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：本文研究了单组分玻色-爱因斯坦凝聚体（BECs）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（LOD）的多尺度有限元方法对相应模型进行空间离散化。尽管这种离散化方法在BECs背景下具有卓越的逼近特性，但如何充分利用其优势而不造成严重计算瓶颈仍具挑战性。为此，本文提出了两种充分考虑LOD空间结构特性的全离散数值方案：一种用于基态计算，另一种用于动力学计算。本文还重点讨论了对于方法实际实现至关重要的实施细节，特别是探讨了能有效控制内存开销的合适数据结构。最后，通过一维、二维和三维空间中的系列数值实验，验证了方法的收敛率与逼近特性，并与谱方法及标准有限元方法对比展示了其性能与计算效率。