A Hybrid High-Order method for variable-exponent Leray-Lions problems with convergence under minimal regularity

1 Introduction

• The paper identifies a gap: no hybrid nonconforming method had been analyzed for variable-exponent Leray–Lions problems, including minimal-regularity convergence for arbitrary-order polytopal methods and non-variational fluxes.The stated purpose is to address this gap.Source: §1 · p. 2
• The analysis handles mesh-scale exponent variation by comparing cellwise fixed-exponent estimates with the pointwise energy exponent using its oscillation on each cell.Log-Hölder continuity controls the scale, while modular bounds and compactness avoid consistency estimates requiring higher derivatives of the solution.Source: §1 · p. 2
• The mean-exponent norm equivalence supplies mesh-uniform polynomial scaling, inverse-estimate and projector bounds, and a discrete Lebesgue embedding tailored to HHO unknowns.Its hidden constants do not depend on mesh size or the mesh element; its two-exponent form yields the discrete embedding.Source: §1 · pp. 2–3
• The discrete functional toolkit establishes equivalent HHO seminorms, variable-exponent inverse and trace inequalities, and Poincaré–Wirtinger and Sobolev–Poincaré inequalities across the method’s full exponent range.The Sobolev–Poincaré result relies on a variable-exponent bound for the node-averaging operator.Source: §1 · p. 3
• The HHO discretization covers general variable-exponent Leray–Lions fluxes without a variational structure, on polytopal meshes and for arbitrary polynomial degree.It supports equal-order and mixed-order cell–face spaces and nonlinear face stabilization with its own degeneracy parameter.Source: §1 · p. 3
• Discrete potentials converge strongly in every uniformly subcritical variable-exponent Lebesgue space, reconstructed gradients converge strongly in the natural space, and stabilization energy vanishes.The result requires no higher-order regularity of the exact solution and is compatible with general Leray–Lions structure and independent bulk and face degeneracy parameters.Source: §1 · p. 3
• Experiments in dimensions two, three, and four observe convergence for singular solutions and for tested exponent ranges [1.05, 2.95] and [2, 6], for degenerate and regularized fluxes.For singular solutions, attainable rates are limited by solution regularity.Source: §1 · p. 3

2.2 Weak formulation

• The weak problem seeks a potential in the homogeneous-boundary energy space whose flux balances the volumetric source term.The analysis treats homogeneous Dirichlet data; manufactured tests with nonzero traces require a lifting.Source: §2 · p. 3, §2.2 · p. 5

2.1 Flux function

• The flux depends on position and on the potential through its gradient, and the analysis assumes continuity, strong monotonicity, and vanishing at rest.The growth exponent and degeneracy parameter may vary spatially.Source: §2 · p. 3, §2.1 · p. 4

2 Continuous problem

• For nonzero boundary traces, extending the full convergence and error analyses requires control of the chosen discrete lifting and its consistency terms.The paper does not claim this extension in its theorems.Source: §2 · pp. 3–4

2.1 Flux function

• The admissible fluxes include the variable-exponent p(·)-Laplace law and Carreau–Yasuda-type laws with spatially dependent parameters.For the Carreau–Yasuda example, the local flow consistency and transition behavior indices may vary, and the p(·)-Laplace case is recovered when δ = 0 and μ = 1.Source: §2.1 · pp. 4–5

2.2 Weak formulation

• Under the flux assumptions, the continuous problem has a unique solution and a variable-exponent a priori estimate.Strong monotonicity supplies uniqueness; existence follows from the continuous well-posedness argument.Source: §2.2 · pp. 5–6

3.1 Discrete setting

• Meshes are polytopal and regularly refined, with cell diameters comparable to face scales and a uniformly bounded number of faces per cell.The stated analysis assumes the mesh regularity parameter remains uniformly bounded away from zero.Source: §3.1 · pp. 6–7

