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Maximum-norm stability of the finite element Stokes projection

✍ Scribed by V. Girault; R.H. Nochetto; R. Scott

Publisher: Elsevier Science
Year: 2005
Tongue: English
Weight: 358 KB
Volume: 84
Category: Article
ISSN: 0021-7824
DOI: 10.1016/j.matpur.2004.09.017

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✦ Synopsis

We prove stability of the finite element Stokes projection in the product space W 1,∞ (Ω) × L ∞ (Ω), i.e., the maximum norm of the discrete velocity gradient and discrete pressure are bounded by the sum of the corresponding exact counterparts, independently of the mesh-size. The proof relies on weighted L 2 estimates for regularized Green's functions associated with the Stokes problem and on a weighted inf-sup condition. The domain is a polygon or polyhedron with a Lipschitzcontinuous boundary, satisfying suitable sufficient conditions on the inner angles of its boundary, so that the exact solution is bounded in W 1,∞ (Ω) × L ∞ (Ω). The triangulation is shape-regular and quasi-uniform. The finite element spaces satisfy a super-approximation property, which is shown to be valid for commonly used stable finite element spaces.

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