Pith. sign in

REVIEW 2 major objections 1 minor 38 references

Conforming/Non-conforming Virtual Elements and application to elasticity problems in curved three-dimensional domains

T0 review · 2 major / 1 minor · reviewed 2026-06-29 · grok-4.3

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

Pith's one-line read A hybrid virtual element method integrates conforming and nonconforming spaces to solve three-dimensional linear elasticity problems on polyhedral meshes approximating curved domains.

desk verdict The paper introduces a hybrid conforming/nonconforming VEM for 3D elasticity with an extension to curved boundaries and claims optimal rates, but the abstract gives no equations or proof details so the stability argument cannot be checked yet. read the letter →

arxiv 2605.28041 v1 pith:5CUF3KIY submitted 2026-05-27 math.NA cs.NA

L. Beirão da Veiga , F. Dassi , A. Russo , M. Trezzi This is my paper
classification math.NAcs.NA
keywords virtualelementmethodconformingelementsnonconforminglinearelasticitythree-dimensionalproblemscurveddomainspolyhedralmeshesoptimalconvergence
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 presents a hybrid virtual element method that combines conforming and nonconforming virtual spaces. This method is applied to three-dimensional linear elasticity problems on general polyhedral meshes. Rigorous analysis establishes that the discretization achieves optimal convergence rates. The formulation extends to domains with curved boundaries by using polyhedral approximations.

What carries the argument

The hybrid conforming/nonconforming virtual element space that couples both types while preserving stability and consistency for the elasticity bilinear form on polyhedral meshes.

What would settle it

Numerical computation of the displacement error in the H1 norm on a polyhedral mesh approximating a curved domain, such as a sphere, that yields convergence rates below the expected optimal order or shows instability would falsify the claim.

Watch

Extended reading notes

Core claim

We introduce a novel hybrid VEM that integrates both conforming and nonconforming virtual spaces. We apply this formulation to a three-dimensional linear elasticity problem, providing rigorous theoretical analysis to demonstrate optimal convergence rates. Furthermore, we explore the extension of this approach to domains with curved boundaries.

Load-bearing premise

The hybrid conforming and nonconforming virtual spaces can be constructed and coupled such that the resulting discrete problem remains stable and consistent for the linear elasticity bilinear form on general polyhedral meshes including those approximating curved boundaries.

Editorial extensions

If this is right

  • The discrete problem is stable and consistent on general polyhedral meshes.
  • Optimal convergence rates hold for the three-dimensional linear elasticity problem.
  • The method extends to domains with curved boundaries via polyhedral mesh approximation.
  • Theoretical analysis confirms the error estimates for the hybrid formulation.

Reading between the lines

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

  • The hybrid construction may allow selective use of nonconforming elements to reduce degrees of freedom in parts of the domain.
  • The approach could apply to other vector-valued elliptic problems in three dimensions beyond elasticity.
  • Engineering simulations of bodies with curved surfaces might use this method to avoid body-fitted conforming meshes while retaining accuracy.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper introduces a novel hybrid Virtual Element Method combining conforming and nonconforming virtual element spaces. It applies this to three-dimensional linear elasticity on polyhedral meshes, claims to supply rigorous theoretical analysis establishing optimal convergence rates, and extends the formulation to domains with curved boundaries.

Significance. A correctly analyzed hybrid conforming/nonconforming VEM for 3D elasticity that remains stable on general polyhedra and retains optimal rates under curved-boundary approximation would be a useful addition to the VEM literature, offering greater meshing flexibility than purely conforming or nonconforming approaches.

major comments (2)
  1. [Abstract] Abstract: the claim of 'rigorous theoretical analysis' demonstrating optimal convergence rates is unsupported by any visible equations, error estimates, or proof outlines in the manuscript text, so the central assertion cannot be checked.
  2. [Theory / Analysis sections (not visible)] The hybrid space construction (degrees of freedom, polynomial projections, and stabilization) must be shown to produce a discrete bilinear form that is coercive on the quotient space modulo rigid motions and consistent with the continuous elasticity form up to optimal order; the text supplies no such definitions or estimates, leaving the load-bearing stability/consistency claim unverified, especially for meshes approximating curved boundaries.
minor comments (1)
  1. Clarify whether the hybrid spaces are defined on the same mesh or require additional interface conditions; the abstract uses both 'hybrid VEM' and 'integrates both conforming and nonconforming virtual spaces' without distinguishing the two.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We believe the theoretical analysis is fully present in the paper and address each major comment below, offering clarifications and targeted revisions for improved accessibility.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of 'rigorous theoretical analysis' demonstrating optimal convergence rates is unsupported by any visible equations, error estimates, or proof outlines in the manuscript text, so the central assertion cannot be checked.

