  This paper develops and analyzes some interior penalty discontinuous Galerkin
methods using piecewise linear polynomials for the Helmholtz equation with the
first order absorbing boundary condition in the two and three dimensions. It is
proved that the proposed discontinuous Galerkin methods are stable (hence
well-posed) without any mesh constraint. For each fixed wave number $k$,
optimal order (with respect to $h$) error estimate in the broken $H^1$-norm and
sub-optimal order estimate in the $L^2$-norm are derived without any mesh
constraint. The latter estimate improves to optimal order when the mesh size
$h$ is restricted to the preasymptotic regime (i.e., $k^2 h \gtrsim 1$).
Numerical experiments are also presented to gauge the theoretical result and to
numerically examine the pollution effect (with respect to $k$) in the error
bounds. The novelties of the proposed interior penalty discontinuous Galerkin
methods include: first, the methods penalize not only the jumps of the function
values across the element edges but also the jumps of the normal and tangential
derivatives; second, the penalty parameters are taken as complex numbers of
positive imaginary parts so essentially and practically no constraint is
imposed on the penalty parameters. Since the Helmholtz problem is a
non-Hermitian and indefinite linear problem, as expected, the crucial and the
most difficult part of the whole analysis is to establish the stability
estimates (i.e., a priori estimates) for the numerical solutions. To the end,
the cruxes of our analysis are to establish and to make use of a local version
of the Rellich identity (for the Laplacian) and to mimic the stability analysis
for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}.
