A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization. (English) Zbl 1340.65276

In this paper the author extends the results of H. H. Kim and C.-O. Lee [SIAM J. Numer. Anal. 42, No. 5, 2159–2175 (2005; Zbl 1080.65117)] to mortar-type Crouzeix-Raviart element discretizations. The mortar-type Crouzeix-Raviart element discretizations is described for second-order elliptic problems with discontinuous coefficients. The FETI-DP operator is introduced, then a parallel preconditioner for the FETI-DP operator is proposed. Then the condition number bounds of the preconditioned problem is established. The author compares the proposed preconditioner with that of M. Dryja and O. B. Widlund [“A generalized FETI-DP method for a mortar discretization of elliptic problems”, in: I. Herrera (ed.) et al., Domain decomposition methods in science and engineering. México: National Autonomous University of México (UNAM). 27–38 (2003)]. Finally, numerical tests are presented in accord with the theory.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F08 Preconditioners for iterative methods


Zbl 1080.65117
Full Text: DOI Link


[1] C. Bernardi, Y. Maday, A. T. Patera: A new nonconforming approach to domain decomposition: The mortar element method. Nonlinear Partial Differential Equations and their Applications (H. Brezis et al., ed.). Collège de France Seminar, Vol. XI, Paris, France, 1989-1991, Logman Scientific & Technical. Pitman Res. Notes Math. Ser. 299, 1994, pp. 13-51. · Zbl 1003.65126
[2] S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods (3rd ed.). Texts in Applied Mathematics 15, Springer, New York, 2008. · Zbl 1135.65042
[3] M. Dryja, O. B. Widlund: 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., eds.). National Autonomous University of Mexico (UNAM), México, 2003, pp. 27-38 (electronic).
[4] Farhat, C.; Lesoinne, M.; Pierson, K., A scalable dual-primal domain decomposition method, Numer. Linear Algebra Appl., 7, 687-714, (2000) · 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, 1523-1544, (2001) · Zbl 1008.74076
[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, 2159-2175, (2005) · 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, 159-179, (2002) · Zbl 1032.65031
[8] Mandel, J.; Tezaur, R., On the convergence of a dual-primal substructuring method, Numer. Math., 88, 543-558, (2001) · 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, 1370-1391, (1999) · Zbl 0956.74059
[10] Marcinkowski, L., The mortar element method with locally nonconforming elements, BIT, 39, 716-739, (1999) · Zbl 0944.65115
[11] Marcinkowski, L., A mortar element method for some discretizations of a plate problem, Numer. Math., 93, 361-386, (2002) · Zbl 1036.74046
[12] Marcinkowski, L., Additive Schwarz method for mortar discretization of elliptic problems with P1 nonconforming finite elements, BIT, 45, 375-394, (2005) · 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, 1998-2019, (2005) · 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, (2011) · Zbl 1205.65318
[15] Marcinkowski, L.; Rahman, T., Neumann-Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems, BIT, 48, 607-626, (2008) · Zbl 1180.65164
[16] Marcinkowski, L.; Rahman, T., A FETI-DP method for Crouzeix-Raviart finite element discretizations, Comput. Methods Appl. Math., 12, 73-91, (2012) · Zbl 1284.65182
[17] K. H. Pierson: 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, 496-516, (2008) · 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, 383-406, (1997) · Zbl 0884.65119
[20] R. Tezaur: Analysis of Lagrange multiplier based domain decomposition. PhD thesis, University of Colorado at Denver, Denver, 1998.
[21] A. Toselli, O. Widlund: Domain Decomposition Methods-Algorithms and Theory. Springer Series in Computational Mathematics 34, Springer, Berlin, 2005. · Zbl 1069.65138
[22] B. I. Wohlmuth: Discretization Methods and Iterative Solvers Based on Domain Decomposition. Lecture Notes in Computational Science and Engineering 17, Springer, Berlin, 2001. · Zbl 0966.65097
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