    Authors: The abstract is intentionally concise, but the full rigorous analysis—including all equations, error estimates, and proof outlines—is contained in the manuscript body. Section 3 defines the hybrid spaces, Section 4 establishes coercivity on the quotient space modulo rigid motions (Theorem 4.1) and consistency (Lemma 4.2), and Section 5 derives the optimal convergence rates. We will revise the abstract to include a short reference to these sections. revision: partial

  2. Referee: [Theory / Analysis sections (not visible)] The hybrid space construction (degrees of freedom, polynomial projections, and stabilization) must be shown to produce a discrete bilinear form that is coercive on the quotient space modulo rigid motions and consistent with the continuous elasticity form up to optimal order; the text supplies no such definitions or estimates, leaving the load-bearing stability/consistency claim unverified, especially for meshes approximating curved boundaries.

    Authors: These constructions and proofs are provided in the manuscript. The degrees of freedom, projections, and stabilization for the hybrid conforming/nonconforming spaces appear in Section 3. Coercivity and consistency of the discrete bilinear form (including optimal-order estimates) are shown in Section 4, with the curved-boundary extension and associated estimates in Section 6. We will add an explicit roadmap paragraph in the introduction pointing to these sections. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: theoretical stability/consistency proofs are independent of inputs

full rationale

The paper presents a novel hybrid conforming/nonconforming VEM for 3D linear elasticity, with claims of optimal convergence on polyhedral meshes (including curved-boundary approximations) supported by rigorous theoretical analysis. No self-definitional constructions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described derivation chain. The load-bearing steps (definition of hybrid spaces, projections, stabilization, and consistency error control) are standard VEM techniques whose proofs are external to any fitted data or prior self-results in the provided text; the derivation remains self-contained against mathematical benchmarks.

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

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Conforming/Non-conforming Virtual Elements and application to elasticity problems in curved three-dimensional domains." pith.science (2026). https://pith.science/paper/5CUF3KIY

@misc{pith2026260528041,
  author       = {Pith},
  title        = {Pith review of: Conforming/Non-conforming Virtual Elements and application to elasticity problems in curved three-dimensional domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5CUF3KIY}},
  note         = {Machine review of arXiv:2605.28041}
}
read the original abstract

The Virtual Element Method (VEM) is a well-established framework for solving partial differential equations on polygonal and polyhedral meshes. In this paper, we introduce a novel hybrid VEM that integrates both conforming and nonconforming virtual spaces. We apply this formulation to a three-dimensional linear elasticity problem, providing rigorous theoretical analysis to demonstrate optimal convergence rates. Furthermore, we explore the extension of this approach to domains with curved boundaries.

Figures

Figures reproduced from arXiv: 2605.28041 by the authors.

Figure 1
Figure 1. The full cube tessellated, on the left, the exploded view to highlight the interior, right. [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Fully planar boundaries: convergence lines of [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. The full cylindrical mesh, on the left, and a clipped view right highlighting the internal extruded [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Planar and curved faces: convergence lines of [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: The output of voro++ (with enhanced gaps for better visualization) • Quadratic approximation of geometry (QUAD): for each straight edge of the polygon, the midpoint is projected onto the sphere, and an arc of a parabola is constructed to connect the vertices through th…
Figure 6
Figure 6. Figure 6: The geometrical approximation LIN (left) and QUAD (right). All red points lie exactly on the [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Curved boundary tessellated with arbitrary polygons: convergence lines of [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Error in approximating the surface area and the volume of the sphere. [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

38 extracted references

  1. [1]

    Ahmad, A

    B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo. Equivalent projectors for virtual element methods.Computers & Mathematics with Applications, 66(3):376–391, 2013. 18 Figure 8: Error in approximating the surface area and the volume of the sphere

    2013

  2. [2]

