Effective preconditioning of Uzawa type schemes for a generalized Stokes problem

The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized Stokes problem (the Stokes problem with the additional term αu in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra 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 α. Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are provided.