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Maximum-norm stability of the finite element Stokes projection

Vivette Girault

Laboratoire Jacques-Louis Lions

Universit\'e Pierre et Marie Curie

75252 Paris cedex 05, France.

girault@ann.jussieu.fr
Ricardo H. Nochetto

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
L. Ridgway Scott

Department of Mathematics and the Computation Institute

University of Chicago

Chicago, Illinois 60637--1581, USA.

ridg@cs.uchicago.edu
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Abstract

We prove stability of the finite element Stokes projection in the
product space $W^{1,\infty}(\Om)\times L^\infty(\Om)$, that is
the maximum norm of the velocity gradient and 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 Lipschitz polygon or polyhedron,
satisfying suitable sufficient conditions on the inner angles of
its boundary, so that the exact solution is bounded in
$W^{1,\infty}(\Om)\times L^\infty(\Om)$.
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.