Crank-Nicolson Galerkin approximations to nonlinear Schr\"odinger equations with disorder potentials

Abstract

This paper analyses the numerical solution of a class of non-linear Schr\"odinger 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 weak and strong disorder potentials, the consideration of some general class of non-linearities, and the proof of convergence with rates in \(L^{\infty}(L^2)\) 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.