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# Frequency-explicit approximability estimates for time-harmonic Maxwell's equations

1 ATLANTIS - Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné
Abstract : We consider time-harmonic Maxwell's equations set in an heterogeneous medium with perfectly conducting boundary conditions. Given a divergence-free right-hand side lying in $L^2$, we provide a frequency-explicit approximability estimate measuring the difference between the corresponding solution and its best approximation by high-order Nédélec finite elements. Such an approximability estimate is crucial in both the a priori and a posteriori error analysis of finite element discretizations of Maxwell's equations, but the derivation is not trivial. Indeed, it is hard to take advantage of high-order polynomials given that the righthand side only exhibits $L^2$ regularity. We proceed in line with previously obtained results for the simpler setting of the scalar Helmholtz equation, and propose a regularity splitting of the solution. In turn, this splitting yields sharp approximability estimates generalizing known results for the scalar Helmoltz equation and showing the interest of high-order methods.
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https://hal.inria.fr/hal-03221188
Contributeur : Théophile Chaumont-Frelet Connectez-vous pour contacter le contributeur
Soumis le : mardi 2 août 2022 - 17:13:34
Dernière modification le : jeudi 4 août 2022 - 17:00:05

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chaumontfrelet_vega_2022a.pdf
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### Citation

Théophile Chaumont-Frelet, Patrick Vega. Frequency-explicit approximability estimates for time-harmonic Maxwell's equations. Calcolo, Springer Verlag, 2022, ⟨10.1007/s10092-022-00464-7⟩. ⟨hal-03221188v2⟩

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