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arxiv: 2103.16896 · v4 · submitted 2021-03-31 · 🧮 math.NA · cs.NA

Lowest order stabilization free Virtual Element Method for the 2D Poisson equation

Stefano Berrone , Andrea Borio , Francesca Marcon This is my paper

Pith reviewed 2026-05-24 13:29 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords virtual element methodstabilization freePoisson equationpolygonal mesheserror estimatesenhancement property
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The pith

The first-order Enlarged Enhancement Virtual Element Method solves the 2D Poisson equation with bilinear forms that require no stabilization term.

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

The paper introduces the E²VEM, a lowest-order virtual element method on polygonal meshes for the Poisson problem. It defines the discrete bilinear form directly from higher-order polynomial projections, made available by enlarging the enhancement property of each local virtual space. The projection degree is set according to the number of vertices of the polygon. Well-posedness follows from this construction, and optimal a priori error estimates are proved in the usual energy and L2 norms.

Core claim

The E²VEM allows the definition of bilinear forms that do not require a stabilization term, thanks to the exploitation of higher order polynomial projections made computable by suitably enlarging the enhancement property of local virtual spaces. The polynomial degree of local projections is chosen based on the number of vertices of each polygon. Proof of well-posedness and optimal order a priori error estimates are provided.

What carries the argument

Enlarged enhancement property of local virtual spaces, which renders higher-order polynomial projections computable from the degrees of freedom and thereby removes the need for stabilization.

If this is right

  • The discrete problem is well-posed once the projection degree is chosen from the number of vertices.
  • Optimal-order a priori error estimates hold in the energy norm and in L2.
  • Numerical convergence rates on both convex and non-convex meshes match the theoretical predictions.
  • No stabilization parameter needs to be selected or tuned.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same enlargement strategy could be applied to other linear elliptic problems to obtain stabilization-free schemes.
  • Implementation cost may decrease because the code no longer needs to assemble or tune a stabilization term.
  • The vertex-dependent degree rule suggests a possible route to adaptive choice of local polynomial spaces on highly irregular meshes.

Load-bearing premise

The enlargement of the enhancement property together with the vertex-count rule for projection degree makes the required higher-order projections computable without stabilization.

What would settle it

A computation on a mesh containing polygons with varying numbers of sides that produces either a singular discrete system or convergence rates below the predicted optimal order when the vertex-based projection degrees are used.

Figures

Figures reproduced from arXiv: 2103.16896 by Andrea Borio, Francesca Marcon, Stefano Berrone.

Figure 1
Figure 1. Figure 1: Meshes 6.2.2 Convergence results For the four mesh sequences, we report the trend of the H1 and the L2 errors in Figure 2a and in Figure 2b, respectively, decreasing the maximum diam￾eter of the polygons. In the legends, we report the computed convergence rates with respect to h, denoted by α. We see that we get the expected values for all the meshes, as obtained in (57) and (65). 6.2.3 Convergence of diff… view at source ↗
Figure 2
Figure 2. Figure 2: Logarithmic convergence plots (a) H1 error (b) L2 error [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Logarithmic convergence plots for diffusion-reaction model [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
read the original abstract

We introduce and analyse the first order Enlarged Enhancement Virtual Element Method (E$^2$VEM) for the Poisson problem. The method allows the definition of bilinear forms that do not require a stabilization term, thanks to the exploitation of higher order polynomial projections that are made computable by suitably enlarging the enhancement (from which comes the prefix of the name E$^2$) property of local virtual spaces. The polynomial degree of local projections is chosen based on the number of vertices of each polygon. We provide a proof of well-posedness and optimal order a priori error estimates. Numerical tests on convex and non-convex polygonal meshes confirm the criterium for well-posedness and the theoretical convergence rates.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the first-order Enlarged Enhancement Virtual Element Method (E²VEM) for the 2D Poisson equation. It constructs stabilization-free bilinear forms by enlarging the enhancement property of local virtual spaces, making higher-order polynomial projections computable; the projection degree is selected according to the number of vertices per polygon. Well-posedness and optimal-order a priori error estimates are proved, with numerical tests on convex and non-convex meshes used to confirm the well-posedness criterion and convergence rates.

