A volume integral equation method for periodic scattering problems for anisotropic Maxwell’s equations. (English) Zbl 1329.65318
Summary: This paper presents a volume integral equation method for an electromagnetic scattering problem for three-dimensional Maxwell’s equations in the presence of a biperiodic, anisotropic, and possibly discontinuous dielectric scatterer. Such scattering problem can be reformulated as a strongly singular volume integral equation (i.e., integral operators that fail to be weakly singular). In this paper, we firstly prove that the strongly singular volume integral equation satisfies a Gårding-type estimate in standard Sobolev spaces. Secondly, we rigorously analyze a spectral Galerkin method for solving the scattering problem. This method relies on the periodization technique of Gennadi Vainikko that allows us to efficiently evaluate the periodized integral operators on trigonometric polynomials using the fast Fourier transform (FFT). The main advantage of the method is its simple implementation that avoids for instance the need to compute quasiperiodic Green’s functions. We prove that the numerical solution of the spectral Galerkin method applied to the periodized integral equation converges quasioptimally to the solution of the scattering problem. Some numerical examples are provided for examining the performance of the method.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45A05 | Linear integral equations |
| 35P25 | Scattering theory for PDEs |
| 35Q61 | Maxwell equations |
| 78A45 | Diffraction, scattering |
| 65T50 | Numerical methods for discrete and fast Fourier transforms |
Keywords:
anisotropic Maxwell’s equations; volume integral equations; periodic structures; electromagnetic scattering; rough coefficientsReferences:
| [1] | Abboud, T., Electromagnetic waves in periodic media, (Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (1993), SIAM: SIAM Philadelphia, Newark, DE), 1-9 · Zbl 0815.35106 |
| [2] | Arens, T., Scattering by biperiodic layered media: the integral equation approach (2010), Universität Karlsruhe, Habilitation thesis |
| [3] | Bao, G.; Cowsar, L.; Masters, W., Mathematical Modeling in Optical Science, SIAM Frontiers in Appl. Math. (2001), SIAM: SIAM Philadelphia · Zbl 0964.00050 |
| [4] | Bojarski, N., The k-space formulation of the scattering problem in the time domain, J. Acoust. Soc. Am., 72, 570 (1982) · Zbl 0517.73025 |
| [5] | Bonnet-Bendhia, A.-S.; Starling, F., Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem, Math. Methods Appl. Sci., 17, 305-338 (1994) · Zbl 0817.35109 |
| [6] | Botha, M. M., Solving the volume integral equations of electromagnetic scattering, J. Comput. Phys., 218, 141-158 (2006) · Zbl 1111.78012 |
| [7] | Bruno, O.; Hyde, E. M., Higher-order Fourier approximation in scattering by two-dimensional, inhomogeneous media, SIAM J. Numer. Anal., 42, 2298-2319 (2005) · Zbl 1128.65106 |
| [8] | Colton, D. L.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory (1992), Springer · Zbl 0760.35053 |
| [9] | Costabel, M.; Darrigrand, E.; Koné, E., Volume and surface integral equations for electromagnetic scattering by a dielectric body, J. Comput. Appl. Math., 234, 1817-1825 (2010) · Zbl 1192.78018 |
| [10] | Dobson, D. C., A variational method for electromagnetic diffraction in biperiodic structures, Math. Model. Numer. Anal., 28, 419-439 (1994) · Zbl 0820.65087 |
| [11] | Ewe, W.-B.; Chu, H.-S.; Li, E.-P., Volume integral equation analysis of surface plasmon resonance of nanoparticles, Opt. Express, 15, 18200-18208 (2007) |
| [12] | Girault, V.; Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations (1986), Springer: Springer Berlin · Zbl 0585.65077 |
| [13] | Greengard, L.; Ho, K. L.; Lee, J.-Y., A fast direct solver for scattering from periodic structures with multiple materials interfaces in two dimensions, J. Comput. Phys., 258, 738-751 (2014) · Zbl 1349.74353 |
| [14] | Hohage, T., On the numerical solution of a three-dimensional inverse medium scattering problem, Inverse Probl., 17, 1743-1763 (2001) · Zbl 0989.35141 |
| [15] | Hohage, T., Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem, J. Comp. Phys., 214, 224-238 (2006) · Zbl 1099.78003 |
| [16] | Hyde, E. M.; Bruno, O., A fast, higher-order solver for scattering by penetrable bodies in three dimensions, J. Comput. Phys., 202, 236-261 (2005) · Zbl 1063.65132 |
