Summary
Elliptic variational inequalities with multiple bodies are considered. It is assumed that an active set method is used to handle the nonlinearity of the inequality constraint, which results in auxiliary linear problems. We describe two domain decomposition methods for solving such linear problems, namely, the FETI-FETI (finite element tearing and interconnecting) and hybrid methods, which are combinations of already existing domain decomposition methods. Estimates of the condition numbers of both methods are provided. The FETI-FETI method has a condition number which depends linearly on the number of subdomains across each body and polylogarithmically on the number of element across each subdomain. The hybrid method is a scalable alternative to the FETI-FETI method, and has a condition number with two polylogarithmic factors depending on the number of elements across each subdomain and across each body. We present numerical results confirming these theoretical findings.
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Philip Avery and Charbel Farhat. The FETI family of domain decomposition methods for inequality-constrained quadratic programming: Application to contact problems with conforming and nonconforming interfaces. Computer Methods in Applied Mechanics and Engineering, 198(21-26):1673–1683, 2009. Advances in Simulation-Based Engineering Sciences - Honoring J. Tinsley Oden.
Philip Avery, Gert Rebel, Michel Lesoinne, and Charbel Farhat. A numerically scalable dual-primal substructuring method for the solution of contact problems–part I: the frictionless case. Comput. Methods Appl. Mech. Engrg., 193(23-26):2403–2426, 2004.
Clark R. Dohrmann. A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput., 25(1):246–258, 2003.
Charbel Farhat, Jan Mandel, and François-Xavier Roux. Optimal convergence properties of the FETI domain decomposition method. Comput. Methods Appl. Mech. Engrg., 115(3-4):365–385, 1994.
Charbel Farhat, Michel Lesoinne, Patrick LeTallec, Kendall Pierson, and Daniel Rixen. FETI-DP: a dual-primal unified FETI method. I. A faster alternative to the two-level FETI method. Internat. J. Numer. Methods Engrg., 50(7):1523–1544, 2001.
Axel Klawonn and Oliver Rheinbach. A hybrid approach to 3-level FETI. PAMM Proc. Appl. Math. Mech., 8(1):10841–10843, 2008.
Axel Klawonn and Oliver Rheinbach. Highly scalable parallel domain decomposition methods with an application to biomechanics. ZAMM Z. Angew. Math. Mech., 90(1):5–32, 2010.
Axel Klawonn and Olof B. Widlund. A domain decomposition method with Lagrange multipliers and inexact solvers for linear elasticity. SIAM J. Sci. Comput., 22(4):1199–1219, 2000.
Axel Klawonn and Olof B. Widlund. Dual-primal FETI methods for linear elasticity. Comm. Pure Appl. Math., 59(11):1523–1572, 2006.
Axel Klawonn, Olof B. Widlund, and Maksymilian Dryja. Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM J. Numer. Anal., 40(1):159–179, 2002.
Jungho Lee. Two domain decomposition methods for auxiliary linear problems arising from an active-set based treatment of a multibody elliptic variational inequality. 2010. Accepted with minor revisions, SIAM J. Sci. Comput.
Jungho Lee. A Hybrid Domain Decomposition Method and its Applications to Contact Problems. PhD thesis, Courant Institute of Mathematical Sciences, September 2009.
Jing Li and Olof B. Widlund. FETI-DP, BDDC, and block Cholesky methods. Internat. J. Numer. Methods Engrg., 66(2):250–271, 2006.
Andrea Toselli and Olof Widlund. Domain decomposition methods—algorithms and theory, volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2005.
Acknowledgements
This work was supported by the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract DE-AC02-06CH11357.
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Lee, J. (2013). Domain Decomposition Methods for Auxiliary Linear Problems of an Elliptic Variational Inequality. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_35
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DOI: https://doi.org/10.1007/978-3-642-35275-1_35
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