  In this paper two types of multgrid methods, i.e., the Rayleigh quotient
iteration and the inverse iteration with fixed shift, are developed for solving
the Maxwell eigenvalue problem with discontinuous relative magnetic
permeability and electric permittivity. With the aid of the mixed form of
source problem associated with the eigenvalue problem, we prove the uniform
convergence of the discrete solution operator to the solution operator in
$L^2(\Omega)$ using discrete compactness of edge element space. Then we prove
the asymptotically optimal error estimates for both multigrid methods.
Numerical experiments confirm our theoretical analysis.
