Two-stage Nite Element Method for Stokes Problem

We consider the numerical solution of the planar Stokes boundary value problem in two stages: the rst obtaining an approximation to the velocity by known methods and the second obtaining an approximation to the pressure by means of a formulation of rst order system that is elliptic in the sense of Petrovski i and a Galerkin, least-squares numerical solution of this system. We consider the possibility of decreasing the element size h and simultaneously increasing the degree of approximating polynomials p. We obtain asymptotic rates of convergence for the second stage that are optimal with respect to h as well as p measured in certain relevant norms. Optimal rates are proved in addition for the case of a polygonal domain for the second stage. A comparison of estimates of the condition numbers with those of rival methods is given for an example. We also provide combined, realizable error estimates for the two stages.