<tt>T</tt>-Coercivity for the Maxwell Problem with Sign-Changing Coefficients

Dhia, Anne‐Sophie Bonnet‐Ben; Chesnel, Lucas; Ciarlet, Patrick

doi:10.1080/03605302.2014.892128

Cited by 37 publications

(4 citation statements)

References 33 publications

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“…Le cas général sera discuté dans d'autres articles. Nous traitons aussi des problèmes à changement de signe (c'est-à-dire, des problèmes qui ne sont pas fortement elliptiques) [3,4,10,11,5,6,29]. Le cas des domaines bornés sera discuté dans [27].…”

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Estimations polynomiales pour les problèmes de transmission sur des domaines à bords plats

2022

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Cet article traite des opérateurs elliptiques du second ordre P dont les coefficients sont vus comme des variables aléatoires. Son but est d'obtenir des estimations de leurs solutions qui soient polynomiales dans les coefficients. Ces estimations sont utiles pour la quantification de l'incertitude. Dans cet article, nous traitons le cas dans lequel le bord et l'interface sont parallèles aux hyperplans {x m = 0} ⊂ R m .

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Estimations polynomiales pour les problèmes de transmission sur des domaines à bords plats

2022

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“…The proposed asymptotic expansions approach is valid for arbitrary optical parameter a c (and complex-valued ones to some extent), one could also consider arbitrary double negative optical parameters b c and work with the double-negative PDE − div(a −1 ∇u) − b k 2 u = 0 (e.g. [11,21,3]). Then, to deduce from the quasi-pairs existence the presence of scattering resonances becomes difficult because the operator is no longer self-adjoint.…”

Section: Discussionmentioning

confidence: 99%

Scattering resonances in unbounded transmission problems with sign-changing coefficient

Carvalho

Moitier

2023

IMA Journal of Applied Mathematics

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It is well-known that classical optical cavities can exhibit localized phenomena associated to scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the “quasimodes to resonances” argument from the black box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, for example made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and showing that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial’s properties, scattering resonances situated closed to the real axis are associated to surface plasmons. Examples for several metamaterial cavities are provided.

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“…The T-coercivity approach is an alternative to Banach-Nečas-Babuška theory for the study of well-posedness and numerical approximation of non-coercive problems. T-coercivity was originally introduced for problems involving an invertible operator perturbed by a compact term [28,29] and problems with sign-changing coefficients, see for instance [19,[30][31][32][33]. More recently, it was applied to saddle-point problems [16,20].…”

Section: The T-coercivity Approachmentioning

confidence: 99%

“…Using ( 29), ( 30) and (31) to bound from below the right-hand side of ( 28) and rearranging terms, we obtain…”

Section: Remarkmentioning

confidence: 99%

Numerical analysis of an incompressible soft material poromechanics model using T-coercivity

Barré¹,

Grandmont²,

Moireau³

2024

Comptes Rendus. Mécanique

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This article is devoted to the numerical analysis of the full discretization of a generalized poromechanical model resulting from the linearization of an initial model fitted to soft tissue perfusion. Our strategy here is based on the use of energy-based estimates and T-coercivity methods, so that the numerical analysis benefits from the essential tools used in the existence analysis of the continuous-time and continuous-space formulation. In particular, our T-coercivity strategy allows us to obtain the necessary inf-sup condition for the global system from the inf-sup condition restricted to a subsystem having the same structure as the Stokes problem. This allows us to prove that any finite element pair adapted to the Stokes problem is also suitable for this global poromechanical model regardless of porosity and permeability, generalizing previous results from the literature studying this model.

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`T`-Coercivity for the Maxwell Problem with Sign-Changing Coefficients

Cited by 37 publications

References 33 publications

Estimations polynomiales pour les problèmes de transmission sur des domaines à bords plats

Estimations polynomiales pour les problèmes de transmission sur des domaines à bords plats

Scattering resonances in unbounded transmission problems with sign-changing coefficient

Numerical analysis of an incompressible soft material poromechanics model using T-coercivity

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T-Coercivity for the Maxwell Problem with Sign-Changing Coefficients

Cited by 37 publications

References 33 publications

Estimations polynomiales pour les problèmes de transmission sur des domaines à bords plats

Estimations polynomiales pour les problèmes de transmission sur des domaines à bords plats

Scattering resonances in unbounded transmission problems with sign-changing coefficient

Numerical analysis of an incompressible soft material poromechanics model using T-coercivity

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`T`-Coercivity for the Maxwell Problem with Sign-Changing Coefficients