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

Daniele Di Pietro (1) , Jerome Droniou (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 School of Mathematical Sciences [Clayton]

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 norm scales as $h^{\frac{k+1}{p-1}}$ when $p\ge 2$ and as $h^{(k+1)(p-1)}$ when $p<2$.

Domaines: Analyse numérique [math.NA]

Daniele Di Pietro, Jerome Droniou. $W^{s,p}$-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems. Mathematical Models and Methods in Applied Sciences, 2017, 27 (5), pp.879--908. ⟨10.1142/S0218202517500191⟩. ⟨hal-01326818⟩