Convergence analysis of an operator-compressed multiscale finite element method for Schr\"{o}dinger equations with multiscale potentials

TL;DR

This paper analyzes the convergence of an operator-compressed multiscale finite element method (OC MsFEM) for Schrödinger equations with multiscale potentials, demonstrating its efficiency and robustness in the semiclassical regime.

Contribution

It provides the first convergence analysis of OC MsFEM for Schrödinger equations with multiscale potentials, including exponential decay of basis functions and error estimates.

Findings

First-order convergence in energy norm

Second-order convergence in L2 norm

Super convergence for high-regularity solutions

Abstract

In this paper, we analyze the convergence of the operator-compressed multiscale finite element method (OC MsFEM) for Schr\"{o}dinger equations with general multiscale potentials in the semiclassical regime. In the OC MsFEM the multiscale basis functions are constructed by solving a constrained energy minimization. Under a mild assumption on the mesh size $H$, we prove the exponential decay of the multiscale basis functions so that localized multiscale basis functions can be constructed, which achieve the same accuracy as the global ones if the oversampling size $m = O(\log(1/H))$. We prove the first-order convergence in the energy norm and second-order convergence in the $L^2$ norm for the OC MsFEM and super convergence rates can be obtained if the solution possesses sufficiently high regularity. By analysing the regularity of the solution, we also derive the dependence of the error estimates on the small parameters of the Schr\"{o}dinger equation. We find that the OC MsFEM outperforms the finite element method (FEM) due to the super convergence behavior for high-regularity solutions and weaker dependence on the small parameters for low-regularity solutions in the presence of the multiscale potential. Finally, we present numerical results to demonstrate the accuracy and robustness of the OC MsFEM.