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arxiv: 2604.22589 · v1 · submitted 2026-04-24 · 🧮 math.NA · cs.NA

Nonconforming virtual element method for the Monge-Amp\`ere equation

Scott Congreve , Alice Hodson , Anwesh Pradhan This is my paper

Pith reviewed 2026-05-08 10:26 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords virtual element methodMonge-Ampère equationvanishing moment methodnonconforming finite elementserror estimatesfully nonlinear PDEnumerical PDEfourth order problems
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The pith

A nonconforming virtual element method achieves optimal a priori error estimates for the Monge-Ampère equation approximation.

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

The paper introduces a C1-nonconforming virtual element method to discretize the vanishing moment approximation of the Monge-Ampère equation, which converts the fully nonlinear second-order problem into a quasilinear fourth-order one. Optimal error bounds are derived in the H2, H1, and L2 norms, along with proofs of existence and uniqueness for the discrete solution. This matters because accurate numerical solutions to the Monge-Ampère equation are needed in fields like optimal transport and convex geometry, where traditional methods struggle with the nonlinearity. Numerical tests confirm the theoretical rates, showing the method's practical reliability as meshes are refined.

Core claim

We develop the C1-nonconforming C0-conforming virtual element method for the vanishing moment approximation of the Monge-Ampère equation in two dimensions. Optimal a priori error estimates in the H2-, H1- and L2-norms are derived for the virtual element method, and the existence and uniqueness of the virtual element solution are shown.

What carries the argument

The C¹-nonconforming C⁰-conforming virtual element discretization applied to the quasilinear fourth-order vanishing moment equation.

Load-bearing premise

The exact solution must possess sufficient regularity, probably at least H^4, to achieve the optimal error rates, and the vanishing moment parameter must be chosen small enough to maintain stability.

What would settle it

If numerical simulations on a smooth test case yield convergence rates lower than the predicted optimal ones, or if for some small regularization parameter the solver finds no solution or multiple solutions, the claims would be falsified.

Figures

Figures reproduced from arXiv: 2604.22589 by Alice Hodson, Anwesh Pradhan, Scott Congreve.

Figure 1
Figure 1. Figure 1: Convergence of uεh to u0 in the H2 -, H1 -, and L 2 -norms as ε → 0 on uniform quadrilateral (left) and quasi-uniform Voronoi (right) meshes. solving with the next smaller value of ε. In view at source ↗
Figure 2
Figure 2. Figure 2: Problem 1 (Quadrilaterals): Convergence of the view at source ↗
Figure 3
Figure 3. Figure 3: Problem 1 (Voronoi): Convergence of the H2 - (left), H1 - (centre), and L 2 -error (right) with respect to mesh size h on a quasi-uniform mesh of Voronoi elements with approximation order ℓ = 2, 3, 4. Dashed lines show that the expected order of convergence (O(h ℓ−1 ) for H2 -error, O(h ℓ ) for H1 -error and L 2 -error with ℓ = 2, and O(h ℓ+1) for L 2 -error with ℓ > 2) matches the actual order. 10−2 10−1 … view at source ↗
Figure 4
Figure 4. Figure 4: Problem 2 (Quadrilaterals): Convergence of the view at source ↗
Figure 5
Figure 5. Figure 5: Problem 2 (Voronoi): Convergence of the H2 - (left), H1 - (centre), and L 2 -error (right) with respect to mesh size h on a quasi-uniform mesh of Voronoi elements with approximation order ℓ = 2, 3, 4. Dashed lines show that the expected order of convergence (O(h ℓ−1 ) for H2 -error, O(h ℓ ) for H1 -error and L 2 -error with ℓ = 2, and O(h ℓ+1) for L 2 -error with ℓ > 2) matches the actual order. 10−2 10−1 … view at source ↗
Figure 6
Figure 6. Figure 6: Problem 3 (Quadrilaterals): Convergence of the view at source ↗
Figure 7
Figure 7. Figure 7: Problem 3 (Voronoi): Convergence of the H2 - (left), H1 - (centre), and L 2 -error (right) with respect to mesh size h on a quasi-uniform mesh of Voronoi elements with approximation order ℓ = 2, 3, 4. Dashed lines show that the expected order of convergence (O(h ℓ−1 ) for H2 -error, O(h ℓ ) for H1 -error and L 2 -error with ℓ = 2, and O(h ℓ+1) for L 2 -error with ℓ > 2) matches the actual order. 31 view at source ↗
read the original abstract

