An Investigation of Stabilization Scaling in Finite-Strain Virtual Element Methods for Hyperelasticity

Paulo Akira F. Enabe; Rodrigo Provasi

arxiv: 2603.07218 · v2 · pith:SU5XLGUSnew · submitted 2026-03-07 · 🧮 math.NA · cs.NA

An Investigation of Stabilization Scaling in Finite-Strain Virtual Element Methods for Hyperelasticity

Paulo Akira F. Enabe , Rodrigo Provasi This is my paper

Pith reviewed 2026-05-21 11:49 UTC · model grok-4.3

classification 🧮 math.NA cs.NA MSC 65N3074S05

keywords virtual element methodhyperelasticitystabilizationnearly incompressiblefinite strainkernel modespolygonal meshes

0 comments

The pith

A decoupled kernel stabilization for hyperelastic virtual elements stays shear-scaled as Poisson ratio approaches one-half.

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

Low-order virtual element methods for finite-strain hyperelasticity project the deformation onto polynomials and add stabilization to control the unresolved kernel modes. Existing practice integrates a nonlinear surrogate over an internal triangulation and mixes bulk and shear contributions through modified Lamé parameters, which injects artificial volumetric stiffness into isochoric modes when the material is nearly incompressible. The paper replaces that construction with a submesh-free, kernel-only term that treats the deviatoric and volumetric channels separately: the deviatoric part is scaled only by a local shear modulus and geometry weights, while the volumetric part uses an independent bulk factor that can be capped or removed. Under standard polygon regularity assumptions, the authors prove that this deviatoric stabilization is uniformly equivalent to the scaled H1 seminorm on the kernel, with constants independent of both mesh size and Poisson ratio. Element spectra and Cook-membrane benchmarks confirm that the new scaling removes the bulk-driven pollution observed in classical surrogates.

Core claim

Under standard polygon regularity assumptions the proposed deviatoric stabilization term is uniformly equivalent to μ_E times the squared H1 seminorm on the kernel, with equivalence constants independent of mesh size and of Poisson ratio; the volumetric term is scaled independently by a bulk measure and may be suppressed as ν approaches 1/2.

What carries the argument

Decoupled kernel-only stabilization, with the deviatoric contribution scaled solely by a local shear modulus plus bounded directional weights and the volumetric contribution scaled by an independent bulk modulus that can be capped.

If this is right

Classical surrogate stabilizations assign bulk-driven energy to isochoric kernel modes as ν approaches 1/2, producing artificial stiffening.
The new scaling remains shear-controlled on the kernel for any Poisson ratio, including the incompressible limit.
Kernel spectra and Cook-membrane tests exhibit improved robustness across triangular, quadrilateral and general polygonal meshes.
Newton tangent linearization stays consistent because the stabilization is defined directly on the kernel without auxiliary sub-triangulations.

Where Pith is reading between the lines

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

The same separation of shear and volumetric channels could be tested in three-dimensional polyhedral virtual elements where sub-tessellation costs grow rapidly.
If the directional geometry weights remain bounded under the stated regularity, the construction may extend to other nonlinear constitutive laws that become singular in certain strain modes.
A practical implementation could default the volumetric stabilization to zero when ν exceeds a user-specified threshold without altering the shear part.

Load-bearing premise

The uniform equivalence and stability bounds hold only when the polygonal elements satisfy standard regularity assumptions that control shape and diameter ratios.

What would settle it

Compute the smallest eigenvalue of the deviatoric stabilization matrix divided by μ_E |·|_{1,E}^2 for a sequence of successively refined nearly incompressible polygonal meshes; if this ratio tends to zero or infinity as h decreases or ν approaches 1/2, the uniform equivalence claim is false.

Figures

Figures reproduced from arXiv: 2603.07218 by Paulo Akira F. Enabe, Rodrigo Provasi.

**Figure 1.** Figure 1: Example of an auxiliary triangulation TE used to evaluate the stabilization contribution without introducing additional degrees of freedom. The element load potential is approximated as described in (41). In particular, for k = 1, the Neumann contribution is computable since uh is piecewise linear on boundary edges, whereas the body-force term is evaluated by replacing the virtual field with a computable c… view at source ↗

**Figure 2.** Figure 2: Schematic definition of an ellipse-based aspect ratio for a polygonal element. [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: Single-element isochoric kernel-mode diagnostic: raw stabilization energy [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

**Figure 4.** Figure 4: Single-element isochoric kernel-mode diagnostic for the classical stabilization: normalized stabilization [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗

**Figure 5.** Figure 5: Comparison of classical and decoupled stabilization on an isochoric kernel mode under the physically relevant [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗

