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 ̊

In this work we prove optimal W -approximation estimates (with p P r1,`8s) 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-boundedness result for L-orthogonal projectors on polynomial subspaces. The W -approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these W -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 pΩq for some p P p1,`8q. 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 -like norm scales as h k`1 p ́1 when p ě 2 and as hpk`1qpp ́1q when p ă 2. 2010 Mathematics Subject Classification: 65N08, 65N30, 65N12