Abstract

Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. More recently, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions, and optimized Schwarz methods can be used both with and without overlap for elliptic problems. We show here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell's equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation. Based on this insight, we develop an entire new hierarchy of optimized overlapping and nonoverlapping Schwarz methods for Maxwell's equations with greatly enhanced performance compared to the classical Schwarz method. We also derive for each algorithm asymptotic formulas for the optimized transmission conditions, which can easily be used in implementations of the algorithms for problems with variable coefficients. We illustrate our findings with numerical experiments.

MSC codes

  1. 65M55
  2. 65F10
  3. 65N22

Keywords

  1. Schwarz algorithms
  2. optimized transmission conditions
  3. Maxwell's equations

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
A. Alonso Rodriguez and L. Gerardo-Giorda, New nonoverlapping domain decomposition methods for the harmonic Maxwell system, SIAM J. Sci. Comput., 28 (2006), pp. 102–122.
2.
S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Edquations: First-order Systems and Applications, Oxford Math. Monogr., Oxford Science Publications, New York, 2007.
3.
T. F. Chan and T. P. Mathew, Domain decomposition algorithms, in Acta Numerica 1994, Vol. 3, Cambridge University Press, New York, 1994, pp. 61–143.
4.
P. Charton, F. Nataf, and F. Rogier, Méthode de décomposition de domaine pour l'équation d'advection-diffusion, C. R. Acad. Sci. Paris Ser. I, 313 (1991), pp. 623–626.
5.
P. Chevalier, Méthodes Numériques Pour les Tubes Hyperfréquences. Résolution par Décomposition de Domaine, Ph.D. thesis, Université Paris VI, Paris, 1998.
6.
P. Chevalier and F. Nataf, Symmetrized method with optimized second-order conditions for the Helmholtz equation, in Domain Decomposition Methods 10, The Tenth International Conference on Domain Decomposition Methods (Boulder, CO, 1997), Amer. Math. Soc., Providence, RI, 1998, pp. 400–407.
7.
S. Clerc, Non-overlapping Schwarz method for systems of first order equations, Contemp. Math., 218 (1998), pp. 408–416.
8.
P. Collino, G. Delbue, P. Joly, and A. Piacentini, A new interface condition in the non-overlapping domain decomposition, Comput. Methods Appl. Mech. Engrg., 148 (1997), pp. 195–207.
9.
Q. Deng, Timely communication: An analysis for a nonoverlapping domain decomposition iterative procedure, SIAM J. Sci. Comput., 18 (1997), pp. 1517–1525.
10.
B. Després, Décomposition de domaine et problème de Helmholtz, C. R. Acad. Sci. Paris, 1 (1990), pp. 313–316.
11.
B. Després, Domain decomposition method and the Helmholtz problem. II, in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993), SIAM, Philadelphia, PA, 1993, pp. 197–206.
12.
B. Després, P. Joly, and J. E. Roberts, A domain decomposition method for the harmonic Maxwell equations, in Iterative Methods in Linear Algebra: Proceedings (Brussels, 1991), North–Holland, Amsterdam, 1992, pp. 475–484.
13.
V. Dolean and M. J. Gander, Why classical Schwarz methods applied to hyperbolic systems converge even without overlap, in Domain Decomposition Methods in Science and Engineering XVII, Seventeenth International Conference on Domain Decomposition Methods, Lect. Notes Comput. Sci. Eng. 60, Springer, Berlin, 2007, pp. 467–476.
14.
V. Dolean, S. Lanteri, and F. Nataf, Construction of interface conditions for solving compressible Euler equations by non-overlapping domain decomposition methods, Internat. J. Numer. Methods Fluids, 40 (2002), pp. 1485–1492.
15.
V. Dolean, S. Lanteri, and F. Nataf, Convergence analysis of a Schwarz type domain decomposition method for the solution of the Euler equations, Appl. Numer. Math., 49 (2004), pp. 153–186.
16.
