Existence, uniqueness and finite element approximation of the solution of time‐harmonic electromagnetic boundary value problems involving metamaterials

Abstract

Existence and uniqueness of the solution of time-harmonic electromagnetic boundary value problems is analyzed together with the convergence of Galerkin finite element approximations. Sufficient conditions based on the presence of different types of losses and on the properties of the hermitian symmetric parts of the effective dielectric permittivity and the effective magnetic permeability are provided. Metamaterials such as double-negative, epsilon-negative and mu-negative substances are covered by our analysis since any hypothesis on the positive definiteness of the aforementioned hermitian symmetric parts is avoided on purpose.

Numerical convergence and physical fidelity analysis for Maxwell’s equations in metamaterials

“…which directly leads to the stability estimate (7). h Furthermore, we can prove that the electric and magnetic fields still satisfy the Gauss' law if the initial fields are divergence free.…”

“…Some recent work in this respect can be found in papers such as [1,3,25,26], however, they are exclusively restricted to the simple medium case. Few theoretical analysis has been carried out for Maxwell's equations when metamaterials are involved except the work [7,8] for time-harmonic problems, and our initial effort in finite element error analysis for time-domain problems [17,20,19]. But in all our previous work, we only considered implicit schemes.…”

Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition

“…For the next lemma we need that the Maxwell equations (5) are satisfied at the time t = 0. Thus, if we assume that…”

“…[4,5]) is a first order approximation of the so-called "transparent" boundary condition. Sometimes it is also called Leontovich or impedance BC, cf.…”

Time-discretization scheme for quasi-static Maxwell's equations with a non-linear boundary condition

“…The classical Silver-Müller BC (cf. [2,8]) is a first order approximation to the so-called "transparent" BC. It can be also found under other names in the literature as Leontovich or impedance BC, cf.…”