  We propose to use a hybridizable discontinuous Galerkin (HDG) method combined
with the continuous Galerkin (CG) method to approximate Maxwell's equations. We
make two contributions in this paper. First, even though there are many papers
using HDG methods to approximate Maxwell's equations, to our knowledge they all
assume that the coefficients are smooth (or constant). Here, we derive optimal
convergence estimates for our HGD-CG approximation when the electromagnetic
coefficients are {\em piecewise} smooth. This requires new techniques of
analysis. Second, we use CG elements to approximate the Lagrange multiplier
used to enforce the divergence condition and we obtain a discrete system in
which we can decouple the discrete the Lagrange multiplier. Because we are
using a continuous Lagrange multiplier space, the number of degrees of freedom
devoted to this are less than for other HDG methods. We present numerical
experiments to confirm our theoretical results.
