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arxiv: 2604.08246 · v2 · submitted 2026-04-09 · 🧮 math.NA · cs.NA

Local discontinuous Galerkin FEM for convex minimization

Carsten Carstensen , Ngoc Tien Tran This is my paper

Pith reviewed 2026-05-10 17:29 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords convex minimizationlocal discontinuous Galerkina priori error estimatesdualitytwo-sided p-growtha posteriori error controladaptive mesh refinementfinite element methods
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The pith

Duality relations yield improved a priori convergence rates for local discontinuous Galerkin methods in convex minimization problems.

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

This paper demonstrates that for convex minimization problems with two-sided p-growth, local discontinuous Galerkin finite element schemes achieve improved convergence rates for the error in minimal energies. The improvement arises from duality relations linking a discrete primal problem to a semi-discrete dual problem, where the infinite-dimensional dual creates only a tiny gap that vanishes for polynomial low-order terms. A reader would care because this removes the previous suboptimality of nonconforming schemes relative to conforming ones, supporting sharper a priori bounds, two-energy a posteriori control, and adaptive mesh refinement for nonlinear problems.

Core claim

The innovative point of departure in a refined analysis of two discontinuous Galerkin schemes exploits duality relations between a discrete primal and a semi-discrete dual problem. The infinite-dimensional dual problem leads to a tiny duality gap that even vanishes for polynomial low-order terms. For a class of degenerated convex minimization problems with two-sided p growth, the novel duality provides improved a priori convergence rates for the error in the minimal energies. This closes the misfit of convergence rates for the conforming and nonconforming schemes at least for the local discontinuous Galerkin schemes at hand.

What carries the argument

Duality relations between a discrete primal problem and a semi-discrete dual problem that exploit the infinite-dimensional dual to produce a tiny duality gap.

If this is right

  • The two-energy principle together with Raviart-Thomas post-processing of the dual variable supplies an a posteriori error estimator.
  • The estimator can drive adaptive mesh-refining procedures.
  • Benchmarks show that adaptive refinement produces visibly faster convergence than uniform refinement.
  • The improved rates apply uniformly to the class of degenerated convex problems obeying two-sided p-growth.

Where Pith is reading between the lines

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

  • The same duality construction could be tested on other nonconforming methods to check whether the optimality gap closes there as well.
  • When the low-order term is polynomial the vanishing duality gap may permit exact energy recovery for certain discrete solutions.
  • The approach may extend naturally to time-dependent or quasistatic versions of the same minimization problems.

Load-bearing premise

The minimization problems satisfy two-sided p-growth conditions that allow duality relations to hold between the discrete primal and semi-discrete dual despite the tiny gap introduced by the infinite-dimensional dual.

What would settle it

A concrete convex minimization example with two-sided p-growth where the local discontinuous Galerkin energy-error convergence rate stays strictly suboptimal and does not attain the improved rate given by the duality analysis.

Figures

Figures reproduced from arXiv: 2604.08246 by Carsten Carstensen, Ngoc Tien Tran.

Figure 1
Figure 1. Figure 1: (a) Initial triangulation of the L-shaped domain into 6 triangles and (b) material distribution in the optimal design problem of Subsection 6.2 102 103 104 105 106 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 O(ndof−2/3) O(ndof−1) O(ndof−5/4) ndof k = 1 k = 2 k = 3 k = 4 [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Convergence history plot of η for k = 1, . . . , 4 and (b) adaptive triangulation in the optimal design problem of Subsection 6.2 6.2. Optimal design problem. This model problem seeks the optimal distribu￾tion of two materials with fixed amounts to fill a given domain for maximal torsion stiffness [40, 7]. Given parameters 0 < t1 < t2 and 0 < µ1 < µ2 with t1µ2 = µ1t2, the energy density W(a) := w(|a|),… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Convergence history plot of η for k = 1, . . . , 4 and (b) adaptive triangulation in the 4-Laplace problem of Subsection 6.3 [−1, 0] ∪ [0, 1] × {0}). Since W satisfies (B3) and (4.14), (4.18) holds with δ(∇u, ∇vC ) := ∥∇(u − vC )∥ 4 4 + ∥DW(∇u) − DW(∇vC )∥ 2 4/3 (1 + ∥∇u∥ 4 4 + ∥∇vC ∥ 4 4 ) 1/2 . Furthermore, DW is strongly monotone w.r.t. the quasi norm [2, 3, 27] so that, additionally, the error ∥ √ϱ… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Convergence history plot of η for k = 1, . . . , 4 and adaptive trian￾gulation in the Bingham flow problem of Subsection 6.4. The results are obtained with ε = 10−5 6.4. Bingham flow through a pipe. Given fixed positive parameters µ, g > 0, the modelling of a uni-directional flow through a pipe with cross-section Ω ⊂ R 2 leads to the minimization problem (1.1) with the energy density W(a) := µ|a| 2 /2 … view at source ↗
Figure 5
Figure 5. Figure 5: Convergence history plot of η for k = 2 and ε = 10−3 , . . . , 10−6 in Subsection 6.4 cf. [30, 17], and ΓD = ∂Ω. An explicit computation [46] shows that W∗ (α) = ( 0 if |α| ≤ g (|α| − g) 2/(2µ) if |α| > g. The strict convexity of W leads to a unique the minimizer u of E in V . Al￾though W is not differentiable, there exists σ ∈ H(div, Ω) = W 2 (div, Ω) such that σ ∈ ∂W(∇u) and div σ = −f pointwise a.e. in … view at source ↗
read the original abstract

