A CRANK-NICOLSON FINITE ELEMENT METHOD AND THE OPTIMAL ERROR ESTIMATES FOR THE MODIFIED TIME-DEPENDENT MAXWELL-SCHRODINGER EQUATIONS

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-infinity norm of the numerical solutions and get rid of certain restrictions like Delta t <= 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.