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Journal Articles SIAM Journal on Mathematical Analysis Year : 2023

Analysis of time-harmonic Maxwell impedance problems in anisotropic media

Abstract

We consider the time-harmonic Maxwell's equations in anisotropic media. The problem to be solved is an approximation of the diffraction problem, or scattering from bounded objects, that is usually set in some exterior domain in $\mathbb{R}^3$. We consider perfectly conducting objects, so the equations are supplemented with a Dirichlet boundary condition on those objects, and we truncate the exterior domain by imposing an impedance condition on an artificial boundary, to model an approximate radiation condition. The resulting problem is then posed in a bounded domain, with Dirichlet and impedance boundary conditions. In this work, we focus on the mathematical meaning of the impedance condition, precisely in which function space it holds. This relies on a careful analysis of the regularity of the traces of electromagnetic fields, which can be derived thanks to the study of the regularity of the solution to second-order surface PDEs. Then, we prove well-posedness of the model, and we determine the a priori regularity of the fields in the domain and on the boundaries, depending on the geometry, the coefficients and the data. Finally, the discretization of the formulations is presented, with an approximation based on edge finite elements. Error estimates are derived, and a benchmark is provided to discuss those estimates.
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Dates and versions

hal-03613183 , version 1 (18-03-2022)
hal-03613183 , version 2 (17-11-2022)
hal-03613183 , version 3 (08-02-2023)

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Damien Chicaud, Patrick Ciarlet. Analysis of time-harmonic Maxwell impedance problems in anisotropic media. SIAM Journal on Mathematical Analysis, 2023, 55 (3), pp.1969-2000. ⟨10.1137/22M1485413⟩. ⟨hal-03613183v3⟩
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