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Journal Articles Calcolo Year : 2022

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

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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Dates and versions

hal-03221188 , version 1 (07-05-2021)
hal-03221188 , version 2 (02-08-2022)

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