arXiv subjects

André Harnist

Publications and source records attributed to André Harnist.

6 recordsLinked to original sources

arXiv2026

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

We consider a homogeneous Dirichlet problem for nonlinear elliptic equations in divergence form with Leray--Lions fluxes of variable growth. The class includes the variable-exponent $p(\cdot)$-Laplacian and regularized Carreau--Yasuda-type laws. We propose and analyze a Hybrid High-Order (HHO) discretization on general polytopal meshes, a hybrid nonconforming method based on cell and face polynomial unknowns of arbitrary degree. We develop a variable-exponent polynomial and discrete functional toolkit centered on a mean-exponent norm equivalence, together with inverse, trace, Poincaré--Wirtinger, Sobolev--Poincaré, discrete Lebesgue embedding, interpolation, and compactness estimates adapted to HHO unknowns. We then establish well-posedness of the nonlinear discrete problem and convergence without assuming additional regularity of the exact solution beyond its natural energy space. The discrete potential converges strongly in every uniformly subcritical variable-exponent Lebesgue space, the reconstructed gradient converges strongly in the natural variable-exponent space, and the stabilization energy vanishes. Numerical experiments in two, three, and four space dimensions confirm convergence for singular solutions and for exponent fields ranging from close to $1$ up to $6$. Quantitative a priori error estimates for the same method are derived in a companion paper.

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arXiv2026

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

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.

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arXiv2026

Virtual Element methods for non-Newtonian shear-thickening fluid flow problems

In this work, we present a comprehensive theoretical analysis for Virtual Element discretizations of incompressible non-Newtonian flows governed by the Carreau-Yasuda constitutive law, in the shear-thickening regime (r > 2) including both degenerate (delta = 0) and non-degenerate (delta > 0) cases. The proposed Virtual Element method features two distinguishing advantages: the construction of an exactly divergence-free discrete velocity field and compatibility with general polygonal meshes. The analysis presented in this work extends a previous work, where only shear-thinning behavior (1 < r < 2) was considered. Indeed, the theoretical analysis of the shear-thickening setting requires several novel analytical tools, including: an inf-sup stability analysis of the discrete velocity-pressure coupling in non-Hilbertian norms, a stabilization term specifically designed to address the nonlinear structure as the exponent r > 2; and the introduction of a suitable discrete norm tailored to the underlying nonlinear constitutive relation. Numerical results demonstrate the practical performance of the proposed formulation.

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arXiv2022

Improved error estimates for Hybrid High-Order discretizations of Leray-Lions problems

We derive novel error estimates for Hybrid High-Order (HHO) discretizations of Leray-Lions problems set in W^(1,p) with p in (1,2]. Specifically, we prove that, depending on the degeneracy of the problem, the convergence rate may vary between (k+1)(p-1) and (k+1), with k denoting the degree of the HHO approximation. These regime-dependent error estimates are illustrated by a complete panel of numerical experiments.

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arXiv2022

A Hybrid High-Order method for creeping flows of non-Newtonian fluids

In this paper, we design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed method has several appealing features including the support of general meshes and high-order, unconditional inf-sup stability, and orders of convergence that match those obtained for scalar Leray-Lions problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau-Yasuda models. Numerical examples complete the exposition.

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arXiv2022

A Hybrid High-Order method for incompressible flows of non-Newtonian fluids with power-like convective behaviour

In this work, we design and analyze a Hybrid High-Order (HHO) discretization method for incompressible flows of non-Newtonian fluids with power-like convective behaviour. We work under general assumptions on the viscosity and convection laws, that are associated with possibly different Sobolev exponents r > 1 and s > 1. After providing a novel weak formulation of the continuous problem, we study its well-posedness highlighting how a subtle interplay between the exponents r and s determines the existence and uniqueness of a solution. We next design an HHO scheme based on this weak formulation and perform a comprehensive stability and convergence analysis, including convergence for general data and error estimates for shear-thinning fluids and small data. The HHO scheme is validated on a complete panel of model problems.

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