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Analysis of variational formulations and low-regularity solutions for time-harmonic electromagnetic problems in complex anisotropic media

Damien Chicaud 1 Patrick Ciarlet 1 Axel Modave 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
CNRS - Centre National de la Recherche Scientifique : UMR7231, UMA - Unité de Mathématiques Appliquées, Inria Saclay - Ile de France
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 functional 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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Submitted on : Friday, December 17, 2021 - 11:54:01 AM
Last modification on : Wednesday, May 11, 2022 - 12:06:05 PM

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Damien Chicaud, Patrick Ciarlet, Axel Modave. Analysis of variational formulations and low-regularity solutions for time-harmonic electromagnetic problems in complex anisotropic media. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2021, 53 (3), pp.2691-2717. ⟨10.1137/20M1344111⟩. ⟨hal-02651682v3⟩

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