  A robust multilevel preconditioner based on the hybridizable discontinuous
Galerkin method for the Helmholtz equation with high wave number is presented
in this paper. There are two keys in our algorithm, one is how to choose a
suitable intergrid transfer operator, and the other is using GMRES smoothing on
coarse grids. The multilevel method is performed as a preconditioner in the
outer GMRES iteration. To give a quantitative insight of our algorithm, we use
local Fourier analysis to analyze the convergence property of the proposed
multilevel method. Numerical results show that for fixed wave number, the
convergence of the algorithm is mesh independent. Moreover, the performance of
the algorithm depends relatively mildly on wave number.
