Preconditioning High-Order Discontinuous Galerkin Discretizations of Elliptic Problems

In recent years, attention has been devoted to the development of efficient iterative solvers for the solution of the linear system of equations arising from the discontinuous Galerkin (DG) discretization of a range of model problems. In the framework of two level preconditioners, scalable non-overlapping Schwarz methods have been proposed and analyzed for the h–version of the DG method in the articles [1, 2, 6, 7, 9]. Recently, in [3] it has been proved that the non-overlapping Schwarz preconditioners can also be successfully employed to reduce the condition number of the stiffness matrices arising from a wide class of high–order DG discretizations of elliptic problems. In this article we aim to validate the theoretical results derived in [3] for the multiplicative Schwarz preconditioner and for its symmetrized variant by testing their numerical performance. This article is organized as follows. In Section 2 we introduce the model problem and its DG approximation. In Section 3 we construct the Schwarz preconditioners, and recall the main theoretical results shown in [3]. Finally, in Section 4 we present some numerical results obtained with the multiplicative Schwarz preconditioner and its symmetrized variant.