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In this paper, based on a domain decomposition (DD) method, we shall proposean efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) methodfor solving the algebraic eigenvalue problem resulting from the edge elementapproximation of the Maxwell eigenvalue problem. In order to eliminate thecomponents in orthogonal complement space of the eigenvalue, we shall solve aparallel preconditioned system and a Helmholtz projection system together infine space. After one coarse space correction in each iteration and minimizingthe Rayleigh quotient in a small dimensional Davidson space, we finally get theerror reduction of this two-level PHJD method as$\gamma=c(H)(1-C\frac{\delta^{2}}{H^{2}})$, where $C$ is a constant independentof the mesh size $h$ and the diameter of subdomains $H$, $\delta$ is theoverlapping size among the subdomains, and $c(H)$ decreasing as $H\to 0$, whichmeans the greater the number of subdomains, the better the convergence rate.Numerical results supporting our theory shall be given.

A Two-Level Preconditioned Helmholtz-Jacobi-Davidson Method for the Maxwell Eigenvalue Problem