Multigrid methods for saddle point systems using constrained smoothers

Abstract

The constrained smoother for solving the saddle point system arising from the constrained minimization problem is a relaxation scheme such that the iteration remains in the constrained subspace. A multigrid method using constrained smoothers for saddle point systems is analyzed in this paper. Uniform convergence of two-level and W-cycle multigrid methods, with sufficient many smoothing steps and full regularity assumptions, are obtained for some stable finite element discretization of Stokes equations. For Braess–Sarazin smoother, a convergence theory using only partial regularity assumption is also developed.