Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

Abstract

Abstract. We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h), where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal. 20 (1983) 345-357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.Mathematics Subject Classification. 65N30, 65N55.

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

“…Our starting point is that, since we want to enrich the discrete space either by refining the mesh size or increasing the local polynomial approximation order, the preconditioner has to be efficient in both regimes, namely the h-and p-versions. We extend to p-DGFEMs the results shown in [24,3,5] for non-overlapping Schwarz methods in the context of the hversion DGFEM. Working in a quite general setting, we prove spectral bounds for the preconditioned stiffness matrix arising in the p-DGFEM of order p 2 .…”

“…[40,32], for example), the question of developing efficient iterative solvers for the solution of the resulting (linear) system of equations has been addressed only recently and only in the framework of the h-version of the DGFEM (h-DGFEM). For example, a wide class of domain decomposition methods for discontinuous Galerkin approximations of elliptic problems has been proposed and analyzed in [24,34,12,27,3,5,21]. We point out that, because of differences in the variational formulation associated with the underlying discontinuous polynomial spaces, the stiffness matrices arising from discontinuous Galerkin approximations possess different sparsity structures compared to those from conforming methods, and indeed, for a given mesh and polynomial degree distribution, the underlying matrix is typically larger in the DGFEM setting.…”

“…On the other hand, the underlying non-conformity of the DGFEM spaces can be successfully exploited in the development of efficient preconditioners leading to situations which have no analogue in the conforming case. For example, in [24,3,5], it has been shown that in the case of h-DGFEMs for elliptic problems, optimal non-overlapping Schwarz preconditioners can be constructed and analyzed leading to spectral bounds of the preconditioned systems of order H/h, where H and h represent the granularity of the coarse and fine meshes, respectively. We remark that, for conforming discretizations, iterative solvers in the classical Schwarz framework can be constructed only by employing overlapping subdomain partitions; indeed, in this setting spectral bounds of order H/h can be obtained when the subdomain partitions possess a minimal overlap.…”

“…In this work, we focus on the p-version of the discontinuous Galerkin finite element method (p-DGFEM), addressing the question of whether the class of non-overlapping Schwarz preconditioners introduced in [24,3,5] for the h-DGFEM can successfully be extended to the p-and hp-versions of the DGFEM. Our starting point is that, since we want to enrich the discrete space either by refining the mesh size or increasing the local polynomial approximation order, the preconditioner has to be efficient in both regimes, namely the h-and p-versions.…”

Preconditioning High–Order Discontinuous Galerkin Discretizations of Elliptic 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.…”