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:在这项工作中,我们考虑的是单成份Bose-Einstein凝聚物(BECs)的地面状态和动态的计算。相应的模型是空间分解的,采用称为局部骨质分解(LOD)的多尺度有限元素法。尽管这种分解在BECs范围内具有突出的近似特性,但充分利用这种近似特性而不产生严重的计算瓶颈可能很困难。因此,我们在本文件中提出了两种完全分解的数字方法,其拟订方式特别考虑到LOD空间的结构。一种是计算地面状态,另一种是计算动态的另一种方法。本文的中心重点是讨论对实际实现方法非常重要的执行方面。特别是,我们讨论了使用适当的数据结构使记忆成本保持经济性。本文件最后以1d、2d和3d的各种数字实验结束,这些实验旨在调查方法的汇合率和近似特性,并表明其性能和计算效率,也与光谱和标准定点要素方法相比较。