**Figure 6.** Figure 6: Anisotropic scaling of the kernel-only deviatoric stabilization. The designed directional ratio [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗

**Figure 7.** Figure 7: Deviatoric/volumetric separation on ker(Π∇ E ) under the decoupled stabilization. Kernel-restricted eigenvalues are shown as the bulk scale κE is varied: a subset of modes remains constant (deviatoric content), whereas modes with volumetric content increase with κE [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗

**Figure 8.** Figure 8: Conditioning of the kernel-restricted stabilization operator across element shapes. The spectral condition [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗

**Figure 9.** Figure 9: Geometry and boundary conditions for Cook’s membrane benchmark. [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗

**Figure 10.** Figure 10: Representative meshes for Cook’s membrane at refinement level [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗

**Figure 11.** Figure 11: Cook’s membrane on the Regular quadrilateral mesh family and nearly-incompressible regime (ν = 0.499). Each column corresponds to a refinement level (h = 0.5000, 0.2500, 0.1250, 0.0625); the top row shows results with the proposed decoupled term and the bottom row with the classic term. The color field represents the displacement magnitude |u|, and the tip displacement uy at the upper-right corner is repo… view at source ↗

**Figure 12.** Figure 12: Cook’s membrane on the Distorted quadrilateral mesh family and nearly-incompressible regime (ν = 0.499). The decoupled term approaches a mesh-independent tip displacement under refinement (uy : 7.793 → 8.511). The classic term remains significantly lower on coarse meshes and converges slowly toward the decoupled response (uy : 3.045 → 7.312), indicating that moderate geometric distortion does not remove t… view at source ↗

**Figure 13.** Figure 13: Cook’s membrane on the Highly distorted quadrilateral mesh family and nearly-incompressible regime (ν = 0.499). Due to the strong geometric distortion, the decoupled term exhibits an elevated coarse-mesh response that decreases with refinement and approaches the fine-mesh range (uy : 10.668 → 8.989). The classic term again underpredicts the tip displacement on coarse meshes and recovers only gradually und… view at source ↗

**Figure 14.** Figure 14: Cook’s membrane on the Voronoi tessellation mesh family and nearly-incompressible regime (ν = 0.499). The decoupled term exhibits a non-monotone transient on the coarsest levels (uy : 9.022 → 8.326) but converges to a consistent fine-mesh tip displacement (uy : 8.518 → 8.527). The classic term also completes the loading history on this mesh family, yet yields substantially smaller displacements on coarse … view at source ↗

**Figure 15.** Figure 15: Cook’s membrane—tip displacement convergence at [PITH_FULL_IMAGE:figures/full_fig_p038_15.png] view at source ↗

read the original abstract

Low-order virtual element methods (VEM) compute a consistent finite-strain contribution through polynomial projections and rely on stabilization to control the unresolved modes in the projector kernel. In current hyperelastic VEM practice, stabilization is often defined by integrating a nonlinear surrogate energy over an auxiliary sub-triangulation and scaled through modified Lam\'e parameters and incompressibility factors; this can introduce sensitivity to the arbitrary internal tessellation, complicate consistent Newton linearization, and, most critically, inject bulk-dependent proxies into shear-type kernel penalties, artificially stiffening isochoric missing modes in the nearly incompressible regime. This work develops a submesh-free, kernel-only stabilization that decouples deviatoric and volumetric channels and is explicitly designed to scale like the current Newton tangent energy on the kernel: the deviatoric term is scaled solely by a shear measure and enhanced by bounded geometry-driven directional weights, while the volumetric term is scaled by an independent bulk measure and can be capped or suppressed as $\nu\to 1/2$. A spectral framework is established in which the canonical VEM stability requirement on the kernel is characterized by generalized Rayleigh quotients and eigenvalue bounds, and it is shown under standard polygon regularity assumptions that the deviatoric stabilization is uniformly equivalent to $\mu_E|\cdot|_{1,E}^2$ on the kernel with constants independent of mesh size and Poisson ratio. Element-level diagnostics confirm that classical surrogate-based stabilizations assign bulk-driven energy to isochoric kernel modes as $\nu\to 1/2$, whereas the proposed decoupled stabilization remains shear-scaled; kernel spectra and Cook's membrane simulations in the nearly incompressible regime further support improved robustness across polygonal mesh families.

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.

