Preconditioning of discontinuous Galerkin methods for second order elliptic problems

摘要

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 coarsespaces of either continuous piecewise polynomial functions or piecewise constant (discontinuous)functions. We show that both choices give rise to uniform, with respectto the mesh size, preconditioners. We also consider multilevel preconditioners basedon the same two types of coarse spaces. In the case when continuous coarse spacesare used, we prove that a variable V-cycle multigrid algorithm is a uniform preconditioner.We present numerical experiments illustrating the behavior of the consideredpreconditioners when applied to various test problems in three spatial dimensions.The numerical results confirm our theoretical results and in the cases not covered bythe theory show the efficiency of the proposed algorithms.Another approach for preconditioning the SIPG method that we consider is analgebraic multigrid algorithm using coarsening based on element agglomeration whichis suitable for unstructured meshes. We also consider an improved version of the algorithmusing a smoothed aggregation technique. We present numerical experimentsusing the proposed algorithms which show their efficiency as uniform preconditioners.For the method of Baumann and Oden we construct a preconditioner based onan orthogonal splitting of the discrete space into piecewise constant functions and functions with zero average over each element. We show that the preconditioneris uniformly spectrally equivalent to an appropriate symmetrization of the discreteequations when quadratic or higher order finite elements are used. In the case of linearelements we give a characterization of the kernel of the discrete system and presentnumerical evidence that the method has optimal convergence rates in both L2 andH1 norms. We present numerical experiments which show that the convergence ofthe proposed preconditioning technique is independent of the mesh size.