Efficient multiscale methods for the semiclassical Schr\"odinger equation with time-dependent potentials

TL;DR

This paper introduces two multiscale finite element methods for efficiently solving the semiclassical Schrödinger equation with time-dependent potentials, significantly improving accuracy over standard methods while maintaining comparable computational complexity.

Contribution

The paper develops two novel multiscale finite element approaches with basis enrichment for better accuracy in solving the semiclassical Schrödinger equation with time-dependent potentials.

Findings

Multiscale methods achieve at least 100 times smaller error than standard finite element methods.

Basis enrichment further reduces approximation error by an additional order of magnitude.

The methods maintain computational complexity similar to standard finite element methods.

Abstract

The semiclassical Schr\"odinger equation with time-dependent potentials is an important model to study electron dynamics under external controls in the mean-field picture. In this paper, we propose two multiscale finite element methods to solve this problem. In the offline stage, for the first approach, the localized multiscale basis functions are constructed using sparse compression of the Hamiltonian operator at the initial time; for the latter, basis functions are further enriched using a greedy algorithm for the sparse compression of the Hamiltonian operator at later times. In the online stage, the Schr\"odinger equation is approximated by these localized multiscale basis in space and is solved by the Crank-Nicolson method in time. These multiscale basis have compact supports in space, leading to the sparsity of stiffness matrix, and thus the computational complexity of these two methods in the online stage is comparable to that of the standard finite element method. However, the spatial mesh size in multiscale finite element methods is $ H=\mathcal{O}(\epsilon) $, while $H=\mathcal{O}(\epsilon^{3/2})$ in the standard finite element method, where $\epsilon$ is the semiclassical parameter. By a number of numerical examples in 1D and 2D, for approximately the same number of basis, we show that the approximation error of the multiscale finite element method is at least two orders of magnitude smaller than that of the standard finite element method, and the enrichment further reduces the error by another one order of magnitude.