Desk Editor's Note private letter to a colleague

This paper gives a submesh-free decoupled stabilization for VEM hyperelasticity that keeps shear scaling clean in the nearly incompressible limit, backed by a spectral equivalence proof under standard polygon assumptions.

read the letter

The main thing to know is that the authors replace the usual surrogate-energy stabilization (which often pulls in bulk effects through sub-triangulations) with a kernel-only term that splits deviatoric and volumetric channels. The deviatoric piece is scaled only by a shear measure plus bounded geometry weights, while the volumetric piece uses an independent bulk factor that can be capped as Poisson's ratio approaches one-half. They then set up a spectral framework with generalized Rayleigh quotients and show, under the usual star-shapedness and minimum-angle conditions, that the deviatoric stabilization is equivalent to the shear seminorm on the kernel with constants independent of mesh size and Poisson ratio. Element diagnostics and Cook's membrane runs are presented to illustrate that classical methods inject unwanted bulk energy into isochoric kernel modes while this version stays shear-scaled. That combination is the actual advance. The analysis is direct and the numerical checks line up with the claims without visible internal contradictions. The proofs rest on standard polygon regularity assumptions, which is normal for this area but does mean the result does not automatically cover arbitrarily distorted elements. The abstract indicates the stabilization is built to match the Newton tangent on the kernel, which should help with consistent linearization, though readers will want the full implementation details to confirm ease of coding. This work is aimed at people already using or extending virtual element methods for nonlinear solid mechanics, especially those dealing with incompressible or nearly incompressible hyperelasticity. It supplies a concrete alternative to existing surrogate approaches and gives enough analysis and targeted tests to be worth examining. I would send it to peer review; the central stability argument looks grounded and the practical motivation is clear.

Referee Report

1 major / 3 minor

Summary. The manuscript develops a submesh-free stabilization for low-order virtual element methods in finite-strain hyperelasticity. It decouples deviatoric and volumetric channels, scaling the deviatoric term by a shear measure with bounded geometry-driven weights and the volumetric term by an independent bulk measure that can be suppressed as ν→1/2. A spectral framework using generalized Rayleigh quotients characterizes kernel stability, and the central result proves that, under standard polygon regularity assumptions, the deviatoric stabilization is uniformly equivalent to μ_E |·|_{1,E}^2 on the kernel with constants independent of mesh size and Poisson ratio. Element diagnostics, kernel spectra, and Cook's membrane results are presented in support.

Significance. If the uniform equivalence holds, the work provides a targeted improvement over classical surrogate stabilizations that suffer from bulk contamination and artificial stiffening in the nearly incompressible regime. The spectral framework offers a systematic, reusable tool for analyzing VEM kernel stability, and the decoupling strategy directly mitigates a known practical difficulty in hyperelastic VEM. The combination of the parameter-free scaling, independence from internal tessellation, and supporting numerics strengthens the case for adoption in robust large-deformation simulations on polygonal meshes.

major comments (1)

[Spectral framework section] Spectral framework section: the proof that the equivalence constants are independent of ν as ν→1/2 relies on the boundedness of the geometry-driven directional weights; an explicit estimate showing that these weights remain O(1) uniformly for all admissible star-shaped polygons (with the stated minimum-angle condition) would make the load-bearing uniformity claim fully transparent.

minor comments (3)

[Spectral framework section] The definition of the generalized Rayleigh quotient should include a short remark on how the material measures (μ_E, bulk modulus) enter the numerator and denominator so that readers can immediately see the shear-only scaling.
[Numerical results] In the Cook's membrane experiments, the figure captions or text should state the precise sequence of Poisson ratios (e.g., 0.3, 0.49, 0.499) and the number of refinement levels used, to allow direct visual confirmation of ν-independence.
[Introduction] A brief sentence in the introduction contrasting the new stabilization with the classical surrogate approach (sub-triangulation, modified Lamé parameters, incompressibility factors) would help readers locate the novelty without consulting the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestion to strengthen the transparency of the uniformity claim in the spectral framework. We address the major comment below.

read point-by-point responses

Referee: [Spectral framework section] Spectral framework section: the proof that the equivalence constants are independent of ν as ν→1/2 relies on the boundedness of the geometry-driven directional weights; an explicit estimate showing that these weights remain O(1) uniformly for all admissible star-shaped polygons (with the stated minimum-angle condition) would make the load-bearing uniformity claim fully transparent.

