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Lower Bounds in Domain Decomposition

  • Susanne C. Brenner
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 55)

Abstract

An important indicator of the efficiency of a domain decomposition preconditioner is the condition number of the preconditioned system. Upper bounds for the condition numbers of the preconditioned systems have been the focus of most analyses in domain decomposition [21, 20, 23]. However, in order to have a fair comparison of two preconditioners, the sharpness of the respective upper bounds must first be established, which means that we need to derive lower bounds for the condition numbers of the preconditioned systems.

Keywords

Domain Decomposition Piecewise Linear Function Domain Decomposition Method Courant Institute Auxiliary Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    P. E. Bjørstad and J. Mandel, On the spectra of sums of orthogonal projections with applications to parallel computing, BIT, 31 (1991), pp. 76–88.zbMATHCrossRefGoogle Scholar
  2. 2.
    J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring, I, Math. Comp., 47 (1986), pp. 103–134.zbMATHCrossRefGoogle Scholar
  3. 3.
    S. C. Brenner, Lower bounds for two-level additive Schwarz preconditioners with small overlap, SIAM J. Sci. Comput., 21 (2000), pp. 1657–1669.zbMATHCrossRefGoogle Scholar
  4. 4.
    S. C. Brenner, Analysis of two-dimensional FETI-DP preconditioners by the standard additive Schwarz framework, Electron. Trans. Numer. Anal., 16 (2003), pp. 165–185.zbMATHGoogle Scholar
  5. 5.
    S. C. Brenner and Q. He, Lower bounds for three-dimensional nonoverlapping domain decomposition algorithms, Numerische Mathematik, (2003).Google Scholar
  6. 6.
    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, second ed., 2002.zbMATHGoogle Scholar
  7. 7.
    S. C. Brenner and L.-Y. Sung, Lower Bounds for Two-Level Additive Schwarz Preconditioners for Nonconforming Finite Elements, vol. 202 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker AG, New York, 1999, pp. 585–604.Google Scholar
  8. 8.
    S. C. Brenner, Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions, Math. Comp., 69 (2000), pp. 1319–1339.zbMATHCrossRefGoogle Scholar
  9. 9.
    M. Dryja, B. F. Smith, and O. B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), pp. 1662–1694.zbMATHCrossRefGoogle Scholar
  10. 10.
    M. Dryja and O. B. Widlund, An additive variant of the Schwarz alternating method in the case of many subregions, Tech. Rep. 339, Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, New York, 1987.Google Scholar
  11. 11.
    M. Dryja, Domain decomposition algorithms with small overlap, SIAM J. Sci.Comput., 15 (1994), pp. 604–620.zbMATHCrossRefGoogle Scholar
  12. 12.
    M. Dryja, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121–155.zbMATHCrossRefGoogle Scholar
  13. 13.
    C. Farhat, M. Lesoinne, P. LeTallec, K. Pierson, and D. Rixen, FETIDP: A Dual-Primal unified FETI method - part I: A faster alternative to the twolevel FETI method, Internat. J. Numer. Methods Engrg., 50 (2001), pp. 1523–1544.zbMATHCrossRefGoogle Scholar
  14. 14.
    M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numerische Mathematik, 70 (1995), pp. 163–180.zbMATHCrossRefGoogle Scholar
  15. 15.
    J. Li and O. B. Widlund, FETI-DP, BDDC, and block Cholesky methods, Tech. Rep. 857, Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, New York, 2004.Google Scholar
  16. 16.
    J. Mandel and C. R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl., 10 (2003), pp. 639–659.zbMATHCrossRefGoogle Scholar
  17. 17.
    J. Mandel, C. R. Dohrmann, and R. Tezaur, An algebraic theory for primal and dual substructuring methods by constraints, Appl. Numer. Math., 54 (2005), pp. 167–193.zbMATHCrossRefGoogle Scholar
  18. 18.
    J. Mandel and R. Tezaur, On the convergence of a dual-primal substructuring method, Numer. Math., 88 (2001), pp. 543–558.zbMATHCrossRefGoogle Scholar
  19. 19.
    A. M. Matsokin and S. V. Nepomnyaschikh, A Schwarz alternating method in a subspace, Soviet Mathematics, 29 (1985), pp. 78–84.zbMATHGoogle Scholar
  20. 20.
    A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999.Google Scholar
  21. 21.
    B. F. Smith, P. E. Bjørstad, and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 1996.Google Scholar
  22. 22.
    D. Stefanica, Lower bounds for additive Schwarz methods with mortars, C.R. Math. Acad. Sci. Paris, 339 (2004), pp. 739–743.zbMATHGoogle Scholar
  23. 23.
    A. Toselli and O. B. Widlund, Domain Decomposition Methods - Algorithms and Theory, vol. 34 of Series in Computational Mathematics, Springer, 2005.Google Scholar
  24. 24.
    X. Tu, Three-level BDDC in two dimensions, Tech. Rep. 856, Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, New York, 2004.Google Scholar
  25. 25.
    J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review, 34 (1992), pp. 581–613.zbMATHCrossRefGoogle Scholar
  26. 26.
    X. Zhang, Studies in Domain Decomposition: Multilevel Methods and the Biharmonic Dirichlet Problem, PhD thesis, Courant Institute, New York University, September 1991.Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Susanne C. Brenner
    • 1
  1. 1.Center for Computation and Technology, Johnston HallLouisiana State UniversityBaton RougeUSA

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