Mesh requirements for the finite element approximation of problems with sign-changing coefficients

Abstract

Transmission problems with sign-changing coefficients occur in electromagnetic theory in the presence of negative materials surrounded by classical materials. For general geometries, establishing Fredholmness of these transmission problems is well-understood thanks to the T-coercivity approach. Moreover, for a plane interface, there exist meshing rules that guarantee an optimal convergence rate for the finite element approximation. We propose here a new treatment at the corners of the interface which allows to design meshing rules for an arbitrary polygonal interface and then recover standard error estimates. This treatment relies on the use of simple geometrical transforms to define the meshes. Numerical results illustrate the importance of this new design.

A generalized finite element method for problems with sign-changing coefficients

“…It guarantees the wellposedness of Problem (2.1), and the stability constant then depends on ⋆ and ‖Y‖ ℒ( 1 0 (Ω)) . In the context of problems with a sign-changing coefficient, a particular form of T-coercivity based on symmetrization has been developed [4,7]. The key idea is to design an operator T that "flips" the sign of its argument in Ω − .…”

“…When discretizing these problems with the standard finite element method, the questions of existence and uniqueness of the discrete solution as well as its convergence toward the continuous solution immediately arise. Simply speaking, they have been answered positively in two different scenarios, namely (a) if the mesh satisfies certain symmetry properties around the interface Γ, which is denoted as T-conformity [7], or (b) if the contrast | + |/| − | is outside an enlarged critical interval̃︀ := [1/̃︀,̃︀], wherẽ︀ > ( [11], Sect. 5.1).…”

An optimization-based numerical method for diffusion problems with sign-changing coefficients

“…La résolution numérique efficace de ce genre de problèmes est importante pour de nombreuses applications (e.g., super-lentilles, invisibilité), mais les méthodes existantes ne sont pour l'instant pas satisfaisantes. Dans [6], les deux approches envisagées reposent (i) sur la discrétisation d'uneéquation stabilisée, pour laquelle les taux de convergence obtenus sont sous-optimaux, ou (ii) sur des hypothèses de symétrie du maillage autour de l'interface où la conductivité change de signe, exigences pouvant s'avérer très contraignantes pour des interfaces générales (voir [3]) ou en 3D. La méthode numérique introduite ici, qui utilise une reformulation du modèle initial en un problème de transmission, ne repose pas sur l'ajout de dissipationà l'équation, et nous montrons sa convergence pour des problèmes elliptiques présentant un changement de signe sans aucune hypothèse de symétrie sur le maillage.…”

“…The first approach is based on a simplicial discretization T h of Ω, that respects the interface Γ and the construction of a conforming finite element space V 0 pT h q that is stable by the operator T. Well-posedness and optimal convergence rates can be shown for this approach. In practice, T-stability is achieved by means of symmetric meshes near the interface, whose construction is a nontrivial task for complicated interfaces (see [3]) or 3D problems. On general meshes, two main approaches have been investigated by Chesnel and Ciarlet Jr.…”

Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients

“…Moreover, the same technique can be applied at the discrete level. Conception of numerical approximations, their well-posedness, and a priori error estimates have been addressed in [9,20] in the conforming finite element context and in [21] in the nonconforming finite element and discontinuous Galerkin context. A posteriori error analysis for problems of type (1.1) has likewise been started recently.…”