Unconditional convergence and optimal error estimates of the Euler semi-implicit scheme for a generalized nonlinear Schrödinger equation

In this paper, we focus on a linearized backward Euler scheme with a Galerkin finite element approximation for the time-dependent nonlinear Schrodinger equation. By splitting an error estimate into two parts, one from the spatial discretization and the other from the temporal discretization, we obtain unconditionally optimal error estimates of the fully-discrete backward Euler method for a generalized nonlinear Schrodinger equation. Numerical results are provided to support our theoretical analysis and efficiency of this method.