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arxiv: 2604.03751 · v1 · submitted 2026-04-04 · 🧮 math.NA · cs.NA

Virtual element approximation of eigenvalue problems: is the stabilization of the right hand side necessary?

Daniele Boffi , Francesca Gardini , Lucia Gastaldi This is my paper

Pith reviewed 2026-05-13 17:11 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords virtual element methodeigenvalue problemsstabilizationmass matrixelliptic eigenvalue problemsfinite element approximationnumerical analysis
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The pith

Lower order standard virtual element spaces do not require mass matrix stabilization for elliptic self-adjoint eigenvalue problems.

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

The paper establishes that stabilization of the right-hand side mass matrix can be omitted in virtual element approximations of elliptic self-adjoint eigenvalue problems when using standard lower-order spaces. This follows from specific algebraic properties of those spaces. The result simplifies the method by removing the need to tune stabilization parameters for the mass term. Numerical experiments indicate the same holds for higher-order schemes across different mesh sequences.

Core claim

For elliptic self-adjoint eigenvalue problems discretized with lower-order standard virtual element spaces, the stabilization of the mass matrix is not necessary. The proof relies on the algebraic structure of these spaces, allowing the discrete problem to be formulated without it while maintaining accuracy. This is confirmed by numerical tests for higher orders.

What carries the argument

Standard lower-order virtual element spaces whose algebraic properties permit skipping stabilization of the mass matrix in the eigenvalue formulation.

If this is right

  • The eigenvalue problem can be discretized using only left-hand side stabilization.
  • Implementation complexity is reduced since no stabilization parameter needs tuning for the right-hand side.
  • The method converges for the considered class of problems without mass stabilization.
  • Numerical results suggest applicability to higher-order VEM on various meshes.

Where Pith is reading between the lines

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

  • This simplification may extend to other linear problems sharing similar symmetry properties.
  • Checking the property for non-standard or higher-order spaces without numerical evidence would require new analysis.
  • Adopting this in software libraries could lead to faster eigenvalue solvers for VEM-based simulations.

Load-bearing premise

The eigenvalue problems are elliptic and self-adjoint, and the virtual element spaces are the standard lower-order ones.

What would settle it

Finding a specific lower-order standard VEM discretization of an elliptic self-adjoint eigenvalue problem where eigenvalues computed without mass stabilization fail to converge to the true values.

Figures

Figures reproduced from arXiv: 2604.03751 by Daniele Boffi, Francesca Gardini, Lucia Gastaldi.

Figure 1
Figure 1. Figure 1: Coarsest meshes of different types: T triangles, S squares, V Voronoi, H hexagons In the virtual element discretization we employ four different mesh types, labeled by T , S, V, H and reported in [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Coarsest dyadic mesh D [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
read the original abstract

The VEM approximation of eigenvalue problems usually involves the appropriate tuning of stabilization parameters, unless self-stabilizing or stabilization-free VEM are used. In this paper we prove that for elliptic self-adjoint eigenvalue problems the stabilization of the mass matrix is not necessary when lower order standard VEM spaces are adopted. Numerical evidence shows that also for higher order schemes the same result is true on various mesh sequences.

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

1 major / 1 minor

Summary. The paper proves that stabilization of the mass matrix is unnecessary in Virtual Element Method (VEM) discretizations of elliptic self-adjoint eigenvalue problems when using lower-order standard VEM spaces. It supplies a theoretical argument based on algebraic properties of these spaces and presents numerical evidence that the same conclusion holds for higher-order schemes on various mesh sequences.

Significance. If the central claim holds, the result removes the need to tune stabilization parameters for the right-hand side in a common class of VEM eigenvalue problems, simplifying implementation and reducing computational overhead while preserving consistency and stability. The extension suggested by numerics for higher orders could broaden the practical utility of stabilization-free VEM in spectral computations.

major comments (1)
  1. [Proof of the main result (likely §3)] The proof for the low-order case is stated to rely on specific algebraic properties of standard VEM spaces that permit dropping the mass-matrix stabilization term; the manuscript should explicitly identify these properties (e.g., the exact polynomial degree and the form of the projection operators) and show how they guarantee the required consistency and stability estimates without the stabilization term.
minor comments (1)
  1. [Numerical experiments section] Numerical results for higher-order schemes are described qualitatively; adding tabulated convergence rates or error norms against mesh size would make the evidence more quantitative and easier to compare with stabilized variants.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive report and the constructive comment. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The proof for the low-order case is stated to rely on specific algebraic properties of standard VEM spaces that permit dropping the mass-matrix stabilization term; the manuscript should explicitly identify these properties (e.g., the exact polynomial degree and the form of the projection operators) and show how they guarantee the required consistency and stability estimates without the stabilization term.

