A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations

In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier–Stokes equations based on preconditioned Krylov methods. These include physicsbased methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in [25]. This taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflow/outflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code. ∗This work was partially supported by the DOE Office of Science MICS Program and by the ASC Program at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. †This work was partially supported by the DOE Office of Science MICS Program and by the ASC Program at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. ‡This work was partially supported by the DOE Office of Science MICS Program and by the ASC Program at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. §Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, [email protected]. The work of this author was supported by the Department of Energy under grant DOEG0204ER25619. ¶Sandia National Laboratories, PO Box 969, MS 9159 Livermore, CA 94551, [email protected]. ‖Sandia National Laboratories, PO Box 5800, MS 1111, Albuquerque, NM 87185, [email protected]. ∗∗Applied Mathematics and Scientific Computing Program and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742. [email protected] ††Sandia National Laboratories, PO Box 969, MS 9159, Livermore, CA 94551, [email protected].