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Recent Developments on Optimized Schwarz Methods

  • Frédéric Nataf
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 55)

Abstract

The classical Schwarz method [31] is based on Dirichlet boundary conditions. Overlapping subdomains are necessary to ensure convergence. As a result, when overlap is small, typically one mesh size, convergence of the algorithm is slow. A first possible remedy is the introduction of Neumann boundary conditions in the coupling between the local solutions. This idea has led to the development of the Dirichlet-Neuman algorithm [10], Neumann-Neumann method [3] and FETI methods [8]. These methods are widely used and have been the subject of many studies, improvements and extensions to various scalar or systems of partial differential equations, see for instance the following books [32], [27], [37] and [35] and references therein. A second cure to the slowness of the original Schwarz method is to use more general interface conditions, Robin conditions were proposed in [19] and pseudo-differential ones in [17]. These methods are well-suited for indefinite problems [5] and as we shall see to heterogeneous problems.

Keywords

Interface Condition Domain Decomposition Helmholtz Equation Iteration Count Domain Decomposition Method 
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.
    Y. Achdou, C. Japhet, Y. Maday, and F. Nataf, A new cement to glue non-conforming grids with Robin interface conditions: the finite volume case, Numer. Math., 92 (2002), pp. 593–620.zbMATHCrossRefGoogle Scholar
  2. 2.
    J. D. Benamou and B. Després, A domain decomposition method for the Helmholtz equation and related optimal control, J. Comp. Phys., 136 (1997), pp. 68–82.zbMATHCrossRefGoogle Scholar
  3. 3.
    J.-F. Bourgat, R. Glowinski, P. Le Tallec, and M. Vidrascu, Variational formulation and algorithm for trace operator in domain decomposition calculations, in Domain Decomposition Methods, T. Chan, R. Glowinski, J. Périaux, and O. Widlund, eds., Philadelphia, PA, 1989, SIAM, pp. 3–16.Google Scholar
  4. 4.
    L. C. Cowsar, J. Mandel, and M. F. Wheeler, Balancing domain decomposition for mixed finite elements, Math. Comp., 64 (1995), pp. 989–1015.zbMATHCrossRefGoogle Scholar
  5. 5.
    B. Després, Décomposition de domaine et problème de Helmholtz, C.R. Acad.Sci. Paris, 1 (1990), pp. 313–316.Google Scholar
  6. 6.
    B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), pp. 629–651.zbMATHCrossRefGoogle Scholar
  7. 7.
    I. Faille, F. Nataf, L. Saas, and F. Willien, Finite volume methods on non-matching grids with arbitrary interface conditions and highly heterogeneous media, in Proceedings of the 15th international conference on Domain Decomposition Methods, R. Kornhuber, R. H. W. Hoppe, J. Péeriaux, O. Pironneau, O. B. Widlund, and J. Xu, eds., Springer-Verlag, 2004, pp. 243–250. Lecture Notes in Computational Science and Engineering.Google Scholar
  8. 8.
    C. Farhat and F. X. Roux, An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 379–396.zbMATHCrossRefGoogle Scholar
  9. 9.
    E. Flauraud and F. Nataf, Optimized interface conditions in domain decomposition methods. Application at the semi-discrete and at the algebraic level to problems with extreme contrasts in the coefficients., Tech. Rep. R.I. 524, CMAP, Ecole Polytechnique, 2004.Google Scholar
  10. 10.
    D. Funaro, A. Quarteroni, and P. Zanolli, An iterative procedure with interface relaxation for domain decomposition methods, SIAM J. Numer. Anal., 25 (1988), pp. 1213–1236.zbMATHCrossRefGoogle Scholar
  11. 11.
    M. J. Gander and G. H. Golub, A non-overlapping optimized Schwarz method which converges with an arbitrarily weak dependence on h, in Fourteenth International Conference on Domain Decomposition Methods, 2002.Google Scholar
  12. 12.
    M. J. Gander, F. Magoulès, and F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput., 24 (2002), pp. 38–60.zbMATHCrossRefGoogle Scholar
  13. 13.
    M. Genseberger, Domain decomposition in the Jacobi-Davidson method for Eigenproblems, PhD thesis, Utrecht University, September 2001.Google Scholar
  14. 14.
    L. G. Giorda and F. Nataf, Optimized Schwarz methods for unsymmetric layered problems with strongly discontinuous and anisotropic coefficients, Tech. Rep. 561, CMAP, CNRS UMR 7641, Ecole Polytechnique, France, 2004. Submitted.Google Scholar
  15. 15.
    D. Givoli, Numerical methods for problems in infinite domains, Elsevier, 1992.Google Scholar
  16. 16.
