$H^1$-Stability of the $L^2$-Projection onto Finite Element Spaces on Adaptively Refined Quadrilateral Meshes

Abstract

The $L^2$-orthogonal projection ${\Pi }_h:L^2(\Omega )\to {\mathbb{V}}_h$ onto a finite element (FE) space ${\mathbb{V}}_h$ is called $H^1$-stable iff $\Vert \nabla {\Pi }_{h}u{\Vert }_{L^2(\Omega )}\le C\Vert u{\Vert }_{H^1(\Omega )}$, for any $u\in H^1(\Omega )$ with a positive constant $C\ne C(h)$ independent of the mesh size $h>0$. In this work, we discuss local criteria for the $H^1$-stability of adaptively refined meshes. We show that adaptive refinement strategies for quadrilateral meshes in 2D (Q-RG and Q-RB), introduced originally in Bank et al. 1982 and Kobbelt 1996, are $H^1$-stable for FE spaces of polynomial degree $p=2,\dots ,9$.

19 pages, 4 figures, 6 tables

References in corpus (1)

• On the Sobolev and $L^p$-Stability of the $L^2$-projection