Adaptive numerical solution of a discontinuous Galerkin method for a Helmholtz problem in low-frequency regime

• We derive an a posteriori error estimator, that is reliable and locally quasi-efficient.
• The efficiency and reliability constants have been characterized with respect to the physical parameter ωω.
• We include numerical examples that validate our theoretical results, recognizing singularities and inner layers, for moderate values of ωω.

We develop an a posteriori error analysis for Helmholtz problem using the local discontinuous Galerkin (LDG for short) approach. For the sake of completeness, we give a description of the main a priori results of this method. Indeed, 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) the approach is convergent, with the expected convergence rates as long as the meshsize hh is small enough. We give precise information on how small hh has to be in terms of the size of the wave number and its distance to the set of eigenvalues for the same boundary value problem for the Laplacian. After that, we present a reliable and efficient a posteriori error estimator with detailed information on the dependence of the constants on the wave number. 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.