Browsing by Author "Shadid, John"
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Item Least Squares Preconditioners for Stabilized Discretizations of the Navier-Stokes Equations(2006-04-20T16:09:59Z) Elman, Howard; Howle, Victoria E.; Shadid, John; Silvester, David; Tuminaro, RayThis paper introduces two stabilization schemes for the Least Squares Commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth and Tuminaro [SIAM J. Sci. Comput., 27, 2006, pp. 1651--1668] for the incompressible Navier-Stokes equations. This preconditioning methodology is one of several choices that are effective for Navier-Stokes equations, and it has the advantage of being defined from strictly algebraic considerations. It has previously been limited in its applicability to div-stable discretizations of the Navier-Stokes equations. This paper shows how to extend the same methodology to stabilized low-order mixed finite element approximation methods.Item A Parallel Block Multi-level Preconditioner for the 3D Incompressible Navier--Stokes Equations(2002-10-25) Elman, Howard; Howle, V. E.; Shadid, John; Tuminaro, RayThe development of robust and efficient algorithms for both steady-state simulations 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