3.2 Hybrid space and norms

• The HHO method uses cell and face polynomial unknowns of degree k, with cell degree k or k + 1 for equal- or mixed-order formulations.The face unknowns have degree k, and the method allows k ≥ 0.Source: §3.2 · p. 7

3.3 Operator reconstructions

• The local gradient reconstruction mimics integration by parts, using face unknowns in the role of the cell-boundary trace.Seeking the reconstructed gradient in the full vector polynomial space yields the stated commutation property.Source: §3.3 · p. 8
• The potential reconstruction enters the stabilization to provide higher-order consistency, while the boundary residual measures the trace mismatch scaled as a gradient.The residual vanishes on interpolates of degree k + 1 polynomials.Source: §3.3 · p. 8

3.4 Stabilization function

• Face stabilization uses a flux function playing the role of the diffusion flux for face residuals, and its weight may depend on polynomial degree.The stabilization is a numerical device and may use a degeneracy parameter different from that of the model flux.Source: §3.4 · p. 9

3.5 Discrete problem

• The discrete problem seeks a homogeneous-boundary HHO unknown whose discrete diffusion operator balances the source term.The operator combines the reconstructed diffusion contribution and the stabilization.Source: §3.5 · p. 9

4 Variable-exponent discrete norms and inequalities

• The section defines variable-exponent discrete seminorms and develops inverse, trace, and embedding inequalities for the convergence analysis.Source: §4 · p. 9

4.1 Equivalence between discrete seminorms

• The local and global discrete seminorms are equivalent to element-mean-exponent seminorms, with constants controlled by mesh and exponent parameters.Source: §4.1 · pp. 10–11

4.2 Variable-exponent discrete inverse and trace inequalities

• For log-Hölder exponents, the section establishes variable-exponent discrete inverse and trace inequalities, alongside a continuous local trace inequality.Source: §4.2 · p. 11

4.3 Variable-exponent discrete Poincaré–Wirtinger inequality

• A discrete Poincaré–Wirtinger inequality bounds mean-zero element polynomials using the variable-exponent discrete seminorm.Source: §4.3 · p. 12

4.4 Variable-exponent discrete Sobolev–Poincaré inequality

• The Sobolev–Poincaré inequality is generalized to allow exponent fields that may exceed the spatial dimension.Source: §4.4 · p. 12

F PFK

• The section states a discrete Sobolev–Poincaré inequality for HHO unknowns when the target exponent r is bounded by the pointwise Sobolev exponent q*.Source: F PFK · p. 14
• For d = 1, the proof uses the fundamental theorem of calculus successively from a boundary endpoint where the face unknown vanishes.This is the direct one-dimensional argument for the discrete estimate.Source: F PFK · p. 14
• For d ≥ 2, the proof controls jumps across interior faces using the single-valued face unknown and bounds the resulting face terms by the HHO norm.The argument combines the jump control with the face estimate to obtain the critical-exponent case.Source: F PFK · p. 14
• The discrete Sobolev–Poincaré estimate holds for every admissible r up to the variable-exponent critical exponent q*.The proof first establishes the estimate at q* and then uses the embedding into L^r for r ≤ q*.Source: F PFK · p. 14

5.1 Variable-exponent approximation properties of the projectors

• The section recalls bounded-projector approximation and stability results, with constants controlled by exponent bounds, log-Hölder continuity, polynomial degree, and geometric regularity.The cited projector results are established in and are used for approximation properties developed in the companion a priori analysis.Source: §5.1 · p. 15
• Variable-exponent projector approximation holds for local Sobolev functions, with uniform mesh-family constants under the stated exponent and mesh bounds.The results include approximation and scaled trace estimates; bounded-projector approximation also assumes a polynomial inverse inequality.Source: §5.1 · p. 15

5.2 Properties of the discrete operators

• The HHO interpolator is bounded in the variable-exponent discrete seminorm by the local W^1,q(·) seminorm.The discrete operators also have polynomial consistency properties.Source: §5.2 · pp. 15–16

