Effective preconditioning of Uzawa type schemes for a generalized Stokes problem

The Schur complement of a model problem is considered as a preconditioner f or the Uzawa type schemes for the generalized Stokes problem (the Stokes pr oblem with the additional term alpha u in the motion equation). The impleme ntation of the preconditioned method requires for each iteration only one e xtra solution of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence is proved. in particular, it is established that the method converges with a rate that is bounded by some constant independent of alpha. Some finite difference a nd finite element methods are discussed. Numerical results for finite diffe rence MAC scheme are provided. Mathematics Subject Classification (1991): 6 5N30, 65F10.