Convergence Analysis of the Localized Orthogonal Decomposition Method for the Semiclassical Schrödinger Equations with Multiscale Potentials

Abstract

<p>We provide a convergence analysis of the localized orthogonal decomposition (LOD) method for Schrödinger equations with general multiscale potentials in the semiclassical regime. We focus on the influence of the semiclassical parameter and the multiscale structure of the potential on the numerical scheme. We construct localized multiscale basis functions by solving a constrained energy minimization problem in the framework of the LOD method. We show that localized multiscale basis functions with a smaller support can be constructed as the semiclassical parameter approaches zero. We obtain the first-order convergence in the energy norm and second-order convergence in the L^2 norm for the LOD method and super convergence rates can be obtained if the solution possesses sufficiently high regularity. We analyse the temporal and spatial regularity of the solution and find that the spatial derivatives are more oscillatory than the time derivatives in the presence of a multiscale potential. By combining the regularity analysis, we are able to derive the dependence of the error estimates on the small parameters of the Schrödinger equation. Moreover, we find that the LOD method outperforms the finite element method in the presence of a multiscale potential due to the super convergence for high-regularity solutions and more relaxed dependence of the error estimates on the small parameters for low-regularity solutions. Finally, we present numerical results to demonstrate the accuracy and robustness of the proposed method.<br></p>