Abstract
We study here optimized Schwarz domain decomposition methods for solving the time-harmonic Maxwell equations discretized by a discontinuous Galerkin (DG) method. Due to the particularity of the latter, a discretization of a more sophisticated Schwarz method is not straightforward. A strategy of discretization is shown in the framework of a DG weak formulation, and the equivalence between multi-domain and single-domain solutions is proved. The proposed discrete framework is then illustrated by some numerical results through the simulation of two-dimensional propagation problems.
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References
Alonso-Rodriguez, A., Gerardo-Giorda, L.: New nonoverlapping domain decomposition methods for the harmonic Maxwell system. SIAM J. Sci. Comput. 28(1), 102–122 (2006)
Arnold, D., Brezzi, F., Cockburn, B., Marini, L.: Unified analysis of Discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)
Chevalier, P., Nataf, F.: 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, Boulder, pp. 400–407. AMS, Providence (1997)
Collino, P., Delbue, G., Joly, P., Piacentini, A.: A new interface condition in the non-overlapping domain decomposition for the Maxwell equations. Comput. Methods Appl. Mech. Eng. 148, 195–207 (1997)
Després, B.: Décomposition de domaine et problème de Helmholtz. C.R. Acad. Sci. Paris 1(6), 313–316 (1990)
Després, B., Joly, P., Roberts, J.: A domain decomposition method for the harmonic Maxwell equations. In: Iterative Methods in Linear Algebra, pp. 475–484. North-Holland, Amsterdam (1992)
Dolean, V., Fol, H., Lanteri, S., Perrussel, R.: Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods. J. Comput. Appl. Math. 218(2), 435–445 (2008)
Dolean, V., Lanteri, S., Perrussel, R.: A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods. J. Comput. Phys. 227(3), 2044–2072 (2008)
Dolean, V., Gerardo-Giorda, L., Gander, M.J.: Optimized Schwarz methods for Maxwell equations. SIAM J. Sci. Comput. 31(3), 2193–2213 (2009)
Dolean, V., Bouajaji, M.E., Gander, M.J., Lanteri, S., Perrussel, R.: Discontinuous Galerkin discretizations of Optimized Schwarz methods for solving the time-harmonic Maxwell equations (2014, in preparation)
El Bouajaji, M., Dolean, V., Gander, M.J., Lanteri, S.: Optimized Schwarz methods for the time-harmonic Maxwell equations with damping. SIAM J. Sci. Comput. 34(4), 2048–2071 (2012)
Gander, M.J., Magoulès, F., Nataf, F.: Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24(1), 38–60 (2002)
Hesthaven, J., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications. Springer, Berlin (2008)
Lee, S.C., Vouvakis, M., Lee, J.F.: A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays. J. Comput. Phys. 203(1), 1–21 (2005)
Peng, Z., Lee, J.F.: Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics. J. Comput. Phys. 229(16), 5615–5629 (2010)
Peng, Z., Rawat, V., Lee, J.F.: One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems. J. Comput. Phys. 229(4), 1181–1197 (2010)
Rawat, V., Lee, J.F.: Nonoverlapping domain decomposition with second order transmission condition for the time-harmonic Maxwell’s equations. SIAM J. Sci. Comput. 32(6), 3584–3603 (2010)
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Bouajaji, M.E., Dolean, V., Gander, M.J., Lanteri, S., Perrussel, R. (2014). DG Discretization of Optimized Schwarz Methods for Maxwell’s Equations. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_18
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DOI: https://doi.org/10.1007/978-3-319-05789-7_18
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