Convergence analysis of an operator-compressed multiscale finite element method for Schrödinger equations with multiscale potentials

by   Zhizhang Wu, et al.

In this paper, we analyze the convergence of the operator-compressed multiscale finite element method (OC MsFEM) for Schrödinger 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 Schrödinger 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.


page 1

page 2

page 3

page 4


Efficient multiscale methods for the semiclassical Schrödinger equation with time-dependent potentials

The semiclassical Schrödinger equation with time-dependent potentials is...

Super-localised wave function approximation of Bose-Einstein condensates

This paper presents a novel spatial discretisation method for the reliab...

Split representation of adaptively compressed polarizability operator

The polarizability operator plays a central role in density functional p...

Wavelet-based Edge Multiscale Finite Element Methods for Singularly Perturbed Convection-Diffusion Equations

We propose a novel efficient and robust Wavelet-based Edge Multiscale Fi...

The role of mesh quality and mesh quality indicators in the Virtual Element Method

Since its introduction, the Virtual Element Method (VEM) was shown to be...

Convergence analysis of nonconform H(div)-finite elements for the damped time-harmonic Galbrun's equation

We consider the damped time-harmonic Galbrun's equation, which is used t...

Super-localization of elliptic multiscale problems

Numerical homogenization aims to efficiently and accurately approximate ...

Please sign up or login with your details

Forgot password? Click here to reset