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

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 trunc...

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Bibliographic Details
Published in:Mathematical problems in engineering 2021, Vol.2021, p.1-11
Main Authors: Lin, Fubiao, Cao, Junying, Liu, Zhixin
Format: Article
Language:English
Subjects:
Algorithms
Boundary value problems
Coordinate transformations
Efficiency
Eigenvalues
Eigenvectors
Estimates
Finite element method
Mathematical analysis
Methods
Numerical analysis
Numerical methods
Polar coordinates
Singularities
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Summary: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.
ISSN:1024-123X
1563-5147
DOI:10.1155/2021/1049917