Optimized Schwarz Methods for the Time-Harmonic Maxwell Equations with Damping
Abstract
In a previous paper, two of the authors have proposed and analyzed an entire hierarchy of optimized Schwarz methods for Maxwell's equations in both the time-harmonic and the time-domain case. The optimization process has been performed in a particular situation where the electric conductivity was neglected. Here, we take into account this physical parameter which leads to a fundamentally different analysis and a new class of algorithms for this more general case. From the mathematical point of view, the approach is different, since the algorithm does not encounter the pathological situations of the zero-conductivity case and thus the optimization problems are different. We analyze one of the algorithms in this class in detail and provide asymptotic results for the remaining ones. We illustrate our analysis with numerical results.
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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.
I. M. Babuška and S. A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal., 34 (1997), pp. 2392--2423.
3.
P. Chevalier and F. Nataf, An OO2 (Optimized Order 2) method for the Helmholtz and Maxwell equations, in 10th International Conference on Domain Decomposition Methods in Science and in Engineering, AMS, Providence, RI, 1997, pp. 400--407.
4.
P. Collino, G. Delbue, P. Joly, and A. Piacentini, A new interface condition in the non-overlapping domain decomposition for the Maxwell equations, Comput. Methods Appl. Mech. Engrg., 148 (1997), pp. 195--207.
5.
B. Després, Décomposition de domaine et problème de Helmholtz, C.R. Acad. Sci. Paris, Sér. I Math., 311 (1990), pp. 313--316.
6.
B. Després, P. Joly, and J. Roberts, A domain decomposition method for the harmonic Maxwell equations, in Iterative Methods in Linear Algebra, North-Holland, Amsterdam, 1992, pp. 475--484.
7.
V. Dolean, M. El Bouajaji, M. J. Gander, S. Lanteri, and R. Perrussel, Domain decomposition methods for electromagnetic wave propagation problems in heterogeneous media and complex domains, in Domain Decomposition Methods in Science and Engineering XIX, Lect. Notes Comput. Sci. Eng. 78, Springer, Heidelberg, 2011, pp. 15--26.
8.
V. Dolean and M. J. Gander, Can the discretization modify the performance of Schwarz methods?, in Domain Decomposition Methods in Science and Engineering XIX, Lect. Notes Comput. Sci. Eng. 78, Springer, Heidelberg, 2011, pp. 117--124.
9.
V. Dolean, M. J. Gander, and L. Gerardo-Giorda, Optimized Schwarz methods for Maxwells equations, SIAM J. Sci. Comput., 31 (2009), pp. 2193--2213.
10.
V. Dolean, S. Lanteri, and R. Perrussel, A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods, J. Comput. Phys., 227 (2008), pp. 2044--2072.
11.
V. Dolean, S. Lanteri, and R. Perrussel, Optimized Schwarz algorithms for solving time-harmonic Maxwell's equations discretized by a discontinuous Galerkin method, IEEE Trans. Magn., 44 (2008), pp. 954--957.
12.
M. El Bouajaji, V. Dolean, M. J. Gander, and S. Lanteri, Comparison of a one and two parameter family of transmission conditions for Maxwell’'s equations with damping, in Domain Decomposition Methods in Science and Engineering XX, Springer, Heidelberg, to appear.
13.
M. El Bouajaji, S. Lanteri, and M. Yedlin, Discontinuous Galerkin frequency domain forward modelling for the inversion of electric permittivity in the 2D case, Geophys. Prosp., 59 (2011), pp. 920--933.
14.
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.
15.
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.
16.
S.-C. Lee, M. Vouvakis, and J.-F. Lee, A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays, J. Comput. Phys., 203 (2005), pp. 1--21.
17.
Z. Peng and J.-F. Lee, Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics, J. Comput. Phys., 229 (2010), pp. 5615--5629.
18.
Z. Peng, V. Rawat, and J.-F. Lee, One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems, J. Comput. Phys., 229 (2010), pp. 1181--1197.
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Published In
SIAM Journal on Scientific Computing
Pages: A2048 - A2071
DOI: 10.1137/110842995
ISSN (online): 1095-7197
Copyright
© 2012, Society for Industrial and Applied Mathematics.
History
Submitted: 1 August 2011
Accepted: 15 May 2012
Published online: 17 July 2012
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