Developing finite element methods for maxwell's equations in a cole-cole dispersive medium

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.