Error estimates of the Crank-Nicolson Galerkin method for the time-dependent Maxwell-Schr\"{o}dinger equations under the Lorentz gauge

In this paper we study the numerical method and the convergence for solving the time-dependent Maxwell-Schr"{o}dinger equations under the Lorentz gauge. An alternating Crank-Nicolson finite element method for solving the problem is presented and the optimal error estimate for the numerical algorithm is obtained by a mathematical inductive method. Numerical examples are then carried out to confirm the theoretical results.