This study numerically considers a class of four efficient iterative solvers for the linear indefinite system of equations arising from the discretization of the Stokes equations. The numerical tests are based on the finite element approximation of 2-D domain using uniform rectangular meshes. The solution methods are based on the suitable choices of preconditioners and smoothers. These solvers are preconditioned minimum residual, preconditioned conjugate gradient, inexact Uzawa and multigrid as well as their combinations as smoother and preconditioner approximations. The comparison is made in terms of iterative counts and computational time. The results obtained indicate that all the methods are robust in terms of changes in mesh size although multigrid solver with the Braess-Sarazin smoother is more efficient since it requires fewer iterations and less computing times. Among the other methods, the inexact Uzawa solver with multigrid preconditioner is also robust and efficient, the preconditioned minimum residual is slightly slower than the preconditioned conjugate gradient but it has an advantage that it is free of any iterative parameter estimates. We discretize the problem using stable Hood-Taylor  pair of finite rectangular elements and present results of the numerical experiments.

Stokes problem, mixed finite element method, multigrid (MGM), smoother, pre-conditioner, non-standard inner product, preconditioned conjugate gradient method (PCG), inexact Uzawa (IUzawa), preconditioned minimum residual (PMINRES).