Comptes Rendus
Functional analysis, Numerical analysis
Stability of a weighted L2 projection in a weighted Sobolev norm
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 757-766.

We prove the stability of a weighted L 2 projection operator onto piecewise linear finite elements spaces in a weighted Sobolev norm. Namely, we consider the orthogonal projections π N,ω from L 2 (𝔻,1/ω(x)dx) to 𝒳 N , where 𝔻 2 is the unit disk, ω(x)=1-|x| 2 and the spaces (𝒳 N ) N consist of piecewise linear functions on a family of shape-regular and quasi-uniform triangulations of 𝔻. We show that π N,ω is stable in a weighted Sobolev norm, and prove an upper bound on the stability constant that does not depend on N. The result also holds when the disk 𝔻 is replaced by a more general surface Γ 3 , replacing the weight ω by ω Γ (x):=d(x,Γ), the square root of the distance from x to the manifold boundary of Γ.

On démontre la stabilité dans une norme de Sobolev à poids, de la projection orthogonale par rapport au produit scalaire d’un espace L 2 à poids, sur une famille d’éléments finis linéaires par morceaux. Plus précisément, soit π N,ω , de L 2 (𝔻,1/ω(x)dx) dans 𝒳 N , où 𝔻 2 est le disque unité, ω(x)=1-|x| 2 et les espaces (𝒳 N ) N sont des espaces de fonctions continues et linéaires par morceaux sur une famille de triangulations régulière de 𝔻. On montre que π N,ω est stable dans une norme de Sobolev à poids, avec une borne supérieure sur la constante de stabilité qui ne dépend pas de N. Le résultat s’étend au cas de surfaces plus générales Γ 3 , en remplaçant le poids ω par ω Γ (x):=d(x,Γ), la racine carrée de la distance de x à Γ, le bord de Γ.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.426
Classification: 46E35, 65N12, 65N38

Martin Averseng 1

1 Seminar for Applied Mathematics, ETH Zurich
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Stability of a weighted {L2} projection in a weighted {Sobolev} norm},
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Martin Averseng. Stability of a weighted L2 projection in a weighted Sobolev norm. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 757-766. doi : 10.5802/crmath.426. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.426/

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