The ground states of Bose-Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross-Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the $L^2$- and $H^1$-norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler-Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to error identities that are ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.
翻译:Bose-Einstein 凝结点在旋转框架中的地面状态可被描述为以角动力值表示的Gross-Pitaevskii能源功能的制约最小化因素。 在本文中,我们考虑了在任意多元顺序的Lagrange有限元素空间中相应的离散最小化问题,并调查了离散地面状态的近距离特性。 特别是,我们证明了一个先验错误估计,即美元2美元和1美元- 诺尔姆的最佳顺序,以及地面状态能量和相应的化学潜力。 在分析这一问题时,一个核心问题是缺少地面状态的独特性,这主要是在复杂的阶段变化中能源功能的不稳定性造成的。 因此,我们的错误分析基于一种Euler-Lagrange功能,我们将它限制在我们具有地方独特性地表的某些相近距离空间。 这导致了错误识别,最终被用来得出预期的先验误差估计数。 我们还进行了数字实验,以说明问题结构的各个方面。</s>