We propose a positivity preserving finite element discretization for the nonlinear Gross-Pitaevskii eigenvalue problem. The method employs mass lumping techniques, which allow to transfer the uniqueness up to sign and positivity properties of the continuous ground state to the discrete setting. We further prove that every non-negative discrete excited state up to sign coincides with the discrete ground state. This allows one to identify the limit of fully discretized gradient flows, which are typically used to compute the discrete ground state, and thereby establish their global convergence. Furthermore, we perform a rigorous a priori error analysis of the proposed non-standard finite element discretization, showing optimal orders of convergence for all unknowns. Numerical experiments illustrate the theoretical results of this paper.

翻译：本文针对非线性Gross-Pitaevskii特征值问题，提出了一种保持正性的有限元离散化方法。该方法采用集中质量技术，使得连续基态在符号意义下的唯一性及正性得以保持到离散情形。我们进一步证明，每个非负离散激发态（在符号意义下）均与离散基态重合。这一性质使得我们可以识别完全离散化梯度流（通常用于计算离散基态）的极限，从而建立其全局收敛性。此外，我们对所提出的非标准有限元离散化进行了严格的先验误差分析，证明了所有未知量均达到最优收敛阶。数值实验验证了本文的理论结果。