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http://dx.doi.org/10.4134/JKMS.j190632

PRECONDITIONERS FOR A COUPLED PROBLEM BY A PENALTY TERM ARISEN IN AN AUGMENTED LAGRANGIAN METHOD  

Lee, Chang-Ock (Department of Mathematical Sciences KAIST)
Park, Eun-Hee (Division of Liberal Studies Kangwon National University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.5, 2020 , pp. 1267-1286 More about this Journal
Abstract
We pay attention to a coupled problem by a penalty term which is induced from non-overlapping domain decomposition methods based on augmented Lagrangian methodology. The coupled problem is composed by two parts mainly; one is a problem associated with local problems in non-overlapping subdomains and the other is a coupled part over all subdomains due to the penalty term. For the speedup of iterative solvers for the coupled problem, we propose two different types of preconditioners: a block-diagonal preconditioner and an additive Schwarz preconditioner as overlapping domain decomposition methods. We analyze the coupled problem and the preconditioned problems in terms of their condition numbers. Finally we present numerical results which show the performance of the proposed methods.
Keywords
Coupled problem; penalty term; domain decomposition; preconditioners; additive Schwarz;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 O. Axelsson and B. Polman, Block preconditioning and domain decomposition methods. II, J. Comput. Appl. Math. 24 (1988), no. 1-2, 55-72. https://doi.org/10.1016/0377-0427(88)90343-3   DOI
2 A. T. Barker, S. C. Brenner, E.-H. Park, and L.-Y. Sung, Two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method, J. Sci. Comput. 47 (2011), no. 1, 27-49. https://doi.org/10.1007/s10915-010-9419-5   DOI
3 S. C. Brenner, The condition number of the Schur complement in domain decomposition, Numer. Math. 83 (1999), no. 2, 187-203. https://doi.org/10.1007/s002110050446   DOI
4 S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, third edition, Texts in Applied Mathematics, 15, Springer, New York, 2008. https://doi.org/10.1007/978-0-387-75934-0
5 M. Brezina and P. Vanek, A black-box iterative solver based on a two-level Schwarz method, Computing 63 (1999), no. 3, 233-263. https://doi.org/10.1007/s006070050033   DOI
6 M. Dryja and O. B. Widlund, An additive variant of the Schwarz alternating method in the case of many subregions, Technical Report 339, Department of Computer Science, Courant Institute, 1987.
7 M. Dryja and O. B. Widlund, Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput. 15 (1994), no. 3, 604-620. https://doi.org/10.1137/0915040   DOI
8 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 (2001), no. 4, 1343-1365. https://doi.org/10.1137/S0036142900378480   DOI
9 R. Glowinski and P. Le Tallec, Augmented Lagrangian interpretation of the nonover-lapping Schwarz alternating method, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), 224-231, SIAM, Philadelphia, PA, 1990.
10 J. Koko and T. Sassi, Augmented Lagrangian domain decomposition method for bonded structures, in Domain decomposition methods in science and engineering XXII, 551-558, Lect. Notes Comput. Sci. Eng., 104, Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-18827-0_56
11 O. A. Bauchau, Parallel computation approaches for flexible multibody dynamics simulations, J. Franklin Inst. 347 (2010), no. 1, 53-68. https://doi.org/10.1016/j.jfranklin.2009.10.001   DOI
12 J. Kwak, T. Chun, S. Shin, and O. A. Bauchau, Domain decomposition approach to flexible multibody dynamics simulation, Comput. Mech. 53 (2014), no. 1, 147-158. https://doi.org/10.1007/s00466-013-0898-8   DOI
13 P. Le Tallec and T. Sassi, Domain decomposition with nonmatching grids: augmented Lagrangian approach, Math. Comp. 64 (1995), no. 212, 1367-1396. https://doi.org/10.2307/2153360   DOI
14 C.-O. Lee and E.-H. Park, A dual iterative substructuring method with a penalty term, Numer. Math. 112 (2009), no. 1, 89-113. https://doi.org/10.1007/s00211-008-0202-6   DOI
15 C.-O. Lee and E.-H. Park, A dual iterative substructuring method with a penalty term in three dimensions, Comput. Math. Appl. 64 (2012), no. 9, 2787-2805. https://doi.org/10.1016/j.camwa.2012.04.011   DOI
16 C.-O. Lee and E.-H. Park, A dual iterative substructuring method with a small penalty parameter, J. Korean Math. Soc. 54 (2017), no. 2, 461-477. https://doi.org/10.4134/JKMS.j160061   DOI
17 Y. Saad, Iterative Methods for Sparse Linear Systems, second edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. https://doi.org/10.1137/1.9780898718003
18 A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer Series in Computational Mathematics, 34, Springer-Verlag, Berlin, 2005. https://doi.org/10.1007/b137868