Stability of the finite element Stokes projection in W1

Abstract We prove stability of the finite element Stokes projection in the product space W 1,∞ (Ω)×L ∞ (Ω) . The proof relies on weighted L2 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,∞ (Ω)×L ∞ (Ω) . The family of triangulations 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. To cite this article: V. Girault et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).