Maximum-norm stability of the finite element Stokes projectionV. Girault∗R.H. Nochetto†R. Scott‡March 30, 2004AbstractWe prove stability of the finite element Stokes projection in the product space W1,∞(Ω) × L∞(Ω),i.e., the maximum norm of the discrete velocity gradient and discrete pressure are bounded by the sumof the corresponding exact counterparts, independently of the mesh-size. The proof relies on weightedL2estimates for regularized Green’s functions as sociated with the Stokes problem and on a weighted inf-sup condition. The domain is a polygon or polyhedron with a Lipschitz-continuous boundary, satisfyingsuitable sufficient conditions on the inner angles of its boundary, so that the exact solution is boundedin W1,∞(Ω) ×L∞(Ω). The triangulation is shape-regular and quasi-uniform. The finite element spacessatisfy a super-approximation property, which is shown to be valid for commonly used stable finiteelement s paces .R´esum´e: Nous d´emontrons la stabilit´e dans W1,∞(Ω) × L∞(Ω) de l’approximation par ´e l´ements finisdu probl`eme de Stokes, i.e., la norme du maximum du gradient de la vitesse et celle de la pression,calcul´es par des m´ethodes d’´el´ements finis usuelles pour discr´etiser le probl`eme de Stokes, sont born´eesind´ependemment du pas de la discr´etisation. La d´emonstration est bas´ee sur des estimations `a poidsdans L2pour des fonctions de Green associ´ees au probl`eme de Stokes et sur une condition inf-sup`a poids. Le domaine est un polygone ou un poly`edre `a fronti`ere Lipschitz dont les angles int´erieurssatisfont des conditions suffisantes convenables p our assurer que la solution exacte est aussi born´ee dansW1,∞(Ω) × L∞(Ω). La triangulation est uniform´eme nt r´eguli`ere.Keywords: Stokes problem, finite element method, Green’s function, duality argument, weighted errorestimates, weighted inf-sup condition, local interpolationAMS Classification: 65N15, 65N30, 76D070 IntroductionThis article is devoted to the proof of estimates, in the maximum norm, for the gradient of the velocity ofthe discrete Stokes projection and its associated pressure in a variety of finite-element spaces. We considera polygonal or polyhedral domain Ω, in two or three dimensions d, a given velocity vector u in H10(Ω)d, withzero divergence, and a pressure p in L20(Ω), i.e. with zero mean value. Then we c onsider a triangulationThof Ω, where h is the global mesh-size, a pair of finite-element spaces on Th, namely Xh⊂ H10(Ω)dand∗Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, 75252 Paris cedex 05, France.†Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park,Maryland 20742–4015, USA. (Partially supported by NSF grant D MS-0204670.)‡Department of Mathematics and the Computation Institute, University of Chicago, Chicago, Illinois 60637–1581, USA.1Mh⊂ L20(Ω), with appropriate approximation properties and stable in the sense that it satisfies a uniformdiscrete inf-sup condition. We define uh∈ Xhand ph∈ Mh, solution of:ZΩ∇uh: ∇vhdx −ZΩphdiv vhdx =ZΩ∇u : ∇vhdx −ZΩp div vhdx ∀vh∈ Xh, (0.1)ZΩqhdiv uhdx = 0 ∀qh∈ Mh. (0.2)Under suitable sufficient res trictions on the angles of the domain and on the triangulation, we shall provethat if the velocity u belongs to W1,∞(Ω)dand the pressure p be longs to L∞(Ω), then there exists aconstant C independent of h, u and p, such thatk∇uhkL∞(Ω)+ kphkL∞(Ω)≤ Ck∇ukL∞(Ω)+ kpkL∞(Ω). (0.3)This result has many important applications. For instance, it is crucial for extending to Navier-Stokesfree surface flows the numerical analysis done by Saavedra & Scott [33] for the discrete Laplace equationwith a free surface. Another application is the numerical analysis of finite-element schemes for highlynon-linear flows such as non-Newtonian flows. Analyzing such flows often requires a W1,∞bound for theexact velocity; thus the numerical analysis of their finite-element schemes requires a similar bound for thediscrete velocity. One example is the numerical analysis of finite-element schemes for a grade-two fluidflow in three dimensions. In two dimensions, (0.3) is not required, cf. Girault & Scott [21], but this isexceptional and (0.3) is essential in three dimensions.It is well-known thatk∇uhkL2(Ω)≤ k∇ukL2(Ω)+ kpkL2(Ω), (0.4)andkphkL2(Ω)≤1β?k∇ukL2(Ω)+ kpkL2(Ω), (0.5)where β?is the constant of the uniform discrete inf-sup condition:supvh∈XhRΩqhdiv vhdxk∇vhkL2(Ω)≥ β?kqhkL2(Ω)∀qh∈ Mh. (0.6)Therefore, by interpolating between (0.3) and (0.4) or (0.5), we obtain for any number r > 2:k∇uhkLr(Ω)+ kphkLr(Ω)≤ Crk∇ukLr(Ω)+ kpkLr(Ω), (0.7)with a constant Crthat depends on r, but not on h, u and p.0.1 Some backgroundThe result we present here is base d essentially on the proof of two results: maximum norm estimates for(the gradient of) finite-element discretizations of the Laplace equation due to Rannacher & Scott [32] andBrenner & Scott [7], and a family of weighted estimates for the inverse of the divergence due to Dur´an &Muschietti [15]. The reader will find in the recent work by Schatz [34], page 878, a good summary of thehistory of maximum norm estimates for the Laplace equation.In 1988, Dur´an, Nochetto & Wang [16] addressed the discrete Stokes problem in two dimensions bymeans of weighted norms with the weight function introduced by Natterer [28]:σ(x) =|x − x0|2+ (κ h)21/2, (0.8)2where x0is a point close to that where the maximum is attained and κ > 1 is a well-chosen parameterindependent of h. But their estimate was not uniform: their constant C had the factor |log h|1/2. Thisdifficulty in proving W1,∞-stability is not due to the degree of the finite elements, as experienced by Ciarlet& Raviart [11], Scott [35] or Nitsche [30, 31] when dealing with the Laplace equation. It is caused by thepresence of the discrete pressure in the discrete equations, even for estimating the velocity. Indeed, inthe absence of weights, the discrete pressure can be eliminated by using test functions with discrete zerodivergence: this is how (0.4) is derived. But in the presence of weights multiplying the test functions, thepressure cannot be eliminated since the discrete divergence of the product is no longer zero. Unfortunately,the weighted inf-sup condition for