Numerical analysis of Vanka-Type solvers for steady stokes and Navier-Stokes flows

In this paper we consider Vanka-type smoothers for solving Stokes and Navier-Stokes problem. In each iteration step this smoother requires the solutions of several small local subproblems over finite element blocks. It is shown that for particular blocks of finite elements the algorithm converges always to the solution of the Stokes problem and under suitable conditions to the solution of the Navier-Stokes problem. The convergence properties are analyzed for the finite elements formulation and numerical examples are presented.