Over the past several years, considerable effort has been placed on developing efficient solution algorithms for the incompressible Navier-Stokes equations. The effectiveness of these methods requires that the solution techniques for the linear subproblems generated by these algorithms exhibit robust and rapid convergence; These methods should be insensitive to problem parameters such as mesh size and Reynolds number. This study concerns a preconditioner derived from a block factorization of the coefficient matrix generated in a Newton nonlinear iteration for the primitive variable formulation of the system. This preconditioner is based on the approximation of the Schur complement operator using a technique proposed by Kay, Loghin, and Wathen [1] and Silvester, Elman, Kay, and Wathen [2]. It is derived using subsidiary computations (solutions of pressure Poisson and convection-diffusion-like subproblems) that are significantly easier to solve than the entire coupled system, and a solver can be built using tools, such as smooth aggregation multigrid for the subproblems.

We discuss a computational study performed using MPSalsa, a stabilized finite element code, in which parallel versions of these preconditioners from the pressure convection-diffusion preconditioners are compared with an overlapping Schwarz domain decomposition preconditioner. Our results show nearly ideal convergence rates for a wide range of Reynolds numbers on two-dimensional problems with both enclosed and in/out flow boundary conditions on both structured and unstructured meshes.

[1] D. Kay, D. Loghin, and A. J. Wathen, A preconditioner for the steady-state Navier-Stokes equations, SIAM J. on Sci. Comp. 24 (2002) 237-256.

[2] D. Silvester, H. Elman, D. Kay, A. Wathen, Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow, J. Comp. Appl. Math. 128 (2001) 261-279.