Efficient preconditioning for the discontinuous Galerkin finite element method by low-order elements

We derive and analyze a block diagonal preconditioner for the linear problems arising from a discontinuous Galerkin finite element discretization. The method can be applied to second-order self-adjoint elliptic boundary value problems and exploits the natural decomposition of the discrete function space into a global low-order subsystem and components of higher polynomial degree. Similar to results for the p-version of the conforming FEM, we prove for the interior penalty discontinuous Galerkin discretization that the condition number of the preconditioned system is uniformly bounded with respect to the mesh size of the triangulation. Numerical experiments demonstrate the performance of the method.