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Well-posedness and finite element approximability of time-harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials. (English) Zbl 1205.78058

Summary: A boundary value problem for the time harmonic Maxwell system is investigated through a variational formulation which is shown to be equivalent to it and well-posed if and only if the original problem is. Different bianisotropic materials and metamaterials filling subregions of the problem domain with Lipschitz continuous boundaries are allowed. Well-posedness and finite element approximability of the variational problem are proved by Lax-Milgram and Strang lemmas for a class of material configurations involving bianisotropic materials and metamaterials. Belonging to this class is not necessary, yet, for well-posedness and finite element approximability. Nevertheless, the material configurations of many radiation or scattering problems and many models of microwave components involving bianisotropic materials or metamaterials belong to the above class. Moreover, none of the other available tools commonly used to prove well-posedness seems to be able to cope with the material configurations left out by our treatment.

MSC:

78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78A25 Electromagnetic theory (general)
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35Q61 Maxwell equations
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