Multilevel methods for discontinuous Galerkin FEM on locally refined meshes

We review shortly the analytical results on multilevel preconditioning for DG methods for elliptic problems. An algorithm is presented which does not loose efficiency under local refinement. It uses subspace smoothing and does not require integration of additional matrices. This algorithm is applied to the interior penalty method and the local discontinuous Galerkin method. A feasible way of implementing the scheme is presented.