(401c) Assessing the State-of-the-Art in Preconditioned Iterative Solvers for Strongly Convected, Three-Dimensional, Incompressible Flows

The computation of three-dimensional, incompressible flows continues to pose great challenges for development of effective preconditioners for projection-based iterative solvers such as GMRES. It is well known that increasing the size of the linear system, by refining the discretization, and increasing the strength of convection, by raising the Reynolds or Rayleigh number, both severely degrade convergence of iterative solvers. The impact of these effects on convergence of strongly convected flows is multiplicative, because with the formation of boundary layers comes a need to refine the discretization. Despite recent progress in preconditioning, these difficulties still limit our ability to compute accurately resolved, strongly convected flows.

In this presentation, we compare results of various preconditioners for solving steady-state, laminar, strongly coupled, buoyant flows at high Rayleigh number. We use our Galerkin-finite element code for solving three-dimensional problems in coupled transport to test the effectiveness of diagonal and ILU(k) preconditioners. We also assess recent reports on other, more sophisticated, preconditioners, focusing in particular on the relative difficulty of the flow problems used to test convergence of these preconditioners. We identify a particular challenge that enclosed, recirculating flows pose for convergence, which is the slow rate that information from the boundaries is conveyed to the interior of the domain by the iterative solver. We discuss the pitfalls in making fair and meaningful comparisons of different preconditioner types, and show in particular that diagonal preconditioning compares more favorably to start-of-the-art preconditioners than is perhaps generally recognized.