    Authors: We agree that greater explicitness will improve readability. In the revised manuscript we will add a dedicated paragraph at the beginning of Section 3 that identifies the key algebraic properties: for the lowest-order standard VEM space (k=1) the local space contains P_1 exactly, both the L^2-projection Π^0 and the gradient projection Π^∇ reproduce polynomials of degree ≤1, and the stabilization form is orthogonal to this polynomial kernel. We will then show directly that these properties imply exact consistency for the mass term (without stabilization) and that the discrete bilinear form remains coercive on the orthogonal complement, yielding the required stability estimates. The updated proof will contain the corresponding error bounds. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proof relies on algebraic properties of VEM spaces

full rationale

The paper presents a mathematical proof that stabilization of the mass matrix is unnecessary for lower-order standard VEM spaces in elliptic self-adjoint eigenvalue problems, based on specific algebraic properties of those spaces. This is a direct derivation from the problem assumptions and VEM definitions rather than any fitted input, self-definition, or load-bearing self-citation. Numerical results for higher-order cases are presented as supporting evidence but are not part of the core proof chain. No steps reduce by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard assumptions of VEM theory for elliptic problems and the algebraic structure of low-order virtual element spaces; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The eigenvalue problem is elliptic and self-adjoint
    The proof is stated only for this class of problems.
  • standard math Standard lower-order VEM spaces satisfy the usual consistency and stability properties
    Invoked implicitly for the mass-matrix analysis.

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Ahmad, A

    B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo . Equivalent projectors for virtual element methods. Comput. Math. Appl. , 66(3):376–391, 2013. 22 DANIELE BOFFI, FRANCESCA GARDINI, AND LUCIA GASTALDI Table 23. First 10 eigenvalues on D with k = 2 Exact Errors (rate) 2 8.6e-04 4.8e-05 (4.16) 2.9e-06 (4.04) 1.8e-07 (4.01) 1.1e-08 (4.00) 5 3.3e-0...

    work page 2013

  2. [2]

    On the stabilization of a virtual element method for an acoustic vibration problem

    Linda Alzaben, Daniele Boffi, Andreas Dedner, and Lucia Ga staldi. On the stabilization of a virtual element method for an acoustic vibration problem. Math. Models Methods Appl. Sci. , 35(3):655–701, 2025

    work page 2025

  3. [3]

    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(1):199–214, 2013

    work page 2013

  4. [4]

    Donatella Marini, and Alessandro Russo

    Louren¸ co Beir˜ ao da Veiga, Franco Brezzi, L. Donatella Marini, and Alessandro Russo. The virtual element method. Acta Numer. , 32:123–202, 2023

    work page 2023

  5. [5]

    Lo west order stabilization free virtual element method for the 2D Poisson equation

    Stefano Berrone, Andrea Borio, and Francesca Marcon. Lo west order stabilization free virtual element method for the 2D Poisson equation. Comput. Math. Appl. , 177:78–99, 2025

    work page 2025

  6. [6]

    A first-order stabilization-free virtual element method

    Stefano Berrone, Andrea Borio, Francesca Marcon, and Gi oana Teora. A first-order stabilization-free virtual element method. Appl. Math. Lett. , 142:Paper No. 108641, 6, 2023

    work page 2023

  7. [7]

    Finite element approximation of eigenvalu e problems

    Daniele Boffi. Finite element approximation of eigenvalu e problems. Acta Numer. , 19:1–120, 2010

    work page 2010

  8. [8]

    App roximation of PDE eigenvalue problems involving parameter dependent matrices

    Daniele Boffi, Francesca Gardini, and Lucia Gastaldi. App roximation of PDE eigenvalue problems involving parameter dependent matrices. Calcolo, 57(4):Paper No. 41, 21, 2020

    work page 2020

  9. [9]

