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Electromagnetic scattering at composite objects : a novelmulti-trace boundary integral formulation

Published online by Cambridge University Press:  31 May 2012

Xavier Claeys  and
Ralf Hiptmair
Xavier Claeys
Affiliation:
Universitéde Toulouse, ISAE, 10 Avenue Edouard-Belin, 31055 Toulouse, France. [email protected]
Ralf Hiptmair
Affiliation:
Seminar of Applied Mathematics, ETHZ, 8092 Zürich, Switzerland; [email protected]
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Abstract

Since matrix compression has paved the way for discretizing the boundary integralequation formulations of electromagnetics scattering on very fine meshes, preconditionersfor the resulting linear systems have become key to efficient simulations. Operatorpreconditioning based on Calderón identities has proved to be a powerful device fordevising preconditioners. However, this is not possible for the usual first-kind boundaryformulations for electromagnetic scattering at general penetrable composite obstacles. Wepropose a new first-kind boundary integral equation formulation following the reasoningemployed in [X. Clayes and R. Hiptmair, Report 2011-45, SAM, ETH Zürich (2011)] foracoustic scattering. We call it multi-trace formulation, because itsunknowns are two pairs of traces on interfaces in the interior of the scatterer. We give acomprehensive analysis culminating in a proof of coercivity, and uniqueness and existenceof solution. We establish a Calderón identity for the multi-trace formulation, which formsthe foundation for operator preconditioning in the case of conforming Galerkin boundaryelement discretization.

Keywords

Integral equationsboundary element methoddomain decompositionMaxwell’s equations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1421 - 1445
Copyright
© EDP Sciences, SMAI, 2012

