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.

翻译：本文提出了一种新颖的空间离散化方法，用于可靠且高效地模拟由Gross-Pitaevskii方程及其对应的非线性特征向量问题描述的玻色-爱因斯坦凝聚体。该方法将数值均匀化方法的高精度特性与一种用于计算基函数的新型超局域化方法相结合。严格的数值分析表明，即使在低正则性条件下，与经典多项式及多尺度有限元方法相比，该方法具有超收敛性。数值实验揭示了该方法在捕捉极端条件下的关键物理效应（如快速旋转势阱中的涡旋晶格形成）方面的竞争力，可与谱方法相媲美。通过从莫特绝缘体到玻色-爱因斯坦凝聚体的相变动力学模拟，进一步凸显了该方法在可靠探索物理现象方面的潜力。