  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.
