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Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomee Criterion for H1-Stability of the L2-Projection onto Finite Element Spaces
Suppose $\S\subset H^1(\Omega)$ be a finite-dimensional linear space based on a triangulation $\T$ of a domain $\Omega$ and let $\Pi:L^2(\Omega)\to L^2(\Omega)$ denote the $L^2$-projection onto $\S$. Provided the mass matrix of each element $T\in\T$ 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 Thom{\'e}e, $\Pi$ is $H^1$-stable: For all $u\in H^1(\Omega)$ we have $\norm{\Pi\,u}{ H^1(\Omega)}\le C\,\norm{u}{ H^1(\Omega)}$ 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-Steinbach criterion for $H^1$-stability on an abstract level. In its general version, the criterion is (i) applicable to {\em all} kind of finite element spaces and yields, in particular, $H^1$-stability for nonconforming schemes on arbitrary (shape-regular) meshes; (ii) it is {\em weaker than} (i.e., implied by) {\em either} the Bramble-Pasciak-Steinbach {\em or} the Crouzeix-Thom{\'e}e criterion for regular triangulations into triangles; (iii) it guarantees $H^1$-stability of $\Pi$ even for the class of {\em adaptively-refined} triangulations into right isosceles triangles.
Mathematics Subject Classification (1991):
65N30, 65R20, 73C50
Bibliographical note: accepted for publication in Math. Comp.
Keywords: finite element method, $L^2$-projection, $H^1$-stability, adaptive algorithm, nonconforming finite element method
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