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arxiv: 2604.19708 · v2 · submitted 2026-04-21 · 🧮 math.NA · cs.NA

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Proximal Discontinuous Galerkin Methods for Variational Inequalities

Alexandre Ern, Beatrice Riviere, Brendan Keith, Dohyun Kim, Rami Masri

Pith reviewed 2026-05-10 01:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords proximal point methoddiscontinuous Galerkinvariational inequalitiesobstacle problemhybrid high-order methodsBregman divergenceerror estimatesnonconforming finite elements
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The pith

Proximal discontinuous Galerkin methods for variational inequalities are analyzed in a unified framework that establishes existence, uniqueness, energy dissipation, and error estimates, with the hybrid high-order variant achieving higher-

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

The authors develop a family of proximal discontinuous Galerkin methods by applying four different nonconforming finite element discretizations to the Bregman proximal point algorithm for solving variational inequalities. They focus on the obstacle problem and provide a common analysis proving existence and uniqueness of solutions, discrete energy dissipation, and error estimates for both the solution and the Lagrange multiplier. The key new result is that the hybrid high-order method using constant cell unknowns and affine facet unknowns delivers higher-order convergence, something not previously obtained for proximal Galerkin methods.

Core claim

We introduce a family of proximal discontinuous Galerkin methods for variational inequalities obtained by applying nonconforming discretizations to the Bregman proximal point method. For the obstacle problem we prove existence and uniqueness of the discrete solutions, discrete energy dissipation, and error estimates for the primal variable and the dual variable. In particular the proximal hybrid high-order method with piecewise constant cell unknowns and piecewise affine facet unknowns yields higher-order convergence rates.

What carries the argument

The Bregman proximal point iteration discretized via nonconforming finite element methods and analyzed uniformly for the obstacle problem, with the hybrid high-order scheme using piecewise constant cell unknowns and piecewise affine facet unknowns as the variant delivering the new convergence result.

If this is right

  • Discrete solutions exist and are unique for each proximal subproblem under the unified framework.
  • A discrete energy functional dissipates monotonically at every proximal iteration step.
  • Error estimates hold for both the primal solution and the dual multiplier variable across the four methods.
  • The hybrid high-order variant realizes higher-order convergence in the energy norm for the first time among proximal Galerkin schemes.

Where Pith is reading between the lines

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

  • The proximal discretization approach could be carried over to other variational inequalities, such as those modeling contact or phase transitions.
  • The dual-variable error control might improve active-set identification in practical constrained optimization codes.
  • Time-dependent or evolutionary versions of the obstacle problem could be treated by applying the same proximal iteration at each time step.

Load-bearing premise

The unified analysis framework applies without additional restrictions to the four chosen nonconforming discretizations when applied to the Bregman proximal point iteration for the obstacle problem.

What would settle it

Numerical experiments in which the proximal hybrid high-order method fails to attain convergence rates higher than first order on a smooth obstacle problem would falsify the higher-order convergence claim.

Figures

Figures reproduced from arXiv: 2604.19708 by Alexandre Ern, Beatrice Riviere, Brendan Keith, Dohyun Kim, Rami Masri.

Figure 1
Figure 1. Figure 1: (a) The solution and the obstacle and (b) the corresponding La￾grange multiplier λ. 10-3 10-2 10-1 Mesh Size 10-6 10-5 10-4 10-3 10-2 10-1 ||u-u h|| L 2 (Ω) 2 MINI DG EG HIP HHO 10-3 10-2 10-1 Mesh Size 10-2 10-1 |u-u h | H 1(T h ) 1 MINI DG EG HIP HHO 10-3 10-2 10-1 Mesh Size 10 || 0 λ-λ h|| H-1 (Ω) 0.5 0.5 MINI DG EG HIP HHO (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence history of (a) ku − uhkL2(Ω); (b) |u − uh|H1(Th) ; (c) kλ − λhkL2(Ω). All methods show optimal convergence in H1 -norm as expected from Theorem 5.13. The solution also converges optimally in the L 2 -norm and the approximate multiplier λh converges in the L 2 -norm with a rate O(h 1/2 ). While all methods show a similar tendency, the HHO method provides the best accuracy, probably due to the us… view at source ↗
Figure 3
Figure 3. Figure 3: Proximal HHO(ℓ, r) method: Convergence history of (a) ku − R(uh)kL2(Ω); (b) |u − R(uh)|H1(Th) . minimal regularity assumptions. We also showed optimal error estimates in the energy norm under mild additional assumptions, demonstrating that the optimization error is decoupled from the spatial discretization error. A notable outcome is that the HHO method with piecewise constant cell unknowns and piecewise l… view at source ↗
read the original abstract

