Abstract
It has been shown in [4] that block Jacobi iterates of a discretization obtained from hybridizable discontinuous Galerkin methods (HDG) can be viewed as non-overlapping Schwarz methods with Robin transmission condition. The Robin parameter is exactly the penalty parameter μ of the HDG method. There is a stability constraint on the penalty parameter and the usual choice of μ results in slow convergence of the Schwarz method. In this paper we show how to overcome this problem without changing μ.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
B. Cockburn, J. Gopalakrishnan, R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)
R.E. Ewing, J. Wang, Y. Yang, A stabilized discontinuous finite element method for elliptic problems. Numer. Linear Algebra Appl. 10(1–2), 83–104 (2003)
M.J. Gander, Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)
M.J. Gander, S. Hajian, Block Jacobi for discontinuous Galerkin discretizations: no ordinary Schwarz methods, in Domain Decomposition Methods in Science and Engineering XXI, ed. by J. Erhel, M.J. Gander, L. Halpern, G. Pichot, T. Sassi, O. Widlund. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2013)
M.J. Gander, S. Hajian, Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization. SIAM J. Numer. Anal. 53(1), 573–597 (2015)
C. Lehrenfeld, Hybrid discontinuous Galerkin methods for incompressible flow problems. Master’s thesis, RWTH Aachen, 2010
Acknowledgement
The author thanks Martin J. Gander for his useful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Hajian, S. (2016). An Optimized Schwarz Algorithm for a Discontinuous Galerkin Method. In: Dickopf, T., Gander, M., Halpern, L., Krause, R., Pavarino, L. (eds) Domain Decomposition Methods in Science and Engineering XXII. Lecture Notes in Computational Science and Engineering, vol 104. Springer, Cham. https://doi.org/10.1007/978-3-319-18827-0_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-18827-0_24
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18826-3
Online ISBN: 978-3-319-18827-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)