Multilevel preconditioning of Stokes discretizations with divergence-free elements

Dr. Peter Oswald

Lucent Technologies, BL0112740, Murray Hill, NJ 07974

In this talk, we first survey some difficulties connected with using divergence-free finite element discretizations for Stokes and related problems. Concerning the design of multigrid/multilevel solvers, the difficulties are similar to those occuring for nonconforming finite elements. Then we introduce an optimal multilevel preconditioner for the divergence-free part of a modified conforming P1-P0 discretization of the two-dimensional Stokes problem which contains a novel prolongation operator preserving the discrete divergence-free property. The proofs utilize the eqivalence to a discretization of the biharmonic Dirichlet problem by Powell-Sabin macro-triangles via a discrete stream-function. We also show how the basic preconditioner can be applied to other Stokes discretizations with discontinuous pressure elements, e.g., to the nonconforming P1-P0 finite element discretization. Finally, we make a few remarks about the adaptive version of our multilevel scheme and on related research.