  We apply the local discontinuous Galerkin (LDG for short) method to solve a
mixed boundary value problems for the Helmholtz equation in bounded polygonal
domain in 2D. Under some assumptions on regularity of the solution of an
adjoint problem, we prove that: (a) the corresponding indefinite discrete
scheme is well posed; (b) there is convergence with the expected convergence
rates as long as the meshsize h is small enough. We give precise information on
how small h has to be in terms of the size of the wavenumber and its distance
to the set of eigenvalues for the same boundary value problem for the
Laplacian. We also present an a posteriori error estimator showing both the
reliability and efficiency of the estimator complemented with detailed
information on the dependence of the constants on the wavenumber. We finish
presenting extensive numerical experiments which illustrate the theoretical
results proven in this paper and suggest that stability and convergence may
occur under less restrictive assumptions than those taken in the present work.
