We propose a novel a posteriori error estimator for the N\'ed\'elec finite element discretization of time-harmonic Maxwell's equations. After the approximation of the electric field is computed, we propose a fully localized algorithm to reconstruct approximations to the electric displacement and the magnetic field, with such approximations respectively fulfilling suitable divergence and curl constraints. These reconstructed fields are in turn used to construct an a posteriori error estimator which is shown to be reliable and efficient. Specifically, the estimator controls the error from above up to a constant that tends to one as the mesh is refined and/or the polynomial degree is increased, and from below up to constant independent of $p$. Both bounds are also fully-robust in the low-frequency regime. The properties of the proposed estimator are illustrated on a set of numerical examples.

翻译：我们针对时谐麦克斯韦方程组的Nédélec有限元离散提出了一种新颖的后验误差估计器。在计算电场近似解之后，我们提出了一种完全局域化的算法来重构电位移矢量和磁场的近似解，这些近似解分别满足适当的散度和旋度约束。这些重构场随后被用于构建后验误差估计器，并证明该估计器具有可靠性和有效性。具体而言，该估计器从上方控制误差，其常数随网格加密和/或多项式阶数提高而趋近于1；从下方控制误差，其常数与$p$无关。两种界限在低频区域均具有完全鲁棒性。通过一组数值算例验证了所提出估计器的性能。