Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation

The main aim of this paper is to apply the conforming bilinear finite element to solve the nonlinear Schrödinger equation (NLSE). Firstly, the stability and convergence for time discrete scheme are proved. Secondly, through a new estimate approach, the optimal order error estimates and superclose properties in H1-norm are obtained with Backward Euler (B-E) and Crank-Nicolson (C-N) fully-discrete schemes, the global superconvergence results are deduced with the help of interpolation postprocessing technique. Finally, some numerical examples are provided to verify the theoretical analysis.