### 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 |
---|---|

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

### Cite this

^{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*,

*27*(5), 879-908. https://doi.org/10.1142/S0218202517500191

}

^{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*, vol. 27, no. 5, pp. 879-908. https://doi.org/10.1142/S0218202517500191

**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.** / Di Pietro, Daniele A.; Droniou, Jerome.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Ws,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

AU - Di Pietro, Daniele A.

AU - Droniou, Jerome

PY - 2017

Y1 - 2017

N2 - In this work, we prove optimal Ws,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 Lp-boundedness result for L2-orthogonal projectors on polynomial subspaces. The Ws,p-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these Ws,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 W1,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 W1,p-like discrete norm scales as hk+1p−1 when p ≥ 2 and as h(k+1)(p−1) when p < 2.

AB - In this work, we prove optimal Ws,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 Lp-boundedness result for L2-orthogonal projectors on polynomial subspaces. The Ws,p-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these Ws,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 W1,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 W1,p-like discrete norm scales as hk+1p−1 when p ≥ 2 and as h(k+1)(p−1) when p < 2.

KW - (Formula presented.)-approximation properties of elliptic projector on polynomials

KW - (Formula presented.)-Laplacian

KW - error estimates

KW - Hybrid High-Order methods

KW - nonlinear elliptic equations

UR - http://www.scopus.com/inward/record.url?scp=85017173602&partnerID=8YFLogxK

U2 - 10.1142/S0218202517500191

DO - 10.1142/S0218202517500191

M3 - Article

VL - 27

SP - 879

EP - 908

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

SN - 0218-2025

IS - 5

ER -

^{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):879-908. https://doi.org/10.1142/S0218202517500191