Crank-Nicolson Galerkin approximations to nonlinear Schr'odinger equations with rough potentials

摘要

This paper analyses the numerical solution of a class of non-linearSchr\"odinger equations by Galerkin finite elements in space and a mass- andenergy conserving variant of the Crank-Nicolson method due to Sanz-Serna intime. The novel aspects of the analysis are the incorporation of rough,discontinuous potentials in the context of weak and strong disorder, theconsideration of some general class of non-linearities, and the proof ofconvergence with rates in $L^{\infty}(L^2)$ under moderate regularityassumptions that are compatible with discontinuous potentials. For sufficientlysmooth potentials, the rates are optimal without any coupling condition betweenthe time step size and the spatial mesh width.