Numerical convergence and physical fidelity analysis for Maxwell’s equations in metamaterials

In this paper, we develop a leap-frog mixed finite element method for solving Maxwell’s equations resulting from metamaterials. Our scheme is similar to the popular Yee’s FDTD scheme used in electrical engineering community, and is preferable for three dimensional large scale modeling since no storage of the large coefficient matrix is needed. Our scheme is proved to obey the Gauss’s law automatically if the initial fields satisfy that. Furthermore, the conditional stability and optimal error estimate for the proposed scheme are proved. To our best knowledge, we are unaware of any other publications devoted to the convergence analysis of this leap-frog explicit scheme for Maxwell’s equations even in a simple medium, while our results for metamaterials automatically reduce to the standard Maxwell’s equations in vacuum by dropping some terms resulting from the constitutive equations. Numerical results confirming our analysis are presented.