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...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-05-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.426/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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 $. |
|---|---|
| ISSN: | 1778-3569 |