Stability of a weighted L2 projection in a weighted Sobolev norm

Abstract

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 $.