Significance. If the central claims hold, the work supplies an explicit stabilization-free construction for lowest-order VEM on general polygonal meshes together with a well-posedness proof and optimal a priori estimates. These elements constitute a concrete technical contribution to the VEM literature; the provision of both analysis and supporting numerical experiments on non-convex meshes strengthens the result.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'confirm the criterium for well-posedness' is imprecise; the abstract should state the explicit well-posedness criterion that is being verified.
  2. [§3] The dependence of the local projection degree on the number of vertices is central to the construction; a short table or explicit formula in §3 would improve readability.
  3. [Introduction] Notation for the enlarged enhancement spaces and the associated projection operators could be introduced with a brief comparison to standard VEM spaces already in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report, so we have no points to address point-by-point at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained construction plus independent analysis

full rationale

The paper defines the E²VEM via an explicit enlargement of the local virtual space enhancement property, selects the projection degree as a function of polygon vertices, and then supplies a separate well-posedness proof together with optimal-order a priori error estimates. These steps are presented as direct mathematical constructions and external analysis rather than reductions of a fitted quantity or a self-citation chain back to the target result. Numerical experiments are described only as confirmation of the already-proven criterion. No load-bearing step equates a claimed prediction or uniqueness result to its own input by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central construction rests on the assumption that enlarging the enhancement space makes higher-order projections computable; the only explicit choice is the per-polygon polynomial degree rule.

free parameters (1)
  • Polynomial degree selection rule
    Degree chosen according to the number of vertices of each polygon to ensure computability of projections.
axioms (1)
  • domain assumption Enlargement of the enhancement property renders higher-order polynomial projections computable without stabilization
    Invoked to eliminate the stabilization term while preserving consistency.

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Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    Ahmad, A

    B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo. Equiva- lent projectors for virtual element methods. Computers & Mathematics with Applications, 66:376–391, September 2013. 25 (a) H1 error (b) L2 error Figure 2: Logarithmic convergence plots (a) H1 error (b) L2 error Figure 3: Logarithmic convergence plots for diffusion-reaction model

    work page 2013

  2. [2]

    P. F. Antonietti, S. Berrone, A. Borio, A. D’Auria, M. Verani, and S. Weisser. Anisotropic a posteriori error estimate for the virtual ele- ment method. IMA Journal of Numerical Analysis , 02 2021

    work page 2021

  3. [3]

    P. F. Antonietti, L. Mascotto, and M. Verani. A multigrid algorithm for the p-version of the virtual element method. ESAIM: Mathematical Modelling and Numerical Analysis , 52:337–364, 03 2017

    work page 2017

  4. [4]

    Babuˇ ska, E

    I. Babuˇ ska, E. G. Podnos, and G. J. Rodin. New fictitious domain methods: formulation and analysis. Mathematical Models and Methods in Applied Sciences, 15(10):1575–1594, 2005

    work page 2005

  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. Mathemat- ical Models and Methods in Applied Sciences , 23(01):199–214, 2013. 26

    work page 2013

  6. [6]

    Beir˜ ao da Veiga, F

    L. Beir˜ ao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. Virtual element methods for general second order elliptic problems on polyg- onal meshes. Mathematical Models and Methods in Applied Sciences , 26(04):729–750, 2015

    work page 2015

  7. [7]

    Beir˜ ao da Veiga, F

    L. Beir˜ ao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. Mixed virtual element methods for general second order elliptic problems on polygo- nal meshes. ESAIM: Mathematical Modelling and Numerical Analysis , 50(3):727–747, 2016

    work page 2016

  8. [8]

    Beir˜ ao da Veiga, K

    L. Beir˜ ao da Veiga, K. Lipnikov, and G. Manzini. The mimetic finite difference method for elliptic problems . Ms&A Modeling, simulation and applications 11. Springer, Cham, 2014

    work page 2014

  9. [9]

    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. Com- puter Methods in Applied Mechanics and Engineering , 295:327–346, 2015

    work page 2015

  10. [10]

    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

    work page 2017

  11. [11]

    Beir˜ ao da Veiga and G

    L. Beir˜ ao da Veiga and G. Manzini. Residual a posteriori error esti- mation for the virtual element method for elliptic problems. ESAIM: M2AN, 49(2):577–599, 2015

    work page 2015

  12. [12]

    M. F. Benedetto, S. Berrone, and A. Borio. The Virtual Element Method for underground flow simulations in fractured media. In Ad- vances in Discretization Methods, volume 12 of SEMA SIMAI Springer Series, pages 167–186. Springer International Publishing, Switzerland, 2016

    work page 2016

  13. [13]

