The modified transmission eigenvalue (mTE) problem, introduced by Cogar et al. (Inverse Probl. 33, 125002, 2017) and Audibert et al. (Inverse Probl. 33, 125011, 2017), plays a fundamental role in the theoretical and numerical investigations of the inverse medium problem. In this paper, we study the discontinuous Galerkin discretization for the mTE problem with absorbing media and present a complete a priori and a posteriori error analysis. We then design and implement an adaptive algorithm based on the a posteriori error estimator, and provide numerical experiments to show that the proposed method is effective and the approximate eigenvalues can reach the optimal convergence order on both polygonal domains and the disk domain.
