@Article{CiCP-24-1073,
author = {Shuangping Li and Zhiwen Zhang},
title = {Computing Eigenvalues and Eigenfunctions of Schrödinger Equations Using a Model Reduction Approach},
journal = {Communications in Computational Physics},
year = {2018},
volume = {24},
number = {4},
pages = {1073--1100},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.2018.hh80.08},
url = {http://global-sci.org/intro/article_detail/cicp/12319.html}
}