On Ordering Strategies in a Multigrid Algorithm

Introduction. We present some numerical results for the influence of grid renumbering strategies on multigrid smoothers applied to the Stokes equations. Using discrete divergence--free finite elements we get positive definite stiffness matrices, which allow the use of standard smoothers such as Jacobi-, Gauß--Seidel- and ILU--relaxation. For different ordering strategies (finite element two level ordering, geometrical ordering (row- and linewise), Cuthill--McKee--algorithm) we show the costs of the renumbering and the convergence rates for solving the Stokes equations on several types of domains and subdivisions. These convergence results carry over to an algorithm for solving the stationary and instationary Navier--Stokes equations. Let \Omega be a bounded domain in R 2 with boundary @\Omega . We consider the usual linear Stokes problem Find a pair fu; pg , such that \