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A priori error estimates for a Hybrid High-Order method on variable-exponent Leray-Lions problems

André Harnist, Romain Mottier

arXiv:2610.21611v1math.NAcs.NAmath.AP

TL;DR

Quantitative a priori estimates for hybrid nonconforming discretizations with variable growth have been lacking, despite prior numerical work on variable-exponent problems. This paper analyzes an HHO method under additional regularity and derives localized and regime-dependent bounds. The estimates recover order k+1 in the subquadratic range when the non-degeneracy scale is resolved, and experiments corroborate the rates while showing the relevance of flux regularity.

  • Problem

    Quantitative a priori estimates for hybrid nonconforming discretizations retaining pointwise variable exponents in bulk and face contributions have not previously been established.

    Source: §1 · pp. 1–2

  • Method

    The paper analyzes the HHO discretization using variable-exponent approximation results for polynomial projectors and discrete tools for HHO seminorms.

    Source: §1 · p. 3

  • Results

    When the local approximation scale resolves the non-degeneracy scale, the subquadratic refinement recovers order k+1, and numerical tests corroborate predicted rates across regimes.

    Source: §1 · p. 3, §8 · p. 28

  • Takeaways & Limitations

    Smooth non-degenerate tests with δ=1 show order k+1 in mixed and superquadratic regimes, while limited flux regularity can prevent higher polynomial degree from improving the asymptotic rate.

    Source: §8 · p. 28

    The stated results do not cover nonhomogeneous Dirichlet data, for which quantitative estimates require control of the discrete lifting and additional consistency terms.

    Source: §2 · p. 4

Abstract

from arXiv · show

We derive a priori error estimates for the Hybrid High-Order (HHO) discretization of variable-exponent Leray--Lions problems introduced in a companion convergence study. HHO methods belong to the family of hybrid nonconforming methods, based on cell and face polynomial unknowns of arbitrary degree. Under additional broken regularity of the exact solution and its flux, we establish a localized error estimate that retains the individual element sizes and a global estimate whose rate distinguishes the purely subquadratic, mixed subquadratic--superquadratic, and purely superquadratic regimes. In the subquadratic range, we also derive a regime-dependent refinement that compares the local approximation scale with the local non-degeneracy scale and recovers the optimal polynomial order when that scale is resolved by the mesh. The analysis relies on variable-exponent approximation properties of polynomial projectors, mean-exponent tools for discrete polynomial spaces, and weighted monotonicity estimates. The framework also covers regularized and mildly non-symmetric anisotropic Carreau--Yasuda-type fluxes. Numerical experiments on Cartesian and Voronoi meshes support the predicted rates and show that the regularity assumption on the flux cannot be dispensed with.

1 Introduction

This paper develops quantitative a priori error estimates for an HHO discretization of variable-exponent Leray–Lions problems, extending prior qualitative convergence results under additional regularity assumptions. It provides localized and regime-sensitive global bounds, a subquadratic refinement, and numerical evidence supporting the predicted rates.

1 Introduction

  • The framework includes regularized and mildly non-symmetric anisotropic Carreau–Yasuda-type fluxes.The paper verifies that the anisotropic example satisfies the analysis’s structural assumptions.Source: §1 · pp. 1, 3
  • The paper addresses a gap in quantitative error estimates for hybrid nonconforming methods that retain the pointwise exponent in both bulk and face contributions.Earlier work includes conforming finite elements, Crouzeix–Raviart discretization, and other schemes, but the stated hybrid-method estimates were not previously established.Source: §1 · p. 2
  • The analysis develops variable-exponent projector approximation and stability results, a uniform John-domain construction, and a discrete Lebesgue embedding for HHO seminorms.These tools support quantitative estimates under additional regularity, complementing the companion paper’s well-posedness and qualitative convergence results under minimal regularity.Source: §1 · pp. 1, 3
  • A localized estimate in the natural discrete W^{1,p(·)}-type norm retains element sizes h_T instead of replacing them with the largest mesh size.Under uniform control of the stated regularity seminorms, the global estimate distinguishes the subquadratic, mixed, and superquadratic regimes.Source: §1 · p. 3
  • The estimates retain pointwise growth dependence across purely subquadratic, mixed, and superquadratic regimes; for constant exponents, the rates are (k+1)p/2 for 1 < p ≤ 2 and (k+1)/(p−1) for p ≥ 2.In the subquadratic range, the robust exponent p/2 improves on the earlier HHO exponent p−1 under the paper’s regularity hypotheses.Source: §1 · p. 3
  • When the local approximation scale resolves the non-degeneracy scale, the subquadratic refinement recovers the optimal order k+1.The refinement compares local approximation and bulk and face non-degeneracy scales; otherwise, the robust estimate remains available.Source: §1 · p. 3
  • Cartesian and Voronoi experiments reach at least the predicted rates for smooth solutions, while limited flux regularity can make higher polynomial degree yield no gain.For non-degenerate tests with δ=1, order k+1 is observed in all three regimes.Source: §1 · pp. 3–4

