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

Abstract

The semiclassical Schrödinger 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 Schrödinger equation is approximated by these localized multiscale basis functions in space and is solved by the Crank–Nicolson method in time. These multiscale basis functions have compact supports in space, leading to the sparsity of the stiffness matrix, and thus the computational complexity of these two methods is comparable to that of the standard finite element method. The spatial mesh size in the multiscale finite element methods is O(ε), while the mesh size in the standard finite element method is o(ε), where ε is the semiclassical parameter. Through 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.