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

Daniele A. Di Pietro, Jerome Droniou

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17 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)879-908
Number of pages30
JournalMathematical Models and Methods in Applied Sciences
Volume27
Issue number5
DOIs
Publication statusPublished - 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

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title = "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",
abstract = "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.",
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author = "{Di Pietro}, {Daniele A.} and Jerome Droniou",
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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.

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