Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Stability and interpolation estimates of Hellinger-Reissner virtual element spaces

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that two explicit stabilizations for Hellinger–Reissner virtual element spaces satisfy two-sided stability bounds and interpolation estimates, with constants depending only on the mesh regularity parameter and the…

desk verdict The main result is probably right, but the proof's lower bound for α_* inherits a μ-dependence from Theorem A.7, so the claimed μ-independence is not established. read the letter →

arxiv 2502.06286 v1 pith:2VJBVIBR submitted 2025-02-10 math.NA cs.NA

Michele Botti , Lorenzo Mascotto , Giuseppe Vacca , Michele Visinoni This is my paper · ORCID
classification math.NAcs.NA MSC 65N1265N30
keywords virtualelementmethodHellinger-Reissnerprinciplestabilityestimateinterpolationlinearelasticitypolytopicmeshessymmetricstresstensors
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper targets Hellinger–Reissner virtual element methods, mixed finite-element schemes for linear elasticity whose stress unknowns are defined implicitly through polynomial projections rather than in closed form. Because of that implicitness, the discrete bilinear forms must be built from a consistency term plus a stabilization term, and up to now the correct scaling of that stabilization was mostly heuristic. The paper proves that two concrete stabilizations, one based on projections and one based on the degrees of freedom themselves, satisfy two-sided stability bounds of the form (20) on the projection kernel, with constants depending only on the mesh regularity parameter $\rho$ and the polynomial degree $p$. It then derives interpolation estimates for the stress and its divergence with the same explicit dependence. These bounds are what turn a plausible mixed method into one with rigorous error control on general polytopic meshes.

What carries the argument

The central object is the kernel of the projector $\Pi^T_p$, where $T_p(K)$ is the space of polynomial stress tensors of the form $C\nabla^S(q_{p+1})$ contained in the Hellinger–Reissner space; the stabilization acts on the complement of this kernel, and the whole argument rests on controlling $\|\tau_h\|^2_{0,K}$ by the stabilization terms. The load-bearing identity is $\nabla^S(\operatorname{div}\tau_h)=\nabla^S(\Pi^\perp_{\mathrm{RM}}\operatorname{div}\tau_h)$ for $\tau_h$ in the Hellinger–Reissner space, which follows because symmetric gradients of rigid body motions vanish and which converts a divergence norm into boundary tractions plus a computable projection. Around that identity the proof assembles polynomial inverse inequalities, the $\mathrm{H}(\operatorname{div})$ trace inequality, and the local mixed well-posedness result of the appendix, and Proposition 3.2 transfers the result from the projection-based stabilization to the dofi-dofi one.

What would settle it

For a fixed polynomial degree $p$, compute the smallest and largest eigenvalues of the generalized eigenproblem (40) for a family of elements that satisfy the stated regularity assumptions but approach its boundary, such as facets whose diameter shrinks while the element diameter stays fixed; if the ratio of largest to smallest eigenvalue is unbounded, the claimed $\rho$-only dependence fails. The tables in Section 5 already perform this computation approximately on hourglass-shaped and trapezoidal elements, so a refined degenerating sequence with explicitly controlled $\rho$ would settle the claim.

Watch

Extended reading notes

Core claim

On shape-regular polytopic three-dimensional meshes, the stabilization $S_K$ in (22) and the stabilization $\widetilde S_K$ in (25) satisfy the stability bounds (20): there are constants $\alpha_*$ and $\alpha^*$ depending only on the mesh regularity parameter $\rho$ and the degree $p$ such that $\mu^{-1}|K|\alpha_*\|\tau_h\|^2_{0,K}\le S_K(\tau_h,\tau_h)\le \mu^{-1}|K|\alpha^*\|\tau_h\|^2_{0,K}$ for every $\tau_h$ in the kernel of the projector $\Pi^T_p$, and the bounds hold even beyond that kernel. The proof of the lower bound runs through an integration-by-parts identity, the fact that the symmetric gradient of a rigid body motion vanishes, polynomial inverse inequalities, and a local Hellinger–Reissner well-posedness result with constants explicit in the star-shapedness ratio; the upper bound uses polynomial inverse and trace inequalities. Combining these bounds with polynomial approximation results yields the interpolation estimates (36) and (37), with constants depending only on shape regularity and polynomial degree. The numerical experiments in Section 5 track the constants on badly shaped elements and show degradation in degenerating geometries, which is consistent with the $\rho$-dependence in the theory.

