# Article

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Keywords:
FETI-DP; Crouzeix-Raviart element; nonstandard mortar condition; preconditioner
Summary:
In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by \$(1+\log (H/h))^2\$, where \$H\$ and \$h\$ are mesh sizes. Finally, numerical tests are presented to verify the theoretical results.
References:
[1] Bernardi, C., Maday, Y., Patera, A. T.: A new nonconforming approach to domain decomposition: The mortar element method. H. Brezis et al. Nonlinear Partial Differential Equations and their Applications Collège de France Seminar, Vol. XI, Paris, France, 1989-1991 Logman Scientific & Technical. Pitman Res. Notes Math. Ser. 299 (1994), 13-51. MR 1268898 | Zbl 0797.65094
[2] Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods (3rd ed.). Texts in Applied Mathematics 15 Springer, New York (2008). MR 2373954 | Zbl 1135.65042
[3] Dryja, M., Widlund, O. B.: A generalized FETI-DP method for a mortar discretization of elliptic problems. Domain Decomposition Methods in Science and Engineering I. Herrera, D. Keyes et al. National Autonomous University of Mexico (UNAM), México (2003), 27-38 (electronic). MR 2093732
[4] 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
[5] Farhat, C., Lesoinne, M., LeTallec, P., Pierson, K., Rixen, D.: FETI-DP: a dual-primal unified FETI method---part I: A faster alternative to the two-level FETI method. Int. J. Numer. Methods Eng. 50 (2001), 1523-1544. DOI 10.1002/nme.76 | MR 1813746
[6] Kim, H. H., Lee, C.-O.: A preconditioner for the FETI-DP formulation with mortar methods in two dimensions. SIAM J. Numer. Anal. 42 (2005), 2159-2175. DOI 10.1137/S0036142903423381 | MR 2139242 | Zbl 1080.65117
[7] Klawonn, A., Widlund, O. B., Dryja, M.: Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM J. Numer. Anal. 40 (2002), 159-179 (electronic). DOI 10.1137/S0036142901388081 | MR 1921914 | Zbl 1032.65031
[8] Mandel, J., Tezaur, R.: On the convergence of a dual-primal substructuring method. Numer. Math. 88 (2001), 543-558. DOI 10.1007/s211-001-8014-1 | MR 1835470 | Zbl 1003.65126
[9] Mandel, J., Tezaur, R., Farhat, C.: A scalable substructuring method by Lagrange multipliers for plate bending problems. SIAM J. Numer. Anal. 36 (1999), 1370-1391 (electronic). DOI 10.1137/S0036142997289896 | MR 1706770 | Zbl 0956.74059
[10] Marcinkowski, L.: The mortar element method with locally nonconforming elements. BIT 39 (1999), 716-739. DOI 10.1023/A:1022343324625 | MR 1735101 | Zbl 0944.65115
[11] Marcinkowski, L.: A mortar element method for some discretizations of a plate problem. Numer. Math. 93 (2002), 361-386. DOI 10.1007/s002110100389 | MR 1941401 | Zbl 1036.74046
[12] Marcinkowski, L.: Additive Schwarz method for mortar discretization of elliptic problems with \$P_1\$ nonconforming finite elements. BIT 45 (2005), 375-394. DOI 10.1007/s10543-005-7123-x | MR 2176199 | Zbl 1080.65118
[13] Marcinkowski, L.: A mortar finite element method for fourth order problems in two dimensions with Lagrange multipliers. SIAM J. Numer. Anal. 42 (2005), 1998-2019 (electronic). DOI 10.1137/S0036142902387574 | MR 2139234 | Zbl 1076.74055
[14] Marcinkowski, L.: A preconditioner for a FETI-DP method for mortar element discretization of a 4th order problem in 2D. ETNA, Electron. Trans. Numer. Anal. 38 1-16 electronic only (2011). MR 2871856 | Zbl 1205.65318
[15] Marcinkowski, L., Rahman, T.: Neumann-Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems. BIT 48 (2008), 607-626. DOI 10.1007/s10543-008-0167-y | MR 2447988 | Zbl 1180.65164
[16] Marcinkowski, L., Rahman, T.: A FETI-DP method for Crouzeix-Raviart finite element discretizations. Comput. Methods Appl. Math. 12 (2012), 73-91. DOI 10.2478/cmam-2012-0005 | MR 3041002 | Zbl 1284.65182
[17] Pierson, K. H.: A family of domain decomposition methods for the massively parallel solution of computational mechanics problems. PhD thesis, University of Colorado at Boulder, Aerospace Engineering Sciences (2001).
[18] Rahman, T., Bjørstad, P., Xu, X.: The Crouzeix-Raviart FE on nonmatching grids with an approximate mortar condition. SIAM J. Numer. Anal. 46 (2008), 496-516. DOI 10.1137/060663593 | MR 2377273 | Zbl 1160.65344
[19] Sarkis, M.: Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math. 77 (1997), 383-406. DOI 10.1007/s002110050292 | MR 1469678 | Zbl 0884.65119
[20] Tezaur, R.: Analysis of Lagrange multiplier based domain decomposition. PhD thesis, University of Colorado at Denver, Denver (1998). MR 2697697
[21] Toselli, A., Widlund, O.: Domain Decomposition Methods-Algorithms and Theory. Springer Series in Computational Mathematics 34 Springer, Berlin (2005). MR 2104179 | Zbl 1069.65138
[22] Wohlmuth, B. I.: Discretization Methods and Iterative Solvers Based on Domain Decomposition. Lecture Notes in Computational Science and Engineering 17 Springer, Berlin (2001). MR 1820470 | Zbl 0966.65097

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