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

Abstract

Abstract The $L^2$-orthogonal projection $ {\varPi }_h:L^2(\varOmega )\rightarrow {\mathbb{V}}_h$ onto a finite element (FE) space $ {\mathbb{V}}_h$ is called $H^1$-stable iff $ \|{ {\varPi }_h u}\|_{H^{1}}(\varOmega )\leq C\|{u}\|_{H^{1}}(\varOmega ) $ for any $u\in H^1(\varOmega )$ with a positive constant $C\neq C(h)$ independent of the mesh size $h>0$. Stability of this projection in the $H^1$-norm is an important component in the numerical analysis of FE methods, and it has been extensively studied before for triangular meshes in two dimensions and tetrahedral meshes in three dimensions. In this work, we analyse $H^1$-stability of adaptively refined quadrilateral meshes in two dimensions. We show that the refinement strategies Q-RG and Q-RB, introduced originally in Bank et al. (1983, Some refinement algorithms and data structures for regular local mesh refinement. Sci. Comput., 1, 3–17) and Kobbelt (1996, Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Comput. Graph Forum, 15, 409–420), are $H^1$-stable for FE spaces of polynomial degree $p=2,\ldots ,9$.