Comptes Rendus
no. 6
Partial differential equations/Numerical analysis
Eigenvalue problems with sign-changing coefficients
[Problèmes aux valeurs propres en présence de coefficients changeant de signe]

Présenté par : Olivier Pironneau

Camille Carvalho1 ; Lucas Chesnel2 ; Patrick Ciarlet3
1 Applied Mathematics Unit, School of Natural Sciences, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA
2 INRIA/Centre de mathématiques appliquées, École polytechnique, Université Paris-Saclay, route de Saclay, 91128 Palaiseau cedex, France
3 POEMS, UMR 7231 CNRS–INRIA–ENSTA, 828, boulevard des Maréchaux, 91762 Palaiseau cedex, France
Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 671-675.

Nous étudions, d'un point de vue théorique et numérique, des problèmes aux valeurs propres mettant en jeu des coefficients dont le signe change sur le domaine d'intérêt.

We consider a class of eigenvalue problems involving coefficients changing sign on the domain of interest. We describe the main spectral properties of these problems according to the features of the coefficients. Then, under some assumptions on the mesh, we explain how one can use classical finite element methods to approximate the spectrum as well as the eigenfunctions while avoiding spurious modes. We also prove localisation results of the eigenfunctions for certain sets of coefficients.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.05.002
Camille Carvalho 1 ; Lucas Chesnel 2 ; Patrick Ciarlet 3

1 Applied Mathematics Unit, School of Natural Sciences, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA
2 INRIA/Centre de mathématiques appliquées, École polytechnique, Université Paris-Saclay, route de Saclay, 91128 Palaiseau cedex, France
3 POEMS, UMR 7231 CNRS–INRIA–ENSTA, 828, boulevard des Maréchaux, 91762 Palaiseau cedex, France 
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     author = {Camille Carvalho and Lucas Chesnel and Patrick Ciarlet},
     title = {Eigenvalue problems with sign-changing coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {671--675},
     publisher = {Elsevier},
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     number = {6},
     year = {2017},
     doi = {10.1016/j.crma.2017.05.002},
     language = {en},
}
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Camille Carvalho; Lucas Chesnel; Patrick Ciarlet. Eigenvalue problems with sign-changing coefficients. Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 671-675. doi : 10.1016/j.crma.2017.05.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.05.002/

[1] A. Abdulle; M. Huber; S. Lemaire An optimization-based numerical method for diffusion problems with sign-changing coefficients, C. R. Acad. Sci. Paris, Ser. I, Volume 355 (2017) no. 4, pp. 472-478 | DOI

[2] I. Babuška; J. Osborn Eigenvalue problems, Handbook of Numerical Analysis, vol. II, North-Holland, Amsterdam, 1991, pp. 641-787 in: P. G. Ciarlet, J.-L. Lions (Eds.)

[3] A.-S. Bonnet-Ben Dhia; L. Chesnel; P. Ciarlet T-coercivity for scalar interface problems between dielectrics and metamaterials, ESAIM: M2AN, Volume 46 (2012), pp. 1363-1387

[4] A.-S. Bonnet-Ben Dhia; C. Carvalho; P. Ciarlet Mesh requirements for the finite element approximation of problems with sign-changing coefficients, 2016 (preprint version 1) | HAL

[5] A.-S. Bonnet-Ben Dhia; P. Ciarlet; C.M. Zwölf Time harmonic wave diffraction problems in materials with sign-shifting coefficients, J. Comput. Appl. Math., Volume 234 (2010), pp. 1912-1919 (corrigendum p. 2616)

[6] L. Chesnel; P. Ciarlet Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials, SIAM J. Math. Anal., Volume 43 (2011), pp. 2150-2169

[7] L. Chesnel; P. Ciarlet T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients, Numer. Math., Volume 124 (2013), pp. 1-29

[8] M. Delfour; J.-P. Zolésio Shapes and Geometries – Metrics, Analysis, Differential Calculus and Optimization, vol. 22, SIAM, 2011

[9] J. Fleckinger; M.L. Lapidus Eigenvalues of elliptic boundary value problems with an indefinite weight function, Trans. Amer. Math. Soc., Volume 295 (1986) no. 1, pp. 305-324

[10] S. Nicaise; J. Venel A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients, J. Comput. Appl. Math., Volume 235 (2011) no. 14, pp. 4272-4282

[11] E. Séré; M. Lewin Spectral pollution and how to avoid it (with applications to Dirac and periodic Schrödinger operators), Proc. Lond. Math. Soc., Volume 100 (2010), pp. 864-900

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