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.

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA

L. Ridgway Scott
Department of Mathematics and the Computation Institute
University of Chicago
Chicago, Illinois 60637--1581, USA.


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.