  In this work, we investigate the propagation of electromagnetic waves in the
Cole-Cole dispersive medium by using the discontinuous Galerkin (DG) method to
solve the coupled time-domain Maxwell's equations and polarization equation. We
define a new and sharpened total energy function for the Cole-Cole model, which
better describes the behaviors of the energy than what is available in the
current literature. A major theme in the time-domain numerical modeling of this
problem has been tackling the difficulty of handling the nonlocal term involved
in the time-domain polarization equation. Based on the diffusive representation
and the quadrature formula, we derive an approximate system, where the
convolution kernel is replaced by a finite number of auxiliary variables that
satisfy local-in-time ordinary differential equations. To ensure the resulted
approximate system is stable, a nonlinear constrained optimization numerical
scheme is established to determine the quadrature coefficients. By a special
choice of the numerical fluxes and projections, we obtain {for the constant
coefficient case } an optimal-order convergence result for the semi-discrete DG
scheme. The temporal discretization is achieved by the standard two-step
backward difference formula and a fast algorithm with linear complexity is
constructed. Numerical examples are provided for demonstrating the efficiency
of the proposed algorithm, validating the theoretical results and illustrating
the behaviors of the energy.
