An Efficient Multigrid Approach for the Oseen Problem Based on the HSS Iteration

Steven Hamilton, Emory University

In computational fluid dynamics, the Oseen problem arises by performing a fixed point linearization of the Navier-Stokes equations. The HSS iteration, first proposed by Bai, Golub and Ng in 2001, consists of splitting a matrix into its symmetric and skew-symmetric components and alternating between solving the resulting problem with one component or the other held constant. For certain discretizations of the Oseen problem, this proves to be a very natural decomposition with properties very amenable to numerical solution. Previous studies have analyzed various properties of the iteration, including its use as a preconditioning technique. This study will investigate the potential use of the HSS iteration as a smoother within a multigrid framework.

A theoretical assessment in the form of a Local Fourier Analysis (LFA) of the iteration will be presented which demonstrates the excellent properties of the HSS iteration as a multigrid smoother and ultimately aids in the selection of a free parameter inherent to the method. A description of the overall multigrid approach will be given, including detailed descriptions of the solution strategies for the subproblems. Numerical results for standard test problems will be presented and compared to results from other leading methods. For simplicity, the analysis and results are presented for 2-D finite differences, though the theory is equally valid (and perhaps even better) for 3-D geometries and can be generalized to more complicated finite element approaches. Finally, a description of the current challenges of the method will be presented, along with improvements that are necessary to arrive at a truly competitive approach.

Abstract Author(s): Steven Hamilton<br /> Michele Benzi<br />Eldad Haber