    M. F. Benedetto, S. Berrone, A. Borio, S. Pieraccini, and S. Scial` o. A hybrid mortar virtual element method for discrete fracture network simulations. J. Comput. Phys. , 306:148–166, 2016

    work page 2016

  14. [14]

    M. F. Benedetto, S. Berrone, A. Borio, S. Pieraccini, and S. Scial` o. Or- der preserving SUPG stabilization for the virtual element formulation of advection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 311:18 – 40, 2016

    work page 2016

  15. [15]

    M. F. Benedetto, S. Berrone, and S. Scial` o. A globally conforming method for solving flow in discrete fracture networks using the virtual element method. Finite Elem. Anal. Des. , 109:23–36, 2016. 27

    work page 2016

  16. [16]

    Berrone and A

    S. Berrone and A. Borio. A residual a posteriori error estimate for the virtual element method. Mathematical Models and Methods in Applied Sciences, 27(08):1423–1458, 2017

    work page 2017

  17. [17]

    Berrone, A

    S. Berrone, A. Borio, and G. Manzini. SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations. Computer Methods in Applied Mechanics and Engineering , 340:500 – 529, 2018

    work page 2018

  18. [18]

    Berrone, A

    S. Berrone, A. Borio, and F. Marcon. Comparison of standard and sta- bilization free virtual elements on anisotropic elliptic problems. Applied Mathematics Letters, 129:107971, 2022

    work page 2022

  19. [19]

    Berrone, S

    S. Berrone, S. Pieraccini, and S. Scial` o. On simulations of discrete frac- ture network flows with an optimization-based extended finite element method. SIAM J. Sci. Comput. , 35(2):A908–A935, 2013

    work page 2013

  20. [20]

    Berrone, S

    S. Berrone, S. Pieraccini, and S. Scial` o. A PDE-constrained optimiza- tion formulation for discrete fracture network flows.SIAM J. Sci. Com- put., 35(2):B487–B510, 2013

    work page 2013

  21. [21]

    Berrone, S

    S. Berrone, S. Pieraccini, and S. Scial` o. An optimization approach for large scale simulations of discrete fracture network flows. J. Comput. Phys., 256:838–853, 2014

    work page 2014

  22. [22]

    D. Boffi, F. Brezzi, and M. Fortin. Approximation of Saddle Point Problems, chapter 5, pages 265–335. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013

    work page 2013

  23. [23]

    S. C. Brenner and L. Sung. Virtual element methods on meshes with small edges or faces. Mathematical Models and Methods in Applied Sciences, 28(07):1291–1336, 2018

    work page 2018

  24. [24]

    Brezzi, K

    F. Brezzi, K. Lipnikov, and V. Simoncini. A family of mimetic finite difference methods on polygonal and polyhedral meshes. Mathematical Models and Methods in Applied Sciences , 15(10):1533–1551, 2005

    work page 2005

  25. [25]

    Brezzi and L

    F. Brezzi and L. D. Marini. Virtual element methods for plate bending problems. Computer Methods in Applied Mechanics and Engineering , 253:455–462, 2013

    work page 2013

  26. [26]

    Burman, S

    E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. Cut- FEM: Discretizing geometry and partial differential equations. Interna- tional Journal for Numerical Methods in Engineering , 104(7):472–501, 2015

    work page 2015

  27. [27]

    Cangiani, E

    A. Cangiani, E. H. Georgoulis, T. Pryer, and O. J. Sutton. A posteriori error estimates for the virtual element method. Numerische Mathe- matik, 137(4):857–893, Dec 2017. 28

    work page 2017

  28. [28]

    Cangiani, G

    A. Cangiani, G. Manzini, and O. J. Sutton. Conforming and noncon- forming virtual element methods for elliptic problems. IMA Journal of Numerical Analysis, 37(3):1317–1354, 08 2016

    work page 2016

  29. [29]

    Desiderio, S

    L. Desiderio, S. Falletta, and L. Scuderi. A virtual element method coupled with a boundary integral non reflecting condition for 2d exte- rior helmholtz problems. Computers & Mathematics with Applications , 84:296–313, 2021

    work page 2021

  30. [30]

    D. A. Di Pietro and A. Ern. Mathematical Aspects of Discontinuous Galerkin Methods. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012

    work page 2012

  31. [31]