Authors: We agree that an explicit estimate would improve clarity and make the load-bearing uniformity claim fully transparent. In the revised manuscript we will add a short lemma in the Spectral framework section establishing that the geometry-driven directional weights are bounded above and below by positive constants depending only on the uniform star-shapedness parameter and the minimum-angle condition. The proof follows directly from the normalization of the weights together with the uniform control on angles and aspect ratios guaranteed by the standard polygon regularity assumptions; the resulting bounds are independent of mesh size, specific polygon geometry, and Poisson ratio. This addition will render the independence of the equivalence constants from ν explicit without altering the existing arguments. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs a decoupled deviatoric-volumetric stabilization explicitly to scale with the Newton tangent energy on the VEM kernel using standard projectors and material measures, then proves uniform equivalence to μ_E |·|_{1,E}^2 under independent polygon regularity assumptions (star-shapedness and minimum angle conditions). This is a direct mathematical argument via generalized Rayleigh quotients and eigenvalue bounds, not a reduction of the central claim to a fitted parameter, self-definition, or self-citation chain. The abstract and analysis contain no load-bearing self-citations, ansatz smuggling, or renaming of known results; the result is self-contained against external benchmarks and standard VEM theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard assumptions from the VEM literature for polygonal elements; no free parameters or new entities are introduced in the abstract description.

axioms (1)

domain assumption standard polygon regularity assumptions
Invoked to establish that the deviatoric stabilization remains uniformly equivalent to the seminorm with constants independent of mesh size and Poisson ratio.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the deviatoric stabilization is uniformly equivalent to μ_E |·|_{1,E}^2 on the kernel with constants independent of mesh size and Poisson ratio

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Sharma, Himanshu, and A

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J. Qiu and J. Meng. The Interior Penalty Virtual Element Method for Two Kinds of Variational Inequal- ities. InJournal of Scientific Computing, V olume 103, Article 45, 2025. https://doi.org/10.1007/ s10915-025-02871-8

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Wriggers, B

P. Wriggers, B. D. Reddy, W. Rust, and others. Efficient virtual element formulations for compressible and incompressible finite deformations. InComputational Mechanics, V olume 60, pages 253–268, 2017. https: //doi.org/10.1007/s00466-017-1405-4

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Wriggers and B

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Hudobivnik, F

B. Hudobivnik, F. Aldakheel, and P. Wriggers. A low order 3D virtual element formulation for finite elasto–plastic deformations. InComputational Mechanics, V olume 63, pages 253–269, 2019. https://doi.org/10.1007/ s00466-018-1593-6

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van Huyssteen and B

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Mascotto

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S. Berrone, A. Borio, and F. Marcon. A stabilization-free Virtual Element Method based on divergence-free projections. InComputer Methods in Applied Mechanics and Engineering, V olume 424, pages 116885, 2024. https://doi.org/10.1016/j.cma.2024.116885

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Berrone, A

S. Berrone, A. Borio, D. Fassino, and F. Marcon. Stabilization-free Virtual Element Method for 2D second order elliptic equations. InComputer Methods in Applied Mechanics and Engineering, V olume 438, Part A, pages 117839, 2025.https://doi.org/10.1016/j.cma.2025.117839

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S. Berrone, A. Borio, D. Fassino, and F. Marcon. A residual a posteriori error estimate for the stabilization- free virtual element method. InJournal of Computational Physics, V olume 553, pages 114704, 2026. https: //doi.org/10.1016/j.jcp.2026.114704

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[1] [1]

B.B. Xu, F. Peng, P. Junker, et al. Virtual element method with non-matching and adaptive meshes for phase field fracture. InComputational Mechanics, V olume 76, pages 1621–1634, 2025. https://doi.org/10.1007/ s00466-025-02665-7

work page 2025

[2] [2]

Sharma, Himanshu, and A

S. Sharma, Himanshu, and A. Ramaswamy. Fine to coarse mesh transition in phase-field fracture simulations using the virtual element method. InFinite Elements in Analysis and Design, V olume 249, pages 104371, 2025. https://doi.org/10.1016/j.finel.2025.104371

work page doi:10.1016/j.finel.2025.104371 2025

[3] [3]

Qiu and J

J. Qiu and J. Meng. The Interior Penalty Virtual Element Method for Two Kinds of Variational Inequal- ities. InJournal of Scientific Computing, V olume 103, Article 45, 2025. https://doi.org/10.1007/ s10915-025-02871-8

work page 2025

[4] [4]

Wriggers, B

P. Wriggers, B. Hudobivnik, and O. Allix. On two simple virtual Kirchhoff–Love plate elements for isotropic and anisotropic materials. InComputational Mechanics, V olume 69, pages 615–637, 2022. https://doi.org/10. 1007/s00466-021-02106-1

work page 2022

[5] [5]

Wang and Y

G. Wang and Y . He. An interface-fitted virtual element method for Stokes moving interface problem. InComputer Methods in Applied Mechanics and Engineering, V olume 454, pages 118861, 2026.https://doi.org/10.1016/ j.cma.2026.118861

work page arXiv 2026

[6] [6]