    Vir tual element approximation of eigen- value problems

    Daniele Boffi, Francesca Gardini, and Lucia Gastaldi. Vir tual element approximation of eigen- value problems. In The virtual element method and its applications , volume 31 of SEMA SIMAI Springer Ser. , pages 275–320. Springer, Cham, [2022] ©2022

    work page 2022

  10. [10]

    Alvin Chen and N. Sukumar. Stabilization-free virtual element method for plane elasticity. Comput. Math. Appl. , 138:88–105, 2023

    work page 2023

  11. [11]

    On spe ctral approximation

    Jean Descloux, Nabil Nassif, and Jacques Rappaz. On spe ctral approximation. I. The problem of convergence. RAIRO Anal. Num´ er., 12(2):97–112, iii, 1978

    work page 1978

  12. [12]

    On spe ctral approximation

    Jean Descloux, Nabil Nassif, and Jacques Rappaz. On spe ctral approximation. II. Error estimates for the Galerkin method. RAIRO Anal. Num´ er., 12(2):113–119, iii, 1978

    work page 1978

  13. [13]

    Benchmarking stabi- lized and self-stabilized p-virtual element methods with variable coefficients

    Paola Pia Foligno, Daniele Boffi, Fabio Credali, and Ricc ardo Vescovini. Benchmarking stabi- lized and self-stabilized p-virtual element methods with variable coefficients. Comput. Meth- ods Appl. Mech. Engrg. To appear

    work page

  14. [14]

    The nonconforming virtual element method for eigenvalue problems

    Francesca Gardini, Gianmarco Manzini, and Giuseppe Va cca. The nonconforming virtual element method for eigenvalue problems. ESAIM Math. Model. Numer. Anal. , 53(3):749– 774, 2019

    work page 2019

  15. [15]

    Virtual element method for second-order elliptic eigenvalue problems

    Francesca Gardini and Giuseppe Vacca. Virtual element method for second-order elliptic eigenvalue problems. IMA J. Numer. Anal. , 38(4):2026–2054, 2018

    work page 2026

  16. [16]

    A Hu-Washizu variational approach to self-stabi lized virtual elements: 2D linear elastostatics

    Andrea Lamperti, Massimiliano Cremonesi, Umberto Per ego, Alessandro Russo, and Carlo Lovadina. A Hu-Washizu variational approach to self-stabi lized virtual elements: 2D linear elastostatics. Comput. Mech. , 71(5):935–955, 2023

    work page 2023

  17. [17]

    A stabilization-free virtual element method for the convection-diffusion eigenproblem

    Francesca Marcon and David Mora. A stabilization-free virtual element method for the convection-diffusion eigenproblem. J. Sci. Comput. , 102(2):Paper No. 46, 33, 2025

    work page 2025

  18. [18]

    Stabilization-free virtual element method for the transmission eigenvalue problem on a nisotropic media

    Jian Meng, Lei Guan, Xu Qian, Songhe Song, and Liquan Mei . Stabilization-free virtual element method for the transmission eigenvalue problem on a nisotropic media. J. Comput. Math., 44(1):103–134, 2026

    work page 2026

  19. [19]

    A lowest- order free-stabilization virtual element method for the Laplacian eigenvalue problem

    Jian Meng, Xue W ang, Linlin Bu, and Liquan Mei. A lowest- order free-stabilization virtual element method for the Laplacian eigenvalue problem. J. Comput. Appl. Math. , 410:Paper No. 114013, 11, 2022

    work page 2022

  20. [20]

    A v irtual element method for the Steklov eigenvalue problem

    David Mora, Gonzalo Rivera, and Rodolfo Rodr ´ ıguez. A v irtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. , 25(8):1421–1445, 2015

    work page 2015

  21. [21]

    F. Casorati

    Team at University of Milano-Bicocca. Virtual element @ bicocca. https://https://sites.google.com/view/vembic/home, 2016. 24 DANIELE BOFFI, FRANCESCA GARDINI, AND LUCIA GASTALDI King Abdullah University of Science and Technology (KAUST), Sa udi Arabia, Dipar- timento di Matematica “F. Casorati”, Universit `a di Pavia, Italy, IMATI-CNR “Enrico Magenes”, Pa...

    work page 2016