Load-bearing premise

The proof assumes every mesh element is star-shaped with respect to a ball of radius at least $\rho$ times its diameter, with facets star-shaped with respect to disks of radius at least $\rho$ times the facet diameter and edges of comparable size relative to their facets; if this geometric control is lost, the claimed dependence of the constants only on $\rho$ and $p$ collapses.

Editorial extensions

If this is right

  • The discrete Hellinger–Reissner method's local bilinear form is coercive and continuous on the projector kernel with constants independent of the Lamé parameters, so well-posedness of the linear system no longer relies on a heuristic choice of stabilization.
  • The interpolation estimates (36) and (37) give optimal-order error bounds in the $L^2$ norm and in the $L^2$ norm of the divergence, with constants depending only on $\rho$ and $p$ on polytopic meshes.
  • Since Proposition 3.2 proves the two stabilizations are equivalent, either the projection-based or the dofi-dofi form can be used on shape-regular meshes with the same stability guarantees.
  • The stability bounds hold for all tensors in the Hellinger–Reissner space, not only those in the projection kernel, which is stronger than the minimal requirement in the virtual element stability framework.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same divergence-control argument should extend to two dimensions, to the lowest-order Hellinger–Reissner element with no interior degrees of freedom, and to natural or mixed boundary conditions, since the proof only uses local identities and inverse inequalities; the paper states the two-dimensional case is a minor modification but does not prove these other variants.
  • Editorial inference: the numerical degradation on hourglass-shaped and trapezoidal elements suggests that mesh-quality indicators based on the star-shapedness ratio could predict when the stabilization constants worsen, a connection the paper does not explore.
  • Editorial inference: a testable extension is whether replacing the standard projection-based stabilization with a mesh-adapted one restores uniform constants on degenerating families, which is the direction the conclusion identifies as future work.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves stability and interpolation estimates for Hellinger–Reissner virtual element spaces in three dimensions (with the two-dimensional case said to follow similarly). Two stabilizations are considered, a projection-based one in (22) and a dofi-dofi one in (25), and the authors claim that both satisfy the scaling bounds (20) with constants depending only on the mesh-regularity parameter ρ and the polynomial degree p, not on the Lamé parameters or the element size. These stability bounds are then used in Theorem 4.3 to derive interpolation estimates in the L2 norm and in the divergence norm. The paper also contains a numerical investigation of the stability constants on sequences of badly shaped polytopes and with increasing polynomial degree. Appendix A develops well-posedness and a priori estimates for the mixed linear elasticity problem with essential boundary conditions, with constants made partially explicit in terms of the aspect ratio of the domain.

Significance. If the central claims are correct, the paper fills a real gap in the virtual element literature: the correct scaling of HR virtual element stabilizations, previously supported only by heuristic arguments, would be rigorously established on shape-regular polytopic meshes, with constants tracked in terms of the mesh regularity parameter and the polynomial degree. The treatment of both a projection-based and a dofi-dofi stabilization, the explicit dependence on ρ and p, and the honest numerical experiments on degenerate geometries are clear strengths. The reliance on external results [11] and [23] for the inf-sup, Korn, and Bogovskii-type inequalities is legitimate and does not constitute circularity, since the main stability and interpolation arguments use those results as tools rather than reducing to them. However, a load-bearing gap in the proof of Proposition 3.1 prevents the paper, as written, from establishing the advertised μ-independence of the stability constants; the numerical experiments fix μ=1 and cannot detect this issue. The overall framework and most of the technical chain are nevertheless sound and the defect appears repairable.