2 Continuous problem

The continuous problem is a variable-exponent Leray–Lions model with a position-dependent flux, encompassing the p(x)-Laplacian and Carreau–Yasuda-type laws. The paper formulates its weak problem for homogeneous Dirichlet data and relies on structural conditions on the flux.

2 Continuous problem

  • The model uses a Leray–Lions flux that acts on the solution gradient and may vary with position.Source: §2 · p. 4
  • Quantitative error estimates are stated for zero boundary data; extending them to nonzero data requires control of the discrete lifting and added consistency terms.Source: §2 · p. 4

2.1 Flux function

  • The flux assumptions impose variable-exponent growth, continuity, strong monotonicity, and zero-at-rest behavior.Source: §2.1 · p. 4
  • The admissible fluxes include the p(x)-Laplacian and regularized or non-symmetric anisotropic Carreau–Yasuda-type laws.Source: §2.1 · p. 5

2.2 Weak formulation

  • The weak formulation is posed in the homogeneous-Dirichlet potential space, and the continuous problem has a unique solution under the flux assumptions.Source: §2.2 · pp. 5–6

3 HHO discretization

The HHO discretization uses cell and face polynomial unknowns on a shape-regular polytopal mesh, with local gradient and potential reconstructions and stabilization assembled into a discrete problem. Its discrete formulation is well-posed under the stated assumptions.

3.1 Discrete setting

  • The mesh sequence is refining and uniformly regular, controlling cell and face scales and the number of faces per cell.Source: §3.1 · p. 6

3.2 Hybrid space and norms

  • HHO unknowns are polynomials on cells and faces, with homogeneous boundary values imposed on boundary-face degrees of freedom.Source: §3.2 · p. 7

3.4 Stabilization function

  • Stabilization controls the discrepancy between cell traces and face unknowns, and its flux need not reproduce the bulk constitutive flux.Source: §3.3 · pp. 7–8, §3.4 · p. 8

3.3 Operator reconstructions

  • The reconstructed gradient uses cell and face values in a discrete integration-by-parts relation, while the potential reconstruction has degree k+1.Source: §3.3 · p. 7

3.5 Discrete problem

  • The discrete diffusion form combines reconstructed-gradient and stabilization contributions cell by cell before assembling the global problem.Source: §3.5 · pp. 8–9
  • The discrete problem has a unique solution under Assumptions 2.2 and 3.3, with a stability bound uniform in mesh size and degeneracy parameters.Source: §3.5 · p. 9

4 Variable-exponent discrete norms and inequalities

This section develops discrete norms and inequalities for the variable-exponent analysis. It establishes equivalence among discrete seminorms, a discrete Lebesgue embedding, and a local trace bound.

