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 against 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. Preconditioners for this problem have been studied before in [Bramble,Pasciak 94], however our approach is different. In essence they constructed the preconditioner by assuming only that the finite elements satisfy the usual Babuska-Brezzi condition for Stokes problem, requiring that the time parameter k < h^2. Our preconditioner is based on the observation that the continuous time-dependent Stokes problem satisfy another Babuska-Brezzi condition, which also the Mini element satisfy. We are therefore able to make a rather simple preconditioner, where the time stepping parameter only enters the preconditioner as a constant. The efficiency of this preconditioner will be demonstrated by numerical experiments done in parallel with Diffpack, a C++ toolbox for finite element simulations, on a Beowulf cluster having roughly 50 CPU's. It is established that the preconditioner works for the Mini element. Other elements like the Crouzeix-Raviart element and the P2-P1 element will be considered.

This is joint work with Hans Petter Langtangen and Ragnar Winther.

An extended abstract can be found here.