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.
Mots clés : 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 }, pages = {1363--1387}, publisher = {EDP-Sciences}, volume = {46}, number = {6}, year = {2012}, doi = {10.1051/m2an/2012006}, zbl = {1276.78008}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012006/} }
TY - JOUR AU - Dhia, Anne-Sophie Bonnet-Ben AU - Chesnel, Lucas AU - Ciarlet, Patrick TI - $T$-coercivity for scalar interface problems between dielectrics and metamaterials JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1363 EP - 1387 VL - 46 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012006/ DO - 10.1051/m2an/2012006 LA - en ID - M2AN_2012__46_6_1363_0 ER -
%0 Journal Article %A Dhia, Anne-Sophie Bonnet-Ben %A Chesnel, Lucas %A Ciarlet, Patrick %T $T$-coercivity for scalar interface problems between dielectrics and metamaterials %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1363-1387 %V 46 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012006/ %R 10.1051/m2an/2012006 %G en %F 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 , Tome 46 (2012) no. 6, pp. 1363-1387. doi : 10.1051/m2an/2012006. http://www.numdam.org/articles/10.1051/m2an/2012006/
[1] Radiation condition for a non-smooth interface between a dielectric and a metamaterial [hal-00651008]. | Zbl
, and ,[2] 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 | Zbl
, and ,[3] 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 | Zbl
, and ,[4] Analyse spectrale et singularités d'un problème de transmission non coercif. C.R. Acad. Sci. Paris, Ser. I 328 (1999) 717-720. | MR | Zbl
, and ,[5] Mixed and hybrid finite element methods. Springer-Verlag (1991). | MR | Zbl
and ,[6] Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials. SIAM J. Math. Anal. 43 (2011) 2150-2169. | MR | Zbl
and ,[7] Analyse asymptotique et numérique de la diffraction d'ondes par des fils minces. Ph.D. thesis, Université Versailles - Saint-Quentin (2008) (in French).
,[8] A direct boundary integral method for transmission problems. J. Math. Anal. Appl. 106 (1985) 367-413. | MR | Zbl
and ,[9] 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).
and ,[10] Partial Differential Equations, Graduate studies in mathematics. Americain Mathematical Society 19 (1998). | Zbl
,[11] 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 | Zbl
and ,[12] Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs. Americain Mathematical Society 52 (1997). | MR | Zbl
, and ,[13] Problèmes aux limites non homogènes et applications. Dunod (1968). | Zbl
and ,[14] Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR | Zbl
,[15] Elliptic problems in domains with piecewise smooth boundaries, Expositions in Mathematics. De Gruyter 13 (1994). | MR | Zbl
and ,[16] General interface problems-I. Math. Meth. Appl. Sci. 17 (1994) 395-429. | MR | Zbl
and ,[17] General interface problems-II. Math. Meth. Appl. Sci. 17 (1994) 431-450. | MR | Zbl
and ,[18] 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 | Zbl
and ,[19] A warning about metamaterials for users of frequency-domain numerical simulators. IEEE Trans. Antennas Propag. 56 (2008) 792-798. | MR
and ,[20] Another approach to elliptic boundary problems. Commun. Pure Appl. Math. 14 (1961) 711-731. | MR | Zbl
,[21] Ill-posed waveguide discontinuity problem involving metamaterials with impedance boundary conditions on the two ports. IET Sci. Meas. Technol. 1 (2007) 232-239.
,[22] Lignes supraconductrices : analyse mathématique et numérique. Ph.D. thesis, Université Paris 6 (1999) (in French).
,[23] Wave scattering by metamaterial wedges and interfaces. Int. J. Numer. Model. 19 (2006) 105-117. | Zbl
, and ,[24] Surface modes of negative-parameter interfaces and the importance of rounding sharp corners. Metamaterials 2 (2008) 113-121.
, and ,[25] Partial Differ. Equ. Cambridge Univ. Press (1987).
,[26] 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).
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