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Fast iterative solution of stabilised Stokes systems. II: Using general block preconditioners. (English) Zbl 0810.76044
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. Part I of this work [ibid. 30, No. 3, 630-649 (1993; Zbl 0776.76024)] described a conjugate gradient- like method (the method of preconditioned conjugate residuals) which is applicable to symmetric indefinite problems. 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.

76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
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