An Optimal Iterative Method for the Time-Dependent Stokes Problem

K. A. Mardal, R. Winther, and H. P. Langtangen
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 Q2-Q1 and P2-P1 elements.