4.1 Equivalence between discrete seminorms

  • The discrete seminorms based on element means, variable exponents, and reconstructed gradients are locally and globally equivalent.Source: §4.1 · p. 10

4.2 HHO discrete Lebesgue embedding

  • The HHO discrete Lebesgue embedding bounds the discrete q-seminorm by the p-seminorm when the stated element-mean exponents satisfy the required relation.Source: §4.2 · p. 10

4.3 Continuous local trace inequality

  • A continuous local trace inequality controls the boundary norm by the cell norm and a mesh-scaled gradient norm.Source: §4.3 · p. 10

5 Projector approximation and discrete operators

The section establishes variable-exponent approximation bounds for polynomial projectors and properties of the local discrete operators. The approximation analysis applies to cells with uniformly controlled geometry.

5.1 Variable-exponent approximation properties of the projectors

  • The projector estimates cover star-shaped domains and unions of a bounded number of comparable, shape-regular simplices with connected full-facet adjacency.Source: §5.1 · p. 11
  • Polynomial projectors satisfy variable-exponent approximation and scaled trace bounds on mesh cells, with rates determined by the smoothness order and cell size.Source: §5.1 · p. 13

5.2 Properties of the discrete operators

  • The local interpolator is bounded in variable-exponent norms, the gradient reconstruction commutes, and the boundary residual vanishes on interpolated polynomials of degree at most k+1.Source: §5.2 · p. 15

6 A priori error estimate

This section establishes localized and global a priori error estimates, interprets their convergence rates across exponent regimes, and completes the global estimates’ proofs by verifying local approximation hypotheses. It also refines elementwise contributions in the subquadratic regime according to whether the local non-degeneracy scale is resolved.

6.1 Error estimates and convergence rates

  • Under broken regularity of the solution and flux, Theorem 6.1 gives localized and global discrete-norm error estimates with constants independent of mesh size and individual elements.The estimates assume u ∈ W^{k+2,p(·)}(T_h) and σ(·,∇u) ∈ W^{1,p'(·)}(Ω) ∩ W^{k+1,p'(·)}(T_h).Source: §6.1 · p. 15
  • For constant p, the rate is (k + 1)p/2 for 1 < p ≤ 2 and (k + 1)/(p − 1) for p ≥ 2.The general variable-exponent rates distinguish purely subquadratic, mixed subquadratic–superquadratic, and purely superquadratic regimes.Source: §6.1 · p. 16
  • For p(·) ≤ 2, Theorem 6.3 refines the estimate using η_T, which compares local approximation and non-degeneracy scales; resolving that scale recovers order k + 1.If the non-degenerate regime is unresolved, Theorem 6.1 still gives the robust rate (k + 1)p−/2 when p+ ≤ 2.Source: §6.1 · p. 16
  • The broken-gradient error is bounded by the discrete error plus a projection-approximation term, which can limit the rate to order k when r_k = k.When r_k = k + 1, the projection term h^{r_k} does not reduce the rate.Source: §6.1 · p. 16

6.2 Proofs and auxiliary lemmas

  • The auxiliary estimates extend three-point perturbation bounds to variable exponents and general fluxes, with cell and skeleton modular bounds and local-aggregation Hölder control.The three-point estimate also applies to the scalar face flux; the bounds support the error-estimate proof.Source: §6.2 · pp. 16–19
  • The local approximation hypotheses yield the consistency estimate with h^(k+1) extracted at the level of Luxemburg norms before applying the variable-exponent powers.The proof verifies the local hypotheses for the flux-face, solution-cell, and solution-face approximation errors.Source: §6.2 · p. 21
  • For Theorem 6.3, the elementwise projection contribution is treated separately in degenerate and non-degenerate cases, using the local scale D_T.When D_T > 0, the proof changes exponent on the discrete error, then uses variable-exponent Hölder and the HHO discrete Lebesgue embedding; for η_T ≥ 1 or D_T = 0, it uses the degenerate bound.Source: §6.2 · p. 22
  • The stabilization contribution is bounded analogously, with weighted boundary norms in the non-degenerate case and the discrete embedding applied to e_T.The stabilization difference is the interpolated face residual, and the non-degenerate argument uses δ_F ≥ D_T.Source: §6.2 · p. 22
  • Combining the flux, consistency, and stabilization bounds proves the global estimate (6.9).The proof cancels one factor of the discrete error norm; the zero-error case is immediate.Source: §6.2 · p. 22
  • The projection term in Corollary 6.5 is of order h^rk in the broken W^1,p(·) seminorm.This follows by applying the triangle inequality with the stated interpolation and approximation bounds.Source: §6.2 · p. 22

7 Numerical results

The numerical study tests the error estimates across exponent regimes, flux-regularity conditions, and Cartesian and Voronoi meshes. Observed rates support the estimates when the stated regularity holds and deteriorate when it fails.

