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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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Preprints, Working Papers, ...
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https://hal.inria.fr/hal-03221188
Contributor : Théophile Chaumont-Frelet Connect in order to contact the contributor
Submitted on : Friday, May 7, 2021 - 7:02:22 PM
Last modification on : Friday, February 4, 2022 - 3:08:41 AM
Long-term archiving on: : Sunday, August 8, 2021 - 7:03:16 PM

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chaumontfrelet_vega_2021a.pdf
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### Identifiers

• HAL Id : hal-03221188, version 1
• ARXIV : 2105.03393

### Citation

Théophile Chaumont-Frelet, Patrick Vega. Frequency-explicit approximability estimates for time-harmonic Maxwell's equations. 2021. ⟨hal-03221188⟩

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