# A note on optimal $H^1$-error estimates for Crank-Nicolson approximations to the nonlinear Schr\"odinger equation

## TL;DR

This paper establishes optimal $H^1$-error estimates for a mass- and energy-conserving Crank-Nicolson scheme applied to nonlinear Schr"odinger equations, filling a gap in theoretical analysis and providing practical implementation insights.

## Contribution

The paper proves optimal $H^1$-error bounds for the Crank-Nicolson method in both semi-discrete and fully-discrete settings, and introduces an efficient fixed point iteration for solving the nonlinear system.

## Key findings

- Proved optimal $L^{ty}(H^1)$-error estimates for the scheme.
- Demonstrated the efficiency of the fixed point iteration.
- Validated theoretical results with numerical experiments.

## Abstract

In this paper we consider a mass- and energy--conserving Crank-Nicolson time discretization for a general class of nonlinear Schr\"odinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal $L^{\infty}(H^1)$-error estimates is still open, both in the semi-discrete Hilbert space setting, as well as in fully-discrete finite element settings. This paper aims at closing this gap in the literature. We also suggest a fixed point iteration to solve the arising nonlinear system of equations that makes the method easy to implement and efficient. This is illustrated by numerical experiments.