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We apply the local discontinuous Galerkin (LDG for short) method to solve amixed boundary value problems for the Helmholtz equation in bounded polygonaldomain in 2D. Under some assumptions on regularity of the solution of anadjoint problem, we prove that: (a) the corresponding indefinite discretescheme is well posed; (b) there is convergence with the expected convergencerates as long as the meshsize h is small enough. We give precise information onhow small h has to be in terms of the size of the wavenumber and its distanceto the set of eigenvalues for the same boundary value problem for theLaplacian. We also present an a posteriori error estimator showing both thereliability and efficiency of the estimator complemented with detailedinformation on the dependence of the constants on the wavenumber. We finishpresenting extensive numerical experiments which illustrate the theoreticalresults proven in this paper and suggest that stability and convergence mayoccur under less restrictive assumptions than those taken in the present work.

On the discontinuous Galerkin method for solving boundary value problems for the Helmholtz equation: A priori and a posteriori error analyses