Crank–Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials

Abstract

This paper analyzes the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass and energy conserving variant of the Crank–Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of nonlinearities, and the proof of convergence with rates in [Formula: see text] under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.