We consider residual-based a posteriori error estimators for Galerkin-type discretizations of time-harmonic Maxwell's equations. We focus on configurations where the frequency is high, or close to a resonance frequency, and derive reliability and efficiency estimates. In contrast to previous related works, our estimates are frequency-explicit. In particular, our key contribution is to show that even if the constants appearing in the reliability and efficiency estimates may blow up on coarse meshes, they become independent of the frequency for sufficiently refined meshes. Such results were previously known for the Helmholtz equation describing scalar wave propagation problems and we show that they naturally extend, at the price of many technicalities in the proofs, to Maxwell's equations. Our mathematical analysis is performed in the 3D case, and covers conforming N\'ed\'elec discretizations of the first and second family, as well as first-order (and hybridizable) discontinuous Galerkin schemes. We also present numerical experiments in the 2D case, where Maxwell's equations are discretized with N\'ed\'elec elements of the first family. These illustrating examples perfectly fit our key theoretical findings, and suggest that our estimates are sharp.
翻译:我们考虑的是Galerkin型时间-和谐Maxwell方程式的后端误差测算器。 我们侧重于频率高或接近共振频率的配置, 并得出可靠性和效率估计。 与以往相关作品相比, 我们的估算值是清晰的。 特别是, 我们的主要贡献是显示, 即使可靠性和效率估计中出现的常数可能会在粗糙的 meshes 上爆炸, 它们也变得独立于充分精细的 meshes 的频率。 这些结果以前在描述卡拉波传播问题的Helmholtz方程式中是已知的。 我们以证据中许多技术要素的价格, 自然地将这些结果延伸到Maxwell的等式。 我们的数学分析是在3D 案例中进行的, 覆盖符合第一和第二家族的“ 电子分解”, 以及第一阶( 混合的) Galerkin 计划。 我们还在2D 中进行了数字实验, Maxwell 方程式中的第一道, 我们的直截面的模型与 N\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\