On the convergence of a linearly implicit finite element method for the nonlinear Schrödinger equation

Abstract

We consider a model initial- and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second-order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal-order error estimates in the $L^2$ and the $H^1$ norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a Courant-Friedrichs-Lewy (CFL) condition between the space mesh size and the time step sizes.