Preconditioning Techniques for the Generalized Stokes Problem

Vivek Sarin

Department of Computer Science,
Texas A&M University,
College Station, TX 77843-3112


Abstract

A discrete divergence-free basis for fluid velocity has the advantage of automatically satisfying the continuity constraint in the Navier-Stokes equations for incompressible fluids. When using local solenoidal functions to represent such a basis, the resulting linear system can be very ill-conditioned, in part due to the conditioning of the basis itself. In an earlier paper [1], we have described an algebraic multilevel technique to construct a well-conditioned hierarchical basis that implicitly preconditions the linear system. A careful analysis of this approach has provided further insights into nature of the resulting linear system. In this paper, we outline a preconditioning technique that exploits the structure of the linear system to achieve near-optimal convergence for each instance of the generalized Stokes problem. We present experiments to show that the convergence of the preconditioned conjugate gradients method is almost insensitive to the parameters of the problem. In conjunction with the algebraic multilevel scheme for a well-conditioned divergence-free basis, the proposed technique appears to be a competitive approach for fluid problems.

[1] ``An Efficient Iterative Method for the Generalized Stokes Problem'', Vivek Sarin and Ahmed Sameh, SIAM J. Sci. Comput., Vol. 19, No. 1, pp. 206-226.