SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2197-2217, January 2021.
This paper proposes a fully discrete method called the symplectic discontinuous Galerkin (dG) full discretization for stochastic Maxwell equations driven by additive noises, based on a stochastic symplectic method in time and a dG method with the upwind fluxes in space. A priori $H^k$-regularity ($k\in\{1,2\}$) estimates for the solution of stochastic Maxwell equations are presented, which have not been reported before to the best of our knowledge. These $H^k$-regularities are vital to making the assumptions of the mean-square convergence analysis on the initial fields, the noise, and the medium coefficients, but not on the solution itself. The convergence order of the symplectic dG full discretization is shown to be $k/2$ in the temporal direction and $k-1/2$ in the spatial direction. Meanwhile we reveal the small noise asymptotic behaviors of the exact and numerical solutions via the large deviation principle, and show that the fully discrete method preserves the divergence relations in a weak sense.

中文翻译：

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随机麦克斯韦方程的辛不连续伽辽金完全离散化

SIAM 数值分析杂志，第 59 卷，第 4 期，第 2197-2217 页，2021 年 1 月。
本文基于时间上的随机辛方法和空间上风通量的 dG 方法，针对加性噪声驱动的随机麦克斯韦方程提出了一种称为辛不连续伽辽金 (dG) 完全离散化的完全离散方法。给出了随机麦克斯韦方程解的先验 $H^k$-正则性 ($k\in\{1,2\}$) 估计值，据我们所知，之前还没有报道过。这些 $H^k$ 正则性对于对初始场、噪声和介质系数进行均方收敛分析的假设至关重要，但对解本身却不是。辛 dG 完全离散化的收敛顺序在时间方向上为 $k/2$，在空间方向上为 $k-1/2$。