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Journal of Scientific Computing

, Volume 24, Issue 1, pp 45–78 | Cite as

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

  • James W. Lottes
  • Paul F. FischerEmail author
Article

Abstract

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.

Keywords

Multigrid Schwarz methods domain decomposition spectral element methods p-version finite element 

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References

  1. 1.
    Beuchler, S. 2002Multigrid solver for the inner problem in domain decomposition methods for p-femSIAM J. Numer. Anal.40928944CrossRefGoogle Scholar
  2. 2.
    Casarin, M.A. 1997Diagonal edge preconditioners in p-version and spectral element methodsSIAM J. Sci. Comput.18610620CrossRefGoogle Scholar
  3. 3.
    Casarin, M.A. 1997Quasi-optimal Schwarz methods for the conforming spectral element discretizationSIAM J. Numer. Anal.3424822502CrossRefGoogle Scholar
  4. 4.
    Casarin, M.A. 1998Schwarz preconditioners for the spectral element Stokes and Navier-Stokes discretizationsBjørstad, P.Espedal, M.Keyes, D. eds. Domain Decomposition 9 Proc.WileyNew York7279Google Scholar
  5. 5.
    Couzy, W. (1995). Spectral Element Discretization of the Unsteady Navier-Stokes Equations and its Iterative Solution on Parallel Computers, Ph.D. thesis, Swiss Federal Institute of Technology-Lausanne. (1994). Thesis nr. 1380Google Scholar
  6. 6.
    Couzy, W., Deville, M.O. 1994Spectral-element preconditioners for the Uzawa pressure operator applied to incompressible flowsJ. Sci. Comput.9107112CrossRefGoogle Scholar
  7. 7.
    Couzy, W., Deville, M.O. 1995A fast Schur complement method for the spectral element discretization of the incompressible Navier-Stokes equationsJ. Comput Phys.116135142CrossRefGoogle Scholar
  8. 8.
    Deville, M.O., Fischer, P.F., Mund, E.H. 2002High-order Methods for Incompressible Fluid FlowCambridge University PressCambridgeGoogle Scholar
  9. 9.
    Deville, M.O., Mund, E.H. 1985Chebyshev pseudospectral solution of second-order elliptic equations with finite element preconditioningJ. Comp. Phys.60517533CrossRefGoogle Scholar
  10. 10.
    Deville, M.O., Mund, E.H. 1990Finite element preconditioning for pseudospectral solutions of elliptic problems SIAM JSci. Stat. Comput.11311342CrossRefGoogle Scholar
  11. 11.
    Dryja, M., and Widlund, O.B. (1987). An additive variant of the Schwarz alternating method for the case of many subregions. Technical Report TR 339, Dept. Comp. Sci., Courant Inst., NYUGoogle Scholar
  12. 12.
    Fischer P.F. (1996). Parallel multi-level solvers for spectral element methods. In: Ilin A.V., Scott L.R. (ed). Third Int. Conference on Spectral and High Order Methods, Houston Journal of Mathematics, pp. 595–604Google Scholar
  13. 13.
    Fischer P.F. (1997). An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations. J. Comput. Phys. 133, 84–101Google Scholar
  14. 14.
    Fischer, P.F., Kruse, G.W., Loth, F. 2002Spectral element methods for transitional flows in complex geometriesJ. Sci. Comput.178198CrossRefMathSciNetGoogle Scholar
  15. 15.
    Fischer, P.F., Miller, N.I., Tufo, H.M. 2000An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flowsBjørstad, P.Luskin, M. eds. Parallel Solution of Partial Differential Equations.SpringerBerlin158180Google Scholar
  16. 16.
    Fischer, P.F., Rönquist, E.M. 1994Spectral element methods for large scale parallel Navier-Stokes calculationsComput. Methods Appl. Mech. Engrg.1166976CrossRefGoogle Scholar
  17. 17.
    Golub, G., Van Loan, C.F. 1996Matrix ComputationsJohns Hopkins University PressBaltimoreGoogle Scholar
  18. 18.
    Heinrichs, W. 1988Line relaxation for spectral multigrid methodsJ. Comput. Phys.77166182CrossRefGoogle Scholar
  19. 19.
    Heinrichs, W. 1989Improved condition number for spectral methodsMath. Comp.53103119Google Scholar
  20. 20.
    Lynch, R.E., Rice, J.R., Thomas, D.H. 1964Direct solution of partial difference equations by tensor product methodsNumer. Math.6185199CrossRefGoogle Scholar
  21. 21.
    Maday, Y., and Muñoz, R. (1989). Numerical analysis of a multigrid method for spectral approximations. In Hussaini, M Y., Dwoyer, D. L., and Voigt, R. G. (eds.), Lecture Notes in Physics, Volume 323: Proc. of the 11th Int. Conf. on Numerical Methods in Fluid Dynamics, Springer, Berlin, pp. 389–394Google Scholar
  22. 22.
