Department of Informatics

University of Oslo

P.O. Box 1080 Blindern

0316 Oslo, Norway

Abstract

In this paper we consider an optimal multigrid/domain decomposition
preconditioner for the time-dependent Stokes problem. Preconditioners for
this problem arise when using fully implicit time stepping schemes for the
Navier-Stokes equations. However, as the time stepping parameter decreases
towards 0, the problem to be solved at each time step changes from the Stokes
problem to the mixed formulation of the Poisson equation. The same
preconditioning techniques do not work in both cases, even the finite elements
typically used for Stokes are not considered stable for the mixed Poisson
equation. We will show that some typical Stokes elements are in fact stable
also for the Poisson equation in another norm, this leads us to a proper
preconditioner working uniformly in the time stepping parameter. The
efficiency of this preconditioner will be demonstrated by numerical
experiments done in with Diffpack, a C++ toolbox for finite element
simulations. It is established that the preconditioner works well for the
Mini element. Numerical experiments indicate that this preconditioner also
works for the Q_{2}-Q_{1} and P_{2}-P_{1}
elements.