introduction

The paper addresses thickness-robust Reissner–Mindlin discretisation on flexible polygonal meshes by constructing a scheme from a discrete BGG twisted complex. It relates the continuous plate equations to a Hodge–Laplacian and develops a modified stabilisation for which coercivity can be proved.

1.1 Continuous PDEs

  • The continuous Reissner–Mindlin model describes moderately thick-plate bending with transverse displacement and rotation unknowns, accounting for transverse shear through first-order derivatives.Its weak formulation has bending and shear contributions and a loading term.Source: §1 · p. 1, §1.1 · p. 3

1 Introduction

  • A central design challenge is stability and accuracy uniform in thickness, since failure to reproduce the limiting shear constraint can cause shear locking.The constraint relates rotation to the gradient of transverse displacement as thickness tends to zero.Source: §1 · p. 1
  • Polygonal meshes allow flexibility such as local refinement with hanging nodes and mesh agglomeration, while prior polygonal Reissner–Mindlin methods have pursued thin-plate robustness.Existing approaches include VEM, Weak Galerkin, DDR–HHO, and HDG methods.Source: §1 · pp. 1–2
  • The first scheme based on a BGG construction for Reissner–Mindlin plates uses a polygonal discrete twisted complex whose operators contain the model’s shear and bending operators.The complex combines a discrete Stokes complex with a tensorised DDR complex.Source: §1 · p. 2
  • The direct Hodge–Laplacian formulation stabilises the discrete gradient of rotation, but its coercivity analysis needs a discrete Korn inequality that is not straightforward to establish.The modified formulation instead stabilises rotation degrees of freedom directly, enabling an adapted discrete Korn inequality and a coercivity proof, though it loses the strict Hodge–Laplacian interpretation.Source: §1 · p. 2
  • At lowest order, the modified formulation has an error estimate uniform in plate thickness, establishing locking-free behavior; the paper also derives arbitrary-order consistency and convergence estimates.Numerical experiments compare both formulations in the thin-plate regime and compare the BGG construction with DDR–HHO.Source: §1 · p. 2

1.2 From twisted complex to Reissner–Mindlin equations

  • The twisted complex links the Reissner–Mindlin weak formulation to a Hodge–Laplacian: with a suitable weighted product and source, the associated problem recovers that formulation.The paper uses the first space of the twisted complex for this connection.Source: §1.2 · p. 4

2 Discrete setting

The discrete setting builds polygonal-mesh spaces and operators around a higher-regularity BGG complex. It uses a discrete H2 counterpart for transverse displacement, motivated by the thin-plate limit and the Hessian energy of the limiting Kirchhoff–Love model.

2.1 Mesh and notation

  • The mesh is a polygonal partition with element, edge, and vertex sets; its compatibility assumptions allow a straight boundary portion to be divided into several mesh edges, as in non-conforming local refinement.A regular mesh sequence is assumed, with hidden constants independent of meshsize and plate thickness.Source: §2.1 · pp. 4–5

2.2 Polynomial spaces

  • Polynomial spaces comprise restrictions of two-variable polynomials of total degree at most ℓ, alongside entitywise continuous and broken mesh polynomial spaces.The convention sets P_ℓ(Y) to zero for ℓ ≤ −1.Source: §2.2 · p. 5

2.3 Discrete spaces and operators

  • The BGG construction supplies a discrete counterpart of a higher-regularity twisted complex, with discrete H2 and H1 vector spaces playing roles of the continuous H1 and Hrot spaces.The construction recalls spaces and operators relevant to the Reissner–Mindlin scheme.Source: §2.3 · p. 5
  • The discrete H2 displacement space is motivated by expected H2 regularity on convex domains and by the thin-plate limit, where the Kirchhoff constraint makes the limiting energy involve the Hessian of displacement.The discrete scalar products combine L2 products of potential reconstructions with consistent stabilisation terms.Source: §2.3 · p. 6, §2.4 · p. 9
  • The discrete operators include gradients and potential reconstructions built from local and entity degrees of freedom, with interpolators representing smooth functions by polynomial vectors.The construction also adopts the non-serendipity complex; its simpler operators suffice because only the first part of the twisted complex is used.Source: §2.3 · pp. 6–8

2.4 Discrete scalar products

  • Discrete scalar products combine L2 products of potential reconstructions with consistent stabilisation terms, whose global forms sum local contributions.The scalar products are defined for the discrete spaces used in the construction.Source: §2.4 · p. 9

3 Discrete schemes

The section presents a Hodge–Laplacian scheme and a variant that changes only the bending stabilisation. The variant retains consistency while providing stabilisation suited to the available discrete Korn inequality for coercivity analysis.

3.1 Discrete scheme as a Hodge–Laplacian problem

  • The Hodge–Laplacian scheme has the expected convergence order k + 1, but its coercivity analysis requires a discrete Korn inequality not established for its gradient stabilisation.The required estimate remains an open question.Source: §3.1 · pp. 9–10

3.2 Variant of the bilinear form 𝑎ℎ

  • The variant changes only the stabilisation component of the bending form; its spaces, consistent bending term, shear form, and load remain unchanged.Source: §3.2 · p. 10
  • Replacing the stabilisation preserves the consistency error order because both stabilisations provide the required approximation order and the other contributions are identical.Source: §3.2 · p. 10
  • The variant stabilises the rotation degrees of freedom, enabling the discrete Korn inequality used for coercivity, but it loses the strict Hodge–Laplacian interpretation.Source: §3.2 · p. 10

4 Analysis

The analysis establishes a discrete Korn inequality for the DDR spaces and uses it to support coercivity and convergence analysis of the modified scheme. The convergence result is stated under additional solution regularity assumptions.

