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A Symmetry-based Decomposition Approach to Eigenvalue Problems: Formulation, Discretization, and Implementation

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.