  Two local discontinuous Galerkin (LDG) methods using some non-standard
numerical fluxes are developed for the Helmholtz equation with the first order
absorbing boundary condition in the high frequency regime. It is shown that the
proposed LDG methods are absolutely stable (hence well-posed) with respect to
both the wave number and the mesh size. Optimal order (with respect to the mesh
size) error estimates are proved for all wave numbers in the preasymptotic
regime. To analyze the proposed LDG methods, they are recasted and treated as
(non-conforming) mixed finite element methods. The crux of the analysis is to
establish a generalized {\em inf-sup} condition, which holds without any mesh
constraint, for each LDG method. The generalized {\em inf-sup} conditions then
easily infer the desired absolute stability of the proposed LDG methods. In
return, the stability results not only guarantee the well-posedness of the LDG
methods but also play a crucial role in the derivation of the error estimates.
Numerical experiments, which confirm the theoretical results and compare the
proposed two LDG methods, are also presented in the paper.
