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A preconditioned conjugate gradient Uzawa‐type method for the solution of the stokes problem by mixed Q1–P0 stabilized finite elements
Author(s) -
Vincent C.,
Boyer R.
Publication year - 1992
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650140304
Subject(s) - preconditioner , mathematics , conjugate gradient method , finite element method , discretization , diagonal , context (archaeology) , bounded function , convergence (economics) , matrix (chemical analysis) , hexahedron , type (biology) , mathematical analysis , block (permutation group theory) , condition number , block matrix , diagonally dominant matrix , linear system , mathematical optimization , geometry , pure mathematics , invertible matrix , eigenvalues and eigenvectors , physics , paleontology , economic growth , materials science , ecology , composite material , biology , quantum mechanics , thermodynamics , economics
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.