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Maximum-norm stability of the finite element Stokes projection



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Maximum norm stability of the finite element Stokes projection V Girault R H Nochetto R Scott March 30 2004 Abstract We prove stability of the finite element Stokes projection in the product space W 1 L 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 L2 estimates for regularized Green s functions associated with the Stokes problem and on a weighted infsup condition The domain is a polygon or polyhedron with a Lipschitz continuous boundary satisfying suitable sufficient conditions on the inner angles of its boundary so that the exact solution is bounded in W 1 L 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 Re sume Nous de montrons la stabilite dans W 1 L de l approximation par e le ments finis du proble me de Stokes i e la norme du maximum du gradient de la vitesse et celle de la pression calcule s par des me thodes d e le ments finis usuelles pour discre tiser le proble me de Stokes sont borne es inde pendemment du pas de la discre tisation La de monstration est base e sur des estimations a poids dans L2 pour des fonctions de Green associe es au proble me de Stokes et sur une condition inf sup a poids Le domaine est un polygone ou un polye dre a frontie re Lipschitz dont les angles inte rieurs satisfont des conditions suffisantes convenables pour assurer que la solution exacte est aussi borne e dans W 1 L La triangulation est uniforme ment re gulie re Keywords Stokes problem finite element method Green s function duality argument weighted error estimates weighted inf sup condition local interpolation AMS Classification 65N15 65N30 76D07 0 Introduction This article is devoted to the proof of estimates in the maximum norm for the gradient of the velocity of the discrete



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