# 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

## Abstract

We prove stability of the finite element Stokes projection in the product space $W^{1,\infty}(\Om)\times L^\infty(\Om)$. 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 a 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 $W^{1,\infty}(\Om)\times L^\infty(\Om)$. 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.