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T-coercivity for scalar interface problems between dielectrics and metamaterials

Published online by Cambridge University Press:  11 April 2012

Anne-Sophie Bonnet-Ben Dhia ,
Lucas Chesnel  and
Patrick Ciarlet Jr.
Anne-Sophie Bonnet-Ben Dhia
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France. anne-sophie.bonnet-bendhia@ensta-paristech.fr; lucas.chesnel@ensta-paristech.fr; patrick.ciarlet@ensta-paristech.fr
Lucas Chesnel
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France. anne-sophie.bonnet-bendhia@ensta-paristech.fr; lucas.chesnel@ensta-paristech.fr; patrick.ciarlet@ensta-paristech.fr
Patrick Ciarlet Jr.
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France. anne-sophie.bonnet-bendhia@ensta-paristech.fr; lucas.chesnel@ensta-paristech.fr; patrick.ciarlet@ensta-paristech.fr
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Abstract

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.

Keywords

Metamaterialsinterface problemT-coercivity
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1363 - 1387
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

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].
Bonnet-Ben Dhia, A.-S., Ciarlet, P. Jr. and Zwölf, C.M., 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. Google Scholar
Bonnet-Ben Dhia, A.-S., Ciarlet, P. Jr. and Zwölf, C.M., 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. Google Scholar
Bonnet-Ben Dhia, A.-S., Dauge, M. and Ramdani, K., Analyse spectrale et singularités d’un problème de transmission non coercif. C.R. Acad. Sci. Paris, Ser. I 328 (1999) 717–720. Google Scholar
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag (1991).
Chesnel, L. and Ciarlet, P. Jr., Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials. SIAM J. Math. Anal. 43 (2011) 2150–2169. Google Scholar
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).
Costabel, M. and Stephan, E., A direct boundary integral method for transmission problems. J. Math. Anal. Appl. 106 (1985) 367–413. Google Scholar
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).
L.D. Evans, Partial Differential Equations, Graduate studies in mathematics. Americain Mathematical Society 19 (1998).
Fernandes, P. and Raffetto, M., 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. Google Scholar
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).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1968).
W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000).
S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, Expositions in Mathematics. De Gruyter 13 (1994).
Nicaise, S. and Sändig, A.M., General interface problems-I. Math. Meth. Appl. Sci. 17 (1994) 395–429. Google Scholar
Nicaise, S. and Sändig, A.M., General interface problems-II. Math. Meth. Appl. Sci. 17 (1994) 431–450. Google Scholar
Nicaise, S. and Venel, J., A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients. J. Comput. Appl. Math. 235 (2011) 4272–4282. Google Scholar
Oliveri, G. and Raffetto, M., A warning about metamaterials for users of frequency-domain numerical simulators. IEEE Trans. Antennas Propag. 56 (2008) 792–798. Google Scholar
Peetre, J., Another approach to elliptic boundary problems. Commun. Pure Appl. Math. 14 (1961) 711–731. Google Scholar
Raffetto, M., Ill-posed waveguide discontinuity problem involving metamaterials with impedance boundary conditions on the two ports. IET Sci. Meas. Technol. 1 (2007) 232–239. Google Scholar
K. Ramdani, Lignes supraconductrices : analyse mathématique et numérique. Ph.D. thesis, Université Paris 6 (1999) (in French).
Sukhorukov, A.A., Shadrivov, I.V. and Kivshar, Y.S., Wave scattering by metamaterial wedges and interfaces. Int. J. Numer. Model. 19 (2006) 105–117. Google Scholar
Wallén, H., Kettunen, H. and Sihvola, A., Surface modes of negative-parameter interfaces and the importance of rounding sharp corners. Metamaterials 2 (2008) 113–121. Google Scholar
J. Wloka, Partial Differ. Equ. Cambridge Univ. Press (1987).
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).