Computing Eigenvalues and Eigenfunctions of Schrödinger Equations Using a Model Reduction Approach (Communications in Computational Physics, 2020)
We present a model reduction approach to construct problem dependent basisfunctions and compute eigenvalues and eigenfunctions of stationary Schrödingerequations. The basis functions are defined on coarse meshes and obtained throughsolving an optimization problem. We shall show that the basis functions span a low-dimensionalgeneralized finite element space that accurately preserves the lowermosteigenvalues and eigenfunctions of the stationary Schrödinger equations. Therefore,our method avoids the application of eigenvalue solver on fine-scale discretization andoffers considerable savings in solving eigenvalues and eigenfunctions of Schrödingerequations. The construction of the basis functions are independent of each other; thusour method is perfectly parallel. We also provide error estimates for the eigenvaluesobtained by our new method. Numerical results are presented to demonstrate the accuracyand efficiency of the proposed method, especially Schrödinger equations withdouble well potentials are tested.