The heart of the a priori and a posteriori error control in convex minimization problems is the sharp control of the differences of discrete and exact minimal energy. Conforming finite element discretizations for p-Laplace type minimization problems provide upper bounds of the energy difference with optimal convergence rates. Even for smooth solutions, known convergence rates for higher-order non-conforming finite element discretizations for the same problem class with $2 < p < \infty$, however, are exclusively suboptimal. Thus the popular a posteriori error control within the two-energy principle, that generalize hyper-circle identities, appears unbalanced. The innovative point of departure in a refined analysis of two discontinuous Galerkin (dG) schemes exploits duality relations between a discrete primal and a semi-discrete dual problem. The infinite-dimensional dual problem leads to a tiny duality gap that even vanishes for polynomial low-order terms. For a class of degenerated convex minimization problems with two-sided $p$ growth, the novel duality provides improved a priori convergence rates for the error in the minimal energies. This closes the misfit of convergence rates for the conforming and nonconforming schemes at least for the local discontinuous Galerkin schemes at hand. The motivating two-energy principle and some post-processing for a Raviart-Thomas dual variable provides an a posteriori error control, that also may drive adaptive mesh-refining. Computational benchmarks provide striking numerical evidence for improved convergence rates of the adaptive beyond uniform mesh-refining.

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 manuscript develops a duality-based analysis for local discontinuous Galerkin (LDG) discretizations of convex minimization problems with two-sided p-growth. It relates the discrete primal LDG problem to a semi-discrete dual problem, claiming that the resulting duality gap is tiny (and vanishes for polynomial low-order terms), which yields improved a priori convergence rates for the minimal-energy error that close the gap with conforming schemes. It also derives an a posteriori estimator from the two-energy principle with Raviart-Thomas post-processing and presents numerical benchmarks showing benefits of adaptive refinement over uniform meshes.

Significance. If the duality-gap control is rigorous, the work would resolve a known discrepancy between optimal conforming rates and suboptimal nonconforming rates for p-Laplace-type problems, providing a general tool for energy-error analysis and adaptive algorithms in nonlinear convex minimization.

major comments (1)
  1. [Duality relations and main a priori result] The central claim rests on the duality gap from the infinite-dimensional dual being of strictly higher order than the improved energy-error rate. The abstract asserts the gap is 'tiny' and vanishes for polynomial low-order terms under two-sided p-growth, yet no explicit bound is visible in the provided description showing that the gap term is absorbed without interaction with the nonconformity measure degrading the rate back to the known suboptimal bound; this must be verified in the main theorem.
minor comments (1)
  1. [Computational benchmarks] The numerical benchmarks are described as 'striking,' but the manuscript should include a direct comparison table of energy-error rates for the LDG scheme versus a conforming reference on the same meshes to quantify the improvement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comment on the duality-gap control is well taken, and we have revised the presentation to make the absorption argument fully explicit while preserving the original analysis.

read point-by-point responses

  1. Referee: [Duality relations and main a priori result] The central claim rests on the duality gap from the infinite-dimensional dual being of strictly higher order than the improved energy-error rate. The abstract asserts the gap is 'tiny' and vanishes for polynomial low-order terms under two-sided p-growth, yet no explicit bound is visible in the provided description showing that the gap term is absorbed without interaction with the nonconformity measure degrading the rate back to the known suboptimal bound; this must be verified in the main theorem.

    Authors: We agree that the absorption step requires explicit verification. In the full manuscript the main a priori result is Theorem 3.4, whose proof begins from the two-energy identity relating the discrete primal energy to the semi-discrete dual. Lemma 3.3 supplies the explicit bound on the infinite-dimensional duality gap: under the two-sided p-growth assumption the gap is controlled by C(h^{r} + nonconformity term), where the exponent r is strictly larger than the target energy-error rate. Because the nonconformity contribution enters with a positive coefficient, a standard Young inequality absorbs the gap into the left-hand side without reducing the leading order. For polynomial low-order terms the gap vanishes identically (Remark 3.5). To address the referee’s concern we have inserted a new paragraph immediately after Theorem 3.4 that isolates this absorption argument and confirms that the nonconformity measure does not degrade the rate. The abstract has also been updated to point to Theorem 3.4. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard duality relations without reduction to inputs by construction

full rationale

The paper claims improved a priori energy-error rates for LDG schemes on degenerated convex minimization problems by exploiting duality between a discrete primal problem and a semi-discrete dual, with the infinite-dimensional dual producing only a tiny gap that vanishes for low-order polynomial terms. This step rests on established convex-analysis duality relations (two-sided p-growth conditions allowing duality between discrete primal and semi-discrete dual) rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equation or theorem in the provided abstract or description reduces the target convergence-rate improvement to a quantity defined in terms of itself or to a prior result whose validity is presupposed by the present work. The argument therefore remains self-contained against external benchmarks in convex minimization theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on convexity of the energy functional, two-sided p-growth conditions for the class of problems, and standard duality relations in variational calculus; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The minimization functional is convex and satisfies two-sided p-growth bounds for 2 < p < infinity
    Invoked to obtain the improved rates and vanishing duality gap for the class of degenerated problems.
  • standard math Duality relations exist between the discrete primal problem and the infinite-dimensional dual problem
    Core of the refined analysis; standard in convex optimization but applied here to close the rate gap.

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