This paper investigates numerical methods for approximating the ground state of Bose--Einstein condensates (BECs) by introducing two relaxed formulations of the Gross--Pitaevskii energy functional. These formulations achieve first- and second-order accuracy with respect to the relaxation parameter \( \tau \), and are shown to converge to the original energy functional as \( \tau \to 0 \). A key feature of the relaxed functionals is their concavity, which ensures that local minima lie on the boundary of the concave hull. This property prevents energy increases during constraint normalization and enables the development of energy-dissipative algorithms. Numerical methods based on sequential linear programming are proposed, accompanied by rigorous analysis of their stability with respect to the relaxed energy. To enhance computational efficiency, an adaptive strategy is introduced, dynamically refining solutions obtained with larger relaxation parameters to achieve higher accuracy with smaller ones. Numerical experiments demonstrate the stability, convergence, and energy dissipation of the proposed methods, while showcasing the adaptive strategy's effectiveness in improving computational performance.

翻译：本文通过引入Gross-Pitaevskii能量泛函的两种松弛形式，研究了玻色-爱因斯坦凝聚体基态计算的数值逼近方法。这些松弛泛函关于松弛参数τ具有一阶和二阶精度，并证明当τ→0时收敛于原始能量泛函。松弛泛函的关键特征在于其凹性，该性质保证局部极小值位于凹包边界上。这一特性可防止约束归一化过程中的能量增长，从而能够构建能量耗散算法。本文提出了基于序列线性规划的数值方法，并对其在松弛能量下的稳定性进行了严格分析。为提升计算效率，引入自适应策略动态优化较大松弛参数获得的解，从而以较小松弛参数实现更高精度。数值实验验证了所提方法的稳定性、收敛性与能量耗散特性，同时展示了自适应策略在提升计算性能方面的有效性。