    D. A. Di Pietro and A. Ern. A hybrid high-order locking-free method for linear elasticity on general meshes. Computer Methods in Applied Mechanics and Engineering, 283:1–21, 2015

    work page 2015

  32. [32]

    D. A. Di Pietro and A. Ern. Hybrid high-order methods for variable- diffusion problems on general meshes. Comptes Rendus Mathematique, 353(1):31–34, 2015

    work page 2015

  33. [33]

    D. A. Di Pietro, A. Ern, and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Computational Methods in Applied Mathematics, 14(4):461–472, 2014

    work page 2014

  34. [34]

    Droniou, R

    J. Droniou, R. Eymard, T. Gallou et, C. Guichard, and R. Herbin. The Gradient Discretisation Method . Math´ ematiques et Applications. Springer, Cham, 2018

    work page 2018

  35. [35]

    Droniou, R

    J. Droniou, R. Eymard, T. Gallou et, and R. Herbin. Gradient Schemes: a generic framework for the discretisation of linear, nonlinear and nonlo- cal elliptic and parabolic equations. Mathematical Models and Methods in Applied Sciences, 23(13):2395–2432, 2013

    work page 2013

  36. [36]

    Glowinski, T.-W

    R. Glowinski, T.-W. Pan, and J. Periaux. A fictitious domain method for Dirichlet problem and applications. Computer Methods in Applied Mechanics and Engineering, 111(3):283–303, 1994

    work page 1994

  37. [37]

    S Hesthaven and T

    J. S Hesthaven and T. Warburton. Nodal Discontinuous Galerkin meth- ods: algorithms, analysis, and applications . Texts in applied mathemat- ics 54. Springer Science & Business Media, New York, 2008

    work page 2008

  38. [38]

    Hudobivnik, F

    B. Hudobivnik, F. Aldakheel, and P. Wriggers. A low order 3D virtual element formulation for finite elasto–plastic deformations. Computa- tional Mechanics, 63:253–269, 02 2019

    work page 2019

  39. [39]

    Mo¨ es, J

    N. Mo¨ es, J. Dolbow, and T. Belytschko. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46(1):131–150, 1999. 29

    work page 1999

  40. [40]

    C. S. Peskin. The immersed boundary method. Acta Numerica , 11:479–517, 2002

    work page 2002

  41. [41]

    Rivi` ere

    B. Rivi` ere. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Society for Industrial and Applied Mathematics, 2008

    work page 2008

  42. [42]

    Strouboulis, I

    T. Strouboulis, I. Babuˇ ska, and K. Copps. The design and analysis of the Generalized Finite Element Method. Computer Methods in Applied Mechanics and Engineering, 181(1):43–69, 2000

    work page 2000

  43. [43]

    Strouboulis, K

    T. Strouboulis, K. Copps, and I. Babuˇ ska. The generalized finite el- ement method: an example of its implementation and illustration of its performance. International Journal for Numerical Methods in En- gineering, 47(8):1401–1417, 2000

    work page 2000

  44. [44]

    Strouboulis, K

    T. Strouboulis, K. Copps, and I. Babuˇ ska. The generalized finite ele- ment method. Computer Methods in Applied Mechanics and Engineer- ing, 190(32):4081–4193, 2001

    work page 2001

  45. [45]

    Sukumar and A

    N. Sukumar and A. Tabarraei. Conforming polygonal finite ele- ments. International Journal for Numerical Methods in Engineering , 61(12):2045–2066, 2004

    work page 2045

  46. [46]

    Tishkin, A

    V. Tishkin, A. A. Samarskii, A. P. Favorskii, and M. Shashkov. Opera- tional finite-difference schemes. Differential Equations, 17:854–862, 07 1981

    work page 1981

  47. [47]

    G. Vacca. Virtual element methods for hyperbolic problems on polyg- onal meshes. Computers & Mathematics with Applications , 74(5):882 – 898, 2017. SI: SDS2016 – Methods for PDEs

    work page 2017

  48. [48]

    Vacca and L

    G. Vacca and L. Beir˜ ao da Veiga. Virtual element methods for parabolic problems on polygonal meshes. Numerical Methods for Partial Differ- ential Equations, 31(6):2110–2134, 2015. A Supplementary materials A.1 Proof of Lemma 4 In order to show the proof, we have to present a preliminary result. Lemma 9. Let ¯q∈RQ(E). Then ∃C > 0, independent of hE, such ...

    work page 2015