# On the Relation Between Optimized Schwarz Methods and Source Transfer

## 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.

## Notes

### Acknowledgements

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

## References

- 1.J.-P. Berenger, A perfectly matched layer for absorption of electromagnetic waves. J. Comput. Phys.
**114**, 185–200 (1994)MathSciNetCrossRefMATHGoogle Scholar - 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.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.B. Engquist, L. Ying, Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Model. Simul.
**9**(2), 686–710 (2011)MathSciNetCrossRefMATHGoogle Scholar - 5.M.J. Gander, Optimized Schwarz methods. SIAM J. Numer. Anal.
**44**, 699–731 (2006)MathSciNetCrossRefMATHGoogle Scholar - 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.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)MathSciNetCrossRefMATHGoogle Scholar - 8.M.J. Gander, F. Nataf, An incomplete preconditioner for problems in acoustics. J. Comput. Acoust.
**13**, 455–476 (2005)MathSciNetCrossRefMATHGoogle Scholar - 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)MathSciNetCrossRefMATHGoogle Scholar - 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)MATHGoogle Scholar - 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 (ddm.org, Bergen, 1998)Google Scholar - 12.F. Nataf, F. Nier, Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains. Numer. Math.
**75**, 357–377 (1997)MathSciNetCrossRefMATHGoogle Scholar - 13.F. Nataf, F. Rogier, E. de Sturler, Optimal interface conditions for domain decomposition methods. Technical report, Polytechnique (1994)MATHGoogle Scholar
- 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.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)MathSciNetCrossRefMATHGoogle Scholar - 16.C. Stolk, A rapidly converging domain decomposition method for the Helmholtz equation. J. Comput. Phys.
**241**, 240–252 (2013)CrossRefGoogle Scholar - 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