In this paper, we consider the time dependent Maxwell’s equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell’s equations when dispersive media are involved.

In this paper, we consider the time-dependent Maxwell’s equations when Cole–Cole dispersive medium is involved. The Cole–Cole model contains a fractional time derivative term, which couples with the standard Maxwell’s equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1–26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572–591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533–1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12–20], we propose two fully discrete mixed finite element schemes for the Cole–Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.

In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectangle cloaking structures. Numerical results using these material parameters are presented to demonstrate the cloaking effect.

In this paper, we consider an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. In Huang and Li (J. Sci. Comput., 42 (2009), 321–340), for both semi and fully discrete DG schemes, we proved error estimates which are optimal in the energy norm, but sub-optimal in the L2-norm. Here by filling this gap, we show that these schemes are optimally convergent in the L2-norm on quasi-uniform tetrahedral meshes if the solution is sufficiently smooth.

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