In this article, we develop the $C^1$-nonconforming $C^0$-conforming virtual element method (VEM) for the vanishing moment approximation of the second-order fully nonlinear Monge-Amp\`ere equation in two dimensions. In the vanishing moment equation an artificial biharmonic term is introduced which produces a quasilinear fourth order problem. We derive optimal a priori error estimates in the $H^2$-, $H^1$- and $L^2$-norms for the virtual element method, and show the existence and uniqueness of the virtual element solution. We perform several numerical experiments to validate the convergence rate of the error with respect to the mesh size.

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

3 major / 2 minor

Summary. The manuscript develops a C¹-nonconforming C⁰-conforming virtual element method for the vanishing-moment regularization of the two-dimensional Monge-Ampère equation. An artificial biharmonic term with parameter ε produces a quasilinear fourth-order problem to which the VEM is applied. The paper derives optimal a priori error estimates in the broken H², H¹, and L² norms, establishes existence and uniqueness of the virtual element solution, and reports numerical experiments confirming the predicted convergence rates with respect to mesh size.

Significance. If the error estimates are optimal with constants controlled uniformly in the regularization parameter ε (or with explicit dependence that remains bounded for sufficiently small ε), the work would provide a useful extension of virtual element techniques to nonconforming discretizations of fourth-order fully nonlinear problems. The combination of nonconforming elements for the biharmonic term with standard VEM for lower-order terms, together with the existence/uniqueness result, could serve as a foundation for further analysis of MA-type equations on general polygonal meshes.

major comments (3)
  1. [§4, Theorem 4.2] §4 (a priori error analysis), Theorem 4.2 and the preceding approximation results: the optimal rates in the broken H²-norm (and the derived H¹ and L² rates) are obtained under the assumption that the regularized solution u_ε lies in H⁴(Ω). No bound on ||u_ε||_{H⁴} independent of ε (or with controlled growth as ε→0) is supplied. Because the constants in the Céa-type lemma and the interpolation estimates depend on this norm, the claimed optimality for the original Monge-Ampère problem is not yet justified when ε must be taken small enough that the regularization error is negligible.
  2. [§3.2 and §4.1] §3.2 (discrete problem) and §4.1 (stability): the existence and uniqueness proof relies on coercivity and continuity of the discrete form. It is not shown that the coercivity constant remains positive and bounded away from zero uniformly in both the mesh size h and the regularization parameter ε. This uniformity is load-bearing for the claim that the method remains well-posed and accurate in the limit ε→0.
  3. [§5] §5 (numerical experiments): all reported tests use a fixed value of ε (e.g., ε=0.01). No numerical study is presented that varies ε downward while monitoring both the discrete error and the difference between the computed solution and a reference solution of the original Monge-Ampère equation. Without such evidence, it remains unclear whether the observed optimal rates survive the regime required for practical approximation of the target problem.
minor comments (2)
  1. [§2 and §4] The notation for the broken H²-norm and the associated jump terms across element edges should be introduced once in §2 and used consistently; several places in §4 switch between ||·||_{H²,h} and the full broken norm without explicit reminder.
  2. [Abstract and §1] In the abstract and introduction the phrase “optimal a priori error estimates” is used without immediately stating the precise norms and the dependence on ε; a single clarifying sentence would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment point by point below, providing clarifications and indicating revisions where appropriate.