    Maday, Y., Muñoz, R., Patera, A.T., and Rønquist, E.M. (1992). Spectral element multigrid methods. In de Groen, P., and Beauwens, R. (eds.), Proc. of the IMACS Int. Symposium on Iterative Methods in Linear Algebra, Brussels, 1991, Elsevier, Amsterdam, pp. 191–201Google Scholar
  23. 23.
    Mandel, J. 1990Two-level domain decomposition preconditioning for the p-version finite element method in three dimensionsInt. J. Numer. Methods Eng.2910951108CrossRefGoogle Scholar
  24. 24.
    Mandel, J., Lett, G.S. 1991Domain decomposition preconditioning for p-version finite elements with high aspect ratiosAppl. Numer. Math.8411425CrossRefGoogle Scholar
  25. 25.
    Mansfield, L. 1988On the use of deflation to improve the convergence of conjugate gradient iterationComm. in Appl. Numer. Meth.4151156CrossRefGoogle Scholar
  26. 26.
    Nicolaides, R.A. 1987Deflation of conjugate gradients with application to boundary value problemsSIAM J. Numer. Anal.24355365CrossRefGoogle Scholar
  27. 27.
    Orszag, S.A. 1980Spectral methods for problems in complex geometryJ. Comput. Phys.377092CrossRefGoogle Scholar
  28. 28.
    Pahl, S.S. (1993). Schwarz type domain decomposition methods for spectral element discretizations. Master’s thesis, Dept. of Computational and Applied Math., Univ. of Witwatersrand, Johannesburg, South AfricaGoogle Scholar
  29. 29.
    Parter, S.V., Rothman, E.E. 1995Preconditioning Legendre spectral collocation approximations to elliptic problemsSIAM J. Numer. Anal.32333385CrossRefGoogle Scholar
  30. 30.
    Pavarino, L.F., Warburton, T. 2000Overlapping Schwarz methods for unstructured spectral elementsJ Comput. Phys.160298317CrossRefMathSciNetGoogle Scholar
  31. 31.
    31 Pavarino, L.F., Widlund, O.B. 1996A polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensionsSIAM J. Numer. Anal.3313031335CrossRefGoogle Scholar
  32. 32.
    Rønquist, E. 1988Optimal Spectral Element Methods for the Unsteady Three- Dimensional Incompressible Navier-Stokes Equations. PhD thesisMassachusetts Institute of TechnologyCambridge, MAGoogle Scholar
  33. 33.
    Rønquist, E.M. (1992). A domain decomposition method for elliptic boundary value problems: application to unsteady incompressible fluid flow. In Keyes, D.E., Chan, T.F., Meurant, G., Scroggs, J.S., and Voigt, R.G. (eds.), Fifth Int Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, pp. 545–557Google Scholar
  34. 34.
    Rønquist E.M. (1996). A domain decomposition solver for the steady Navier-Stokes equations. In: Ilin A.V., Scott L.R. (ed). Third Int. Conference on Spectral and High Order Methods, Houston Journal of Mathematics, pp. 469–485Google Scholar
  35. 35.
    Rønquist, E.M., Patera, A.T. 1987Spectral element multigrid, I: Formulation and numerical resultsJ. Sci Comput.2389406CrossRefGoogle Scholar
  36. 36.
    Saad, Y., Schultz, M.H. 1986GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systemsSIAM J. Sci. Stat. Comput.7856869CrossRefGoogle Scholar
  37. 37.
    Shen, J.E., Wang, F., Xu, J. 2000A finite element multigrid preconditioner for chebyshev-collocation methodsAppl. Numer. Math.33471477CrossRefMathSciNetGoogle Scholar
  38. 38.
    Shen J.E. (1996). Efficient Chebyshev-Legendre Galerkin methods for elliptic problems. In Ilin, A.V., and Scott, L.R. (eds.), Third Int. Conference on Spectral and High Order Methods, Houston Journal of Mathematics, pp. 233–239Google Scholar
  39. 39.
    Smith, B., Bjørstad, P., Gropp, W. 1996Domain Decomposition: Parallel Multilevel Methods for Elliptic PDEsCambridge University PressCambridgeGoogle Scholar
  40. 40.
    Thomas, S.J., Dennis, J.M., Tufo, H.M., Fischer, P.F. 2003A Schwarz preconditioner for the cubed-sphereSIAM J. Sci. Comput.25442453CrossRefGoogle Scholar
  41. 41.
    Tufo, H.M., Fischer, P.F. 2001Fast parallel direct solvers for coarse-grid problemsJ. Parallel Distrib Comput.61151177CrossRefGoogle Scholar
  42. 42.
    Zang, T.A., Wong, Y.S., Hussaini, M.Y 1982Spectral multigrid methods for elliptic equationsJ Comput. Phys.48485501CrossRefGoogle Scholar
  43. 43.
    Zang, T.A., Wong, Y.S., Hussaini, M.Y 1984Spectral multigrid methods for elliptic equations IIJ. Comput. Phys.54489507CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Theoretical and Applied MechanicsUniversity of IllinoisUrbanaUSA
  2. 2.Mathematics and Computer Science DivisionArgonne National LaboratoryArgonneUSA

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