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

Suppose S subset of H-1(Omega) is a finite-dimensional linear space based o n a triangulation T of a domain Omega, and let Pi : L-2(Omega) --> L-2(Omeg a) denote the L-2-projection onto S. Provided the mass matrix of each eleme nt T is an element of T and the surrounding mesh-sizes obey the inequalitie s due to Bramble, Pasciak, and Steinbach or that neighboring element-sizes obey the global growth-condition due to Crouzeix and Thomee, Pi is H-1-stab le: For all u is an element of H-1(Omega) we have parallel to Piu parallel to (H1(Omega)) less than or equal to C parallel tou parallel to (H1(Omega)) with a constant C that is independent of, e.g., the dimension of S. This paper provides a more flexible version of the Bramble-Pasciak-Steinbac h 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 y ields, in particular, H-1-stability for nonconforming schemes on arbitrary (shape-regular) meshes; (ii) it is weaker than (i.e., implied by) either th e Bramble-Pasciak-Steinbach or the Crouzeix-Thomee criterion for regular tr iangulations into triangles; (iii) it guarantees H-1-stability of Pi a prio ri for a class of adaptively-refined triangulations into right isosceles tr iangles.