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

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