major comments (2)
  1. [§3.1, proof of Proposition 3.1, Eq. (24)] The estimate (24) is derived by invoking Theorem A.7 for the local problem (12), but Theorem A.7 states that its constant cHR depends on the Lamé parameter μ, not only on the mesh-regularity parameter ρ. Consequently the constant cST in (24) inherits a μ-dependence, and the subsequent chain of inequalities establishes only that the lower stability constant α_* in (20) is independent of λ and h_K, not that it is independent of μ. This is load-bearing, because the abstract and the statement of (20) explicitly promise independence of λ and μ. The proof would need a stress-only a priori estimate of the form ∥τ_h∥_{0,K} ≤ C(ρ,p)(∥div τ_h∥_{0,K} + ∥τ_h n_K∥_{0,∂K}) with C independent of μ for the local problem (12); no such estimate is supplied, and Theorem A.7 as stated cannot provide it. The numerical experiments in Section 5 keep μ=1 throughout and therefore cannot detect this issue.
  2. [Appendix A, Proposition A.6] The statement of Proposition A.6 claims that the coercivity constant α*_0 in (61) is independent of the ratio hΩ/ρΩ, but the proof concludes α*_0 = 1/(2μ C_d), where C_d is the constant from Lemma A.4 and depends on that ratio. The proposition is therefore false as written. If the ratio-independence is not needed for the main argument, the statement should be corrected to record the dependence on the ratio (and on μ); if it is needed, a separate proof must be given. Moreover, the sentence in the proof of Theorem A.7 stating that the relevant constants are independent of hΩ and ρΩ is inconsistent with Proposition A.5, whose inf-sup constant β*_0 depends on the ratio.
minor comments (4)
  1. [§2, Eq. (20)] The notation `µ^{-1} |K α_*` is missing a closing absolute-value bar; it should read μ^{-1}|K| α_* throughout, including in (22) and (25).
  2. [§3.2, proof of Proposition 3.2, upper bound part 2] In the display following (31), the norm ∥Π⊥_RM div τ_h∥_{0,F} should be ∥Π⊥_RM div τ_h∥_{0,K}; the current subscript is inconsistent with the preceding inequality.
  3. [§5] Since the claimed μ-independence of the stability constants is central, the numerical study would be more informative if μ were varied over several orders of magnitude; the present tests fix μ=1 and vary only λ or the geometry.
  4. [§4, Theorem 4.3] The statement of Theorem 4.3 says the constants depend only on the shape of K, while the proof tracks dependence on ρ and p; the wording should be harmonized to avoid an apparent mismatch.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central stability and interpolation derivation is not equivalent to its own inputs; the μ-dependence mismatch in Proposition 3.1 is a correctness gap, not a circular reduction.

full rationale

The derivation chain is self-contained in the relevant sense. Proposition 3.1 obtains the lower stability bound from the a priori estimate (24), which is presented as a consequence of Theorem A.7; Theorem A.7 is proved in the appendix using standard inf-sup, Korn, and Nečas–Lions tools, with cited inequalities from [11], [19], and [23]. These prior results are parameter-free theorems whose assumptions do not include the VEM stability or interpolation claims, so citing them, including the authors' own [11] and coauthor [31], is independent support rather than circularity. The interpolation estimates in Theorem 4.3 use the stability bounds as proved and standard polynomial approximation, without re-imposing the target result. I flag one non-circular correctness risk: Theorem A.7 explicitly says its constant cHR depends on μ, while Proposition 3.1 asserts a constant cST depending only on ρ; the written proof does not display the cancellation needed to justify the μ-independence of α_* in (20). This is an omitted justification, not a self-referential reduction, and it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the constants in the estimates are proven to depend on rho and p only through known inverse, approximation, and trace inequalities plus explicit-constant external results. No new entities, forces, or dimensions are introduced.

assumptions (5)
  • domain assumption Mesh shape regularity: each element K is star-shaped w.r.t. a ball of radius >= rho h_K; each facet is star-shaped w.r.t. a disk of radius >= rho h_F; h_F >= rho h_K; each edge has h_e >= rho h_F.
    Used throughout Section 3 and in Theorem A.7 to control the constants c_inv, c_tr, c_ST. The numerical tests in Section 5 show constants degrade as shapes degenerate.
  • domain assumption Lame parameters lambda and mu are piecewise constant over each mesh (eq. (7)), so the elasticity tensor C is constant on each element K.
    Ensures T_p(K) contains only polynomials up to order p and that the inclusion (13) holds; needed for the projector Pi^T_p and for the interpolation estimates.
  • domain assumption The exact stress sigma has sufficient regularity to define the interpolant in (32), e.g., sigma in H^{1/2+epsilon}(Omega) or sigma in L^s, s>2, div sigma in L^q, q>6/5 (Remark 5).
    The interpolant and the estimates (36)-(37) are only defined under these smoothness hypotheses, which are standard for a priori error analysis.
  • standard math Explicit-constant Bogovskii / generalized Poincare inequality from Guzman-Salgado [23] with constant depending on h_Omega/rho_Omega (used in Lemma A.4, eq. (58), and Lemma A.1).
    Cited as an external theorem; the explicit dependence on the star-shaped radius is essential for the constants in Theorem A.7 and hence in Proposition 3.1.
  • standard math Polynomial inverse inequalities on polytopes with constants depending on rho and p (cited [30], used in Proposition 3.1 and Theorem 4.3).
    Standard tool in finite element analysis; the dependence of the constants on mesh quality and polynomial degree is quoted from the literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Stability and interpolation estimates of Hellinger-Reissner virtual element spaces." pith.science (2026). https://pith.science/paper/2VJBVIBR

