×

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.

MSC:

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

Citations:

Zbl 1080.65117
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.