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：旋转系中玻色-爱因斯坦凝聚体的基态可描述为带有角动量项的Gross-Pitaevskii能量泛函在约束条件下的极小化问题。本文研究任意多项式阶Lagrange有限元空间中对应的离散极小化问题，并探讨离散基态的逼近性质。特别地，我们证明了$L^2$-范数和$H^1$-范数下的最优阶先验误差估计，以及基态能量与相应化学势的误差估计。该问题的分析核心在于基态的非唯一性——这主要由能量泛函在复相移变换下的不变性导致。因此，我们的误差分析基于一个限制在特定切空间上的Euler-Lagrange泛函，在该切空间中基态具有局部唯一性。由此得到的误差恒等式最终被用于推导所需的先验误差估计。我们还通过数值实验展示了问题结构的多个方面。