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arxiv: 2606.07477 · v1 · pith:7FB755XRnew · submitted 2026-06-05 · 🧮 math.NA · cs.NA

A Mixed Virtual Element Method for the p-Laplace equation

Kirubell B. Haile , Giuseppe Vacca This is my paper

Pith reviewed 2026-06-27 21:01 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords mixed virtual element methodp-Laplace equationnon-Hilbertian normsinf-sup stabilitya priori error estimatesnonlinear stabilization
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The pith

A mixed virtual element method for the p-Laplace equation achieves well-posedness and error estimates across all p in (1, infinity) via a nonlinear stabilization term.

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

The paper develops a mixed virtual element discretization for the p-Laplace equation that remains stable when the natural function spaces are not Hilbert spaces. It introduces a nonlinear stabilization that respects the power-law nonlinearity of the operator. Discrete inf-sup conditions are proved in the appropriate norms, which in turn yields existence, uniqueness, and convergence rates for both the solution and its flux. This matters because many nonlinear problems in mechanics and materials science involve p-Laplacians with p far from 2, where standard Hilbert-space tools fail.

Core claim

The central claim is that combining standard mixed virtual element spaces with a novel nonlinear stabilization term produces a discrete problem that is well-posed for every p in (1, infinity). The authors prove that the discrete bilinear form is continuous and coercive in non-Hilbertian norms and that a discrete inf-sup condition holds, from which a priori error bounds for the primal variable and the flux follow directly.

What carries the argument

The novel nonlinear stabilization term that mimics the power-law structure of the continuous p-Laplacian operator.

If this is right

  • The discrete problem is well-posed for arbitrary p in (1, infinity).
  • A priori error estimates hold for the primal variable and the flux.
  • Numerical tests confirm the theoretical convergence rates.

Where Pith is reading between the lines

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

  • The method could extend to other nonlinear elliptic problems whose growth is governed by a power law.
  • Similar stabilization ideas might apply to other virtual element or finite element schemes for non-Hilbertian problems.
  • The approach suggests that virtual element methods can be adapted beyond linear or quadratic cases by tailoring the stabilization to the nonlinearity.

Load-bearing premise

The novel nonlinear stabilization term must be close enough to the continuous power-law structure that the discrete inf-sup condition and coercivity still hold in the non-Hilbertian norms for every p.

What would settle it

A numerical experiment in which the discrete solution fails to converge or the computed inf-sup constant drops to zero for some p not equal to 2 would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.07477 by Giuseppe Vacca, Kirubell B. Haile.

Figure 1
Figure 1. Figure 1: Example of the adopted polygonal meshes. [PITH_FULL_IMAGE:figures/full_fig_p034_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Test 1. Computed errors defined as in (137) as a function of the mesh size (loglog scale), for [PITH_FULL_IMAGE:figures/full_fig_p035_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Test 1. Computed errors defined as in (137) as a function of the mesh size (loglog scale), for [PITH_FULL_IMAGE:figures/full_fig_p036_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Test 1. Computed errors defined as in (137) as a function of the mesh size (loglog scale), for [PITH_FULL_IMAGE:figures/full_fig_p037_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Test 2. Discrete pressures 𝑢ℎ obtained on a Cartesian tessellation of (0, 1) 2 with 64 × 64 elements. The values of 𝑝 ′ are reported in the plot. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_5.png] view at source ↗
read the original abstract

We introduce and analyze a mixed Virtual Element Method for the $p$-Laplace equation in a non-Hilbertian setting, covering the full range $p \in (1, \infty)$. The discrete framework combines standard mixed Virtual Element spaces with a novel non-linear stabilization term designed to mimic the power-law structure of the continuous operator. We establish discrete inf-sup stability under non-Hilbertian norms and rigorously prove the continuity and coercivity of the discrete form. This guarantees the well-posedness of the problem and allows us to derive a priori error estimates for the primal variable and the flux. A set of numerical tests supports the theoretical derivations.

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 paper introduces a mixed Virtual Element Method for the p-Laplace equation in the non-Hilbertian setting for the full range p ∈ (1, ∞). It combines standard mixed VEM spaces with a novel nonlinear stabilization term designed to mimic the continuous power-law structure, establishes discrete inf-sup stability under non-Hilbertian norms, proves continuity and coercivity of the discrete form to guarantee well-posedness, derives a priori error estimates for the primal variable and the flux, and includes numerical tests supporting the theory.

Significance. If the proofs hold, the work provides a valuable extension of the VEM framework to nonlinear elliptic problems in non-Hilbert spaces across the entire p-range. The rigorous treatment of inf-sup stability, coercivity, and error estimates under the novel stabilization, together with numerical validation, strengthens the contribution to numerical analysis for p-Laplace-type equations.

minor comments (3)
  1. [§3] §3 (or the section defining the discrete spaces and stabilization): clarify the precise dependence of the nonlinear stabilization parameter on the local mesh size and polynomial degree to ensure the coercivity constant is independent of these quantities as claimed.
  2. [§5] The error estimate statements (likely in §5) should explicitly state the norms in which the estimates for the primal variable and flux hold, and whether the constants are independent of p.
  3. [Table 1] Table 1 (numerical results): include the observed convergence rates alongside the theoretical predictions for direct comparison; the current presentation makes it harder to verify the a priori estimates numerically.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript on the mixed Virtual Element Method for the p-Laplace equation and for recommending minor revision. No specific major comments were provided in the report, so we have no individual points to address at this stage. We will carefully consider any minor suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation chain consists of defining a mixed VEM discretization with a novel nonlinear stabilization term that mimics the p-Laplace power-law structure, followed by direct proofs of discrete inf-sup stability, continuity, and coercivity in non-Hilbertian norms for p in (1, infinity). These proofs establish well-posedness and enable a priori error estimates for the primal variable and flux. No equations reduce a claimed result to a fitted input or self-citation by construction; the stabilization is introduced as an ansatz but its properties are independently verified rather than assumed to hold tautologically. The analysis is self-contained against standard functional-analytic benchmarks and does not rely on load-bearing self-citations or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

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