Since matrix compression has paved the way for discretizing the boundary integral equation formulations of electromagnetics scattering on very fine meshes, preconditioners for the resulting linear systems have become key to efficient simulations. Operator preconditioning based on Calderón identities has proved to be a powerful device for devising preconditioners. However, this is not possible for the usual first-kind boundary formulations for electromagnetic scattering at general penetrable composite obstacles. We propose a new first-kind boundary integral equation formulation following the reasoning employed in [X. Clayes and R. Hiptmair, Report 2011-45, SAM, ETH Zürich (2011)] for acoustic scattering. We call it multi-trace formulation, because its unknowns are two pairs of traces on interfaces in the interior of the scatterer. We give a comprehensive analysis culminating in a proof of coercivity, and uniqueness and existence of solution. We establish a Calderón identity for the multi-trace formulation, which forms the foundation for operator preconditioning in the case of conforming Galerkin boundary element discretization.
Mots-clés : integral equations, boundary element method, domain decomposition, Maxwell's equations
@article{M2AN_2012__46_6_1421_0,
author = {Claeys, Xavier and Hiptmair, Ralf},
title = {Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1421--1445},
publisher = {EDP-Sciences},
volume = {46},
number = {6},
year = {2012},
doi = {10.1051/m2an/2012011},
mrnumber = {2996334},
zbl = {1277.78032},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2012011/}
}
TY - JOUR AU - Claeys, Xavier AU - Hiptmair, Ralf TI - Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1421 EP - 1445 VL - 46 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2012011/ DO - 10.1051/m2an/2012011 LA - en ID - M2AN_2012__46_6_1421_0 ER -
%0 Journal Article %A Claeys, Xavier %A Hiptmair, Ralf %T Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1421-1445 %V 46 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2012011/ %R 10.1051/m2an/2012011 %G en %F M2AN_2012__46_6_1421_0
Claeys, Xavier; Hiptmair, Ralf. Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1421-1445. doi : 10.1051/m2an/2012011. http://archive.numdam.org/articles/10.1051/m2an/2012011/
[1] , , , , , and , A multiplicative Calderon preconditioner for the electric field integral equation. IEEE Trans. Antennas Propag. 56 (2008) 2398-2412. | MR
[2] , , , and , A Calderón multiplicative preconditioner for the combined field integral equation. IEEE Trans. Antennas Propag. 57 (2009) 3387-3392. | MR
[3] , Numerical analysis of the exterior boundary value problem for time harmonic Maxwell equations by a boundary finite element method. Part 2 : The discrete problem. Math. Comput. 43 (1984) 47-68. | MR | Zbl
[4] , and , A boundary-element solution of the Leontovitch problem. IEEE Trans. Antennas Propag. 47 (1999) 1597-1605. | MR | Zbl
[5] , and , Coupling of a non-overlapping domain decomposition method for a nodal finite element method with a boundary element method. Int. J. Numer. Methods Eng. 73 (2008) 1624-1650. | MR | Zbl
[6] , Remarks on the discretization of some noncoercive operator with applications to the heterogeneous Maxwell equations. SIAM J. Numer. Anal. 43 (2005) 1-18. | MR | Zbl
[7] and , A dual finite element complex on the barycentric refinement. Math. Comput. 76 (2007) 1743-1769. | MR | Zbl
[8] and , On traces for functional spaces related to Maxwell's equations I. An integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24 (2001) 9-30. | MR | Zbl
[9] and , On traces for functional spaces related to Maxwell's equations II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Methods Appl. Sci. 24 (2001) 31-48. | MR | Zbl
[10] and , Galerkin boundary element methods for electromagnetic scattering, in Topics in computational wave propagation. Lect. Notes Comput. Sci. Eng. 31 (2003) 83-124. | MR | Zbl
[11] , and , Boundary element methods for Maxwell's equations on non-smooth domains. Numer. Math. 92 (2002) 679-710. | MR | Zbl
[12] , and , On traces for H(curl,Ω) in Lipschitz domains. J. Math. Anal. Appl. 276 (2002) 845-867. | MR | Zbl
[13] , , and , Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer. Math. 95 (2003) 459-485. | MR | Zbl
[14] and , A surface formulation or characteristic modes of material bodies. IEEE Trans. Antennas Propag. 25 (1977) 789-795.
