Super-localised wave function approximation of Bose-Einstein condensates

TL;DR

This paper introduces a new spatial discretisation technique combining homogenisation and super-localisation for simulating Bose-Einstein condensates, demonstrating high accuracy and efficiency in capturing complex physical phenomena.

Contribution

The paper presents a novel super-localisation approach for basis function calculation, improving accuracy and efficiency over classical methods in Bose-Einstein condensate simulations.

Findings

Superconvergence compared to classical methods.

Effective in extreme physical conditions like vortex formation.

Capable of simulating phase transitions reliably.

Abstract

This paper presents a novel spatial discretisation method for the reliable and efficient simulation of Bose-Einstein condensates modelled by the Gross-Pitaevskii equation and the corresponding nonlinear eigenvector problem. The method combines the high-accuracy properties of numerical homogenisation methods with a novel super-localisation approach for the calculation of the basis functions. A rigorous numerical analysis demonstrates superconvergence of the approach compared to classical polynomial and multiscale finite element methods, even in low regularity regimes. Numerical tests reveal the method's competitiveness with spectral methods, particularly in capturing critical physical effects in extreme conditions, such as vortex lattice formation in fast-rotating potential traps. The method's potential is further highlighted through a dynamic simulation of a phase transition from Mott insulator to Bose-Einstein condensate, emphasising its capability for reliable exploration of physical phenomena.