Comptes Rendus
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

Assyr Abdulle1 ; Martin E. Huber1 ; Simon Lemaire1
1 ANMC, Institut de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Comptes Rendus. Mathématique, Volume 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
Assyr Abdulle 1 ; Martin E. Huber 1 ; Simon Lemaire 1

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

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

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

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