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

Abstract: 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.