Finite Element Methods For Maxwell-schr(?)dinger Equations With Nonlocal Effect In Metamaterials

This paper mainly studies the following two problems.In the first part,a novel system of Maxwell-Schr（?）dinger equations with non-local effect in metamaterials is derived from the Drude model,hydrodynamical model and Schr（?）dinger equation.We assume that free electrons and bound elec-trons are separated,and they play different roles,respectively.By adding two new partial differential equations based on hydrodynamic model,the influence of nonlocal effect will be shown.The governing equation of magnetization will be proved by time harmonic technique.Under the conditions of magnetic Gauss law and temporal gauge field,the connection equation between electric field and vector potential function can be established.Through vector potential function,Schr（?）dinger equation and Maxwell system are coupled together.In this case,the coupling of wave function and vector potential function in Schr（?）dinger equa-tion can describe the quantum current density.In order to give an efficient and decoupled leap-frog finite element discrete scheme,we establish the exact initial approximation conditions.It is proved that the scheme is conditionally stable in the sense of energy norm.The proof of convergence of the discrete scheme is divided into three parts,and we can get the error estimate with the convergence rate of(²+?),whereis the time step size,?is the mesh size andis the order of basis functions defined in finite element spaces.Finally,numerical results are given to verify the theories.In the second part,the governing equation of wave propagation in metamate-rials is non-dimensionalized form,which simplifies the expression of the equation.The complete PML governing equation in the corner region is derived.And a leap-frog mixed finite element method is proposed to solve the PML model.Fur-thermore,the conditional stability and optimal error estimate for the proposed scheme are proved.