no. 4
Numerical analysis
An optimization-based numerical method for diffusion problems with sign-changing coefficients
[Une méthode d'optimisation pour des problèmes de diffusion avec changement de signe]

Présenté par : Olivier Pironneau

Abdulle, Assyr1 ; Huber, Martin E.1 ; Lemaire, Simon1
1 ANMC, Institut de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 472-478.

Nous proposons une nouvelle méthode, basée sur la résolution d'un problème de minimisation, pour l'approximation de problèmes de diffusion avec changement de signe. Cette approche, qui tire profit d'une reformulation du modèle initial sous la forme d'un problème de transmission, ne repose pas sur la discrétisation d'une équation stabilisée, et la convergence de la méthode est obtenue sans hypothèse de symétrie du maillage dans un voisinage de l'interface où la conductivité change de signe.

A new optimization-based numerical method is proposed for the solution to diffusion problems with sign-changing conductivity coefficients. In contrast to existing approaches, our method does not rely on the discretization of a stabilized equation, and the convergence of the scheme can be proved without any symmetry assumption on the mesh near the interface where the conductivity sign changes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.02.010
Abdulle, Assyr 1 ; Huber, Martin E. 1 ; Lemaire, Simon 1

1 ANMC, Institut de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland 
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Abdulle, Assyr; Huber, Martin E.; Lemaire, Simon. An optimization-based numerical method for diffusion problems with sign-changing coefficients. Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 472-478. doi : 10.1016/j.crma.2017.02.010. http://www.numdam.org/articles/10.1016/j.crma.2017.02.010/

[1] A. Abdulle, S. Lemaire, An optimization-based method for sign-changing elliptic PDEs, in preparation.

[2] Bernardi, C.; Maday, Y.; Rapetti, F. Discrétisations variationnelles de problèmes aux limites elliptiques, Mathématiques & Applications, vol. 45, Springer-Verlag, Berlin, 2004

[3] A.-S. Bonnet-Ben Dhia, C. Carvalho, P. Ciarlet Jr., Mesh requirements for the finite element approximation of problems with sign-changing coefficients, submitted for publication, 2016, Preprint hal-01335153, . | HAL

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

[5] Brenner, S.C. Poincaré–Friedrichs inequalities for piecewise H1 functions, SIAM J. Numer. Anal., Volume 41 (2003) no. 1, pp. 306-324

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

[7] Ciarlet, P.G. The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA, 2002 (reprint of the 1978 original [North-Holland, Amsterdam])

[8] Gunzburger, M.D.; Heinkenschloss, M.; Lee, H.K. Solution of elliptic partial differential equations by an optimization-based domain decomposition method, Appl. Math. Comput., Volume 113 (2000) no. 2–3, pp. 111-139

[9] Gunzburger, M.D.; Peterson, J.S.; Lee, H.K. An optimization-based domain decomposition method for partial differential equations, Comput. Math. Appl., Volume 37 (1999) no. 10, pp. 77-93

[10] Nguyen, H.-M. Asymptotic behavior of solutions to the Helmholtz equations with sign-changing coefficients, Trans. Amer. Math. Soc., Volume 367 (2015), pp. 6581-6595

[11] Nguyen, H.-M. Limiting absorption principle and well-posedness for the Helmholtz equation with sign-changing coefficients, J. Math. Pures Appl., Volume 106 (2016) no. 2, pp. 342-374

[12] Nguyen, H.-M. Negative index materials and their applications: recent mathematics progress, Chin. Ann. Math., Ser. B, Volume 38 (2017) no. 2, pp. 601-628

[13] Nicaise, S.; Venel, J. A posteriori error estimates for a finite element approximation of transmission problems with sign-changing coefficients, J. Comput. Appl. Math., Volume 235 (2011), pp. 4272-4282

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