  In [Dai et al, Multi. Model. Simul., 2020], a structure-preserving gradient
flow method was proposed for the ground state calculation in Kohn-Sham density
functional theory, based on which a linearized method was developed in [Hu, et
al, EAJAM, accepted] for further improving the numerical efficiency. In this
paper, a complete convergence analysis is delivered for such a linearized
method for the all-electron Kohn-Sham model. Temporally, the convergence, the
asymptotic stability, as well as the structure-preserving property of the
linearized numerical scheme in the method is discussed following previous
works, while spatially, the convergence of the h-adaptive mesh method is
demonstrated following [Chen et al, Multi. Model. Simul., 2014], with a key
study on the boundedness of the Kohn-Sham potential for the all-electron
Kohn-Sham model. Numerical examples confirm the theoretical results very well.