Beirão da Veiga, F

L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo. Basic principles of virtual element methods. InMathematical Models and Methods in Applied Sciences, V olume 23, Issue 1, pages 199–214, 2013.https://doi.org/10.1142/S0218202512500492

work page doi:10.1142/s0218202512500492 2013

[7] [7]

The hitchhiker’s guide to the virtual element method

L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo. The Hitchhiker’s guide to the virtual element method. InMathematical Models and Methods in Applied Sciences, V olume 2, Issue 8, pages 1541–1573, 2014. https://doi.org/10.1142/s021820251440003x

work page doi:10.1142/s021820251440003x 2014

[8] [8]

Ahmad, A

B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo. Equivalent projectors for virtual element methods. In Computers and Mathematics with Applications, V olume 66, Issue 3, pages 376–391, 2013.http://dx.doi.org/ 10.1016/j.camwa.2013.05.015

work page doi:10.1016/j.camwa.2013.05.015 2013

[9] [9]

Wriggers, B

P. Wriggers, B. D. Reddy, W. Rust, and others. Efficient virtual element formulations for compressible and incompressible finite deformations. InComputational Mechanics, V olume 60, pages 253–268, 2017. https: //doi.org/10.1007/s00466-017-1405-4

work page doi:10.1007/s00466-017-1405-4 2017

[10] [10]

Wriggers and B

P. Wriggers and B. Hudobivnik. A low order virtual element formulation for finite elasto–plastic deformations. InComputer Methods in Applied Mechanics and Engineering, V olume 327, pages 459–477, 2017. https: //doi.org/10.1016/j.cma.2017.08.053

work page doi:10.1016/j.cma.2017.08.053 2017

[11] [11]

Hudobivnik, F

B. Hudobivnik, F. Aldakheel, and P. Wriggers. A low order 3D virtual element formulation for finite elasto–plastic deformations. InComputational Mechanics, V olume 63, pages 253–269, 2019. https://doi.org/10.1007/ s00466-018-1593-6

work page 2019

[12] [13]

van Huyssteen and B

D. van Huyssteen and B. D. Reddy. A virtual element method for transversely isotropic hyperelasticity. In Computer Methods in Applied Mechanics and Engineering, V olume 386, pages 114108, 2021. https://doi. org/10.1016/j.cma.2021.114108

work page doi:10.1016/j.cma.2021.114108 2021

[13] [14]

Mascotto

L. Mascotto. The role of stabilization in the virtual element method: A survey. InComputers and Mathematics with Applications, V olume 151, pages 244–251, 2023.https://doi.org/10.1016/j.camwa.2023.09.045. 40 MARCH10, 2026

work page doi:10.1016/j.camwa.2023.09.045 2023

[14] [15]

doi:10.1016/j.cma

R. Fujimoto and I. Saiki. Study of the stabilization parameter in the virtual element method. InComputer Methods in Applied Mechanics and Engineering, V olume 428, pages 117106, 2024.https://doi.org/10.1016/j.cma. 2024.117106

work page doi:10.1016/j.cma 2024

[15] [16]

Berrone, A

S. Berrone, A. Borio, and F. Marcon. A stabilization-free Virtual Element Method based on divergence-free projections. InComputer Methods in Applied Mechanics and Engineering, V olume 424, pages 116885, 2024. https://doi.org/10.1016/j.cma.2024.116885

work page doi:10.1016/j.cma.2024.116885 2024

[16] [17]

Berrone, A

S. Berrone, A. Borio, D. Fassino, and F. Marcon. Stabilization-free Virtual Element Method for 2D second order elliptic equations. InComputer Methods in Applied Mechanics and Engineering, V olume 438, Part A, pages 117839, 2025.https://doi.org/10.1016/j.cma.2025.117839

work page doi:10.1016/j.cma.2025.117839 2025

[17] [18]

Berrone, A

S. Berrone, A. Borio, D. Fassino, and F. Marcon. A residual a posteriori error estimate for the stabilization- free virtual element method. InJournal of Computational Physics, V olume 553, pages 114704, 2026. https: //doi.org/10.1016/j.jcp.2026.114704

work page doi:10.1016/j.jcp.2026.114704 2026

[18] [19]

L.C. Evans. Partial differential equations. American Mathematical Society, Providence, R.I., 2010

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H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer, New York, NY , 2011. https://doi.org/10.1007/978-0-387-70914-7. 41 MARCH10, 2026 A Functional analysis overview This appendix summarizes the functional-analytic setting and notation used throughout the paper, including the geometric regularity assumptions required...

work page doi:10.1007/978-0-387-70914-7 2011