A PRECONDITIONED CONJUGATE-GRADIENT UZAWA-TYPE METHOD FOR THE SOLUTION OF THE STOKES PROBLEM BY MIXED Q1-P0 STABILIZED FINITE-ELEMENTS

We study the behaviour of a conjugate gradient Uzawa-type method for a stabilized finite element approximation of the Stokes problem. Many variants of the Uzawa algorithm have been described for different finite elements satisfying the well-known Inf-Sup condition of Babuska and Brezzi, but it is surprising that developments for unstable 'low-order' discretizations with stabilization procedures are still missing. Our paper is presented in this context for the popular (so-called) Q1-P0 element. First we show that a simple stabilization technique for this element permits us to retain the property of a convergence factor bounded independently of the discretization mesh size. The second contribution of this work deals with the construction of a less costly preconditioner taking full advantages of the block diagonal structure of the stabilization matrix. Its efficiency is supported by 2D and 3D numerical results.