An error analysis of a mixed discontinuous Galerkin (DG) method with lifting operators as numerical fluxes for the time-harmonic Maxwell equations with minimal smoothness requirements is presented. The key difficulty in the error analysis for the DG method is that due to the low regularity the tangential trace of the exact solution is not well defined on the faces of the computational mesh. This difficulty is addressed by adopting the face-to-cell lifting introduced by Ern & Guermond (2021, Quasi-optimal nonconforming approximation of elliptic PDEs with contrasted coefficients and $H^{1+r}$, $r>0$, regularity. Found. Comput. Math., 1–36). To obtain optimal local interpolation estimates, we introduce Scott–Zhang-type interpolations that are well defined for $H(\textrm {curl})$ and $H(\textrm {div})$ functions with minimal regularity requirements. As a by-product of penalizing the lifting of the tangential jumps, an explicit and easily computable stabilization parameter is given.

中文翻译：

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具有最小平滑度要求的时谐 Maxwell 方程的混合不连续 Galerkin 方法分析

提出了以提升算子为数值通量的混合不连续 Galerkin (DG) 方法的误差分析，该方法适用于具有最小平滑度要求的时谐 Maxwell 方程。DG 方法误差分析的主要困难在于，由于规则性低，精确解的切线轨迹在计算网格的面上没有很好地定义。通过采用 Ern & Guermond（2021, Quasi-optimal nonconforming approximation of elliptic PDEs with contrast coefficients and $H^{1+r}$, $r>0$, regularity . 发现. 计算机. 数学., 1-36)。为了获得最佳的局部插值估计，我们引入了 Scott–Zhang 类型的插值，它为 $H(\textrm {curl})$ 和 $H(\textrm {div})$ 函数定义良好，具有最小的规律性要求。作为惩罚切向跳跃提升的副产品，给出了一个明确且易于计算的稳定参数。