FAST ITERATIVE SOLUTION OF STABILIZED STOKES SYSTEMS PART .2. USING GENERAL BLOCK PRECONDITIONERS

Mixed finite element approximation of the classical Stokes problem describing slow viscous incompressible how 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. Part I of this work described a conjugate gradient-like method (the method of preconditioned conjugate residuals) which is applicable to symmetric indefinite problems [A. J. Wathen and D. J. Silvester, SIAM J. Numer. Anal., 30 (1993), pp. 630-649]. Using simple arguments, estimates for the eigenvalue distribution of the discrete Stokes operator on which the convergence rate of the iteration depends are easily derived. Part I discussed the important case of diagonal preconditioning (scaling). This paper considers the more general class of block preconditioners, where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables. It is shown that, provided the appropriate scaling is used for the blocks corresponding to the pressure variables, the preconditioning of the Laplacian (viscous) terms determines the complete eigenvalue spectrum of the preconditioned Stokes operator. All conventional preconditioners for the Laplacian including (modified) incomplete factorisation, hierarchical basis, multigrid and domain decomposition methods are covered by the analysis. It is shown that if a spectrally equivalent preconditioner is used for the Laplacian terms then the convergence rate of iterative solution algorithms is independent of the mesh-size. The results apply to both locally and globally stabilised mixed approximations as well as to mixed methods that are inherently stable.