Abstract
In this paper, we present two variants of the Additive Schwarz Method (ASM) for a Crouzeix-Raviart finite volume (CRFV) discretization of the second order elliptic problem with discontinuous coefficients, where the discontinuities are only across subdomain boundaries. The resulting system, which is nonsymmetric, is solved using the preconditioned GMRES iteration, where in one variant of the ASM the preconditioner is symmetric while in the other variant it is nonsymmetric. The proposed methods are almost optimal, in the sense that the convergence of the GMRES iteration, in the both cases, depend only poly-logarithmically on the mesh parameters.
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
Similar content being viewed by others
References
S.C. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comput. 65(215), 897–921 (1996)
S.C. Brenner, L.-Y. Sung, Balancing domain decomposition for nonconforming plate elements. Numer. Math. 83(1), 25–52 (1999)
X.-C. Cai, O.B. Widlund, Domain decomposition algorithms for indefinite elliptic problems. SIAM J. Sci. Stat. Comput. 13(1), 243–258 (1992)
P. Chatzipantelidis, A finite volume method based on the Crouzeix-Raviart element for elliptic PDE’s in two dimensions. Numer. Math. 82(3), 409–432 (1999)
S.H. Chou, J. Huang, A domain decomposition algorithm for general covolume methods for elliptic problems. J. Numer. Math. 11(3), 179–194 (2003)
S.C. Eisenstat, H.C. Elman, M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20(2), 345–357 (1983)
A. Loneland, L. Marcinkowski, T. Rahman, Additive average Schwarz method for the Crouzeix-Raviart finite volume element discretization of elliptic problems (2014a, submitted)
A. Loneland, L. Marcinkowski, T. Rahman, Edge based Schwarz methods for the Crouzeix-Raviart finite volume element discretization of elliptic problems (2014b, to appear in ETNA in 2015)
L. Marcinkowski, T. Rahman, Neumann-Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems. BIT Numer. Math. 48(3), 607–626 (2008)
L. Marcinkowski, T. Rahman, J. Valdman, Additive Schwarz preconditioner for the general finite volume element discretization of symmetric elliptic problems. Tech. Report 204, Institute of Applied Mathematics and Mechanics, University of Warsaw (2014) [Published online in arXiv:1405.0185] [math.NA]
M. Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math. 77(3), 383–406 (1997)
A. Toselli, O. Widlund, Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2005)
S. Zhang, On domain decomposition algorithms for covolume methods for elliptic problems. Comput. Methods Appl. Mech. Eng. 196(1–3), 24–32 (2006)
Acknowledgements
This work was partially supported by Polish Scientific Grant 2011/01/B/ ST1/01179 and Chinese Academy of Science Project: 2013FFGA0009 - GJHS20140901004635677.
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
Marcinkowski, L., Loneland, A., Rahman, T. (2016). Schwarz Methods for a Crouzeix-Raviart Finite Volume Discretization of Elliptic Problems. 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_61
Download citation
DOI: https://doi.org/10.1007/978-3-319-18827-0_61
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)