TY - JOUR T1 - Computing Eigenvalues and Eigenfunctions of Schrödinger Equations Using a Model Reduction Approach AU - Shuangping Li & Zhiwen Zhang JO - Communications in Computational Physics VL - 4 SP - 1073 EP - 1100 PY - 2018 DA - 2018/06 SN - 24 DO - http://doi.org/10.4208/cicp.2018.hh80.08 UR - https://global-sci.org/intro/article_detail/cicp/12319.html KW - Schrödinger equation, eigenvalue problems, model reduction, two-level techniques, problem dependent basis functions, computational chemistry. AB -

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.