Nonconforming Finite Element Approximation and Energy Lower Bound Estimation for the Gross--Pitaevskii Energy Functional

The ground state of Bose–Einstein condensates can be described as the minimizer of the Gross–Pitaevskii energy functional subject to a mass conservation constraint. In this paper, we study the corresponding discrete optimization problem in nonconforming finite element spaces and establish a priori error estimates for the discrete ground state energy, the discrete eigenvalue, and the discrete ground state. Specifically, we derive explicit convergence rates for the a priori error in the particular case of the $E{Q}_1^{\mathrm{r}\mathrm{o}\mathrm{t}}$ finite element. Furthermore, we proof that within the $E{Q}_1^{\mathrm{r}\mathrm{o}\mathrm{t}}$ finite element framework, the discrete ground state energy provides a lower bound estimation to the exact energy. Finally, numerical experiments are presented to validate the theoretical analysis.