Hybrid Schwarz-Multigrid Methods for the Spectral Element Method: Extensions to Navier-Stokes
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.
KeywordsMultigrid Method Spectral Element Iteration Count Spectral Element Method Hairpin Vortex
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- 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
- M. O. Deville, P. F. Fischer, and E. H. Mund. High-order methods for incompressible fluid flow. Cambridge University Press, Cambridge, 2002.Google Scholar
- 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
- 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
- J. W. Lottes and P. F. Fischer. Hybrid multigrid/Schwarz algorithms for the spectral element method. J. Sci. Comput., (to appear), 2004.Google Scholar
- 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
- 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
- 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
- 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
- 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
- B. Smith, P. Bjøstad, and W. Gropp. Domain Decomposition: Parallel Multilevel Methods for Elliptic PDEs. Cambridge University Press, Cambridge, 1996.Google Scholar