4.1 Discrete Korn inequality

  • The discrete Korn inequality is obtained by combining local equivalence between discrete gradients and reconstructed-potential gradients with a discrete Korn inequality for those potentials.The analysis uses stabilisation acting directly on rotation degrees of freedom to control nonconformity and element-wise rigid-body modes.Source: §4.1 · p. 11

4.2 Coercivity

  • The discrete Korn inequality is a key ingredient in proving coercivity of the modified scheme.Source: §4.2 · p. 13
  • The convergence estimate is derived from the scheme’s coercivity and consistency using the third Strang lemma, under additional regularity assumptions on displacement and rotation.A remark allows weaker regularity with a modified right-hand-side bound.Source: §4.2 · p. 13

5 Locking-free error estimate

At lowest order, the error estimate is uniform in plate thickness under the stated convex-domain regularity conditions, establishing locking-free behaviour. The proof adapts lifting operators to the discrete spaces used here.

5 Locking-free error estimate

  • At k = 0, the error estimate is locking-free on convex domains: its bound is uniform as plate thickness t tends to zero.The estimate includes t multiplying the rotation’s H1-seminorm term, which remains bounded under the stated standard regularity estimates.Source: §5 · p. 15

5.1 Lifting

  • The proof adapts lifting operators to the present discrete spaces, including a local reduction operator for the lifting construction.The local reduction is described as discarding components corresponding to derivatives.Source: §5 · p. 15, §5.1 · p. 16

6 Numerical results

The numerical tests examine mesh convergence, thin-plate robustness, and the impact of enhanced continuity, using two scheme variants and comparisons with DDR–HHO. Across the reported cases, DDR–BGG shows the expected convergence and robustness, with improved jump behavior in the enhanced-continuity comparisons.

6 Numerical results

  • The tests use a smooth polynomial solution for mesh convergence and a solution with stronger thickness dependence in its shear strain for thin-plate robustness; both scheme variants are tested.Computations are performed on the unit square unless otherwise stated.Source: §6 · p. 17

6.1 Polynomial solution

  • For the smooth polynomial test at fixed thickness t=1, the expected order k+1 convergence rate is observed in the non-HL scheme.Figure 1 presents the error against mesh size for this test.Source: §6.1 · p. 17, Figure 1 · p. 18

6.2 Analytical solution

  • For the thin-plate test at k=0, the modified scheme exhibits robustness with respect to thickness and the established first-order convergence rate.The test solution has bounded L2 shear strain while its higher-order derivatives grow as the plate becomes thinner; Figure 3 reports the results.Source: §6.2 · p. 20, Figure 3 · p. 21

6.3 Numerical impact of the 𝐻2-based construction

  • At k=0, DDR–BGG has an observed energy-error rate of approximately 1.5 on hexagonal and Cartesian meshes, versus a rate close to one for DDR–HHO [18].The comparison applies to both choices of the DDR–BGG scheme.Source: §6.3 · p. 20
  • For triangular analytical meshes at k=1, rotation-jump convergence orders are 2 for DDR–HHO and 3 for DDR–BGG; for hexagonal polynomial meshes at k=0, they are 1 and 2.Figure 4 plots the scaled jump of the reconstructed rotation against the square root of total DOFs.Source: §6.3 · p. 21, Figure 4 · p. 22
  • 0.89 vs 17.65 maximum local scaled jump gives DDR–BGG a reduction of about 20-fold, while its space has 9,882 DOFs versus 3,760 for DDR–HHO.This discontinuous-load test uses k=0 and t=1 on a hexagonal mesh; the smoother DDR–BGG reconstruction is a qualitative comparison, not an error measurement.Source: §6.3 · pp. 21–22

7 Thin-plate limit of the discrete schemes

The section examines the fixed-mesh thin-plate limit of both discrete scheme choices and their relation to Kirchhoff–Love schemes.

7 Thin-plate limit of the discrete schemes

  • The discrete energy is uniformly bounded as the plate thickness tends to zero, with the shear contribution weighted by t^-2.The analysis keeps the mesh size fixed while taking this limit.Source: §7 · p. 22
  • For choice (3.3), the formal limit uses a discrete H2,0(T_h) displacement space and a discrete Hessian operator.The passage identifies the discrete space as a counterpart of H2,0(Ω).Source: §7 · p. 22
  • With C = Id and no serendipity reduction of cell unknowns, scheme (7.1) coincides with the Kirchhoff–Love scheme introduced in [16].Source: §7 · p. 22
  • For choice (3.6), the discrete scheme is a modified scheme for the Kirchhoff–Love model and can be proved to have the same convergence rate as the previous one using techniques of [16].Source: §7 · p. 23
  • The relation between the discrete Reissner–Mindlin and Kirchhoff–Love schemes links the twisted complex to the discrete Hessian complex of [17, Eq. (6.1)].The link is also noted at the continuous level in.Source: §7 · p. 23

A Notation

The notation section identifies a symbol table and discrete differential operators.

A Notation

  • The section labels a table for symbols, their continuous counterparts, and definitions.Source: A Notation · p. 23
  • The section separately labels the discrete differential operators.Source: A Notation · p. 24
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