An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

We provide an effective finite element method to solve the Schrödinger eigenvalue problem with an inverse potential on a spherical domain. To overcome the difficulties caused by the singularities of coefficients, we introduce spherical coordinate transformation and transfer the singularities from th...

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Bibliographic Details
Published in:Advances in difference equations Vol. 2020; no. 1; pp. 1 - 15
Main Authors: Sui, Yubing, Zhang, Donghao, Cao, Junying, Zhang, Jun
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 17.10.2020
SpringerOpen
Subjects:
Analysis
Applications
Difference and Functional Equations
Dimension reduction scheme
Eigenvalue problem
Error estimation
Finite element method
Functional Analysis
Mathematics
Mathematics and Statistics
Methods
Ordinary Differential Equations
Partial Differential Equations
Singularity
Topics in Special Functions and q-Special Functions: Theory
Finite element method
Singularity
65N15
Error estimation
Eigenvalue problem
Dimension reduction scheme
65N30
ISSN:1687-1847, 1687-1847
Online Access:Get full text
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Summary:We provide an effective finite element method to solve the Schrödinger eigenvalue problem with an inverse potential on a spherical domain. To overcome the difficulties caused by the singularities of coefficients, we introduce spherical coordinate transformation and transfer the singularities from the interior of the domain to its boundary. Then by using orthogonal properties of spherical harmonic functions and variable separation technique we transform the original problem into a series of one-dimensional eigenvalue problems. We further introduce some suitable Sobolev spaces and derive the weak form and an efficient discrete scheme. Combining with the spectral theory of Babuška and Osborn for self-adjoint positive definite eigenvalue problems, we obtain error estimates of approximation eigenvalues and eigenvectors. Finally, we provide some numerical examples to show the efficiency and accuracy of the algorithm.
ISSN:1687-1847
1687-1847
DOI:10.1186/s13662-020-03034-9