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arxiv: 2502.01148 · v2 · submitted 2025-02-03 · 🧮 math.NA · cs.NA· math.AP

A Discontinuous Galerkin Method for H(curl)-Elliptic Hemivariational Inequalities

Xiajie Huang , Fei Wang , Weimin Han , Min Ling This is my paper

Pith reviewed 2026-05-23 04:41 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP
keywords discontinuous Galerkin methodhemivariational inequalitiesH(curl) spaceinterior penaltya priori error estimatesnumerical fluxfinite element method
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The pith

An Interior Penalty Discontinuous Galerkin scheme solves H(curl)-elliptic hemivariational inequalities with optimal convergence under regularity assumptions.

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

The paper develops a Discontinuous Galerkin method for H(curl)-elliptic hemivariational inequalities. An Interior Penalty DG scheme is built by choosing a suitable numerical flux. Analysis establishes consistency, boundedness, stability, existence, uniqueness and uniform boundedness of the discrete solutions. A priori error estimates then prove optimal-order convergence assuming the true solution meets suitable regularity conditions. A numerical example verifies the predicted rates.

Core claim

By selecting an appropriate numerical flux, an Interior Penalty Discontinuous Galerkin (IPDG) scheme is constructed for H(curl)-elliptic hemivariational inequalities. The scheme is shown to be consistent, bounded and stable. Existence, uniqueness and uniform boundedness of the numerical solutions are proved. A priori error estimates demonstrate the optimal convergence order under suitable solution regularity assumptions.

What carries the argument

The Interior Penalty Discontinuous Galerkin (IPDG) scheme, which discretizes the hemivariational inequality over a discontinuous finite-element space using a chosen numerical flux together with penalty terms to enforce stability.

If this is right

  • The IPDG scheme supplies a convergent numerical method for H(curl)-elliptic hemivariational inequalities arising in electromagnetics.
  • Mesh size can be chosen in advance to meet a target accuracy using the explicit error bounds.
  • The consistency and stability properties extend directly to related variational problems in the same space.

Where Pith is reading between the lines

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

  • The flux choice may transfer to other DG formulations for hemivariational inequalities without major redesign.
  • If typical physical solutions satisfy the regularity needed, the method delivers reliable accuracy on practical meshes.
  • Extension to time-dependent or nonlinear variants would follow the same stability framework.

Load-bearing premise

The true solution must possess enough smoothness for the a priori error estimates to reach the claimed optimal order.

What would settle it

A numerical experiment on a problem whose solution has insufficient regularity where the observed convergence rate drops below the predicted optimal order.

Figures

Figures reproduced from arXiv: 2502.01148 by Fei Wang, Min Ling, Weimin Han, Xiajie Huang.

Figure 1
Figure 1. Figure 1: Two adjacent tetrahedral elements and a common face [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of the numerical solutions 1 in the energy norm ∥ · ∥h with respect to the grid size h. The numerical convergence orders in the energy norm ∥ · ∥h match the theoretical result established in Theorem 3 [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Streamlines of numerical solutions 24 [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
read the original abstract

In this paper, we develop a Discontinuous Galerkin (DG) method for solving H(curl)-elliptic hemivariational inequalities. By selecting an appropriate numerical flux, we construct an Interior Penalty Discontinuous Galerkin (IPDG) scheme. A comprehensive numerical analysis of the IPDG method is conducted, addressing key aspects such as consistency, boundedness, stability, and the existence, uniqueness, uniform boundedness of the numerical solutions. Building on these properties, we establish a priori error estimates, demonstrating the optimal convergence order of the numerical solutions under suitable solution regularity assumptions. Finally, a numerical example is presented to illustrate the theoretically predicted convergence order and to show the effectiveness of the proposed 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

1 major / 1 minor

Summary. The paper develops an Interior Penalty Discontinuous Galerkin (IPDG) scheme for H(curl)-elliptic hemivariational inequalities by selecting an appropriate numerical flux. It performs a numerical analysis covering consistency, boundedness, stability, existence, uniqueness and uniform boundedness of the discrete solutions. Building on these, a priori error estimates are derived that establish optimal convergence rates under suitable solution regularity assumptions. A numerical example is presented to illustrate the predicted rates.

Significance. If the analysis holds, the work extends DG methods to hemivariational inequalities in the H(curl) setting, which is relevant for applications involving non-monotone nonlinearities such as certain electromagnetic or contact problems. The numerical example provides concrete verification of the convergence claims.

major comments (1)
  1. [Abstract] Abstract (final paragraph): the a priori error estimates claim optimal order under 'suitable solution regularity assumptions,' yet no quantitative conditions are supplied (e.g., smallness of the Lipschitz constant of the hemivariational term or mesh-independent penalty scaling) that would guarantee the Céa-type argument survives when the exact solution exhibits the reduced regularity (limited Sobolev index or jumps in tangential traces) typical of hemivariational inequalities. This assumption is load-bearing for the central optimality claim.
minor comments (1)
  1. The abstract states that 'a comprehensive numerical analysis' is conducted; the manuscript should explicitly reference the sections containing the full proofs of consistency, boundedness and stability rather than summarizing them at a high level.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph): the a priori error estimates claim optimal order under 'suitable solution regularity assumptions,' yet no quantitative conditions are supplied (e.g., smallness of the Lipschitz constant of the hemivariational term or mesh-independent penalty scaling) that would guarantee the Céa-type argument survives when the exact solution exhibits the reduced regularity (limited Sobolev index or jumps in tangential traces) typical of hemivariational inequalities. This assumption is load-bearing for the central optimality claim.

    Authors: We thank the referee for highlighting this point. The quantitative conditions underlying the a priori estimates are stated explicitly in the body of the paper: the penalty parameter is required to be sufficiently large but mesh-independent (see the coercivity and boundedness arguments in Section 3), and the solution regularity is quantified in Theorems 4.3 and 5.2 (u belonging to a space with Sobolev index sufficient for the optimal rate, with tangential traces controlled via standard trace inequalities). The Céa-type argument does not rely on a smallness condition for the Lipschitz constant of the hemivariational term; instead, it uses the monotonicity of the linear part together with the boundedness and consistency properties of the nonlinear term. We agree, however, that the abstract is too terse on these points and will revise it to reference the specific assumptions appearing in the theorems. revision: yes

Circularity Check

0 steps flagged

No circularity detected; standard conditional a priori analysis

full rationale

The provided abstract and description outline a standard DG discretization and error analysis for hemivariational inequalities that proceeds from consistency, boundedness, stability, and existence/uniqueness to conditional a priori estimates under regularity assumptions. No equations, parameters, or claims are shown to reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the regularity hypothesis is an explicit external assumption rather than a derived or renamed quantity. The numerical example merely illustrates the predicted order, without feeding back into the derivation. This is self-contained numerical analysis with no exhibited circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; the paper invokes standard mathematical assumptions on the nonlinear operator and solution regularity that are typical for hemivariational inequalities and DG theory.

axioms (2)
  • domain assumption The nonlinear operator satisfies the standard conditions that guarantee existence and uniqueness for the continuous hemivariational inequality.
    Required for the discrete existence/uniqueness and error analysis statements.
  • domain assumption The exact solution possesses the regularity needed for optimal-order error estimates.
    Explicitly stated in the abstract as the condition under which optimal convergence holds.

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