[15] , Discrete Fredholm properties and convergence estimates for the electric field integral equation. Math. Comput. 73 (2004) 143-167. | MR | Zbl
[16] and , Des préconditionneurs pour la résolution numérique des équations intégrales de frontière de l'acoustique. C. R. Acad. Sci. Paris, Sér. I Math. 330 (2000) 617-622. | Zbl
[17] and , A preconditioner for the electric field integral equation based on Calderón formulas. SIAM J. Numer. Anal. 40 (2002) 1100-1135. | MR | Zbl
[18] , A single trace integral formulation of the second kind for acoustic scattering. Seminar of Applied Mathematics, ETH Zürich, Technical Report 2011-14. Submitted to J. Appl. Math. (2011).
[19] and , Boundary integral formulation of the first kind for acoustic scattering by composite structures. Report 2011-45, SAM, ETH Zürich (2011)
[20] and , Inverse acoustic and electromagnetic scattering theory, 2nd edition. Appl. Math. Sci. 93 (1998). | MR | Zbl
[21] , and , A Calderón, preconditioned PMCHWT equation, in Proc. of the International Conference on Electromagnetics in Advanced Applications, ICEAA'09. Torino, Italy (2009) 521-524.
[22] , Boundary integral operators on Lipschitz domains : elementary results. SIAM J. Math. Anal. 19 (1988) 613-626. | MR | Zbl
[23] , Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237-339. | MR | Zbl
[24] , Coupling of finite elements and boundary elements in electromagnetic scattering. SIAM J. Numer. Anal. 41 (2003) 919-944. | MR | Zbl
[25] , Operator preconditioning. Comput. Math. Appl. 52 (2006) 699-706. | MR | Zbl
[26] and , Multiple traces boundary integral formulation for Helmholtz transmission problems. SAM, ETH Zürich, Report 2010-35 (2010). | Zbl
[27] and , Natural boundary element methods for the electric field integral equation on polyhedra. SIAM J. Numer. Anal. 40 (2002) 66-86. | MR | Zbl
[28] , and , Domain decomposition methods via boundary integral equations, Numerical Analysis VI, Ordinary differential equations and integral equations. J. Comput. Appl. Math. 125 (2000) 521-537. | MR | Zbl
[29] , and , Calderon preconditioned surface integral equations for composite objects with junctions. IEEE Trans. Antennas Propag. 59 (2011) 546-554. | MR
[30] and , Boundary element tearing and interconnecting methods. Computing 71 (2003) 205-228. | MR | Zbl
[31] , Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR | Zbl
[32] and , Computer Techniques for Electromagnetics, in Integral equation solution of three-dimensional scattering problems, Chapter 4, Pergamon, New York (1973) 159-263. | MR
[33] , and , Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans. Antennas Propag. 30 (1986) 409-418.
[34] and , Boundary element methods, Springer Series in Comput. Math. 39 (2011). | MR | Zbl
[35] and , The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math 9 (1998) 191-216. | MR | Zbl
[36] and , Modified combined field integral equations for electromagnetic scattering. SIAM J. Numer. Anal. 47 (2009) 1149-1167. | MR | Zbl
[37] and , Preconditioned electric field integral equation using Calderon identities and dual loop/star basis functions. IEEE Trans. Antennas Propag. 57 (2009) 1274-1279.
[38] , Boundary integral equations for mixed Dirichlet, Neumann and transmission problems. Math. Methods Appl. Sci. 11 (1989) 185-213. | MR | Zbl
[39] , Boundary element tearing and interconnecting methods for acoustic and electromagnetic scattering. Ph.D. thesis, Graz University of Technology (2010).
[40] and , Scattering from arbitrarily-shaped lossy di-electric bodies of revolution. Radio Sci. 12 (1977) 709-718.
[41] , and , A comparative study of Calderón preconditioners for PMCHWT equations. IEEE Trans. Antennas Propag. 58 (2010) 2375-2383. | MR
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