For the ground state of the Gross-Pitaevskii (GP) eigenvalue problem, we consider a fully discretized Sobolev gradient flow, which can be regarded as the Riemannian gradient descent on the sphere under a metric induced by a modified $H^1$-norm. We prove its global convergence to a critical point of the discrete GP energy and its local exponential convergence to the ground state of the discrete GP energy. The local exponential convergence rate depends on the eigengap of the discrete GP energy. When the discretization is the classical second-order finite difference in two dimensions, such an eigengap can be further proven to be mesh independent, i.e., it has a uniform positive lower bound, thus the local exponential convergence rate is mesh independent. Numerical experiments with discretization by high order $Q^k$ spectral element methods in two and three dimensions are provided to validate the efficiency of the proposed method.

翻译：针对Gross-Pitaevskii（GP）特征值问题的基态，本文考虑了一种完全离散的Sobolev梯度流，该梯度流可视为在由修正$H^1$范数诱导的度量下球面上的黎曼梯度下降。我们证明了该梯度流全局收敛于离散GP能量的临界点，并局部指数收敛于离散GP能量的基态。局部指数收敛速率取决于离散GP能量的特征值间隙。当离散化采用二维经典二阶有限差分时，该特征值间隙可进一步证明具有网格无关性，即存在一致正下界，因此局部指数收敛速率与网格无关。通过二维和三维高阶$Q^k$谱元法离散化的数值实验，验证了所提方法的有效性。