@misc{pith2026250206286,
  author       = {Pith},
  title        = {Pith review of: Stability and interpolation estimates of Hellinger-Reissner virtual element spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2VJBVIBR}},
  note         = {Machine review of arXiv:2502.06286}
}
read the original abstract

We prove stability and interpolation estimates for Hellinger-Reissner virtual elements; the constants appearing in such estimates only depend on the aspect ratio of the polytope under consideration and the degree of accuracy of the scheme. We further investigate numerically the behaviour of the constants appearing in the stability estimates on sequences of badly-shaped polytopes and for increasing degree of accuracy.

Figures

Figures reproduced from arXiv: 2502.06286 by the authors.

Figure 1
Figure 1. A sequence of 2D hourglass shaped elements for a compressible material. Second test case. We consider a sequence of elements as in [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. A sequence of 3D hourglass shaped elements for a compressible material. Third test case. We consider a sequence of elements as in [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. A sequence of 2D trapezoidal elements for an incompressible material. Numerical results: minimum and maximum generalized eigenvalues. In [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Fully mixed virtual element schemes for a new model of steady-state poroelastic stress-assisted diffusion in the brain

    math.NA 2025-10 reject novelty 6.0 of 10

    A fully mixed virtual element scheme for poroelastic stress-assisted diffusion is proposed with claimed optimal error estimates, but the central well-posedness proof relies on a false inf-sup condition.

Reference graph

Works this paper leans on

32 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [11]

    Botti and L

    M. Botti and L. Mascotto. A Neˇ cas-Lions inequality with symmetric gradients on star-shaped domains based on a first order Babuˇ ska-Aziz inequality.J. Math. Anal. Appl. , 545(2), 2025

    work page 2025

  2. [23]

    Guzm´ an and A

    J. Guzm´ an and A. J. Salgado. Estimation of the continuity constants for Bogovskii and regularized Poincar´ e integral operators. J. Math. Anal. Appl. , 502(1):Paper No. 125246, 36, 2021

    work page 2021

  3. [1]

    Adams and B

    S. Adams and B. Cockburn. A mixed finite element method for elasticity in three dimensions. J. Sci. Comput. , 25(3):515–521, 2005

    work page 2005

  4. [2]

    D. N. Arnold and G. Awanou. Rectangular mixed finite elements for elasticity. Math. Models Methods Appl. Sci., 15(09):1417–1429, 2005

    work page 2005

  5. [3]

    D. N. Arnold, G. Awanou, and R. Winther. Finite elements for symmetric tensors in three dimensions. Math. Comp., 77(263):1229–1251, 2008. 16

    work page 2008

  6. [4]

    D. N. Arnold and R. Winther. Mixed finite elements for elasticity. Numer. Math. , 92:401–419, 2002

    work page 2002

  7. [5]

    Artioli, S

    E. Artioli, S. De Miranda, C. Lovadina, and L. Patruno. A stress/displacement virtual element method for plane elasticity problems. Comput. Methods Appl. Mech. Engrg. , 325:155–174, 2017

    work page 2017

  8. [6]

    Artioli, S

    E. Artioli, S. De Miranda, C. Lovadina, and L. Patruno. A family of virtual element methods for plane elasticity problems based on the Hellinger–Reissner principle. Comput. Methods Appl. Mech. Engrg. , 340:978–999, 2018

    work page 2018

Show all 32 references
  1. [7]

    A. K. Aziz, editor. The mathematical foundations of the finite element method with applications to partial differential equations. Academic Press, New York-London, 1972

    1972

  2. [8]

    Beir˜ ao da Veiga, F

    L. Beir˜ ao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, and A. Russo. Basic principles of virtual element methods. Math. Models Methods Appl. Sci. , 23(01):199–214, 2013

    2013

  3. [9]

    Beir˜ ao da Veiga, F

    L. Beir˜ ao da Veiga, F. Brezzi, and L. D. Marini. Virtual elements for linear elasticity problems. SIAM J. Numer. Anal., 51(2):794–812, 2013

