Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations
<|reference_start|>Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations: We prove sharp wavenumber-explicit error bounds for first- or second-type-N\'ed\'elec-element (a.k.a. edge-element) conforming discretisations, of arbitrary (fixed) order, of the variable-coefficient time-harmonic Maxwell equations posed in a bounded domain with perfect electric conductor (PEC) boundary conditions. The PDE coefficients are allowed to be piecewise regular and complex-valued; this set-up therefore includes scattering from a PEC obstacle and/or variable real-valued coefficients, with the radiation condition approximated by a perfectly matched layer (PML).   In the analysis of the $h$-version of the finite-element method, with fixed polynomial degree $p$, applied to the time-harmonic Maxwell equations, the $\textit{asymptotic regime}$ is when the meshwidth, $h$, is small enough (in a wavenumber-dependent way) that the Galerkin solution is quasioptimal independently of the wavenumber, while the $\textit{preasymptotic regime}$ is the complement of the asymptotic regime.   The results of this paper are the first preasymptotic error bounds for the time-harmonic Maxwell equations using first-type N\'ed\'elec elements or higher-than-lowest-order second-type N\'ed\'elec elements. Furthermore, they are the first wavenumber-explicit results, even in the asymptotic regime, for Maxwell scattering problems with a non-empty scatterer.<|reference_end|>
@article{chaumont-frelet2024sharp,
    title={Sharp error bounds for edge-element discretisations of the
  high-frequency Maxwell equations},
    author={Th'eophile Chaumont-Frelet, Jeffrey Galkowski, Euan A. Spence},
    journal={arXiv preprint arXiv:2408.04507},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.04507},
    primaryClass={math.NA cs.NA}
}