Stability of a weighted L2 projection in a weighted Sobolev norm
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 $\pi _{N,\omega }$ from $L^2(\mathbb{D},1/\omega (x)\mathrm{d}x)$ to $\mathcal{X}_N$, where $\mathbb{D} \subset \math...
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Académie des sciences
2023-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.426/ |
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| author | Averseng, Martin |
| author_facet | Averseng, Martin |
| author_sort | Averseng, Martin |
| collection | DOAJ |
| description | 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 $\pi _{N,\omega }$ from $L^2(\mathbb{D},1/\omega (x)\mathrm{d}x)$ to $\mathcal{X}_N$, where $\mathbb{D} \subset \mathbb{R}^2$ is the unit disk, $\omega (x) = \sqrt{1 - \vert x\vert ^2}$ and the spaces $(\mathcal{X}_N)_{N \in \mathbb{N}}$ consist of piecewise linear functions on a family of shape-regular and quasi-uniform triangulations of $\mathbb{D}$. We show that $\pi _{N,\omega }$ 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 $\mathbb{D}$ is replaced by a more general surface $\Gamma \subset \mathbb{R}^3$, replacing the weight $\omega $ by $\omega _\Gamma (x) := \sqrt{\mathrm{d}(x,\partial \Gamma )}$, the square root of the distance from $x$ to the manifold boundary of $\Gamma $. |
| format | Article |
| id | doaj-art-799c51f7cdfe4557bcc61a861861f8f2 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-799c51f7cdfe4557bcc61a861861f8f22025-02-07T11:07:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-05-01361G475776610.5802/crmath.42610.5802/crmath.426Stability of a weighted L2 projection in a weighted Sobolev normAverseng, Martin0Seminar for Applied Mathematics, ETH ZurichWe 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 $\pi _{N,\omega }$ from $L^2(\mathbb{D},1/\omega (x)\mathrm{d}x)$ to $\mathcal{X}_N$, where $\mathbb{D} \subset \mathbb{R}^2$ is the unit disk, $\omega (x) = \sqrt{1 - \vert x\vert ^2}$ and the spaces $(\mathcal{X}_N)_{N \in \mathbb{N}}$ consist of piecewise linear functions on a family of shape-regular and quasi-uniform triangulations of $\mathbb{D}$. We show that $\pi _{N,\omega }$ 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 $\mathbb{D}$ is replaced by a more general surface $\Gamma \subset \mathbb{R}^3$, replacing the weight $\omega $ by $\omega _\Gamma (x) := \sqrt{\mathrm{d}(x,\partial \Gamma )}$, the square root of the distance from $x$ to the manifold boundary of $\Gamma $.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.426/ |
| spellingShingle | Averseng, Martin Stability of a weighted L2 projection in a weighted Sobolev norm Comptes Rendus. Mathématique |
| title | Stability of a weighted L2 projection in a weighted Sobolev norm |
| title_full | Stability of a weighted L2 projection in a weighted Sobolev norm |
| title_fullStr | Stability of a weighted L2 projection in a weighted Sobolev norm |
| title_full_unstemmed | Stability of a weighted L2 projection in a weighted Sobolev norm |
| title_short | Stability of a weighted L2 projection in a weighted Sobolev norm |
| title_sort | stability of a weighted l2 projection in a weighted sobolev norm |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.426/ |
| work_keys_str_mv | AT aversengmartin stabilityofaweightedl2projectioninaweightedsobolevnorm |