Numer. Math. 33, 397-424 (1979) Numerische MathemaUk 9 by Springer-Verlag 1979 On a Mixed Finite Element Approximation of the Stokes Problem (I) Convergence of the Approximate Solutions R. Glowinski and O. Pironneau I.R.I.A.-Laboria, Domaine de Voluceau, Rocquencourt, 78150 Le Chesnay, France Summary. We study in this paper a new mixed finite element approximation of the Stokes' problem in the velocity pressure formulation. This approxima- tion which is based on a new variational principle allows the use of low order Lagrange elements and leads to optimal order of convergence for the velocity and the pressure. Iterative and direct methods for the solution of the approximate problems will be discussed in a forthcoming paper. Subject Classifications. AMS(MOS): 65N30; CR: 5.17. 1. Introduction The Stokes problem mainly occurs in the two following instances: (i) The simulation of low Reynolds number incompressible viscous flows. (ii) As a step in the numerical solution of the Navier-Stokes equations for incompressible, viscous fluids. It is mostly for the second case that a good code for the numerical solution of the Stokes problem may be needed. It is shown moreover (see e.g. Jamet-Raviart [1], Temam [2], Girault- Raviart [3], Raviart [4], Le Tallec [5]) that Stokes and Navier-Stokes equations lead to discretization errors of the same order, if the same type of approximation is used for both problems; in fact it seems that a good understanding of the Stokes problem is a necessary preliminary step for the theoretical and numerical analysis of Navier-Stokes equations. The approximation of the Stokes (and Navier-Stokes) problem by Finite Element Methods has motivated many papers; it is not possible to mention all of them. Let us mention however [1-5] and [6-11], Fortin [12], Crouzeix-Raviart [13], Hood-Taylor [14], Argyris-Dunne [15], Zienkiewicz [16], Bercovier [17], Hughes-Taylor-Levy [18], Fortin-Thomasset [19], Johnson [20], Glowinski- Pironneau [21] .... (see also the references therein). The above references deal with pressure-velocity or stream function-vorticity formulations. 0029-599 X/79/0033/0397/$05.60