Developments in Overlapping Schwarz Preconditioning of High-Order Nodal Discontinuous Galerkin Discretizations

  • Luke N. Olson
  • Jan S. Hesthaven
  • Lucas C. Wilcox
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 55)

Abstract

Recent progress has been made to more robustly handle the increased complexity of high-order schemes by focusing on the local nature of the discretization. This locality is particularly true for many Discontinuous Galerkin formulations and is the focus of this paper. The contributions of this paper are twofold. First, novel observations regarding various flux representations in the discontinuous Galerkin formulation are highlighted in the context of overlapping Schwarz methods. Second, we conduct additional experiments using high-order elements for the indefinite Helmholtz equation to expose the impact of overlap.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749–1779.MATHCrossRefGoogle Scholar
  2. 2.
    X.-C. Cai, A family of overlapping Schwarz algorithms for nonsymmetric and indefinite elliptic problems, in Domain-based parallelism and problem decomposition methods in computational science and engineering, D. E. Keyes, Y. Saad, and D. G. Truhlar, eds., SIAM, Philadelphia, PA, 1995, pp. 1–19.Google Scholar
  3. 3.
    X.-C. Cai, M. A. Casarin, F. W. Elliott Jr., and O. B. Widlund, Overlapping Schwarz algorithms for solving Helmholtz's equation, in Domain decomposition methods, 10 (Boulder, CO, 1997), vol. 218 of Contemp. Math., AMS, Providence, RI, 1998, pp. 391–399.Google Scholar
  4. 4.
    X.-C. Cai and O. B. Widlund, Domain decomposition algorithms for indefi- nite elliptic problems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 243–258.MATHCrossRefGoogle Scholar
  5. 5.
    H. C. Elman, O. G. Ernst, and D. P. O'Leary, A multigrid method enhanced by Krylov subspace iteration for discrete Helmhotz equations, SIAM J. Sci. Comput., 23 (2001), pp. 1291–1315.MATHCrossRefGoogle Scholar
  6. 6.
    J. S. Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Numer. Anal., 35 (1998), pp. 655–676.MATHCrossRefGoogle Scholar
  7. 7.
    R. M. Kirby, Toward dynamic spectral/hp refinement: algorithms and applications to flow-structure interactions, PhD thesis, Brown University, May 2003.Google Scholar
  8. 8.
    C. Lasser and A. Toselli, Overlapping preconditioners for discontinuous Galerkin approximations of second order problems, in Thirteenth international conference on domain decomposition, N. Debit, M. Garbey, R. Hoppe, J. Pèriaux, D. Keyes, and Y. Kuznetsov, eds., ddm.org, 2001, pp. 78–84.Google Scholar
  9. 9.
    J. W. Lottes and P. F. Fischer, Hybrid multigrid/Schwarz algorithms for the spectral element method, Tech. Rep. ANL/MCS-P1052–0403, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, May 2003.Google Scholar
  10. 10.
    A. Toselli and O. B. Widlund, Domain Decomposition Methods - Algorithms and Theory, vol. 34 of Series in Computational Mathematics, Springer, 2005.Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Luke N. Olson
    • 1
  • Jan S. Hesthaven
    • 1
  • Lucas C. Wilcox
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations