Preconditioning of discontinuous Galerkin methods for second order elliptic problems

Abstract

We consider algorithms for preconditioning of two discontinuous Galerkin (DG) methods for second order elliptic problems, namely the symmetric interior penalty (SIPG) method and the method of Baumann and Oden. For the SIPG method we first consider two-level preconditioners using coarse spaces of either continuous piecewise polynomial functions or piecewise constant (discontinuous) functions. We show that both choices give rise to uniform, with respect to the mesh size, preconditioners. We also consider multilevel preconditioners based on the same two types of coarse spaces. In the case when continuous coarse spaces are used, we prove that a variable V-cycle multigrid algorithm is a uniform preconditioner. We present numerical experiments illustrating the behavior of the considered preconditioners when applied to various test problems in three spatial dimensions. The numerical results confirm our theoretical results and in the cases not covered by the theory show the efficiency of the proposed algorithms. Another approach for preconditioning the SIPG method that we consider is an algebraic multigrid algorithm using coarsening based on element agglomeration which is suitable for unstructured meshes. We also consider an improved version of the algorithm using a smoothed aggregation technique. We present numerical experiments using the proposed algorithms which show their efficiency as uniform preconditioners. For the method of Baumann and Oden we construct a preconditioner based on an orthogonal splitting of the discrete space into piecewise constant functions and functions with zero average over each element. We show that the preconditioner is uniformly spectrally equivalent to an appropriate symmetrization of the discrete equations when quadratic or higher order finite elements are used. In the case of linear elements we give a characterization of the kernel of the discrete system and present numerical evidence that the method has optimal convergence rates in both L2 and H1 norms. We present numerical experiments which show that the convergence of the proposed preconditioning technique is independent of the mesh size.