H 1-stability of the L2-projection onto finite element spaces on adaptively refined quadrilateral meshes

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