  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.
