On the Relation Between Optimized Schwarz Methods and Source Transfer

  • Zhiming Chen
  • Martin J. Gander
  • Hui ZhangEmail author
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


Optimized Schwarz methods (OS) use Robin or higher order transmission conditions instead of the classical Dirichlet ones. An optimal Schwarz method for a general second-order elliptic problem and a decomposition into strips was presented in [13]. Here optimality means that the method converges in a finite number of steps, and this was achieved by replacing in the transmission conditions the higher order operator by the subdomain exterior Dirichlet-to-Neumann (DtN) maps. It is even possible to design an optimal Schwarz method that converges in two steps for an arbitrary decomposition and an arbitrary partial differential equation (PDE), see [6], but such algorithms are not practical, because the operators involved are highly non-local. Substantial research was therefore devoted to approximate these optimal transmission conditions, see for example the early reference [11], or the overview [5] which coined the term “optimized Schwarz method”, and references therein. In particular for the Helmholtz equation, Gander et al. [9] presents optimized second-order approximations of the DtN, Toselli [17] (improperly) and Schädle and Zschiedrich [14] (properly) tried for the first time using perfectly matched layers (PML, see [1]) to approximate the DtN in OS.



This work was supported by the Université de Genève. HZ thanks the International Science and Technology Cooperation Program of China (2010DFA14700).


  1. 1.
    J.-P. Berenger, A perfectly matched layer for absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Z. Chen, X. Xiang, A source transfer domain decomposition method for Helmholtz equations in unbounded domain. SIAM J. Numer. Anal. 51(4), 2331–2356 (2013a)Google Scholar
  3. 3.
    Z. Chen, X. Xiang, A source transfer domain decomposition method for Helmholtz equations in unbounded domain part II: extensions. Numer. Math. Theory Methods Appl. 6(3), 538–555 (2013b)Google Scholar
  4. 4.
    B. Engquist, L. Ying, Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Model. Simul. 9(2), 686–710 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M.J. Gander, Optimized Schwarz methods. SIAM J. Numer. Anal. 44, 699–731 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M.J. Gander, F. Kwok, Optimal interface conditions for an arbitrary decomposition into subdomains, in Domain Decomposition Methods in Science and Engineering XIX, ed. by Y. Huang, R. Kornhuber, O.B. Widlund, J. Xu (Springer, Heidelberg, 2011), pp. 101–108CrossRefGoogle Scholar
  7. 7.
    M.J. Gander, F. Nataf, AILU: a preconditioner based on the analytic factorization of the elliptic operator. Numer. Linear Algebra Appl. 7, 505–526 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M.J. Gander, F. Nataf, An incomplete preconditioner for problems in acoustics. J. Comput. Acoust. 13, 455–476 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M.J. Gander, F. Magoulès, F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24, 38–60 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    C. Geuzaine, A. Vion, Double sweep preconditioner for Schwarz methods applied to the Helmholtz equation, in Domain Decomposition Methods in Science and Engineering XXII (Springer, Heidelberg, 2015)zbMATHGoogle Scholar
  11. 11.
    C. Japhet, Optimized Krylov-Ventcell method. Application to convection-diffusion problems, in Ninth International Conference on Domain Decomposition Methods, ed. by P.E. Bjorstad, M.S. Espedal, D.E. Keyes (, Bergen, 1998)Google Scholar
  12. 12.
    F. Nataf, F. Nier, Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains. Numer. Math. 75, 357–377 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    F. Nataf, F. Rogier, E. de Sturler, Optimal interface conditions for domain decomposition methods. Technical report, Polytechnique (1994)zbMATHGoogle Scholar
  14. 14.
    A. Schädle, L. Zschiedrich, Additive Schwarz method for scattering problems using the PML method at interfaces, in Domain Decomposition Methods in Science and Engineering XVI, ed. by O.B. Widlund, D.E. Keyes (Springer, Heidelberg, 2007), pp. 205–212CrossRefGoogle Scholar
  15. 15.
    A. St-Cyr, M.J. Gander, S.J. Thomas, Optimized multiplicative, additive, and restricted additive Schwarz preconditioning. SIAM J. Sci. Comput. 29, 2402–2425 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    C. Stolk, A rapidly converging domain decomposition method for the Helmholtz equation. J. Comput. Phys. 241, 240–252 (2013)CrossRefGoogle Scholar
  17. 17.
    A. Toselli, Overlapping methods with perfectly matched layers for the solution of the Helmholtz equation, in Eleventh International Conference on Domain Decomposition Methods, ed. by C.-H. Lai, P. Bjorstad, M. Cross, O.B. Widlund (1999), pp. 551–558Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.LSEC, Institute of Computational Mathematics, Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina
  2. 2.Section de MathématiquesUniversité de GenèveGenève 4Switzerland
  3. 3.Department of MathematicsZhejiang Ocean UniversityZhoushanChina

Personalised recommendations