MATHEMATICS OF COMPUTATION VOLUME 51, NUMBER 184 OCTOBER 1988, PAGES 491-506 Sharp Maximum Norm Error Estimates for Finite Element Approximations of the Stokes Problem in 2 - D By R. Duran, R. H. Nochetto and Junping Wang* Abstract. This paper deals with finite element approximations of the Stokes equations in a plane bounded domain Cl, using the so-called velocity-pressure mixed formulation. Quasi-optimal error estimates in the maximum norm are derived for the velocity, its gradient and the pressure fields. The analysis relies on abstract properties which are in turn a consequence of the eixstence of a local projection operator 11/, satisfying / div(v-nhv)9dx = 0, Vve[i/¿(n)]2, VqeMh, Jn where M^ is the finite element space associated with the pressure. Several examples for which this operator can be constructed locally illustrate the theory. 1. Introduction. We consider the Stokes problem arising in fluid dynamics, which describes the flow of a viscous incompressible fluid. In its simplest form, we have to solve Í- Au +Vp =f in 0, divu = 0 infi, u = 0 on dû, where H is a bounded domain in R2, u represents the velocity of the fluid, p its pressure and f a given external force. Several finite element spaces have been considered to approximate the solu- tion of problem (1.1) using the following velocity-pressure formulation: find u E [Ho(V)}2, P S Lg(fi), such that f (Vu, Vv) - (p,divv) = (f,Y), Vv E [m(Q)}2, I (<7,divu)=0, V9eL2(iî), where (•,■) denotes the inner product in L2(Q) and Lq(H) is the space of L2- functions having mean value zero. It is known that this weak formulation is equiv- alent to a saddle point problem. The approximation by finite elements of this kind of problems has been studied in an abstract form by F. Brezzi [4], M. Fortin [13] and R. S. Falk and J. E. Osborn [12]. Received April 13, 1987. 1980 Mathematics SubjectClassification (1985 Revision). Primary 65N30, 65N15, 65N50,65B05, 76D07. Key words and phrases. Finite element method, Stokes equation. "This research was supported in part by the Institute for Mathematics and Its Applications at the University of Minnesota with funds provided by the National Science Foundation. ©1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page 491 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use