$W^{s,p}$-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems

TL;DR

This paper establishes optimal $W^{s,p}$-approximation estimates for elliptic projectors on polynomial spaces and applies these results to analyze the error of a Hybrid High-Order discretization for Leray-Lions problems, relevant in fluid and material modeling.

Contribution

It introduces new abstract lemmas for approximation and boundedness of projectors, leading to novel $W^{s,p}$-approximation estimates applicable to numerical methods and error analysis of complex PDE discretizations.

Findings

Derived $W^{s,p}$-approximation estimates for elliptic projectors.

Established error bounds for Hybrid High-Order discretizations of Leray-Lions problems.

Demonstrated applicability to problems in glacier motion, turbulent flows, and airfoil design.

Abstract

In this work we prove optimal $W^{s,p}$-approximation estimates (with $p\in[1,+\infty]$) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont--Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an $L^p$-boundedness result for $L^2$-orthogonal projectors on polynomial subspaces. The $W^{s,p}$-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these $W^{s,p}$-estimates to derive novel error estimates for a Hybrid High-Order discretization of Leray--Lions elliptic problems whose weak formulation is classically set in $W^{1,p}(\Omega)$ for some $p\in(1,+\infty)$. This kind of problems appears, e.g., in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by $h$ the meshsize, we prove that the approximation error measured in a $W^{1,p}$-like discrete norm scales as $h^{\frac{k+1}{p-1}}$ when $p\ge 2$ and as $h^{(k+1)(p-1)}$ when $p<2$.