Multiscale algorithms and computations for the time-dependent Maxwell--Schrödinger system in heterogeneous nanostructures

Abstract

In this paper, we discuss the multiscale computations of the time-dependent Maxwell-Schrödinger system with rapidly oscillating discontinuous coefficients. The multiscale asymptotic method for the system is presented. We propose a novel multiscale asymptotic expansion for the vector potential to capture the oscillations caused by the quantum current density. To solve the homogenized Maxwell-Schrödinger system, we present an alternating Crank-Nicolson finite element method. The stability estimates and the solvability of the discrete system are established. An iteration algorithm together with its convergence analysis is given. Numerical examples are carried out to demonstrate the efficiency and accuracy of the multiscale algorithms.