Year: 2018
Communications in Computational Physics, Vol. 24 (2018), Iss. 4 : pp. 1073–1100
Abstract
We present a model reduction approach to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Schrödinger equations. The basis functions are defined on coarse meshes and obtained through solving an optimization problem. We shall show that the basis functions span a low-dimensional generalized finite element space that accurately preserves the lowermost eigenvalues and eigenfunctions of the stationary Schrödinger equations. Therefore, our method avoids the application of eigenvalue solver on fine-scale discretization and offers considerable savings in solving eigenvalues and eigenfunctions of Schrödinger equations. The construction of the basis functions are independent of each other; thus our method is perfectly parallel. We also provide error estimates for the eigenvalues obtained by our new method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method, especially Schrödinger equations with double well potentials are tested.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.2018.hh80.08
Communications in Computational Physics, Vol. 24 (2018), Iss. 4 : pp. 1073–1100
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 28
Keywords: Schrödinger equation eigenvalue problems model reduction two-level techniques problem dependent basis functions computational chemistry.