Preasymptotic error estimates of higher-order EEM for the time-harmonic Maxwell equations with large wave number

Abstract

The time-harmonic Maxwell equations with impedance boundary condition and large wave number are discretized using the second-type N\'{e}d\'{e}lec's edge element method (EEM). Preasymptotic error bounds are derived, showing that, under the mesh condition $\kappa^{2p+1}h^{2p}$ being sufficiently small, the error of the EEM of order $p$ in the energy norm is bounded by $\mathcal{O}\big(\kappa^{p}h^p + \kappa^{2p+1}h^{2p}\big)$, while the error in the $\kappa$-scaled $\boldsymbol{L}^2$ norm is bounded by $\mathcal{O}\big((\kappa h)^{p+1} + \kappa^{2p+1} h^{2p}\big)$. Here, $\kappa$ is the wave number and $h$ is the mesh size. Numerical tests are provided to illustrate our theoretical results.