A leap-frog finite element method for wave propagation of Maxwell-Schrödinger equations with nonlocal effect in metamaterials

Abstract

Abstract In this paper, a novel system of Maxwell–Schrodinger equations with nonlocal effect in metamaterials is derived from the Drude model, hydrodynamical model and Schrodinger equation. A leap-frog finite element scheme, which can be solved one by one efficiently, is constructed by presenting a group of initial values. This scheme is proved to be stable conditionally in energy norm. It is confirmed that the error convergent rate is O ( τ 2 + h r ) by splitting the proof into three parts, where τ is the time step, h is the mesh size and r is the maximum total degree of polynomials in finite element spaces. Finally, some numerical results are given to verify the theories.