A preconditioned conjugate gradient Uzawa‐type method for the solution of the stokes problem by mixed Q1–P0 stabilized finite elements
- 15 February 1992
- journal article
- research article
- Published by Wiley in International Journal for Numerical Methods in Fluids
- Vol. 14 (3) , 289-298
- https://doi.org/10.1002/fld.1650140304
Abstract
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 Babuška 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.Keywords
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