Abstract
We consider time-harmonic Maxwell’s equations set in a 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 right-hand 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 Helmholtz equation and showing the interest of high-order methods.
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Notes
The authors believe it is of interest to explicitly mention \(c_s\) proofs, since at least in principle, the regularity splitting results may apply in cases where \(c_s\) is not obtain via Theorem 1.
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Chaumont-Frelet, T., Vega, P. Frequency-explicit approximability estimates for time-harmonic Maxwell’s equations. Calcolo 59, 22 (2022). https://doi.org/10.1007/s10092-022-00464-7
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DOI: https://doi.org/10.1007/s10092-022-00464-7