V. Dolean, G. Rapin, and F. Nataf, Deriving a new domain decomposition method for the Stokes equations using the Smith factorization, Math. Comp., 78 (2009), pp. 789–814.
17.
B. Engquist and H.-K. Zhao, Absorbing boundary conditions for domain decomposition, Appl. Numer. Math., 27 (1998), pp. 341–365.
18.
E. Faccioli, F. Maggio, A. Quarteroni, and A. Tagliani, Spectral domain decomposition methods for the solution of acoustic and elastic wave propagation, Geophys., 61 (1996), pp. 1160–1174.
19.
E. Faccioli, F. Maggio, A. Quarteroni, and A. Tagliani, 2d and 3d elastic wave propagation by pseudo-spectral domain decomposition method, J. Seismology, 1 (1997), pp. 237–251.
20.
M. J. Gander, Optimized Schwarz methods, SIAM J. Numer. Anal., 44 (2006), pp. 699–731.
21.
M. J. Gander and L. Halpern, Méthodes de relaxation d'ondes pour l'équation de la chaleur en dimension 1, C. R. Acad. Sci. Paris Sér. I, 336 (2003), pp. 519–524.
22.
M. J. Gander, L. Halpern, and F. Magoulès, An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation, Internat. J. Numer. Methods Fluids, 55 (2007), pp. 163–175.
23.
M. J. Gander, L. Halpern, and F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation, SIAM J. Numer. Anal., 41 (2003), pp. 1643–1681.
24.
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.
25.
T. Hagstrom and S. Lau, Radiation boundary conditions for Maxwell's equations: A review of accurate time-domain formulations, J. Comput. Math, 25 (2007), pp. 305–336.
26.
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), pp. 299–302.
27.
C. Japhet, F. Nataf, and F. Rogier, The optimized order 2 method. Application to convection-diffusion problems, Future Generation Comput. Systems, 18 (2001), pp. 17–30.
28.
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, Houston, TX, 1989, T. F. Chan, R. Glowinski, J. Périaux, and O. Widlund, eds., SIAM, Philadelphia, PA, 1990, pp. 202–223.
29.
F. Nataf and F. Rogier, Factorization of the convection-diffusion operator and the Schwarz algorithm, Math. Models Methods Appl. Sci., 5 (1995), pp. 67–93.
30.
J.-C. Nedelec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Appl. Math. Sci. 144, Springer, New York, 2001.
31.
A. Quarteroni, Domain decomposition methods for systems of conservation laws: Spectral collocation approximations, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 1029–1052.
32.
A. Quarteroni and L. Stolcis, Homogeneous and heterogeneous domain decomposition methods for compressible fluid flows at high Reynolds numbers, CFD Rev., II (1998), pp. 1064–1078.
33.
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, New York, 1999.
34.
B. F. Smith, P. E. Bjørstad, and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, 1996.
35.
I. Sofronov, Nonreflecting inflow and outflow in a wind tunnel for transonic time-accurate simulation, J. Math. Anal. Appl., 221 (1998), pp. 92–115.
36.
H. Sun and W.-P. Tang, An overdetermined Schwarz alternating method, SIAM J. Sci. Comput., 17 (1996), pp. 884–905.
37.
W. P. Tang, Generalized Schwarz splittings, SIAM J. Sci. Stat. Comp., 13 (1992), pp. 573–595.
38.
A. Toselli, Overlapping Schwarz methods for Maxwell's equations in three dimensions, Numer. Math., 86 (2000), pp. 733–752.
39.
A. Toselli and O. Widlund, Domain Decomposition Methods - Algorithms and Theory, Springer Ser. Comput. Math. 34, Springer, New York, 2004.
40.
J. Wloka, B. Rowley, and B. Lawruk, Boundary Value Problems for Elliptic Systems, Cambridge University Press, New York, 1995.
41.
J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), pp. 581–613.
42.
J. Xu and J. Zou, Some nonoverlapping domain decomposition methods, SIAM Rev., 40 (1998), pp. 857–914.

Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 2193 - 2213
ISSN (online): 1095-7197

History

Submitted: 25 June 2008
Accepted: 5 December 2008
Published online: 7 May 2009

MSC codes

  1. 65M55
  2. 65F10
  3. 65N22

Keywords

  1. Schwarz algorithms
  2. optimized transmission conditions
  3. Maxwell's equations

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.