| [17] | Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, vol. 16 (1995), SIAM · Zbl 0832.65046 |
| [18] | Kirsch, A., Diffraction by periodic structures, (Pävarinta, L.; Somersalo, E., Proc. Lapland Conf. on Inverse Problems (1993), Springer), 87-102 · Zbl 0924.35013 |
| [19] | Kirsch, A., An integral equation approach and the interior transmission problem for Maxwell’s equations, Inverse Probl. Imaging, 1, 159-180 (2007) · Zbl 1129.35080 |
| [20] | Kirsch, A., An integral equation for Maxwell’s equations in a layered medium with an application to the Factorization method, J. Integral Equ. Appl., 19, 333-358 (2007) · Zbl 1136.78310 |
| [21] | Kirsch, A.; Lechleiter, A., The operator equations of Lippmann-Schwinger type for acoustic and electromagnetic scattering problems in \(L^2\), Appl. Anal., 88, 6, 807-830 (2009) · Zbl 1179.78053 |
| [22] | Kottmann, J.; Martin, O., Accurate solution of the volume integral equation for high-permittivity scatterers, IEEE Trans. Antennas Propag., 48, 11, 1719-1726 (2000) |
| [23] | Lechleiter, A.; Nguyen, D.-L., Spectral volumetric integral equation methods for acoustic medium scattering in a 3D waveguide, IMA J. Numer. Math., 32, 3, 813-844 (2012) · Zbl 1328.76066 |
| [24] | Lechleiter, A.; Nguyen, D.-L., On uniqueness in electromagnetic scattering from biperiodic structures, ESAIM, Math. Model. Numer. Anal., 47, 1167-1184 (2013) · Zbl 1282.78022 |
| [25] | Lechleiter, A.; Nguyen, D.-L., Volume integral equations for scattering from anisotropic diffraction gratings, Math. Methods Appl. Sci., 36, 262-274 (2013) · Zbl 1267.78027 |
| [26] | Lechleiter, A.; Nguyen, D.-L., A trigonometric Galerkin method for volume integral equations arising in TM grating scattering, Adv. Comput. Math., 40, 1-25 (2014) · Zbl 1304.78010 |
| [27] | Linton, C. M., The Green’s function for the two-dimensional Helmholtz equation in periodic domains, J. Eng. Math., 33, 377-402 (1998) · Zbl 0922.76274 |
| [28] | Millard, X.; Liu, Q. H., A fast volume integral equation solver for electromagnetic scattering from large inhomogeneous objects in planarly layered media, IEEE Trans. Antennas Propag., 51, 2393-2401 (2003) · Zbl 1368.78188 |
| [29] | Monk, P., Finite Element Methods for Maxwell’s Equations (2003), Oxford Science Publications: Oxford Science Publications Oxford · Zbl 1024.78009 |
| [30] | Otani, Y.; Nishimura, N., A periodic FMM for Maxwell’s equations in 3d and its applications to problems related to photonic crystals, J. Comput. Phys., 227, 4630-4652 (2008) · Zbl 1206.78084 |
| [31] | (Petit, R., Electromagnetic Theory of Gratings (1980), Springer) |
| [32] | Pinsky, M.; Stanton, N.; Trapa, P., Fourier series of radial functions in several variables, J. Funct. Anal., 116, 111-132 (1993) · Zbl 0796.42010 |
| [33] | Potthast, R., Electromagnetic scattering from an orthotropic medium, J. Integral Equ. Appl., 11, 179-215 (1999) · Zbl 0980.78004 |
| [34] | Richmond, J., TE-wave scattering by a dielectric cylinder of arbitrary cross-section shape, IEEE Trans. Antennas Propag., 14, 4, 460-464 (1966) |
| [35] | Sandfort, K., The factorization method for inverse scattering from periodic inhomogeneous media (2010), Karlsruher Institut für Technologie, Ph.D. thesis |
| [36] | Saranen, J.; Vainikko, G., Periodic Integral and Pseudodifferential Equations with Numerical Approximation (2002), Springer · Zbl 0991.65125 |
| [37] | Sauter, S.; Schwab, C., Boundary Element Methods (2007), Springer |
| [38] | Schmidt, G., On the diffraction by biperiodic anisotropic structures, Appl. Anal., 82, 75-92 (2003) · Zbl 1037.35086 |
| [39] | Vainikko, G., Fast solvers of the Lippmann-Schwinger equation, (Newark, D., Direct and Inverse Problems of Mathematical Physics. Direct and Inverse Problems of Mathematical Physics, Int. Soc. Anal. Appl. Comput., vol. 5 (2000), Kluwer: Kluwer Dordrecht), 423 · Zbl 0962.65097 |
| [40] | Wang, D.; Yung, E.; Chen, R.; Lau, P.-Y., An efficient volume integral equation solution to EM scattering by complex bodies with inhomogeneous bi-isotropy, IEEE Trans. Antennas Propag., 55, 1970-1980 (2007) · Zbl 1369.78324 |
| [41] | Zwamborn, P.; van den Berg, P., The three dimensional weak form of the conjugate gradient FFT method for solving scattering problems, IEEE Trans. Microw. Theory Tech., 40, 9, 1757-1766 (1992) |
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