On $L^r$ Estimates for Maxwell's Equations with Complex Coefficients in Lipschitz Domains

Xiang, Xingfei

doi:10.1137/19m1246833

Cited by 2 publications

(4 citation statements)

References 15 publications

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“…This condition is slightly more general than the one proposed e.g. by [27], varying only by a phase factor. This allows us to cover a more general class of material properties, such as material tensors coming from plasma theory (e.g.…”

Section: Ellipticity Conditionmentioning

confidence: 72%

“…This allows us to cover a more general class of material properties, such as material tensors coming from plasma theory (e.g. in [6]) which do not satisfy the standard ellipticity condition used in [27], but which are elliptic for well-suited phase factors according to Definition 2.1. Because ξ can be non-Hermitian, the mapping (v, w) → (ξv|w) 0,Ω is, in general, not a scalar product in L 2 (Ω); orthogonality properties are lateralized, in the sense that (ξv|w) 0,Ω = 0 is not equivalent to (ξw|v) 0,Ω = 0.…”

Section: Ellipticity Conditionmentioning

confidence: 99%

“…The well-posedness of variational formulations with non-Hermitian material tensors has been studied e.g. in [3] for complex symmetric tensors, in [6] for particular anisotropic media coming from plasma theory, and in [27] for material tensors with an elliptic real part.…”

Section: Introductionmentioning

confidence: 99%

“…where we used (27) to reach the third line. We conclude by the shift theorem 3.3 that ϕ E ∈ H min(s,r)+1 (Ω), and…”

mentioning

confidence: 99%

See 3 more Smart Citations

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

Chicaud¹,

Ciarlet²,

Modave³

2021

SIAM J. Math. Anal.

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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.

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Section: Ellipticity Conditionmentioning

confidence: 72%

Section: Ellipticity Conditionmentioning

confidence: 99%