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In this paper, an efficient multiscale finite element method via local defect-correction technique is developed. This method is used to solve the Schrödinger eigenvalue problem with three-dimensional domain. First, this paper considers a three-dimensional bounded spherical region, which is the truncation of a three-dimensional unbounded region. Using polar coordinate transformation, we successfully transform the three-dimensional problem into a series of one-dimensional eigenvalue problems. These one-dimensional eigenvalue problems also bring singularity. Second, using local refinement technique, we establish a new multiscale finite element discretization method. The scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. Finally, the error estimates of eigenvalues and eigenfunctions are also proved. Numerical examples show that our numerical method can significantly improve the accuracy of eigenvalues.

The Local and Parallel Finite Element Scheme for Electric Structure Eigenvalue Problems