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  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  for the multiplicative Schwarz preconditioner and for its symmetrized variant by testing their numerical performance.
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- .P. F. Antonietti and B. Ayuso. Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. M2AN Math. Model. Numer. Anal., 41(1):21–54, 2007.Google Scholar
- .P. F. Antonietti and B. Ayuso. Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems. M2AN Math. Model. Numer. Anal., 42(3):443–469, 2008.Google Scholar
- .D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779 (electronic), 2001/02.Google Scholar
- .X. Feng and O. A. Karakashian. Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal., 39(4):1343–1365 (electronic), 2001.Google Scholar
- .A. Toselli and O. Widlund. Domain decomposition methods—algorithms and theory, volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2005.Google Scholar