  We introduce and analyze two-level and multi-level preconditioners for a
family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of
second order elliptic problems with large jumps in the diffusion coefficient.
Our approach to IPDG-type methods is based on a splitting of the DG space into
two components that are orthogonal in the energy inner product naturally
induced by the methods. As a result, the methods and their analysis depend in a
crucial way on the diffusion coefficient of the problem. The analysis of the
proposed preconditioners is presented for both symmetric and non-symmetric IP
schemes; dealing simultaneously with the jump in the diffusion coefficient and
the non-nested character of the relevant discrete spaces presents extra
difficulties in the analysis which precludes a simple extension of existing
results. However, we are able to establish robustness (with respect to the
diffusion coefficient) and nearly-optimality (up to a logarithmic term
depending on the mesh size) for both two-level and BPX-type preconditioners.
Following the analysis, we present a sequence of detailed numerical results
which verify the theory and illustrate the performance of the methods. The
paper includes an Appendix with a collection of proofs of several technical
results required for the analysis.
