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.