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In this paper, we study a Crank--Nicolson finite element method for the modified Maxwell--Schrodinger equations, which are deduced from the time-dependent Maxwell--Schrodinger equations under some assumptions. We present an (almost) unconditionally optimal error estimate for the numerical algorithm which only requires the time step $\Delta t < 1$. The key to our analysis is twofold. For one thing, we use some tricks to tackle the tough nonlinear terms without applying the inverse inequality to estimate the $L^{\infty}$ norm of the numerical solutions and get rid of certain restrictions like $\Delta t \leq C \, h^{\alpha}$ on the time step. For another, we employ a new technique to weaken the time step restriction required by the application of the discrete Gronwall inequality. Numerical tests are carried out to verify the theoretical results.

A Crank--Nicolson Finite Element Method and the Optimal Error Estimates for the Modified Time-Dependent Maxwell--Schrödinger Equations