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.

翻译：我们证明了任意（固定）阶的一类或二类Nédélec元（亦称边元）保形离散化，在具有完美电导体边界条件的有界域中，对变系数时谐Maxwell方程所给出的波数显式尖锐误差界。允许偏微分方程系数为分段正则复值；因此该框架包含完美电导体障碍物散射和/或变实值系数情形，其中辐射条件通过完美匹配层进行近似。在固定多项式次数p的有限元法h版本应用于时谐Maxwell方程的分析中，当网格宽度h足够小（以波数依赖的方式）使得Galerkin解具有与波数无关的拟最优性时，称为渐近区域；而预渐近区域则是渐近区域的补集。本文结果首次给出了一类Nédélec元或高于最低阶的二类Nédélec元求解时谐Maxwell方程的预渐近误差界。此外，即使在渐近区域，这些结果也是针对非空散射体的Maxwell散射问题首次获得的波数显式结论。