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 Thomée, Π is H¹-stable: For all u∈ H¹(Ω ) we have Π u 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
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Carsten Carstensen (2001) studied this question.
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