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

Abstract

AbstractWe 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 and the 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.