Smoothers in Coupled Multigrid Methods for the Stokes and Navier-Stokes Equations

Volker John
Lutz Tobiska
Otto-von-Guericke-Universität Magdeburg, Germany


Coupled multigrid methods have been proven as efficient solvers for the saddle point problems arising in the discretization and linearization of the incompressible Navier-Stokes equations. In our talk, we consider two classes of smoothers for these methods. On the one hand, these are Vanka-type smoothers where in each smoothing step a number of small linear systems has to be solved. On the other hand, we will study smoothers proposed by Braess and Sarazin, which are based on the solution of a global pressure Schur complement system.

First, we consider the Stokes equations which are discretized with the nonconforming P1/P0-finite element method. We will sketched the convergence proof of the coupled multigrid method with Braess-Sarazin-type smoothers. Thus, the results of Braess/Sarazin are extended to nonconforming finite element discretizations. Numerical tests which confirm the theoretical results are presented.

The second part of the talk is devoted to a numerical comparison of Vanka-type and Braess-Sarazin-type smoothers in coupled multigrid methods for the solution of the Navier-Stokes equations. These tests will show the superiority of the Vanka-type smoothers for steady state equations as well as the advantageous behaviour of some Braess-Sarazin-type smoothers for time dependent problems.