Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for $H^1$-stability of the $L^2$-projection onto finite element spaces

Suppose $\mathcal{S}\subset H^1(\Omega)$ is a finite-dimensional linear space based on a triangulation $\mathcal{T}$ of a domain $\Omega$, and let $\Pi:L^2(\Omega)\to L^2(\Omega)$ denote the $L^2$-projection onto $\mathcal{S}$. Provided the mass matrix of each element $T\in\mathcal{T}$ and the surrounding mesh-sizes obey the inequalities due to Bramble, Pasciak, and Steinbach or that neighboring element-sizes obey the global growth-condition due to Crouzeix and Thomée, $\Pi$ is $H^1$-stable: For all $u\in H^1(\Omega)$ we have $\Vert \Pi u\Vert_{H^1(\Omega)}\le C\,\Vert u\Vert_{ H^1(\Omega)}$ with a constant $C$ that is independent of, e.g., the dimension of $\mathcal{S}$.

This paper provides a more flexible version of the Bramble-Pasciak- Steinbach criterion for $H^1$-stability on an abstract level. In its general version, (i) the criterion is applicable to all kind of finite element spaces and yields, in particular, $H^1$-stability for nonconforming schemes on arbitrary (shape-regular) meshes; (ii) it is weaker than (i.e., implied by) either the Bramble-Pasciak-Steinbach or the Crouzeix-Thomée criterion for regular triangulations into triangles; (iii) it guarantees $H^1$-stability of $\Pi$ a priori for a class of adaptively-refined triangulations into right isosceles triangles.