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

TL;DR

This paper develops and analyzes a numerical method combining Galerkin finite elements and a Crank-Nicolson scheme for solving nonlinear Schrödinger equations with rough, discontinuous potentials, ensuring convergence and optimal rates.

Contribution

It introduces a novel analysis accommodating rough potentials and general nonlinearities, proving convergence with rates under minimal regularity assumptions.

Findings

Convergence proven for rough, discontinuous potentials.

Optimal rates achieved for smooth potentials.

Method conserves mass and energy.

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 rough, discontinuous potentials in the context of weak and strong disorder, 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.