A Parallel Block Multi-level Preconditioner for the 3D Incompressible Navier--Stokes Equations

TitleA Parallel Block Multi-level Preconditioner for the 3D Incompressible Navier--Stokes Equations
Publication TypeReports
Year of Publication2002
AuthorsElman H, Howle VE, Shadid J, Tuminaro R
Date Published2002/10/25/
InstitutionInstititue for Advanced Computer Studies, Univ of Maryland, College Park
KeywordsTechnical Report

The development of robust and efficient algorithms for both steady-statesimulations and fully-implicit time integration of the Navier--Stokes
equations is an active research topic. To be effective, the linear
subproblems generated by these methods require solution techniques that
exhibit robust and rapid convergence. In particular, they should be
insensitive to parameters in the problem such as mesh size, time step, and
Reynolds number. In this context, we explore a parallel preconditioner
based on a block factorization of the coefficient matrix generated in an
Oseen nonlinear iteration for the primitive variable formulation of the
system. The key to this preconditioner is the approximation of a certain Schur complement operator by a technique first proposed by Kay, Loghin,
and Wathen [25] and Silvester, Elman, Kay, and Wathen [45]. The resulting
operator entails subsidiary computations (solutions of pressure Poisson
and convection--diffusion subproblems) that are similar to those required
for decoupled solution methods; however, in this case these solutions are
applied as preconditioners to the coupled Oseen system. One important
aspect of this approach is that the convection--diffusion and Poisson
subproblems are significantly easier to solve than the entire coupled
system, and a solver can be built using tools developed for the
subproblems. In this paper, we apply smoothed aggregation algebraic
multigrid to both subproblems. Previous work has focused on demonstrating
the optimality of these preconditioners with respect to mesh size on
serial, two-dimensional, steady-state computations employing geometric
multi-grid methods; we focus on extending these methods to large-scale,
parallel, three-dimensional, transient and steady-state simulations
employing algebraic multigrid (AMG) methods. Our results display nearly
optimal convergence rates for steady-state solutions as well as for
transient solutions over a wide range of CFL numbers on the
two-dimensional and three-dimensional lid-driven cavity problem.
Also UMIACS-TR-2002-95