Abstract
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.
PH acknowledges the financial support of the EPSRC under the grant EP/H005498.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Bibliography
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.
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.
P. F. Antonietti and P. Houston. A class of domain decomposition preconditioners for h p-discontinuous Galerkin finite element methods. J. Sci. Comp., 46(1):124–149, 2011.
D. N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19(4):742–760, 1982.
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.
A. T. Barker, S. C. Brenner, P. Eun-Hee, and L.-Y. Sung. Two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method. J. Sci. Comp., 47:27–49, 2011.
S. C. Brenner and K. Wang. Two-level additive Schwarz preconditioners for C 0 interior penalty methods. Numer. Math., 102(2):231–255, 2005.
S. C. Eisenstat, H. C. Elman, and M. H. Schultz. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 20(2):345–357, 1983.
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.
A. Toselli and O. Widlund. Domain decomposition methods—algorithms and theory, volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2005.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Antonietti, P.F., Houston, P. (2013). Preconditioning High–Order Discontinuous Galerkin Discretizations of Elliptic Problems. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-35275-1_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35274-4
Online ISBN: 978-3-642-35275-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)