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

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- by Carsten Carstensen;
- Math. Comp.
**71**(2002), 157-163 - DOI: https://doi.org/10.1090/S0025-5718-01-01316-3
- Published electronically: May 7, 2001
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## Abstract:

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 $\Vert \Pi u\Vert _{H^1(\Omega )}\le C \|u\|{ H^1(\Omega )}$ 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.

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## Bibliographic Information

**Carsten Carstensen**- Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany
- Email: cc@numerik.uni-kiel.de
- Received by editor(s): January 11, 2000
- Received by editor(s) in revised form: May 30, 2000
- Published electronically: May 7, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Math. Comp.
**71**(2002), 157-163 - MSC (2000): Primary 65N30, 65R20, 73C50
- DOI: https://doi.org/10.1090/S0025-5718-01-01316-3
- MathSciNet review: 1862993