We introduce a family of proximal discontinuous Galerkin methods for variational inequalities, focusing on the obstacle problem as a didactic example. Each member of this family is born from applying a different well-known nonconforming finite element discretization to the Bregman proximal point method. We explicitly treat four examples: the symmetric interior penalty discontinuous Galerkin, the enriched Galerkin, the hybridizable interior penalty and the hybrid high-order methods. We formulate a unified analysis framework for this family of methods and prove the existence and uniqueness of solutions, energy dissipation, and error estimates for both the primal and dual variables. Remarkably, the proximal hybrid high-order method with piecewise constant cell unknowns and piecewise affine facet unknowns leads to the first higher-order convergence result for any proximal Galerkin method.

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

2 major / 3 minor

Summary. The manuscript introduces a family of proximal discontinuous Galerkin methods for variational inequalities, using the obstacle problem as the main example. It combines the Bregman proximal point iteration with four nonconforming discretizations (symmetric interior penalty DG, enriched Galerkin, hybridizable interior penalty, and hybrid high-order methods) and develops a unified analysis framework proving existence and uniqueness of solutions, energy dissipation, and error estimates for both primal and dual variables. The paper emphasizes that the proximal HHO method (piecewise constant cell unknowns and piecewise affine facet unknowns) yields the first higher-order convergence result for any proximal Galerkin method.

Significance. If the central claims hold, the work offers a valuable unified framework for applying proximal methods to nonconforming finite element discretizations of variational inequalities, with explicit proofs of well-posedness and energy stability. The potential for higher-order convergence via the proximal HHO variant would be a notable advance, as it addresses a gap in existing proximal Galerkin literature. The manuscript credits the combination of standard proximal and DG ingredients without introducing new ad-hoc parameters.

major comments (2)
  1. [§4 and Theorem 5.3] §4 (unified analysis framework) and Theorem 5.3 (error estimates for proximal HHO): the claim that the unified framework delivers higher-order convergence for the proximal HHO method (P0 cell + P1 facet) without additional restrictions beyond the other three methods is load-bearing for the headline result. The obstacle problem solution has at most H^2 regularity away from the free boundary; the consistency terms arising from the nonconforming HHO space and the proximal mapping must be controlled without mesh-dependent deterioration or implicit higher-regularity assumptions. The provided proof sketch does not explicitly verify this control for the HHO choice.
  2. [§3.3 and §5] §3.3 (proximal HHO formulation) and the dual-variable error estimate in §5: the analysis treats the proximal mapping applied to the obstacle constraint uniformly across methods, but the HHO facet unknowns (P1) introduce additional consistency terms whose bounding relies on the same regularity as the other methods. If this step invokes a constant that grows with the polynomial degree or mesh size near the free boundary, the stated higher-order rate fails to hold in the claimed generality.
minor comments (3)
  1. [§2] The notation for the Bregman proximal operator and the discrete dual variable is introduced without a dedicated table of symbols; adding one would improve readability when comparing the four methods.
  2. [Figure 1] Figure 1 (schematic of the four discretizations) uses inconsistent line styles for cell vs. facet unknowns; clarifying the legend would help readers distinguish the HHO variant.
  3. [§1 and §4] A few references to prior work on proximal methods for VIs (e.g., on conforming FEM) are cited only in the introduction; moving one or two to the unified analysis section would better contextualize the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our work. We address the major comments point by point below, providing clarifications on the analysis for the proximal HHO method.