7.1 General setting

  • The tests use Carreau–Yasuda fluxes with δ=0 or 1, three exponent regimes, and polynomial degrees k=0,1,2,3; Voronoi meshes supplement the Cartesian grids.For δ=0, the flux is the degenerate p(x)-Laplacian; δ=1 removes degeneracy.Source: §7.1 · p. 23, §7.4 · p. 27

7.2 Convergence rates for smooth solutions

  • For smooth exponential and polynomial solutions, the estimate holds in all three regimes; δ=0 is near order k+1 subquadratically, while δ=1 reaches k+1 in every regime.The observed k+1 rate for δ=1 is not implied by Theorem 6.3 outside p+≤2.Source: §7.2 · p. 24

7.3 Role of the flux regularity assumption

  • For radial solutions, γ=4 retains the estimate in all regimes; with γ=2, rates stop improving at higher degrees in mixed and superquadratic regimes, while the subquadratic estimate still holds.In the γ=2 subquadratic case, the regularity deficit is only logarithmic.Source: §7.3 · pp. 24–25
  • For the sinusoidal solution, critical points at the center and corners cause the regularity assumption to fail from k=1 or 2 depending on regime; rates eventually saturate and the estimate is lost at the highest degrees.The center has exponent p+, while the corners have p−.Source: §7.3 · pp. 25–26
  • When flux regularity fails at critical points with p≠2, rates eventually saturate and may fall below Opk,pq, showing that convergence can become flux-regularity-limited.The estimate holds in configurations where its regularity assumption is satisfied.Source: §7.3 · p. 26

7.4 Polygonal meshes

  • Across Voronoi meshes, the exponential case is optimal in all three exponent regimes, while the sinusoidal case reproduces the rate loss seen on Cartesian meshes.Overall, the results support Theorem 6.1 independently of Cartesian mesh structure.Source: §7.4 · p. 28

8 Conclusion

The paper establishes localized and global a priori error estimates for HHO discretizations of variable-exponent Leray–Lions problems, distinguishing three growth regimes. Numerical tests corroborate the predicted rates while showing that limited flux regularity can prevent higher approximation degree from improving the asymptotic rate.

8 Conclusion

  • The estimates retain pointwise growth structure and distinguish purely subquadratic, mixed, and superquadratic cases.They are established in localized and global forms for the HHO discretization.Source: §8 · p. 28
  • For p+ ≤ 2, the refined estimate recovers order k + 1 when the approximation scale is sufficiently small relative to local non-degeneracy scales.Source: §8 · p. 28
  • The framework accommodates independent degeneracy parameters in the bulk and stabilization, as well as a class of anisotropic constitutive laws.Source: §8 · p. 28
  • Cartesian and Voronoi tests corroborate guaranteed rates and often converge faster; with limited flux regularity, increasing approximation degree alone need not improve asymptotic rates.For smooth non-degenerate tests with δ = 1, rates of k + 1 are observed also in mixed and superquadratic regimes.Source: §8 · p. 28
  • Sharp non-degenerate theory across the full variable-exponent range and direct error bounds for nonhomogeneous Dirichlet data remain natural extensions.Source: §8 · p. 28

A Variable-exponent Lebesgue and Sobolev background

This section collects variable-exponent background used in the error analysis and repeats selected definitions and conventions from the companion convergence study to make the argument self-contained.

A Variable-exponent Lebesgue and Sobolev background

A.1 Lebesgue and Sobolev spaces with variable exponents

The appendix specifies variable-exponent function spaces and the regularity condition on their exponents, including conventions for mesh faces and bounded domains.

