Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
J.-P. Berenger, A perfectly matched layer for absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)
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)
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)
B. Engquist, L. Ying, Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Model. Simul. 9(2), 686–710 (2011)
M.J. Gander, Optimized Schwarz methods. SIAM J. Numer. Anal. 44, 699–731 (2006)
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. 101–108
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)
M.J. Gander, F. Nataf, An incomplete preconditioner for problems in acoustics. J. Comput. Acoust. 13, 455–476 (2005)
M.J. Gander, F. Magoulès, F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24, 38–60 (2002)
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)
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 (ddm.org, Bergen, 1998)
F. Nataf, F. Nier, Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains. Numer. Math. 75, 357–377 (1997)
F. Nataf, F. Rogier, E. de Sturler, Optimal interface conditions for domain decomposition methods. Technical report, Polytechnique (1994)
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. 205–212
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)
C. Stolk, A rapidly converging domain decomposition method for the Helmholtz equation. J. Comput. Phys. 241, 240–252 (2013)
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. 551–558
Acknowledgements
This work was supported by the Université de Genève. HZ thanks the International Science and Technology Cooperation Program of China (2010DFA14700).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Chen, Z., Gander, M.J., Zhang, H. (2016). On the Relation Between Optimized Schwarz Methods and Source Transfer. In: Dickopf, T., Gander, M., Halpern, L., Krause, R., Pavarino, L. (eds) Domain Decomposition Methods in Science and Engineering XXII. Lecture Notes in Computational Science and Engineering, vol 104. Springer, Cham. https://doi.org/10.1007/978-3-319-18827-0_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-18827-0_20
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18826-3
Online ISBN: 978-3-319-18827-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)