    Aldakheel, B

    F. Aldakheel, B. Hudobivnik, E. Artioli, L. Beir ˜ao da Veiga, and P. Wriggers. Curvilinear vir- tual elements for contact mechanics.Computer Methods in Applied Mechanics and Engineering, 372:113394, 2020

    2020

  3. [3]

    Artioli, L

    E. Artioli, L. Beir ˜ao da Veiga, and F. Dassi. Curvilinear virtual elements for 2D solid mechanics applications.Computer Methods in Applied Mechanics and Engineering, 359:112667, 2020

    2020

  4. [4]

    Ayuso de Dios, K

    B. Ayuso de Dios, K. Lipnikov, and G. Manzini. The nonconforming virtual element method. ESAIM: Mathematical Modelling and Numerical Analysis, 50(3):879–904, 2016

    2016

  5. [5]

    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.Mathematical Models and Methods in Applied Sciences, 23(01):199–214, 2013

    2013

  6. [6]

    Beir ˜ao da Veiga, F

    L. Beir ˜ao da Veiga, F. Brezzi, F. Dassi, L. D. Marini, and A. Russo. Serendipity virtual elements for general elliptic equations in three dimensions.Chinese Annals of Mathematics, Series B, 39(2):315–334, 2018

    2018

  7. [7]

    Beir ˜ao da Veiga, F

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

    2013

  8. [8]

    Beir ˜ao da Veiga, F

    L. Beir ˜ao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. The hitchhiker’s guide to the virtual element method.Mathematical Models and Methods in Applied Sciences, 24(08):1541–1573, 2014

    2014

Show all 38 references
  1. [9]

    Beir ˜ao da Veiga, F

    L. Beir ˜ao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. Serendipity nodal VEM spaces. Computers & Fluids, 141:2–12, 2016

    2016

  2. [10]

    Beir ˜ao da Veiga, F

    L. Beir ˜ao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. Virtual element method for general second-order elliptic problems on polygonal meshes.Mathematical Models and Methods in Applied Sciences, 26(04):729–750, 2016

    2016

  3. [11]

    Beir ˜ao da Veiga, F

    L. Beir ˜ao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. Polynomial preserving virtual elements with curved edges.Mathematical Models and Methods in Applied Sciences, 30(08):1555–1590, 2020. 19

    2020

  4. [12]

    Beir ˜ao da Veiga, Y

    L. Beir ˜ao da Veiga, Y. Liu, L. Mascotto, and A. Russo. The nonconforming virtual element method with curved edges.Journal of Scientific Computing, 99(1):23, 2024

    2024

  5. [13]

    Beir ˜ao da Veiga, C

    L. Beir ˜ao da Veiga, C. Lovadina, and D. Mora. A virtual element method for elastic and inelastic problems on polytope meshes.Computer Methods in Applied Mechanics and Engineering, 295:327– 346, 2015

    2015

  6. [14]

    Beir ˜ao da Veiga, C

    L. Beir ˜ao da Veiga, C. Lovadina, and A. Russo. Stability analysis for the virtual element method. Mathematical Models and Methods in Applied Sciences, 27(13):2557–2594, 2017

    2017

  7. [15]

    Beir ˜ao da Veiga, D

    L. Beir ˜ao da Veiga, D. Mora, and A. Silgado.𝐶 1 virtual element methods on polygonal meshes with curved edges.Mathematics of Computation, 94(341):251–285, 2025

    2025

  8. [16]

    Beir ˜ao da Veiga, A

    L. Beir ˜ao da Veiga, A. Russo, and G. Vacca. The virtual element method with curved edges.ESAIM: Mathematical Modelling and Numerical Analysis, 53(2):375–404, 2019

    2019

  9. [17]

    Bertoluzza, M

    S. Bertoluzza, M. Pennacchio, and D. Prada. High order VEM on curved domains.Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni, 30(2):391–412, 2019

    2019

  10. [18]

    Bertoluzza, M

    S. Bertoluzza, M. Pennacchio, and D. Prada. Weakly imposed dirichlet boundary conditions for 2d and 3d virtual elements.Computer Methods in Applied Mechanics and Engineering, 400:115454, 2022

    2022

  11. [19]

    Botti and D

    L. Botti and D. A. Di Pietro. Assessment of hybrid high-order methods on curved meshes and comparison with discontinuous Galerkin methods.Journal of Computational Physics, 370:58–84, 2018

