A symplectic discontinuous Galerkin full discretization for stochastic Maxwell equations

Abstract

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.