read point-by-point responses

  1. Referee: [§4 and Theorem 5.3] §4 (unified analysis framework) and Theorem 5.3 (error estimates for proximal HHO): the claim that the unified framework delivers higher-order convergence for the proximal HHO method (P0 cell + P1 facet) without additional restrictions beyond the other three methods is load-bearing for the headline result. The obstacle problem solution has at most H^2 regularity away from the free boundary; the consistency terms arising from the nonconforming HHO space and the proximal mapping must be controlled without mesh-dependent deterioration or implicit higher-regularity assumptions. The provided proof sketch does not explicitly verify this control for the HHO choice.

    Authors: We appreciate this observation. The unified framework in Section 4 is designed to apply to all four methods, including HHO, by relying on abstract properties of the discrete spaces and the proximal operator. Specifically, the consistency error for the HHO method with P0 cell and P1 facet unknowns is controlled using the standard approximation properties and the H^2 regularity of the solution away from the free boundary, which is the same as assumed for the other methods. The proximal mapping does not introduce additional regularity requirements because the estimates are based on the variational inequality structure and the energy dissipation. We acknowledge that the proof sketch in Theorem 5.3 could be expanded to explicitly detail the HHO case. We will revise the manuscript to include a more detailed verification of the consistency terms for the HHO method in the revised version. revision: partial

  2. Referee: [§3.3 and §5] §3.3 (proximal HHO formulation) and the dual-variable error estimate in §5: the analysis treats the proximal mapping applied to the obstacle constraint uniformly across methods, but the HHO facet unknowns (P1) introduce additional consistency terms whose bounding relies on the same regularity as the other methods. If this step invokes a constant that grows with the polynomial degree or mesh size near the free boundary, the stated higher-order rate fails to hold in the claimed generality.

    Authors: Thank you for pointing this out. In Section 3.3, the proximal HHO formulation is presented, and the dual error estimates in Section 5 follow from the unified framework. The additional consistency terms from the P1 facet unknowns are bounded using trace inequalities and the local regularity, without dependence on the mesh size in a deteriorating way or on the polynomial degree beyond the fixed degree 1. The higher-order convergence is achieved because the HHO space allows for better approximation in the dual variable compared to standard DG methods. We do not invoke constants that grow with mesh size near the free boundary; the estimates are global but localized via the proximal iteration. We maintain that the analysis holds as stated. We will add a remark in the revised manuscript to clarify the bounding of these terms. revision: partial

Circularity Check

0 steps flagged

No significant circularity; unified framework derives error estimates independently

full rationale

The paper constructs a unified analysis for proximal DG methods (SIPG, EG, HDG, HHO) applied to the Bregman proximal point iteration on the obstacle problem. Existence, uniqueness, energy dissipation, and primal/dual error estimates follow from standard nonconforming DG consistency and stability arguments plus proximal mapping properties. The higher-order claim for P0-cell/P1-facet HHO is obtained by plugging the specific approximation properties of that space into the same abstract estimates; no parameter is fitted to data and then renamed a prediction, no self-citation supplies a uniqueness theorem that forces the result, and the derivation chain does not reduce any claimed quantity to itself by definition. The framework remains self-contained against external benchmarks (standard DG theory and proximal-point convergence).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard mathematical assumptions from finite-element theory and proximal-point algorithms but introduces no free parameters, new axioms, or invented entities.

axioms (1)
  • domain assumption Standard well-posedness assumptions for variational inequalities and nonconforming finite-element spaces
    Implicit in the statements of existence, uniqueness, and error estimates for the obstacle problem.

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Forward citations

Cited by 2 Pith papers

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    Proximal Galerkin method exactly enforces the isometry constraint for nonlinear plates at mesh cell barycenters without preprocessing, yielding asymptotically mesh-independent convergence.

  2. Proximal Galerkin for Phase Field Fracture

    math.NA 2026-04 unverdicted novelty 6.0

    The proximal Galerkin method reformulates phase-field fracture constraints into saddle-point problems to enforce physical bounds and irreversibility for static and dynamic cases.

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