We derive a priori and a posteriori error estimates for the discontinuous Galerkin (dG) approximation of the time-harmonic Maxwell's equations. Specifically, we consider an interior penalty dG method, and establish error estimates that are valid under minimal regularity assumptions and involving constants that do not depend on the frequency for sufficiently fine meshes. The key result of our a priori error analysis is that the dG solution is asymptotically optimal in an augmented energy norm that contains the dG stabilization. Specifically, up to a constant that tends to one as the mesh is refined, the dG solution is as accurate as the best approximation in the same norm. The main insight is that the quantities controlling the smallness of the mesh size are essentially those already appearing in the conforming setting. We also show that for fine meshes, the inf-sup stability constant is as good as the continuous one up to a factor two. Concerning the a posteriori analysis, we consider a residual-based error estimator under the assumption of piecewise constant material properties. We derive a global upper bound and local lower bounds on the error with constants that (i) only depend on the shape-regularity of the mesh if it is sufficiently refined and (ii) are independent of the stabilization bilinear form.

翻译：本文推导了时谐麦克斯韦方程间断伽辽金（dG）逼近的先验与后验误差估计。具体而言，我们考虑一种内部惩罚dG方法，并在最小正则性假设下建立误差估计，该估计对足够精细的网格具有与频率无关的常数。我们先验误差分析的关键结果表明：在包含dG稳定项的增强能量范数中，dG解具有渐近最优性。具体来说，当网格加密时，dG解的精度在趋于1的常数倍范围内与相同范数下的最佳逼近相当。核心发现是：控制网格尺寸精细度的量本质上与协调元情形中出现的量相同。我们还证明对于精细网格，inf-sup稳定性常数在因子2范围内与连续情形相当。关于后验分析，我们在分段常数材料属性的假设下考虑基于残差的误差估计器。我们推导了误差的全局上界与局部下界，其常数满足：（i）仅依赖于充分加密网格的形状正则性；（ii）与稳定双线性形式无关。