    I. G. Graham and M. J. Hagger, Unstructured additive Schwarz-CG method for elliptic problems with highly discontinuous coefficients, SIAM J. Sci. Comput., 20 (1999), pp. 2041–2066.zbMATHCrossRefGoogle Scholar
  17. 17.
    T. Hagstrom, R. P. Tewarson, and A. Jazcilevich, Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems, Appl. Math. Lett., 1 (1988).Google Scholar
  18. 18.
    A. Klawonn, O. B. Widlund, and M. Dryja, Dual-Primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients, SIAM J.Numer.Anal., 40 (2002).Google Scholar
  19. 19.
    P.-L. Lions, On the Schwarz alternating method. III: a variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, March 20–22, 1989, T. F. Chan, R. Glowinski, J. Périaux, and O. Widlund, eds., Philadelphia, PA, 1990, SIAM.Google Scholar
  20. 20.
    G. Lube, L. Mueller, and H. Mueller, A new non-overlapping domain decomposition method for stabilized finite element methods applied to the nonstationary Navier-Stokes equations, Numer. Lin. Alg. Appl., 7 (2000), pp. 449–472.zbMATHCrossRefGoogle Scholar
  21. 21.
    R. B. Morgan, GMRES with deflated restarting, SIAM J. Sci. Comput., 24 (2002), pp. 20–37.zbMATHCrossRefGoogle Scholar
  22. 22.
    R. Nabben and C. Vuik, A comparison of deflation and coarse grid correction applied to porous media flow, Tech. Rep. R03-10, Delft University of Technology, 2003.Google Scholar
  23. 23.
    F. Nataf, Interface connections in domain decomposition methods, in Modern methods in scientific computing and applications (Montréal, QC, 2001), vol. 75 of NATO Sci. Ser. II Math. Phys. Chem., Kluwer Acad. Publ., Dordrecht, 2002, pp. 323–364.Google Scholar
  24. 24.
    F. Nataf and F. Rogier, Factorization of the convection-diffusion operator and the Schwarz algorithm, M3AS, 5 (1995), pp. 67–93.zbMATHGoogle Scholar
  25. 25.
    F. Nataf, F. Rogier, and E. de Sturler, Optimal interface conditions for domain decomposition methods, Tech. Rep. 301, CMAP (Ecole Polytechnique), 1994.Google Scholar
  26. 26.
    F. Nier, Remarques sur les algorithmes de décomposition de domains, in Seminaire: équations aux Dérivées Partielles, 1998–1999, école Polytech., 1999, pp. Exp. No. IX, 26.Google Scholar
  27. 27.
    A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, 1999.Google Scholar
  28. 28.
    F.-X. Roux, F. Magoulès, S. Salmon, and L. Series, Optimization of interface operator based on algebraic approach, in Fourteenth International Conference on Domain Decomposition Methods, I. Herrera, D. E. Keyes, O. B. Widlund, and R. Yates, eds., ddm.org, 2003.Google Scholar
  29. 29.
    Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869.zbMATHCrossRefGoogle Scholar
  30. 30.
    Y. Saad, M. Yeung, J. Erhel, and F. Guyomarc'h, A deflated version of the conjugate gradient algorithm, SIAM J. Sci. Comput., 21 (2000), pp. 1909–1926. Iterative methods for solving systems of algebraic equations (Copper Mountain, CO, 1998).zbMATHCrossRefGoogle Scholar
  31. 31.
    H. A. Schwarz, über einen Grenzübergang durch alternierendes Verfahren, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15 (1870), pp. 272–286.Google Scholar
  32. 32.
    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
  33. 33.
    P. L. Tallec and M. Vidrascu, Generalized Neumann-Neumann preconditioners for iterative substructuring, in Domain Decomposition Methods in Sciences and Engineering, P. E. Bjørstad, M. Espedal, and D. Keyes, eds., John Wiley & Sons, 1997. Proceedings from the Ninth International Conference, June 1996, Bergen, Norway.Google Scholar
  34. 34.
    K. H. Tan and M. J. A. Borsboom, On generalized Schwarz coupling applied to advection-dominated problems, in Seventh International Conference of Domain Decomposition Methods in Scientific and Engineering Computing, D. E. Keyes and J. Xu, eds., AMS, 1994, pp. 125–130. Held at Penn State University, October 27–30, 1993.Google Scholar
  35. 35.
    A. Toselli and O. Widlund, Domain Decomposition Methods - Algorithms and Theory, vol. 34 of Springer Series in Computational Mathematics, Springer, 2004.Google Scholar
  36. 36.
    F. Willien, I. Faille, F. Nataf, and F. Schneider, Domain decomposition methods for fluid flow in porous medium, in 6th European Conference on the Mathematics of Oil Recovery, September 1998.Google Scholar
  37. 37.
    B. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, vol. 17 of Lecture Notes in Computational Science and Engineering, Springer, 2001.Google Scholar

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© Springer 2007

Authors and Affiliations

  • Frédéric Nataf
    • 1
  1. 1.Laboratoire J.L. Lions, CNRS UMR 7598Université Pierre et Marie CurieParis Cedex 05France

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