Super-localised wave function approximation of Bose-Einstein condensates

This paper presents a novel spatial discretisation method for 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 of the nonlinear eigenvector problem demonstrates superconvergence of the ideal approach compared to classical polynomial and multiscale finite element methods, even in low regularity regimes. Numerical tests show that the super-localised method is competitive 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.