The Finite Element Method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation

TL;DR

This paper develops and analyzes a finite element method for simulating the time-dependent Gross-Pitaevskii equation with rotation, providing error estimates and demonstrating effectiveness through numerical experiments.

Contribution

It introduces a mass conserving Crank-Nicolson-type discretization for the rotating Gross-Pitaevskii equation and proves optimal error estimates under regularity assumptions.

Findings

Optimal convergence rates achieved in numerical experiments

Error estimates validated for maximum norm in time and energy norm in space

Method effectively simulates rotating Bose-Einstein condensates

Abstract

We consider the time-dependent Gross-Pitaevskii equation describing the dynamics of rotating Bose-Einstein condensates and its discretization with the finite element method. We analyze a mass conserving Crank-Nicolson-type discretization and prove corresponding a priori error estimates with respect to the maximum norm in time and the $L^2$- and energy-norm in space. The estimates show that we obtain optimal convergence rates under the assumption of additional regularity for the solution to the Gross-Pitaevskii equation. We demonstrate the performance of the method in numerical experiments.