5.3 Strong monotonicity of the discrete diffusion function

• The discrete diffusion function is strongly monotone under Assumptions 2.2 and 3.3.The estimate applies to all discrete unknown pairs.Source: §5.3 · p. 16

6 Well-posedness of the discrete problem

• A unique discrete solution exists under Assumptions 2.2 and 3.3, with a variable-exponent stability bound.The bound's hidden constant is independent of h, σ_m, σ_c, and the degeneracy parameters.Source: §6 · p. 17
• The a priori bound follows from strong monotonicity, the discrete Sobolev–Poincaré inequality, and modular–norm relations.Its hidden constants are independent of σ_m.Source: §6 · p. 17

7.1 Variable-exponent discrete compactness

• Bounded discrete sequences are relatively compact in L^1(Ω), and converge strongly in L^q(·)(Ω) when q has a uniform gap below p*(·).Pointwise subcriticality alone need not provide that uniform gap.Source: §7.1 · pp. 17–18
• Discrete compactness identifies the limit as a member of W_0^1,p(·)(Ω), with the weak limit of reconstructed gradients equal to its gradient.The boundary condition follows from the limit identity and integration by parts.Source: §7.1 · p. 19

7.2 Convergence under minimal regularity

• Under Assumptions 2.2 and 3.3 and sup_{h∈H} D_h < ∞, a unique u solves (2.6), and reconstructed gradients converge strongly in L^{p(·)}(Ω) to ∇u.Source: §7.2 · pp. 19–21
• The reconstructed gradients converge strongly in L^p(·)(Ω), and the corresponding fluxes converge strongly in L^p′(·)(Ω).The proof obtains strong gradient convergence and uses continuity and uniform integrability to conclude flux convergence.Source: §7.2 · pp. 20–21

8.1 General setting

• Tests use the Carreau–Yasuda flux on unit hypercubes in dimensions two through four, with δ∈{0,1} and equal-order cell and face degrees k∈{0,1,2,3}.Picard and Newton linearizations are selected according to the exponent range; a continuation in the exponent field is used.Source: §8.1 · pp. 21–22

8.2 Robustness with respect to the regularity of the solution

• For the smooth solution, the HHO-norm error rate keeps improving with polynomial degree k across both exponent fields.The solution’s regularity does not impose a ceiling on k in this test.Source: §8.2 · p. 23, Figure 1 · p. 23
• For γ=1.1 and k≥1, errors scale as h^1.1 for the linear field and h^0.9 for the sinusoidal field, below the moderate-deficit rates h^1.5 and h^1.2.For the moderate deficit, convergence persists but rates stop improving once k exceeds what the solution regularity supports.Source: §8.2 · pp. 23–24

8.3 Robustness with respect to extreme values of p(·)

• Across exponent ranges [1.05, 2.95] and [2, 6], errors decrease for both δ=0 and δ=1, including exponents as high as 6.A missing finest-mesh result and irregular slopes in some low-boundary cases are attributed to the nonlinear solver, not the scheme.Source: §8.3 · pp. 24–25

8.4 Extension to higher dimension

• In dimensions three and four, errors decrease under refinement for every degree on both exponent fields; four-dimensional rates remain variable on necessarily coarse meshes.The higher-dimensional tests use the same scheme and solver without modification.Source: §8.4 · pp. 25–26

