Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients

In this article we develop and analyze two-level and multi-level methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coefficients (exhibiting large jumps across interfaces in the domain). These methods are based on a decomposition of the DG finite element space that inherently hinges on the diffusion coefficient of the problem. Our analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes, and we establish both robustness with respect to the jump in the coefficient and near-optimality with respect to the mesh size. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods.