Unconditional optimal error estimates for BDF2-FEM for a nonlinear Schrödinger equation

Abstract

This paper analyzes unconditional optimal error estimates for a 2-step backward differentiation formula (BDF2) method for a nonlinear Schrödinger equation. In the analysis, we split an error estimate into two parts, one from the spatial discretization and the other from the temporal discretization. We present the boundedness of the solution of the time-discrete system in the certain strong norms, and the error estimates for time discretization. By these boundedness and temporal error estimates, we obtain the L2 error estimates without any conditions on a time step size. Numerical experiments are provided to validate our analysis and check the efficiency of our method.