Previous |  Up |  Next


two-level domain decomposition; hybrid FETI; Schur complement; bounds on the spectrum
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.
[1] Brenner, S. C.: The condition number of the Schur complement in domain decomposition. Numer. Math. 83 (1999), 187-203. DOI 10.1007/s002110050446 | MR 1712684 | Zbl 0936.65141
[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). DOI 10.1016/S0377-0427(99)00368-4 | MR 1806768 | Zbl 0970.65074
[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. DOI 10.1002/cnm.881 | MR 2282408 | Zbl 1107.65104
[4] Dostál, Z., Kozubek, T., Sadowská, M., Vondrák, V.: Scalable Algorithms for Contact Problems. Advances in Mechanics and Mathematics 36, Springer, New York (2016). DOI 10.1007/978-1-4939-6834-3 | MR 3586594 | Zbl 06658822
[5] Farhat, C., Lesoinne, M., Pierson, K.: A scalable dual-primal domain decomposition method. Numer. Linear Algebra Appl. 7 (2000), 687-714. DOI 10.1002/1099-1506(200010/12)7:7/8<687::AID-NLA219>3.0.CO;2-S | MR 1802366 | Zbl 1051.65119
[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. DOI 10.1016/0045-7825(94)90068-X | MR 1285024
[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. DOI 10.1002/nme.1620320604 | MR 3618550 | Zbl 0758.65075
[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. DOI 10.1137/0913020 | MR 1145192 | Zbl 0746.65086
[9] Klawonn, A., Rheinbach, O.: A hybrid approach to 3-level FETI. Proc. Appl. Math. Mech. 8 (2008), 10841-10843. DOI 10.1002/pamm.200810841
[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. DOI 10.1002/zamm.200900329 | MR 2603676 | Zbl 1355.65169
[11] Lee, J.: Two domain decomposition methods for auxiliary linear problems for a multibody variational inequality. SIAM J. Sci. Comput. 35 (2013), A1350--A1375. DOI 10.1137/100783753 | MR 3055241 | Zbl 1276.65037
[12] Lukáš, D., Bouchala, J., Vodstrčil, P., Malý, L.: 2-dimensional primal domain decomposition theory in detail. Appl. Math., Praha 60 (2015), 265-283. DOI 10.1007/s10492-015-0095-5 | MR 3419962 | Zbl 1363.65215
[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). DOI 10.1007/978-3-642-23588-7 | MR 3013465 | Zbl 1272.65100
[14] Toselli, A., Widlund, O. B.: Domain Decomposition Methods---Algorithms and Theory. Springer Series on Computational Mathematics 34, Springer, Berlin (2005). MR 2104179 | Zbl 1069.65138
Partner of
EuDML logo