Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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EXPLICIT BOUNDS FOR THE TWO-LEVEL PRECONDITIONER OF THE P1 DISCONTINUOUS GALERKIN METHOD ON RECTANGULAR MESHES

  • Received : 2009.09.04
  • Accepted : 2009.11.11
  • Published : 2009.12.25
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Abstract

In this paper we investigate a simple two-level additive Schwarz preconditioner for the P1 symmetric interior penalty Galerkin method of the Poisson equation on rectangular meshes. The construction is based on the decomposition of the global space of piecewise linear polynomials into the sum of local subspaces, each of which corresponds to an element of the underlying mesh, and the global coarse subspace consisting of piecewise constants. This preconditioner is a direct combination of the block Jacobi iteration and the cell-centered finite difference method, and thus very easy to implement. Explicit upper and lower bounds for the maximum and minimum eigenvalues of the preconditioned matrix system are derived and confirmed by some numerical experiments.

Keywords

References

  1. P. F. Antonietti and B. Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case, M2AN Math. Model. Numer. Anal. 41 (2007), no. 1, 21–54. https://doi.org/10.1051/m2an:2007006
  2. 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 (2001/02), no. 5, 1749–1779. https://doi.org/10.1137/S0036142901384162
  3. P. Bastian and V. Reichenberger, Multigrid for higher order discontinuous Galerkin finite elements applied to groundwater flow, Technical Report 2000-37, SFB 359, Heidelberg University, 2000.
  4. J. H. Bramble, R. E. Ewing, J. E. Pasciak and J. Shen, The analysis of multigrid algorithms for cell centered finite difference methods, Adv. Comput. Math. 5 (1996), no. 1, 15–29. https://doi.org/10.1007/BF02124733
  5. S. C. Brenner and J. Zhao, Convergence of multigrid algorithms for interior penalty methods, Appl. Numer. Anal. Comput. Math. 2 (2005), no. 1, 3–18. https://doi.org/10.1002/anac.200410019
  6. V. A. Dobrev, R. D. Lazarov, P. S. Vassilevski and L. T. Zikatanov, Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations, Numer. Linear Algebra Appl. 13 (2006), no. 9, 753–770. https://doi.org/10.1002/nla.504
  7. V. A. Dobrev, R. D. Lazarov and L. T. Zikatanov, Preconditioning of symmetric interior penalty discontinuous Galerkin FEM for elliptic problems, In Domain decomposition methods in science and engineering XVII, 33–44, Lect. Notes Comput. Sci. Eng., 60, Springer, Berlin, 2008.
  8. X. Feng and O. A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems, SIAM J. Numer. Anal. 39 (2001), no. 4, 1343–1365. https://doi.org/10.1137/S0036142900378480
  9. M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math. 70 (1995), no. 2, 163–180. https://doi.org/10.1007/s002110050115
  10. J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numer. Math. 95 (2003), no. 3, 527–550. https://doi.org/10.1007/s002110200392
  11. B. T. Helenbrook and H. L. Atkins, Application of p multigrid to discontinuous Galerkin formulations of the Poisson equation, AIAA Journal 44 (2005), no. 3, 566–575.
  12. B. T. Helenbrook and H. L. Atkins, Solving discontinuous Galerkin formulations of Poisson's equation using geometric and p multigrid, AIAA Journal 46 (2008), no. 4, 894–902. https://doi.org/10.2514/1.31163
  13. P. W. Hemker, W. Hoffmann and M. H. van Raalte, Two-level Fourier analysis of a multigrid approach for discontinuous Galerkin discretization, SIAM J. Sci. Comput. 25 (2003), no. 3, 1018–1041. https://doi.org/10.1137/S1064827502405100
  14. D. Y. Kwak, V -cycle multigrid for cell-centered finite differences, SIAM J. Sci. Comput. 21 (1999), no. 2, 552–564. https://doi.org/10.1137/S1064827597327310
  15. D. Y. Kwak and J. S. Lee, Multigrid algorithm for the cell-centered finite difference method. II. Discontinuous coefficient case, Numer. Methods Partial Differential Equations 20 (2004), no. 5, 742–764. https://doi.org/10.1002/num.20001
  16. C. Lasser and A. Toselli, An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems, Math. Comp. 72 (2003), no. 243, 1215–1238. https://doi.org/10.1090/S0025-5718-03-01484-4
  17. B. Riviere, Discontinuous Galerkin methods for solving elliptic and parabolic equations. Theory and implementation, Frontiers in Applied Mathematics, 35. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008.
  18. A. Toselli and O.Widlund, Domain decomposition methods-algorithms and theory. Springer Series in Computational Mathematics, 34. Springer-Verlag, Berlin, 2005.
  19. J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581-613. https://doi.org/10.1137/1034116