Analysis of Variational Formulations and Low-regularity Solutions for Time-harmonic Electromagnetic Problems in Complex Anisotropic Media

Abstract

We consider the time-harmonic Maxwell's equations with physical parameters, namely the electric permittivity and the magnetic permeability, that are complex, possibly non-Hermitian, tensor fields. Both tensor fields verify a general ellipticity condition. In this work, the well-posedness of formulations for the Dirichlet and Neumann problems (i.e. with a boundary condition on the electric field or its curl, respectively) is proven using well-suited function spaces and Helmholtz decompositions. For both problems, the a priori regularity of the solution and the solution's curl is analysed. The regularity results are obtained by splitting the fields and using shift theorems for second-order divergence elliptic operators. Finally, the discretization of the formulations with a H (curl)-conforming approximation based on edge finite elements is considered. An a priori error estimate is derived and verified thanks to numerical results with an elementary benchmark.

Analysis of Time-Harmonic Maxwell Impedance Problems in Anisotropic Media

“…We now express the ellipticity condition on the coefficients ε, µ, α (cf. Definition 3.1.1 in [12]).…”

“…when the artificial boundary is polyhedral with pathological vertices), we obtain that the regular part depends continuously on g γ + g π , π T A C π , and γ T A E γ . An alternate proof, that does not make use of a lifting of α, is proposed in [12] (in this case, one assumes that α ∈ W 2,∞ (Γ A )).…”

“…. Following the steps of Theorem 3.9 in [13], we introduce E 0 := E −E d ∈ H 0 (curl, Ω). Then, one uses a continuous splitting of E 0 into regular and singular part, cf.…”

“…While here we have done it for coercive forms, which allows the use of Céa's lemma, in more general cases, one would have to prove a uniform discrete inf-sup condition, which seems quite tedious in the general anisotropic plus Robin condition context. Finally, this work provides a suitable framework for the mathematical analysis of decomposed problems and domain decomposition methods making use of impedance conditions; this has been sketched in Chapter 7 of [12].…”

“…We recall that the electromagnetic materials are usually characterized by two tensor-valued coefficients, the electric permittivity ε and the magnetic permeability µ (the coefficients will be respectively denoted ε and µ if they are scalar-valued). In [13], with A. Modave, we have proposed the mathematical and numerical analyses of the model for elliptic, anisotropic and possibly non-hermitian coefficients ε and µ, when the time-harmonic Maxwell's equations are supplemented with either a Dirichlet boundary condition, or a Neumann boundary condition. In this manuscript, we consider a model including an impedance boundary condition on (some part of) the boundary.…”