Numerische Mathematik

, Volume 124, Issue 1, pp 1–29 | Cite as

T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients

Article

Abstract

To solve variational indefinite problems, one uses classically the Banach–Nečas–Babuška theory. Here, we study an alternate theory to solve those problems: T-coercivity. Moreover, we prove that one can use this theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma. We apply this theory to solve the indefinite problem \(\text{ div}\sigma \nabla u=f\) set in \(H^1_0\), with \(\sigma \) exhibiting a sign change.

Mathematics Subject Classification

65N30 78A48 35J20 35B65 

References

  1. 1.
    Bonnet-Ben Dhia, A.-S., Chesnel, L., Ciarlet, P. Jr.: \(T\)-coercivity for scalar interface problems between dielectrics and metamaterials. Math. Mod. Num. Anal. 46, 1363–1387 (2012)Google Scholar
  2. 2.
    Bonnet-Ben Dhia, A.-S., Chesnel, L., Claeys, X.: Radiation condition for a non smooth interface between dielectric and metamaterial. Math. Models Meth. Appl. Sci. (to appear)Google Scholar
  3. 3.
    Bonnet-Ben Dhia, A.-S., Ciarlet, P. Jr.: Zwölf, C.-M.: Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234, 1912–1919 (2010). Corrigendum: J. Comput. Appl. Math. 234, 2616, 2010. Eight International Conference on Mathematical and Numerical Aspects of Waves (Waves 2007).Google Scholar
  4. 4.
    Bonnet-Ben Dhia, A.-S., Dauge, M., Ramdani, K.: Analyse spectrale et singularités d’un problème de transmission non coercif. C. R. Acad. Sci. Paris. Ser. I 328, 717–720 (1999)MathSciNetGoogle Scholar
  5. 5.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, New York (1991)Google Scholar
  6. 6.
    Chesnel, L., Ciarlet, P. Jr.: Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials. SIAM J. Math. Anal. 43, 2150–2169 (2011)Google Scholar
  7. 7.
    Chung, E.T., Ciarlet, P. Jr.: Scalar transmission problems between dielectrics and metamaterials: \(T\)-coercivity for the Discontinuous Galerkin approach. J. Comput. Appl. Math. (to appear) Google Scholar
  8. 8.
    Ciarlet, P. Jr.: \(T\)-coercivity: Application to the discretization of Helmholtz-like problems. Comput. Math. Appl. 64, 22–34 (2012)Google Scholar
  9. 9.
    Costabel, M., Stephan, E.: A direct boundary integral method for transmission problems. J. Math. Anal. Appl. 106, 367–413 (1985)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dauge, M., Texier, B.: Problèmes de transmission non coercifs dans des polygones. http://hal.archives-ouvertes.fr/docs/00/56/23/29/PDF/BenjaminT_arxiv.pdf (2010)
  11. 11.
    Engheta, N.: An idea for thin subwavelength cavity resonator using metamaterials with negative permittivity and permeability. IEEE Antennas Wireless Propag. Lett. 1, 10–13 (2002)CrossRefGoogle Scholar
  12. 12.
    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, Berlin (2004)MATHCrossRefGoogle Scholar
  13. 13.
    Genov, D.A., Zhang, S., Zhang, X.: Mimicking celestial mechanics in metamaterials. Nat. Phys. 5(9), 687 (2009)CrossRefGoogle Scholar
  14. 14.
    Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris (1992)MATHGoogle Scholar
  15. 15.
    Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, vol. 52. AMS, Providence (1997)Google Scholar
  16. 16.
    Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications. Dunod (1968)Google Scholar
  17. 17.
    Maystre, D., Enoch, S.: Perfect lenses made with left-handed materials: Alice’s mirror? J. Opt. Soc. Amer. A21, 122–131 (2004)CrossRefGoogle Scholar
  18. 18.
    McLean, W.: Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  19. 19.
    Monk, P.: Finite element methods for Maxwell’s equations. Oxford University Press, New York (2003)MATHCrossRefGoogle Scholar
  20. 20.
    Nicaise, S., Venel, J.: A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients. J. Comput. Appl. Math. 235, 4272–4282 (2011)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85(18), 3966–3969 (2000)CrossRefGoogle Scholar
  22. 22.
    Ramdani, K.: Lignes supraconductrices: analyse mathématique et numérique. Ph.D. thesis, Université Paris 6 (1999)Google Scholar
  23. 23.
    Scott, R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIAENSTA ParisTechPalaiseau CedexFrance

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