$T$-coercivity for scalar interface problems between dielectrics and metamaterials
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, p. 1363-1387

Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of Rd, with d = 2,3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive + compact) framework. For that, we build some criteria, based on the geometry of the interface between the dielectric and the metamaterial. The proofs combine simple geometrical arguments with the approach of T-coercivity, introduced by the first and third authors and co-worker. Furthermore, the use of localization techniques allows us to derive well-posedness under conditions that involve the knowledge of the coefficients only near the interface. When the coefficients are piecewise constant, we establish the optimality of the criteria.

DOI : https://doi.org/10.1051/m2an/2012006
Classification:  35Q60,  35Q61,  35J20
Keywords: metamaterials, interface problem, T-coercivity
@article{M2AN_2012__46_6_1363_0,
author = {Dhia, Anne-Sophie Bonnet-Ben and Chesnel, Lucas and Ciarlet, Patrick},
title = {$T$-coercivity for scalar interface problems between dielectrics and metamaterials},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {46},
number = {6},
year = {2012},
pages = {1363-1387},
doi = {10.1051/m2an/2012006},
zbl = {1276.78008},
language = {en},
url = {http://www.numdam.org/item/M2AN_2012__46_6_1363_0}
}

Dhia, Anne-Sophie Bonnet-Ben; Chesnel, Lucas; Ciarlet, Patrick. $T$-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, pp. 1363-1387. doi : 10.1051/m2an/2012006. http://www.numdam.org/item/M2AN_2012__46_6_1363_0/

[1] A.-S. Bonnet-Ben Dhia, L. Chesnel and X. Claeys, Radiation condition for a non-smooth interface between a dielectric and a metamaterial [hal-00651008]. | Zbl 1283.35135

[2] A.-S. Bonnet-Ben Dhia, P. Ciarlet Jr. and C.M. Zwölf, A new compactness result for electromagnetic waves. Application to the transmission problem between dielectrics and metamaterials. Math. Models Methods Appl. Sci. 18 (2008) 1605-1631. | MR 2446403 | Zbl 1173.35119

[3] A.-S. Bonnet-Ben Dhia, P. Ciarlet Jr. and C.M. Zwölf, Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234 (2010) 1912-1919; Corrigendum J. Comput. Appl. Math. 234 (2010) 2616. | MR 2644187 | Zbl 1202.78026

[4] A.-S. Bonnet-Ben Dhia, M. Dauge and K. Ramdani, Analyse spectrale et singularités d'un problème de transmission non coercif. C.R. Acad. Sci. Paris, Ser. I 328 (1999) 717-720. | MR 1680769 | Zbl 0932.35153

[5] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002

[6] L. Chesnel and P. Ciarlet Jr., Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials. SIAM J. Math. Anal. 43 (2011) 2150-2169. | MR 2837498 | Zbl 1250.78007

[7] X. Claeys, Analyse asymptotique et numérique de la diffraction d'ondes par des fils minces. Ph.D. thesis, Université Versailles - Saint-Quentin (2008) (in French).

[8] M. Costabel and E. Stephan, A direct boundary integral method for transmission problems. J. Math. Anal. Appl. 106 (1985) 367-413. | MR 782799 | Zbl 0597.35021

[9] M. Dauge and B. Texier, Problèmes de transmission non coercifs dans des polygones. Technical Report 97-27, Université de Rennes 1, IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France (1997) http://hal.archives-ouvertes.fr/docs/00/56/23/29/PDF/BenjaminT˙arxiv.pdf (in French).

[10] L.D. Evans, Partial Differential Equations, Graduate studies in mathematics. Americain Mathematical Society 19 (1998). | Zbl 0902.35002

[11] P. Fernandes and M. Raffetto, Well posedness and finite element approximability of time-harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials. Math. Models Methods Appl. Sci. 19 (2009) 2299-2335. | MR 2599662 | Zbl 1205.78058

[12] V.A. Kozlov, V.G. Maz'Ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs. Americain Mathematical Society 52 (1997). | MR 1469972 | Zbl 0947.35004

[13] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1968). | Zbl 0165.10801

[14] W. Mclean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR 1742312 | Zbl 0948.35001

[15] S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, Expositions in Mathematics. De Gruyter 13 (1994). | MR 1283387 | Zbl 0806.35001

[16] S. Nicaise and A.M. Sändig, General interface problems-I. Math. Meth. Appl. Sci. 17 (1994) 395-429. | MR 1274152 | Zbl 0824.35014

[17] S. Nicaise and A.M. Sändig, General interface problems-II. Math. Meth. Appl. Sci. 17 (1994) 431-450. | MR 1257586 | Zbl 0824.35015

[18] S. Nicaise and J. Venel, A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients. J. Comput. Appl. Math. 235 (2011) 4272-4282. | MR 2801447 | Zbl 1220.65167

[19] G. Oliveri and M. Raffetto, A warning about metamaterials for users of frequency-domain numerical simulators. IEEE Trans. Antennas Propag. 56 (2008) 792-798. | MR 2424396

[20] J. Peetre, Another approach to elliptic boundary problems. Commun. Pure Appl. Math. 14 (1961) 711-731. | MR 171069 | Zbl 0104.07303

[21] M. Raffetto, Ill-posed waveguide discontinuity problem involving metamaterials with impedance boundary conditions on the two ports. IET Sci. Meas. Technol. 1 (2007) 232-239.

[22] K. Ramdani, Lignes supraconductrices : analyse mathématique et numérique. Ph.D. thesis, Université Paris 6 (1999) (in French).

[23] A.A. Sukhorukov, I.V. Shadrivov and Y.S. Kivshar, Wave scattering by metamaterial wedges and interfaces. Int. J. Numer. Model. 19 (2006) 105-117. | Zbl 1087.78004

[24] H. Wallén, H. Kettunen and A. Sihvola, Surface modes of negative-parameter interfaces and the importance of rounding sharp corners. Metamaterials 2 (2008) 113-121.

[25] J. Wloka, Partial Differ. Equ. Cambridge Univ. Press (1987).

[26] C.M. Zwölf, Méthodes variationnelles pour la modélisation des problèmes de transmission d'onde électromagnétique entre diélectrique et méta-matériau. Ph.D. thesis, Université Versailles, Saint-Quentin (2008) (in French).