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arxiv: 2308.03223 · v3 · submitted 2023-08-06 · 🧮 math.NA · cs.NA

Discrete weak duality of hybrid high-order methods for convex minimization problems

Ngoc Tien Tran This is my paper

Pith reviewed 2026-05-24 08:06 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords hybrid high-order methodsconvex minimizationdiscrete dualitya priori error estimatesa posteriori error estimatesadaptive mesh refinementpolyhedral meshes
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The pith

A hybrid high-order method for convex minimization admits a matching discrete dual problem whose weak duality produces error estimates.

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

The paper constructs a discrete dual problem that pairs with a given hybrid high-order discretization of a convex minimization problem. The resulting primal-dual pair satisfies a weak convex duality identity that holds on arbitrary polyhedral meshes and for any polynomial degree. This identity directly supplies a priori error estimates that achieve convergence rates once the solution is assumed smooth enough. A new postprocessing step then converts the duality into a posteriori error indicators that are computable on regular simplicial triangulations and that justify an adaptive refinement strategy shown to outperform uniform refinement.

Core claim

For a prototypical hybrid high-order method applied to convex minimization, the paper derives an explicit discrete dual problem on the same mesh. The discrete primal and dual problems obey a weak convex duality relation that follows from the algebraic structure of the discretization. Under additional smoothness the relation produces a priori error estimates with explicit rates. The same duality, after a novel postprocessing of the discrete solution, supplies a posteriori estimates on simplicial meshes and thereby motivates an adaptive algorithm.

What carries the argument

The discrete dual problem constructed from the hybrid high-order primal formulation, together with the weak convex duality identity that it satisfies by direct algebraic construction.

If this is right

  • A priori error estimates with convergence rates hold under smoothness assumptions on general polyhedral meshes and for arbitrary polynomial degrees.
  • A novel postprocessing step produces computable a posteriori error indicators on regular triangulations into simplices.
  • The a posteriori indicators justify an adaptive mesh-refining algorithm that outperforms uniform refinement in the reported experiments.
  • All stated results apply without restriction on mesh topology beyond polyhedral cells or on the polynomial degree of the hybrid high-order spaces.

Where Pith is reading between the lines

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

  • The same discrete-duality construction may apply to other nonconforming methods whose degrees of freedom allow a natural dual formulation.
  • The postprocessed a posteriori estimator could be combined with existing residual-type indicators to produce more robust marking strategies in three dimensions.
  • Because the duality holds for arbitrary degree, the adaptive algorithm could be tested with increasing polynomial degree on fixed meshes to separate h- and p-convergence effects.

Load-bearing premise

The weak duality identity is assumed to follow immediately from the discrete hybrid high-order construction without any further consistency or stability hypotheses.

What would settle it

A numerical test in which the computed duality gap between the discrete primal and dual solutions fails to vanish at the rate predicted by the a priori analysis for a smooth exact solution.

Figures

Figures reproduced from arXiv: 2308.03223 by 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−3/4) O(ndof−5/4) ndof k = 0 k = 1 k = 2 k = 3 k = 4 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Convergence history plot of Eσ0 (v0)−E∗ σ0 (σ0) for various k and (b) adaptive triangulation into 2013 triangles obtained with k = 2 in Subsection 6.2 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence history plot of Eσ ε 0 (v ε 0 ) − E∗ σ ε 0 (σ ε 0 ) for (a) various k and ε = 10−4 and (b) k = 2 and various ε in Subsection 6.3 6.3. Bingham flow in 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 Ψ(a) := µ|a| 2 /2 + g|a| for any a ∈ R 2 , cf. [17]… view at source ↗
Figure 4
Figure 4. Figure 4: Convergence history plot of (a) RHS and (b) ∥ √ϱ∇(u−v0)∥ 2 2 for various k in Subsection 6.4 6.4. p-Laplace problem. In this final benchmark, we consider the 4-Laplace prob￾lem from Remark 4.5 with the exact solution u(r, φ) = r 7/8 sin(7φ/8) ∈ W1,4 (Ω) and the right-hand side f(r, φ) = (7/8)3/4 r −11/8 sin(7φ/8) ∈ L 16/11−ε (Ω) [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence history plot of (a) ∥σ−∇Ψ(∇v0)∥ 2 4/3 and (b) ∥∇(u−v0)∥ 2 4 for various k in Subsection 6.4 for any ε > 0 from [13]. Some remarks are in order due to the inhomogenous Dirichlet data prescribed by u. The discrete problem minimizes (1.4) in the affine space IV u + VD(M). Since the construction of σ0 in Lemma 5.2 only relies on the discrete Euler-Lagrange equations, it applies verbatim. To obtain … view at source ↗
read the original abstract

This paper derives a discrete dual problem for a prototypical hybrid high-order method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds for general polyhedral meshes and arbitrary polynomial degrees of the discretization. A novel postprocessing is proposed and allows for a~posteriori error estimates on regular triangulations into simplices using primal-dual techniques. This motivates an adaptive mesh-refining algorithm, which performs superiorly compared to uniform mesh refinements.

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 derives a discrete dual problem paired with a prototypical hybrid high-order (HHO) discretization of convex minimization problems. It establishes that the discrete primal and dual satisfy a weak convex duality identity that holds identically from the construction, for general polyhedral meshes and arbitrary polynomial degrees. This identity is used to derive a priori error estimates with convergence rates under additional smoothness assumptions. A novel postprocessing is introduced that enables a posteriori error estimates on regular simplicial triangulations via primal-dual techniques, which in turn motivates an adaptive mesh-refinement algorithm shown to outperform uniform refinement.

Significance. If the central derivation holds, the work supplies a useful discrete duality framework for HHO methods on nonlinear convex problems. The mesh- and degree-independent character of the weak duality, together with the postprocessing that yields computable a posteriori bounds, strengthens the analytical toolkit for these methods and supports practical adaptivity. The direct construction without extra consistency hypotheses beyond the standard HHO stability is a positive feature of the approach.

minor comments (3)
  1. The abstract states that the duality 'follows directly from the discrete construction,' but the precise algebraic steps establishing the weak duality identity (presumably in the main theorem) should be highlighted with an explicit equation reference for readers.
  2. Notation for the discrete dual variables and the postprocessing operator should be introduced with a short table or diagram in the preliminaries to improve readability across the a priori and a posteriori sections.
  3. The numerical experiments section would benefit from an additional column or plot quantifying the effect sizes of the adaptive versus uniform refinement (e.g., degrees of freedom versus error) to make the superiority claim more quantitative.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The provided summary accurately captures the derivation of the discrete weak duality, its use for a priori estimates, the postprocessing for a posteriori bounds, and the adaptive algorithm. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs a discrete dual problem paired with the HHO primal discretization for convex minimization problems on general polyhedral meshes. The weak convex duality is presented as following directly from this discrete construction for arbitrary polynomial degrees, without reduction to fitted parameters, self-citations, or ansatzes imported from prior work. Error estimates follow from the duality under extra smoothness assumptions, and the postprocessing for a posteriori estimates is a separate proposal. No load-bearing step equates a claimed result to its inputs by definition; the derivation remains self-contained as an algebraic identity from the discrete formulation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the domain assumption that the continuous problem is convex and that a prototypical hybrid high-order discretization exists for it; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption The continuous minimization problem is convex.
    Required for the notion of weak convex duality to be meaningful in both continuous and discrete settings.
  • domain assumption A prototypical hybrid high-order method can be defined on general polyhedral meshes for arbitrary polynomial degree.
    Stated explicitly as the setting in which the discrete dual and duality identity are derived.

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