BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//NA-M-91-04
       ENTRY:: January 28, 1996
ORGANIZATION:: Stanford University, Department of Computer Science, Numerical
               Analysis Project
       TITLE:: Fast iterative solution of stabilised Stokes systems. Part I:
               Using simple diagonal preconditioners
        TYPE:: Manuscript
      AUTHOR:: Wathen, Andrew J.
      AUTHOR:: Silvester, David J.
        DATE:: October 1991
       PAGES:: 36
    ABSTRACT:: Mixed finite element approximation of the classical Stokes
               problem describing slow viscous incompressible flow gives
               rise to symmetric indefinite systems for the discrete
               velocity and pressure variables. Iterative solution of such
               indefinite systems is feasible and is an attractive approach
               for large problems. The use of stabilisation methods for
               convenient (but unstable) mixed elements introduces
               stabilisation parameters. We show how these can be chosen to
               obtain rapid iterative convergence.

               We propose a conjugate gradient-like method (the method of
               preconditioned conjugate residuals) which is applicable to
               symmetric indefinite problems, describe the effects of
               stabilisation on the algebraic structure of the discrete
               Stokes operator and derive estimates of the eigenvalue
               spectrum of this operator on which the convergence rate of
               the iteration depends. Here we discuss the simple case of
               diagonal preconditioning. Our results apply to both locally
               and globally stabilised mixed elements as well as to elements
               which are inherently stable. We demonstrate that convergence
               rates comparable to that achieved using the diagonally scaled
               conjugate gradient method applied to the discrete Laplacian
               are approachable for the Stokes problem.
       NOTES:: [Adminitrivia V1/Prg/19960128]
         END:: STAN//NA-M-91-04