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

Advances in Difference Equations

Yubing Sui¹  Jun Zhang²  Junying Cao³  Donghao Zhang⁴ 

[1] College of Economics, Shenzhen University, 518060, Shenzhen, China;Guizhou Key Laboratory of Big Data Statistics Analysis, Guizhou University of Finance and Economics, 550025, Guiyang, China;School of Data Science and Information Engineering, Guizhou Minzu University, 550025, Guiyang, China;School of Insurance, Southwestern University of Finance and Economics, 610074, Chendu, China;

关键词: Eigenvalue problem;    Singularity;    Dimension reduction scheme;    Finite element method;    Error estimation;    65N15;    65N30;   

DOI  :  10.1186/s13662-020-03034-9

来源: Springer

【 摘 要 】

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.

【 授权许可】

CC BY