In this work, we consider the numerical computation of ground states and dynamics of single-component Bose-Einstein condensates (BECs). The corresponding models are spatially discretized with a multiscale finite element approach known as Localized Orthogonal Decomposition (LOD). Despite the outstanding approximation properties of such a discretization in the context of BECs, taking full advantage of it without creating severe computational bottlenecks can be tricky. In this paper, we therefore present two fully-discrete numerical approaches that are formulated in such a way that they take special account of the structure of the LOD spaces. One approach is devoted to the computation of ground states and another one for the computation of dynamics. A central focus of this paper is also the discussion of implementation aspects that are very important for the practical realization of the methods. In particular, we discuss the use of suitable data structures that keep the memory costs economical. The paper concludes with various numerical experiments in 1d, 2d and 3d that investigate convergence rates and approximation properties of the methods and which demonstrate their performance and computational efficiency, also in comparison to spectral and standard finite element approaches.

翻译：本文研究单组分玻色-爱因斯坦凝聚态基态与动力学的数值计算问题。我们采用称为局部正交分解的多尺度有限元方法对相应模型进行空间离散化。尽管该离散化方法在玻色-爱因斯坦凝聚态模拟中具有优异的逼近特性，但如何充分利用其优势而不产生严重计算瓶颈仍具挑战性。为此，本文提出两种充分考虑LOD空间结构的全离散数值方法：一种专用于基态计算，另一种用于动力学模拟。本文的核心重点还包括对算法实现中关键实践问题的探讨，特别讨论了采用节省内存开销的高效数据结构。最后通过一系列一维、二维及三维数值实验，验证了方法的收敛阶与逼近特性，并与谱方法及标准有限元方法进行对比，展示了所提方法的性能与计算效率。