  In this paper, we propose a decomposition approach for eigenvalue problems
with spatial symmetries, including the formulation, discretization as well as
implementation. This approach can handle eigenvalue problems with either
Abelian or non-Abelian symmetries, and is friendly for grid-based
discretizations such as finite difference, finite element or finite volume
methods. With the formulation, we divide the original eigenvalue problem into a
set of subproblems and require only a smaller number of eigenpairs for each
subproblem. We implement the decomposition approach with finite elements and
parallelize our code in two levels. We show that the decomposition approach can
improve the efficiency and scalability of iterative diagonalization. In
particular, we apply the approach to solving Kohn--Sham equations of symmetric
molecules consisting of hundreds of atoms.
