  In this paper we consider a class of robust multilevel precontioners for the
Helmholtz equation with high wave number. The key idea in this work is to use
the continuous interior penalty finite element methods (CIP-FEM) studied in
\cite{Wu12,Wu12-hp} to construct the stable coarse grid correction problems.
The multilevel methods, based on GMRES smoothing on coarse grids, are then
served as a preconditioner in the outer GMRES iteration. In the one dimensional
case, convergence property of the modified multilevel methods is analyzed by
the local Fourier analysis. From our numerical results, we find that the
proposed methods are efficient for a reasonable range of frequencies. The
performance of the algorithms depends relatively mildly on wave number. In
particular, only one GMRES smoothing step may guarantee the optimal convergence
of our multilevel algorithm, which remedies the shortcoming of the multilevel
algorithm in \cite{EEO01}.
