| Title:
|
On conditioning of Schur complements of H-TFETI clusters for 2D problems governed by Laplacian (English) |
| Author:
|
Vodstrčil, Petr |
| Author:
|
Bouchala, Jiří |
| Author:
|
Jarošová, Marta |
| Author:
|
Dostál, Zdeněk |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
62 |
| Issue:
|
6 |
| Year:
|
2017 |
| Pages:
|
699-718 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities. (English) |
| Keyword:
|
two-level domain decomposition |
| Keyword:
|
hybrid FETI |
| Keyword:
|
Schur complement |
| Keyword:
|
bounds on the spectrum |
| MSC:
|
34B16 |
| MSC:
|
34C25 |
| idZBL:
|
Zbl 06861552 |
| idMR:
|
MR3745747 |
| DOI:
|
10.21136/AM.2017.0193-17 |
| . |
| Date available:
|
2018-01-02T13:47:19Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147004 |
| . |
| Reference:
|
[1] Brenner, S. C.: The condition number of the Schur complement in domain decomposition.Numer. Math. 83 (1999), 187-203. Zbl 0936.65141, MR 1712684, 10.1007/s002110050446 |
| Reference:
|
[2] Dostál, Z., Neto, F. A. M. Gomes, Santos, S. A.: Duality-based domain decomposition with natural coarse-space for variational inequalities.J. Comput. Appl. Math. 126 397-415 (2000). Zbl 0970.65074, MR 1806768, 10.1016/S0377-0427(99)00368-4 |
| Reference:
|
[3] Dostál, Z., Horák, D., Kučera, R.: Total FETI---an easier implementable variant of the FETI method for numerical solution of elliptic PDE.Commun. Numer. Methods Eng. 22 (2006), 1155-1162. Zbl 1107.65104, MR 2282408, 10.1002/cnm.881 |
| Reference:
|
[4] Dostál, Z., Kozubek, T., Sadowská, M., Vondrák, V.: Scalable Algorithms for Contact Problems.Advances in Mechanics and Mathematics 36, Springer, New York (2016). Zbl 06658822, MR 3586594, 10.1007/978-1-4939-6834-3 |
| Reference:
|
[5] Farhat, C., Lesoinne, M., Pierson, K.: A scalable dual-primal domain decomposition method.Numer. Linear Algebra Appl. 7 (2000), 687-714. Zbl 1051.65119, MR 1802366, 10.1002/1099-1506(200010/12)7:7/8<687::AID-NLA219>3.0.CO;2-S |
| Reference:
|
[6] Farhat, C., Mandel, J., Roux, F.-X.: Optimal convergence properties of the FETI domain decomposition method.Comput. Methods Appl. Mech. Eng. 115 (1994), 365-385. MR 1285024, 10.1016/0045-7825(94)90068-X |
| Reference:
|
[7] Farhat, C., Roux, F.-X.: A method of finite element tearing and interconnecting and its parallel solution algorithm.Int. J. Numer. Methods Eng. 32 (1991), 1205-1227. Zbl 0758.65075, MR 3618550, 10.1002/nme.1620320604 |
| Reference:
|
[8] Farhat, C., Roux, F.-X.: An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems.SIAM J. Sci. Stat. Comput. 13 (1992), 379-396. Zbl 0746.65086, MR 1145192, 10.1137/0913020 |
| Reference:
|
[9] Klawonn, A., Rheinbach, O.: A hybrid approach to 3-level FETI.Proc. Appl. Math. Mech. 8 (2008), 10841-10843. 10.1002/pamm.200810841 |
| Reference:
|
[10] Klawonn, A., Rheinbach, O.: Highly scalable parallel domain decomposition methods with an application to biomechanics.ZAMM, Z. Angew. Math. Mech. 90 (2010), 5-32. Zbl 1355.65169, MR 2603676, 10.1002/zamm.200900329 |
| Reference:
|
[11] Lee, J.: Two domain decomposition methods for auxiliary linear problems for a multibody variational inequality.SIAM J. Sci. Comput. 35 (2013), A1350--A1375. Zbl 1276.65037, MR 3055241, 10.1137/100783753 |
| Reference:
|
[12] Lukáš, D., Bouchala, J., Vodstrčil, P., Malý, L.: 2-dimensional primal domain decomposition theory in detail.Appl. Math., Praha 60 (2015), 265-283. Zbl 1363.65215, MR 3419962, 10.1007/s10492-015-0095-5 |
| Reference:
|
[13] Pechstein, C.: Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems.Lecture Notes in Computational Science and Engineering 90, Springer, Berlin (2013). Zbl 1272.65100, MR 3013465, 10.1007/978-3-642-23588-7 |
| Reference:
|
[14] Toselli, A., Widlund, O. B.: Domain Decomposition Methods---Algorithms and Theory.Springer Series on Computational Mathematics 34, Springer, Berlin (2005). Zbl 1069.65138, MR 2104179 |
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