Parallel Performance of Some Two-Level ASPIN Algorithms

  • Leszek Marcinkowski
  • Xiao-Chuan Cai
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)


In this paper we study the parallel performance of some nonlinear additive Schwarz preconditioned inexact Newton methods for solving large sparse system of nonlinear equations arising from the discretization of partial differential equations. The main idea of nonlinear preconditioning is to replace an ill-conditioned nonlinear system by an equivalent nonlinear system that has more balanced nonlinearities. In addition to balance the nonlinearities through nonlinear preconditioning, we also need to make sure that the multilayered iterative solver is scalable with respect to the number of processors. We focus on some two-level nonlinear additive Schwarz preconditioners, and show numerically that these two-level methods can reduce the nonlinearities and at the same time maintain the parallel scalability. Parallel numerical results for some high Reynolds number incompressible Navier-Stokes equations will be presented.


Initial Guess Coarse Grid Parallel Performance Coarse Solver Domain Decomposition Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. S. Balay, K. Buschelman, W. D. Gropp, D. Kaushik, M. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang. PETSc users manual. Technical Report ANL-95/11-Revision 2.1.5, Argonne National Laboratory, 2002.Google Scholar
  2. X.-C. Cai and D. E. Keyes. Nonlinearly preconditioned inexact Newton algorithms. SIAM J. Sci. Comput., 24(1):183–200, 2002.MathSciNetCrossRefGoogle Scholar
  3. X.-C. Cai, D. E. Keyes, and L. Marcinkowski. Non-linear additive Schwarz preconditioners and application in computational fluid dynamics. Internat. J. Numer. Methods Fluids, 40(12):1463–1470, 2002.MathSciNetCrossRefGoogle Scholar
  4. X.-C. Cai, D. E. Keyes, and D. P. Young. A nonlinear additive Schwarz preconditioned inexact Newton method for shocked duct flow. In N. Debit, M. Garbey, R. Hoppe, J. Periaux, D. Keyes, and Y. Kuznetsov, editors, 13th Int. Conference on Domain Decomposition Methods, (Lyon, France, October 9–12 2000), pages 343–350., Augsburg, 2000.Google Scholar
  5. M. Dryja and W. Hackbusch. On the nonlinear domain decomposition method. BIT, 37(2):296–311, 1997.MathSciNetGoogle Scholar
  6. S. C. Eisenstat and H. F. Walker. Globally convergent inexact Newton methods. SIAM J. Optim., 4(2):393–422, 1994.MathSciNetCrossRefGoogle Scholar
  7. C. Hirsch. Numerical computation of internal and external Flows: computational methods for inviscid and viscous flows. John Wiley & Sons, New York, 1990.Google Scholar
  8. F.-N. Hwang and X.-C. Cai. Improving robustness and parallel scalability of Newton's method through nonlinear preconditioning. These proceedings, 2003a.Google Scholar
  9. F.-N. Hwang and X.-C. Cai. A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations. Preprint, Department of Computer Science, University of Colorado at Boulder, 2003b.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Leszek Marcinkowski
    • 1
  • Xiao-Chuan Cai
    • 2
  1. 1.Department of Mathematics, Informatics and MechanicsWarsaw UniversityWarszawaPoland
  2. 2.Department of Computer ScienceUniversity of Colorado at BoulderBoulderUSA

Personalised recommendations