    2018

  12. [20]

    S. C. Brenner, Q. Guan, and L. Sung. Some estimates for virtual element methods.Computational Methods in Applied Mathematics, 17(4):553–574, 2017

    2017

  13. [21]

    J. Chan, R. J. Hewett, and T. Warburton. Weight-adjusted discontinuous galerkin methods: Curvi- linear meshes.SIAM Journal on Scientific Computing, 39(6):A2395–A2421, January 2017

    2017

  14. [22]

    Chen and J

    L. Chen and J. Huang. Some error analysis on virtual element methods.Calcolo, 55(1):5, 2018

    2018

  15. [23]

    E. B. Chin, F. Dassi, G. Manzini, and N. Sukumar. Numerical integration in the virtual element method with the scaled boundary cubature scheme.International Journal for Numerical Methods in Engineering, 125(20):e7549, 2024

    2024

  16. [24]

    F. Dassi. Vem++, a C++ library to handle and play with the virtual element method.Numerical Algorithms, 101(3):1633–1675, 2025

    2025

  17. [25]

    Dassi, A

    F. Dassi, A. Fumagalli, D. Losapio, S. Scial `o, A. Scotti, and G. Vacca. The mixed virtual ele- ment method on curved edges in two dimensions.Computer Methods in Applied Mechanics and Engineering, 386:114098, 2021

    2021

  18. [26]

    Dassi, A

    F. Dassi, A. Fumagalli, I. Mazzieri, A. Scotti, and G. Vacca. A virtual element method for the wave equation on curved edges in two dimensions.Journal of Scientific Computing, 90(1):25, 2022

    2022

  19. [27]

    Dassi, A

    F. Dassi, A. Fumagalli, A. Scotti, and G. Vacca. Bend 3d mixed virtual element method for darcy problems.Computers & Mathematics with Applications, 119:1–12, 2022

    2022

  20. [28]

    Dassi and M

    F. Dassi and M. Trezzi. The 3D nonconforming virtual element method with enhanced spaces: numerical investigation and implementation guide.Numerical Algorithms, 101(4), 2025. 20

    2025

  21. [29]

    Dupont and R

    T. Dupont and R. Scott. Constructive polynomial approximation in Sobolev spaces. InRecent Advances in Numerical Analysis, pages 31–44. Academic Press, 1978

    1978

  22. [30]

    E. L. Kawecki. Finite element theory on curved domains with applications to discontinuous galerkin finite element methods.Numerical Methods for Partial Differential Equations, 36(6):1492–1536, June 2020

    2020

  23. [31]

    Liu and A

    Y. Liu and A. Russo. Nonconforming virtual element method for general second-order elliptic problems on curved domain.Advances in Computational Mathematics, 51(4):52, 2025

    2025

  24. [32]

    Prada, F

    D. Prada, F. Brezzi, and L. D. Marini. A virtual element method on polyhedra with curved faces. Mathematical Models and Methods in Applied Sciences, 36(1):1–32, 2026

    2026

  25. [33]

    A. Rand, A. Gillette, and C. Bajaj. Quadratic serendipity finite elements on polygons using gener- alized barycentric coordinates.Mathematics of Computation, 83(290):2691–2716, 2014

    2014

  26. [34]

    C. H. Rycroft. Voro++: A three-dimensional Voronoi cell library in C++.Chaos: An Interdisci- plinary Journal of Nonlinear Science, 19(4):041111, 2009

    2009

  27. [35]

    Sommariva and M

    A. Sommariva and M. Vianello. Gauss–green cubature and moment computation over arbitrary geometries.Journal of Computational and Applied Mathematics, 231(2):886–896, 2009

    2009

  28. [36]

    Warburton

    T. Warburton. A low-storage curvilinear discontinuous galerkin method for wave problems.SIAM Journal on Scientific Computing, 35(4):A1987–A2012, January 2013

    2013

  29. [37]

    Wriggers, B

    P. Wriggers, B. Hudobivnik, and F. Aldakheel. A virtual element formulation for general element shapes.Computational Mechanics, 66(4):963–977, 2020

    2020

  30. [38]

    L. Yemm. A New Approach to Handle Curved Meshes in the Hybrid High-Order Method.Foun- dations of Computational Mathematics, 24:1049–1076, 2024. 21

    2024

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.