9 Conclusion

• The mean-exponent norm equivalence yields discrete Lebesgue embedding and, with HHO reconstructions and stabilization, the norm, Sobolev–Poincaré, and compactness tools for variable growth.The embedding includes a mesh-scale factor depending on the mean exponent.Source: §9 · p. 27
• The method handles structurally continuous and monotone Leray–Lions fluxes without variational structure, with equal- or mixed-order formulations for every degree k ≥ 0.The discrete problem is well-posed; potentials converge strongly in uniformly subcritical variable-exponent Lebesgue spaces, reconstructed gradients in Lp(·), and stabilization energy vanishes.Source: §9 · p. 27
• Numerical tests show error decay across polynomial degrees and dimensions, expected rate saturation for reduced regularity, and robustness alongside nonlinear-solver limits in extreme-exponent cases.Figure 8 plots the HHO norm error against mesh size for a smooth solution in d = 4, with separate exponent fields and δ values.Source: §9 · p. 27, Figure 8 · p. 28
• Theorem 7.3 is qualitative; quantitative estimates, including localized variable-exponent rates, are developed in companion work under additional approximation regularity.The companion paper also gives a regime-dependent non-degenerate refinement.Source: §9 · p. 27

A Variable-exponent Lebesgue and Sobolev background

• The appendix collects standard functional-analytic material used throughout the paper.Source: A Variable-exponent Lebesgue and… · p. 28

A.1 Lebesgue and Sobolev spaces with variable exponents

• Variable exponents belong to a locally log-Hölder class, and exponents on mesh faces are restrictions of a continuous extension to the domain.For the bounded domains considered, no additional decay condition at infinity is needed.Source: §A.1 · p. 28
• The Luxemburg norm is needed when the exponent varies over the integration domain; for a constant exponent, it reduces to the modular raised to the reciprocal exponent.Source: §A.1 · p. 28
• The variable-exponent Lebesgue space consists of measurable functions with finite norm, while the Sobolev space adds distributional derivatives on open sets.The definitions also extend componentwise to vector-valued functions.Source: §A.1 · p. 29
• The section recalls variable-exponent Hölder and Young inequalities for exponents satisfying the stated reciprocal-exponent relation.The displayed Hölder bound includes a factor of 2.Source: §A.1 · p. 29

A.2 Bounded variable-exponent class

• The bounded variable-exponent class uses endpoint bounds, with the reflexive class requiring p− > 1 and the bounded class requiring p+ < ∞.Source: §A.2 · p. 29
• The conjugate exponent is p′ = p/(p − 1), and the auxiliary exponents are p = max(p, 2) and q_p = min(p, 2).Source: §A.2 · p. 29
• The section states modular–norm relations and an up-to-constant version with explicit dependence of the constants for bounded exponents.The accompanying lemma relates the Luxemburg norm to a scaled modular through the endpoint exponent bounds.Source: §A.2 · pp. 29–30

• The appendix collects polynomial variable-exponent estimates used in the discrete analysis.Source: B Variable-exponent polynomial tools · p. 30
• On mesh faces, an exponent is evaluated using its unique finite-valued continuous extension to the domain.Log-Hölder continuity alone guarantees a continuous extension of the reciprocal exponent, but not necessarily a finite-valued extension of the exponent itself.Source: B Variable-exponent polynomial tools · p. 30
• Polynomial Lebesgue embeddings on faces use intrinsic (d − 1)-dimensional measure and follow from fixed-exponent arguments in the face’s affine hull.They rely on induced uniform face-regularity assumptions, rather than treating faces as sets of positive d-dimensional measure.Source: B Variable-exponent polynomial tools · p. 30

B.1 Mean exponent equivalence and discrete Lebesgue embedding

• The fixed-exponent polynomial embedding also applies to cells formed from a bounded number of comparable-diameter shape-regular simplices.The associated constants are controlled by submesh regularity.Source: §B.1 · p. 31
• Theorem B.3 gives a mean-exponent equivalence for polynomial functions under the stated geometric assumptions.Source: §B.1 · p. 31
• The estimates retain dependence on the domain because log-Hölder continuity does not uniformly control arbitrarily large diameters.Source: §B.1 · p. 31
• Corollary B.5 generalizes the polynomial Lebesgue embedding to variable exponents, using mean-exponent equivalence with a fixed-exponent embedding.Source: §B.1 · pp. 30, 32