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Suppose S ⊂ H1(Ω) is a finite-dimensional linear space based on a triangulation T of a domain Ω, and let Π : L2(Ω) → L2 (Ω) denote the L2-projection onto S. Provided the mass matrix of each element T ∈ 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 Thomee, Π is H1-stable: For all u ∈ H1 (Ω) we have ||Πu||H1 (Ω) ≤ C ||u||H1 (Ω) 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-Steinbach criterion for H1-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, H1-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-Thomee criterion for regular triangulations into triangles; (iii) it guarantees H1-stability of Π a priori for a class of adaptively-refined triangulations into right isosceles triangles.

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