  This paper proposes a fully discrete method called the symplectic dG full
discretization for stochastic Maxwell equations driven by additive noises,
based on a stochastic symplectic method in time and a discontinuous Galerkin
(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 make 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.
