Hybrid Schwarz-Multigrid Methods for the Spectral Element Method: Extensions to Navier-Stokes

  • Paul F. Fischer
  • James W. Lottes
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)


The performance of multigrid methods for the standard Poisson problem and for the consistent Poisson problem arising in spectral element discretizations of the Navier-Stokes equations is investigated. It is demonstrated that overlapping additive Schwarz methods are effective smoothers, provided that the solution in the overlap region is weighted by the inverse counting matrix. It is also shown that spectral element based smoothers are superior to those based upon finite element discretizations. Results for several large 3D Navier-Stokes applications are presented.


Multigrid Method Spectral Element Iteration Count Spectral Element Method Hairpin Vortex 
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  1. S. Beuchler. Multigrid solver for the inner problem in domain decomposition methods for p-fem. SIAM J. Numer. Anal., 40:928–944, 2002.MATHMathSciNetCrossRefGoogle Scholar
  2. M. A. Casarin. Quasi-optimal Schwarz methods for the conforming spectral element discretization. SIAM J. Numer. Anal., 34:2482–2502, 1997.MATHMathSciNetCrossRefGoogle Scholar
  3. W. Couzy. Spectral Element Discretization of the Unsteady Navier-Stokes Equations and its Iterative Solution on Parallel Computers. PhD thesis, Swiss Federal Institute of Technology-Lausanne, 1995. Thesis nr. 1380.Google Scholar
  4. M. O. Deville, P. F. Fischer, and E. H. Mund. High-order methods for incompressible fluid flow. Cambridge University Press, Cambridge, 2002.Google Scholar
  5. M. Dryja and O. B. Widlund. An additive variant of the Schwarz alternating method for the case of many subregions. Technical Report TR 339, Courant Inst., NYU, 1987. Dept. Comp. Sci.Google Scholar
  6. P. F. Fischer. An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations. J. Comput. Phys., 133:84–101, 1997.MATHMathSciNetCrossRefGoogle Scholar
  7. P. F. Fischer. Projection techniques for iterative solution of Ax = b with successive right-hand sides. Comput. Methods Appl. Mech. Engrg., 163:193–204, 1998.MATHMathSciNetCrossRefGoogle Scholar
  8. P. F. Fischer, N. I. Miller, and H. M. Tufo. An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows. In P. Bjøstad and M. Luskin, editors, Parallel Solution of Partial Differential Equations, pages 158–180, Berlin, 2000. Springer.Google Scholar
  9. A. Frommer and D. B. Szyld. An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms. SIAM Journal on Numerical Analysis, 39:463–479, 2001.MathSciNetCrossRefGoogle Scholar
  10. J. W. Lottes and P. F. Fischer. Hybrid multigrid/Schwarz algorithms for the spectral element method. J. Sci. Comput., (to appear), 2004.Google Scholar
  11. R. E. Lynch, J. R. Rice, and D. H. Thomas. Direct solution of partial difference equations by tensor product methods. Numer. Math., 6:185–199, 1964.MathSciNetCrossRefGoogle Scholar
  12. Y. Maday and R. Muñoz. Spectral element multigrid: Numerical analysis. J. Sci. Comput., 3:323–354, 1988.MathSciNetCrossRefGoogle Scholar
  13. Y. Maday, R. Muñoz, A. T. Patera, and E. M. Røquist. Spectral element multigrid methods. In P. de Groen and R. Beauwens, editors, Proc. of the IMACS Int. Symposium on Iterative Methods in Linear Algebra, Brussels, 1991, pages 191–201, Amsterdam, 1992. Elsevier.Google Scholar
  14. Y. Maday and A. T. Patera. Spectral element methods for the Navier-Stokes equations. In A. K. Noor and J. T. Oden, editors, State-of-the-Art Surveys in Computational Mechanics, pages 71–143. ASME, New York, 1989.Google Scholar
  15. Y. Maday, A. T. Patera, and E. M. Røquist. An operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow. J. Sci. Comput., 5:263–292, 1990.MathSciNetCrossRefGoogle Scholar
  16. S. A. Orszag. Spectral methods for problems in complex geometry. J. Comput. Phys., 37:70–92, 1980.MATHMathSciNetCrossRefGoogle Scholar
  17. S. S. Pahl. Schwarz type domain decomposition methods for spectral element discretizations. Master's thesis, Univ. of Witwatersrand, Johannesburg, South Africa, 1993. Dept. of Computational and Applied Math.Google Scholar
  18. J. B. Perot. An analysis of the fractional step method. J. Comp. Phys., 108:51–58, 1993.MATHMathSciNetCrossRefGoogle Scholar
  19. E. Røquist. Optimal Spectral Element Methods for the Unsteady Three-Dimensional Incompressible Navier-Stokes Equations. PhD thesis, Massachusetts Institute of Technology, 1988. Cambridge, MA.Google Scholar
  20. E. M. Røquist and A. T. Patera. Spectral element multigrid, I:. Formulation and numerical results. J. Sci. Comput., 2:389–406, 1987.MathSciNetCrossRefGoogle Scholar
  21. J. E. Shen. Efficient Chebyshev-Legendre Galerkin methods for elliptic problems. In A. V. Ilin and L. R. Scott, editors, Third Int. Conference on Spectral and High Order Methods, pages 233–239. Houston Journal of Mathematics, 1996.Google Scholar
  22. J. E. Shen, F. Wang, and J. Xu. A finite element multigrid preconditioner for Chebyshev-collocation methods. Appl. Numer. Math., 33:471–477, 2000.MathSciNetCrossRefGoogle Scholar
  23. B. Smith, P. Bjøstad, and W. Gropp. Domain Decomposition: Parallel Multilevel Methods for Elliptic PDEs. Cambridge University Press, Cambridge, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Paul F. Fischer
    • 1
  • James W. Lottes
    • 2
  1. 1.Argonne National LaboratoryMathematics and Computer Science DivisionArgonne
  2. 2.Dept. of Theoretical and Applied MechanicsUniversity of Illinois

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