  In this paper, based on a domain decomposition (DD) method, we shall propose
an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method
for solving the algebraic eigenvalue problem resulting from the edge element
approximation of the Maxwell eigenvalue problem. In order to eliminate the
components in orthogonal complement space of the eigenvalue, we shall solve a
parallel preconditioned system and a Helmholtz projection system together in
fine space. After one coarse space correction in each iteration and minimizing
the Rayleigh quotient in a small dimensional Davidson space, we finally get the
error reduction of this two-level PHJD method as
$\gamma=c(H)(1-C\frac{\delta^{2}}{H^{2}})$, where $C$ is a constant independent
of the mesh size $h$ and the diameter of subdomains $H$, $\delta$ is the
overlapping size among the subdomains, and $c(H)$ decreasing as $H\to 0$, which
means the greater the number of subdomains, the better the convergence rate.
Numerical results supporting our theory shall be given.