A.1 Lebesgue and Sobolev spaces with variable exponents

  • On mesh faces, exponents in surface integrals are restrictions of a continuous extension, while volume exponents are restrictions from Ω.Source: §A.1 · p. 28
  • The exponent class uses a local log-Hölder condition on measurable exponents.The supplied condition bounds the difference of reciprocal exponent values by a logarithmic function of the distance.Source: §A.1 · pp. 28–29
  • For bounded domains, no additional condition controlling the exponent at infinity is needed; Sobolev and trace results still require suitable domain geometry.Source: §A.1 · p. 29
  • The variable-exponent Lebesgue space contains measurable functions with finite Luxemburg norm, and the Sobolev space is defined analogously using weak derivatives.Source: §A.1 · p. 30
  • The stated Hölder and Young estimates apply to exponents satisfying a reciprocal-exponent relation.A parameter-dependent form of Young’s inequality is also introduced.Source: §A.1 · p. 30

A.2 Bounded variable-exponent class

This subsection sets out endpoint and auxiliary-exponent conventions for bounded variable-exponent classes and notes associated modular–norm relations.

A.2 Bounded variable-exponent class

  • The bounded-exponent classes impose p− > 1 and, where specified, p+ < ∞.Source: §A.2 · p. 30
  • The conjugate exponent and auxiliary exponents are defined by p′ = p/(p − 1), p = max(p, 2), and q_p = min(p, 2).Source: §A.2 · p. 30
  • For bounded exponents, the appendix states basic modular–norm relations and cites for a further proposition.Source: §A.2 · pp. 30–31

A.3 Variable-exponent polynomial tools

This section collects the variable-exponent polynomial estimates needed in the discrete analysis, including face estimates interpreted intrinsically on each face.

A.3 Variable-exponent polynomial tools

  • Face embeddings use the affine hull's intrinsic (d − 1)-dimensional measure and follow from a fixed-exponent polynomial argument under uniform face-regularity assumptions.The estimates are collected for use in the discrete analysis.Source: §A.3 · p. 31

A.3.1 Mean exponent equivalence and discrete Lebesgue embedding

This section develops mean-exponent equivalence and discrete polynomial embedding tools for variable exponents. Their applicability relies on suitable geometric embedding assumptions, with constants whose dependencies are specified.

A.3.1 Mean exponent equivalence and discrete Lebesgue embedding

  • The results extend to geometric settings with a uniform fixed-exponent polynomial embedding, including regular simplicial submeshes.For finite exponents in fixed bounded intervals, the embedding constant can be chosen uniformly in the exponents.Source: §A.3.1 · p. 31
  • Mean-exponent equivalence is identified as the main tool for the discrete analysis.The theorem also gives an equivalence for polynomial functions on the set.Source: §A.3.1 · p. 31
  • The discrete Lebesgue embedding applies to polynomial functions on bounded measurable sets satisfying the stated geometric condition.Its hidden constant depends on the domain, polynomial degree, dimension, exponent bounds and regularity parameters.Source: §A.3.1 · p. 32

B Verification of the anisotropic Carreau–Yasuda example

The appendix verifies the structural assumptions for the anisotropic Carreau–Yasuda flux using Jacobian bounds. It concludes that the non-symmetric flux in Example 2.8 satisfies Assumption 2.2.

B Verification of the anisotropic Carreau–Yasuda example

  • The verification uses a Jacobian-bounds criterion that implies the flux's Hölder continuity and strong monotonicity.The criterion imposes derivative bounds wherever the derivative exists, including away from zero when δ = 0.Source: B Verification of the anisotropic… · p. 32
  • For the subquadratic case, the proof handles a segment crossing the origin by splitting the integral, whose singularity is integrable for p > 1.The argument is stated for δ = 0 and 1 < p < 2.Source: B Verification of the anisotropic… · p. 33
  • The anisotropic Carreau–Yasuda flux of Example 2.8 satisfies Assumption 2.2 under its stated hypotheses.The proof applies Jacobian bounds to establish the required Hölder continuity and strong monotonicity structure.Source: B Verification of the anisotropic… · pp. 32–33

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