  In this paper, a novel adaptive finite element method is proposed to solve
the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive
technique and the augmented subspace method. Different from the classical
self-consistent field iterative algorithm which requires to solve the Kohn-Sham
equation directly in each adaptive finite element space, our algorithm
transforms the Kohn-Sham equation into some linear boundary value problems of
the same scale in each adaptive finite element space, and then the
wavefunctions derived from the linear boundary value problems are corrected by
solving a small-scale Kohn-Sham equation defined in a low-dimensional augmented
subspace. Since the new algorithm avoids solving large-scale Kohn-Sham equation
directly, a significant improvement for the solving efficiency can be obtained.
In addition, the adaptive moving mesh technique is used to generate the
nonnested adaptive mesh for the nonnested augmented subspace method according
to the singularity of the approximate wavefunctions. The modified Hessian
matrix of the approximate wavefunctions is used as the metric matrix to
redistribute the mesh. Through the moving mesh adaptive technique, the
redistributed mesh is almost optimal. A number of numerical experiments are
carried out to verify the efficiency and the accuracy of the proposed
algorithm.