    2013

  4. [10]

    Boffi, F

    D. Boffi, F. Brezzi, and M. Fortin. Mixed finite element methods and applications , volume 44. Springer, 2013

    2013

  5. [12]

    Chen and J

    L. Chen and J. Huang. Some error analysis on virtual element methods. Calcolo, 55(1):Paper No. 5, 23, 2018

    2018

  6. [13]

    Chen and X

    L. Chen and X. Huang. A finite element elasticity complex in three dimensions. Math. Comp. , 91(337):2095– 2127, 2022

    2022

  7. [14]

    Chen and Y.-N

    S.-C. Chen and Y.-N. Wang. Conforming rectangular mixed finite elements for elasticity. J. Sci. Comput. , 47:93–108, 2011

    2011

  8. [15]

    S. H. Christiansen, J. Gopalakrishnan, J. Guzm´ an, and K. Hu. A discrete elasticity complex on three- dimensional Alfeld splits. Numer. Math. , 156(1):159–204, 2024

    2024

  9. [16]

    F. Dassi. Vem++, a c++ library to handle and play with the Virtual Element Method. https://arxiv.org/ abs/2310.05748, 2023

    2023 arXiv

  10. [17]

    Dassi, C

    F. Dassi, C. Lovadina, and M. Visinoni. A three-dimensional Hellinger-Reissner virtual element method for linear elasticity problems. Comput. Methods Appl. Mech. Engrg. , 364:112910, 17, 2020

    2020

  11. [18]

    D. A. Di Pietro and A. Ern. Mathematical aspects of discontinuous Galerkin methods , volume 69. Springer Science & Business Media, 2011

    2011

  12. [19]

    R. G. Dur´ an. An elementary proof of the continuity from L2 0(Ω) to H 1 0 (Ω)n of Bogovskii’s right inverse of the divergence. Rev. Un. Mat. Argentina , 53(2):59–78, 2012

    2012

  13. [20]

    Ern and J.-L

    A. Ern and J.-L. Guermond. Finite elements I: Approximation and interpolation , volume 72. Springer Nature, 2021

    2021

  14. [21]

    S. Gong, J. Gopalakrishnan, J. Guzm´ an, and M. Neilan. Discrete elasticity exact sequences on Worsey–Farin splits. ESAIM Math. Model. Numer. Anal. , 57(6):3373–3402, 2023

    2023

  15. [22]

    Gopalakrishnan, J

    J. Gopalakrishnan, J. Guzm´ an, and J. J. Lee. The Johnson-Mercier elasticity element in any dimensions. https://arxiv.org/abs/2403.13189, 2024

    2024 arXiv

  16. [24]

    Hlav´ aˇ cek

    I. Hlav´ aˇ cek. Convergence of an equilibrium finite element model for plane elastostatics.Aplikace Matematiky, 24(6):427–457, 1979

    1979

  17. [25]

    Hu and S

    J. Hu and S. Zhang. A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci. China Math. , 58:297–307, 2015

    2015

  18. [26]

    Johnson and B

    C. Johnson and B. Mercier. Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math. , 30:103–116, 1978

    1978

  19. [27]

    Kˇ r ´ ıˇ zek

    M. Kˇ r ´ ıˇ zek. An equilibrium finite element method in three-dimensional elasticity. Aplikace Matematiky , 27(1):46–75, 1982

    1982

  20. [28]

    Mascotto

    L. Mascotto. The role of stabilization in the virtual element method: a survey. Comput. Math. Appl. , 151:244– 251, 2023

    2023

  21. [29]

    W. C. H. McLean. Strongly elliptic systems and boundary integral equations. Cambridge university press, 2000

    2000

  22. [30]

    Verf¨ urth.A Posteriori Error Estimation Techniques for Finite Element Methods

    R. Verf¨ urth.A Posteriori Error Estimation Techniques for Finite Element Methods . Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2013

    2013

  23. [31]

    Visinoni

    M. Visinoni. A family of three-dimensional virtual elements for Hellinger-Reissner elasticity problems. Comp. Math. Appl. , 155:97–109, 2024

    2024

  24. [32]

    V. B. Watwood Jr. and B. J. Hartz. An equilibrium stress field model for finite element solutions of two- dimensional elastostatic problems. Int. J. Solids Struct. , 4(9):857–873, 1968. 17 A Well-posedness of the linear elasticity problem in mixed formulation with essential bo...

    1968

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.