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References

Andriulli, F.P., Cools, K., Bagci, H., Olyslager, F., Buffa, A., Christiansen, S. and Michielssen, E., A multiplicative Calderon preconditioner for the electric field integral equation. IEEE Trans. Antennas Propag. 56 (2008) 2398–2412. Google Scholar
Bagci, H., Andriulli, F.P., Coolst, K., Olyslager, F. and Michielssen, E., A Calderón multiplicative preconditioner for the combined field integral equation. IEEE Trans. Antennas Propag. 57 (2009) 3387–3392. Google Scholar
Bendali, A., 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. Google Scholar
Bendali, A., Fares, M.B. and Gay, J., A boundary-element solution of the Leontovitch problem. IEEE Trans. Antennas Propag. 47 (1999) 1597–1605. Google Scholar
Boubendir, Y., Bendali, A. and Fares, M.B., 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. Google Scholar
Buffa, A., Remarks on the discretization of some noncoercive operator with applications to the heterogeneous Maxwell equations. SIAM J. Numer. Anal. 43 (2005) 1–18. Google Scholar
Buffa, A. and Christiansen, S.H., A dual finite element complex on the barycentric refinement. Math. Comput. 76 (2007) 1743–1769. Google Scholar
Buffa, A. and Ciarlet, P. Jr., 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. Google Scholar
Buffa, A. and Ciarlet, P. Jr., 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. Google Scholar
Buffa, A. and Hiptmair, R., Galerkin boundary element methods for electromagnetic scattering, in Topics in computational wave propagation. Lect. Notes Comput. Sci. Eng. 31 (2003) 83–124. Google Scholar
Buffa, A., Costabel, M. and Schwab, C., Boundary element methods for Maxwell’s equations on non-smooth domains. Numer. Math. 92 (2002) 679–710. Google Scholar
Buffa, A., Costabel, M. and Sheen, D., On traces for H(curl,Ω) in Lipschitz domains. J. Math. Anal. Appl. 276 (2002) 845–867. Google Scholar
Buffa, A., Hiptmair, R., von Petersdorff, T. and Schwab, C., Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer. Math. 95 (2003) 459–485. Google Scholar
Chang, Y. and Harrington, R., A surface formulation or characteristic modes of material bodies. IEEE Trans. Antennas Propag. 25 (1977) 789–795. Google Scholar
Christiansen, S.H., Discrete Fredholm properties and convergence estimates for the electric field integral equation. Math. Comput. 73 (2004) 143–167. Google Scholar
Christiansen, S.H. and Nédélec, J.-C., 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. Google Scholar
Christiansen, S.H. and Nédélec, J.-C., A preconditioner for the electric field integral equation based on Calderón formulas. SIAM J. Numer. Anal. 40 (2002) 1100–1135. Google Scholar
X. Claeys, 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).
X. Claeys and R. Hiptmair, Boundary integral formulation of the first kind for acoustic scattering by composite structures. Report 2011-45, SAM, ETH Zürich (2011)
D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, 2nd edition. Appl. Math. Sci. 93 (1998).
K. Coolst, F.P. Andriulli and F. Olyslager, A Calderón, preconditioned PMCHWT equation, in Proc. of the International Conference on Electromagnetics in Advanced Applications, ICEAA’09. Torino, Italy (2009) 521–524.
Costabel, M., Boundary integral operators on Lipschitz domains : elementary results. SIAM J. Math. Anal. 19 (1988) 613–626. Google Scholar
Hiptmair, R., Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237–339. Google Scholar
Hiptmair, R., Coupling of finite elements and boundary elements in electromagnetic scattering. SIAM J. Numer. Anal. 41 (2003) 919–944. Google Scholar
Hiptmair, R., Operator preconditioning. Comput. Math. Appl. 52 (2006) 699–706. Google Scholar
R. Hiptmair and C. Jerez-Hanckes, Multiple traces boundary integral formulation for Helmholtz transmission problems. SAM, ETH Zürich, Report 2010-35 (2010).
Hiptmair, R. and Schwab, C., Natural boundary element methods for the electric field integral equation on polyhedra. SIAM J. Numer. Anal. 40 (2002) 66–86. Google Scholar
Hsiao, G.C., Steinbach, O. and Wendland, W.L., Domain decomposition methods via boundary integral equations, Numerical Analysis VI, Ordinary differential equations and integral equations. J. Comput. Appl. Math. 125 (2000) 521–537. Google Scholar
Jäärvenpä, S., Kiminki, S.P. and Ylä-Oijala, P., Calderon preconditioned surface integral equations for composite objects with junctions. IEEE Trans. Antennas Propag. 59 (2011) 546–554. Google Scholar
Langer, U. and Steinbach, O., Boundary element tearing and interconnecting methods. Computing 71 (2003) 205–228. Google Scholar
W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000).
E. Miller and A. Poggio, Computer Techniques for Electromagnetics, in Integral equation solution of three-dimensional scattering problems, Chapter 4, Pergamon, New York (1973) 159–263.
Rao, S.M., Wilton, D.R. and Glisson, A.W., Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans. Antennas Propag. 30 (1986) 409–418. Google Scholar
S.A. Sauter and C. Schwab, Boundary element methods, Springer Series in Comput. Math. 39 (2011).
Steinbach, O. and Wendland, W.L., The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math 9 (1998) 191–216. Google Scholar
Steinbach, O. and Windisch, M., Modified combined field integral equations for electromagnetic scattering. SIAM J. Numer. Anal. 47 (2009) 1149–1167. Google Scholar
Stephanson, M.B. and Lee, J.-F., Preconditioned electric field integral equation using Calderon identities and dual loop/star basis functions. IEEE Trans. Antennas Propag. 57 (2009) 1274–1279. Google Scholar
von Petersdorff, T., Boundary integral equations for mixed Dirichlet, Neumann and transmission problems. Math. Methods Appl. Sci. 11 (1989) 185–213. Google Scholar
M. Windisch, Boundary element tearing and interconnecting methods for acoustic and electromagnetic scattering. Ph.D. thesis, Graz University of Technology (2010).
Wu, T.-K. and Tsai, L.-L., Scattering from arbitrarily-shaped lossy di-electric bodies of revolution. Radio Sci. 12 (1977) 709–718. Google Scholar
Yan, S., Jin, J.-M. and Nie, Z.-P., A comparative study of Calderón preconditioners for PMCHWT equations. IEEE Trans. Antennas Propag. 58 (2010) 2375–2383. Google Scholar