  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.