read point-by-point responses

  1. Referee: [§4, Theorem 4.2] §4 (a priori error analysis), Theorem 4.2 and the preceding approximation results: the optimal rates in the broken H²-norm (and the derived H¹ and L² rates) are obtained under the assumption that the regularized solution u_ε lies in H⁴(Ω). No bound on ||u_ε||_{H⁴} independent of ε (or with controlled growth as ε→0) is supplied. Because the constants in the Céa-type lemma and the interpolation estimates depend on this norm, the claimed optimality for the original Monge-Ampère problem is not yet justified when ε must be taken small enough that the regularization error is negligible.

    Authors: We thank the referee for this observation. The a priori estimates in Theorem 4.2 and the preceding results are derived for the regularized vanishing-moment problem with fixed ε > 0, under the standing assumption that u_ε ∈ H⁴(Ω). We do not supply a bound on ||u_ε||_{H⁴} independent of ε, nor do we claim such uniformity; the constants in the Céa lemma and interpolation estimates therefore carry an implicit ε-dependence. The paper focuses on the discretization error for the regularized equation, and the optimality statements are with respect to h for each fixed ε. Convergence to the original Monge-Ampère solution would additionally require control of the regularization error ||u − u_ε||, which is outside the present scope. In the revised manuscript we will explicitly state that the estimates apply to the regularized problem and add a remark clarifying the distinction from the original equation. revision: partial

  2. Referee: [§3.2 and §4.1] §3.2 (discrete problem) and §4.1 (stability): the existence and uniqueness proof relies on coercivity and continuity of the discrete form. It is not shown that the coercivity constant remains positive and bounded away from zero uniformly in both the mesh size h and the regularization parameter ε. This uniformity is load-bearing for the claim that the method remains well-posed and accurate in the limit ε→0.

    Authors: We agree that uniformity of the coercivity constant with respect to both h and ε is necessary to justify well-posedness in the vanishing-moment limit. The existence and uniqueness proof establishes well-posedness for each fixed pair (h, ε), but the coercivity and continuity constants inherit ε-dependence from the biharmonic term. We have not proved that these constants remain bounded away from zero independently of ε. In the revised version we will add a detailed remark on the ε-dependence of the stability constants and note that full uniformity requires further analysis of the regularized problem, possibly under additional assumptions on the data. revision: partial

  3. Referee: [§5] §5 (numerical experiments): all reported tests use a fixed value of ε (e.g., ε=0.01). No numerical study is presented that varies ε downward while monitoring both the discrete error and the difference between the computed solution and a reference solution of the original Monge-Ampère equation. Without such evidence, it remains unclear whether the observed optimal rates survive the regime required for practical approximation of the target problem.

    Authors: We appreciate the suggestion to strengthen the numerical validation. The experiments in Section 5 confirm the predicted rates for the regularized problem at a representative small value of ε. We will augment the numerical section with additional tests in which ε is successively reduced (e.g., ε = 0.1, 0.01, 0.001) and the discrete solutions are compared against a reference solution of the original Monge-Ampère equation obtained by a different method or with a much smaller regularization parameter. These results will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: error estimates derived from standard VEM theory and approximation arguments

full rationale

The paper develops a C1-nonconforming VEM for the vanishing-moment regularization of the Monge-Ampère equation and derives optimal a priori estimates in broken H2, H1 and L2 norms together with existence/uniqueness. These steps rest on standard virtual-element approximation theory, consistency and stability arguments for fourth-order problems, and explicit regularity assumptions on the regularized solution u_ε (typically H4 or higher). No load-bearing self-citation chain, self-definitional reduction, or fitted parameter renamed as prediction appears; the numerical experiments serve only as validation, not as the source of the claimed rates. While uniform-in-ε control of constants is left open, that is an assumption-strength issue rather than a circularity in the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the vanishing-moment regularization parameter is mentioned but not quantified or fitted in the given text.

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