## Abstract

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 L

^{2}-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

^{s,p}*W*-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

^{s,p}^{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*-like discrete norm scales as

^{1,p}*h*^{k+1}_{p−1} when p ≥ 2 and as h^{(k+1)(p−1)} when p < 2.

Original language | English |
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Pages (from-to) | 879-908 |

Number of pages | 30 |

Journal | Mathematical Models and Methods in Applied Sciences |

Volume | 27 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2017 |

## Keywords

- (Formula presented.)-approximation properties of elliptic projector on polynomials
- (Formula presented.)-Laplacian
- error estimates
- Hybrid High-Order methods
- nonlinear elliptic equations