Efficient numerical method for the Schr\"{o}dinger equation with high-contrast potentials

In this paper, we study the Schr\"{o}dinger equation in the semiclassical regime and with multiscale potential function. We develop the so-called constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM), in the framework of Crank-Nicolson (CN) discretization in time. The localized multiscale basis functions are constructed by addressing the spectral problem and a constrained energy minimization problem related to the Hamiltonian norm. A first-order convergence in the energy norm and second-order convergence in the $L^2$ norm for our numerical scheme are shown, with a relation between oversampling number in the CEM-GMsFEM method, spatial mesh size and the semiclassical parameter provided. Furthermore, we demonstrate the convergence of the proposed Crank-Nicolson CEM-GMsFEM scheme. The convergence requires $H/\sqrt{\Lambda}=O(\varepsilon^{\frac{5}{4}})$, $\Delta t=O(\varepsilon^{\frac{5}{4}})$ if $\varepsilon\leq \delta$; while if $\delta<\varepsilon$, the convergence requires $H/\sqrt{\Lambda}=O(\varepsilon^{\frac{1}{4}}\delta)$, $\Delta t=O(\frac{\delta^2}{\varepsilon^{3/4}})$ (where $H$ represents the maximum diameter of coarse elements, $\Lambda$ is the minimal eigenvalue associated with the eigenvector not included in the auxiliary space, $\Delta t$ is the time step, $0 < \varepsilon\ll 1$ is the Planck constant and $\delta$ describes the multiscale structure of the potential).Several numerical examples including 1D and 2D in space, with high-contrast potential are conducted to demonstrate the efficiency and accuracy of our proposed scheme.