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^s,p-approximation estimates (with p ∈ [1,+∞]) 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 L²-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 (HHO) discretisation of Leray–Lions elliptic problems whose weak formulation is classically set in W^1,p(Ω) for some p ∈ (1,+∞). 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^k+1_p−1 when p ≥ 2 and as h^(k+1)(p−1) when p < 2.