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

  • Authors:

  • Howard Elman;V. E. Howle;John Shadid;Robert Shuttleworth;Ray Tuminaro

  • Affiliations:

  • Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, United States;Sandia National Laboratories, P.O. Box 969, MS 9159 Livermore, CA 94551, United States;Sandia National Laboratories, P.O. Box 5800, MS 1111, Albuquerque, NM 87185, United States;Applied Mathematics and Scientific Computing Program and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, United States;Sandia National Laboratories, P.O. Box 969, MS 9159, Livermore, CA 94551, United States

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2008

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Abstract

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 physics-based 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 [A. Quarteroni, F. Saleri, A. Veneziani, Factorization methods for the numerical approximation of Navier-Stokes equations, Computational Methods in Applied Mechanical Engineering 188 (2000) 505-526]. 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.