Title: Preconditioners Based on Algebraic Commutators for Incompressible Navier-Stokes Equations

Abstract: The bottleneck in implicit numerical solution of the incompressible Navier-Stokes equations is the solution of the linear subproblem. Realistic simulations are large enough that iterative solvers must be used, so in turn an effective preconditioning strategy is required.

In this talk I’ll discuss several strategies for automatically generating block preconditioners for efficient, scalable solution of the incompressible Navier-Stokes equations. The techniques presented are motivated by the “pressure convection-diffusion (PCD)” preconditioners proposed by Kay, Loghin, and Wathen and Silvester, Elman, Kay, and Wathen and the “BFBt” preconditioner of Elman. Numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology. The methods have several drawbacks, however. The PCD method typically out-performs the BFBt method for certain problems, but requires the construction of non-standard operators which are not generally available from simulation codes. Therefore, deployment requires intrusive modifications to the flow simulation software. This need for intrusive modifications has been a barrier to adoption of this method. We consider alternative preconditioning formulations based on algebraic considerations alone. This avenue has led to two new methods, “sparse approximate commutators (SPAC)” and “least squares commutators (LSC),” which are related to the “BFBt” preconditioner.  Neither method requires specialized operators and the LSC method out-performs both the BFBt and PCD methods on many problems.

In cases where stabilized finite elements are used, some additional modifications
are necessary to ensure the stability of the LSC preconditioner. We have also developed a stabilized version of the LSC preconditioner that retains its algebraic character. We demonstrate that these strategies lead to favorable convergence properties, lessening the bottleneck in numerical solution of the incompressible Navier-Stokes equations.

In addition, I will briefly discuss other capabilities and areas of expertise at Sandia that may be of interest to Simbios.