Neumann-Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems. (English) Zbl 1180.65164

This paper proposes two scalable variants of the Neumann-Neumann algorithm for the lowest order Crouzeix-Raviart finite element or the nonconforming \(P_1\) finite element on non matching meshes. The overall discretization is done using a mortar technique which is based on the application of an approximate matching condition for the discrete functions, requiring function values only at the mesh interface nodes. The algorithms are analyzed using the abstract Schwarz framework, proving a convergence which is independent of the jumps in the coefficients of the problem and only depends logarithmically on the ratio between the subdomain size and the mesh size.


65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Full Text: DOI


[1] Y. Achdou, Y. Maday, and O. B. Widlund, Iterative substructuring preconditioners for mortar element methods in two dimensions, SIAM J. Numer. Anal., 36 (1999), pp. 551–580. · Zbl 0931.65110
[2] M. Amara, C. Bernardi, and M.-A. Moussaoui, Handling corner singularities by the mortar spectral element method, Appl. Anal., 46 (1992), pp. 25–44. · Zbl 0788.65103
[3] T. Apel, S. Nicaise, and J. Schöberl, Crouzeix–Raviart type finite elements on anisotropic meshes, Numer. Math., 89 (2001), pp. 193–223. · Zbl 0989.65130
[4] D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates, RAIRO, Modélisation Math. Anal. Numér., 19 (1985), pp. 7–32. · Zbl 0567.65078
[5] F. Ben Belgacem, The mortar finite element method with Lagrange multipliers, Numer. Math., 84 (1999), pp. 173–197 (First published as a technical report in 1994). · Zbl 0944.65114
[6] F. Ben Belgacem and Y. Maday, The mortar element method for three-dimensional finite elements, RAIRO, Modélisation Math. Anal. Numér., 31 (1997), pp. 289–302. · Zbl 0868.65082
[7] F. Ben Belgacem and Y. Maday, Coupling spectral and finite elements for second order elliptic three-dimensional equations, SIAM J. Numer. Anal., 36 (1999), pp. 1234–1263. · Zbl 0942.65132
[8] C. Bernardi and Y. Maday, Mesh adaptivity in finite elements by the mortar mesh, Rev. Eur. Élém. Finis, 9 (2000), pp. 451–465. · Zbl 0954.65081
[9] C. Bernardi, Y. Maday, and A. T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, in Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. XI (Paris, 1989–1991), Pitman Res. Notes Math. Ser., vol. 299, Longman Sci. Tech., Harlow, 1994, pp. 13–51.
[10] S. Bertoluzza and S. Falletta, The mortar method with approximate constraint, in Domain Decomposition Methods in Science and Engineering. 14th International Conference on Domain Decomposition Methods (Cocoyoc, Mexico), I. Herrera, D. E. Keyes, O. B. Widlund, and R. Yates, eds., National Autonomous University of Mexico (UNAM), Mexico City, Mexico, 2003, pp. 357–364.
[11] P. E. Bjørstad, M. Dryja, and T. Rahman, Additive Schwarz methods for elliptic mortar finite element problems, Numer. Math., 95 (2003), pp. 427–457. · Zbl 1036.65107
[12] D. Braess, W. Dahmen, and C. Wieners, A multigrid algorithm for the mortar finite element method, SIAM J. Numer. Anal., 37 (1999), pp. 48–69. · Zbl 0942.65139
[13] D. Braess and R. Verfurth, Multigrid methods for noconforming finite element methods, SIAM J. Numer. Anal., 27 (1990), pp. 979–986. · Zbl 0703.65067
[14] S. C. Brenner, A multigrid algorithm for the lowest-order Raviart–Thomas mixed triangular finite element method, SIAM J. Numer. Anal., 29 (1992), pp. 647–678. · Zbl 0759.65080
[15] S. C. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods, Math. Comp., 65 (1996), pp. 897–921. · Zbl 0859.65124
[16] S. C. Brenner, The condition number of the Schur complement in domain decomposition, Numer. Math., 83 (1999), pp. 187–203. · Zbl 0936.65141
[17] S. C. Brenner, Poincaré–Friedrichs inequalities for piecewise H 1 functions, SIAM J. Numer. Anal., 41 (2003), pp. 306–324 (electronic). · Zbl 1045.65100
[18] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Texts Appl. Math., vol. 15, Springer, New York, 2002. · Zbl 1012.65115
[19] Z. Chen, Analysis of mixed methods using conforming and nonconforming finite element methods, RAIRO, Modélisation Math. Anal. Numér., 27 (1993), pp. 9–34.
[20] Z. Chen, R. E. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second-order elliptic problems, Math. Comp., 65 (1996), pp. 467–490. · Zbl 0849.65088
[21] L. C. Cowsar, Domain decomposition methods for nonconforming finite element spaces of Lagrange type, Tech. Report TR 93-11, Department of Mathematical Sciences, Rice University, Houston, 1993.
[22] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite elements for solving the stationary Stokes equations I., RAIRO, Modélisation Math. Anal. Numér., 7 (1973), pp. 33–76.
[23] M. Dryja, A Neumann–Neumann algorithm for a mortar discetization of elliptic problems with discontinuous coefficients, Numer. Math., 99 (2005), pp. 645–656. · Zbl 1069.65135
[24] M. Dryja and O. B. Widlund, A generalized FETI-DP method for a mortar discretization of elliptic problems, in Domain Decomposition Methods in Science and Engineering, Natl. Auton. Univ. Mex., México, 2003, pp. 27–38 (electronic).
[25] A. Hansbo, P. Hansbo, and M. G. Larson, A finite element method on composite grids based on Nitsche’s method, ESAIM, Math. Model. Numer. Anal., 37 (2003), pp. 495–514. · Zbl 1031.65128
[26] R. H. W. Hoppe and B. Wohlmuth, Adaptive multilevel iterative techniques for nonconforming finite element discretizations, East-West J. Numer. Math., 3 (1995), pp. 179–197. · Zbl 0836.65127
[27] H. H. Kim and O. B. Widlund, Two-level Schwarz algorithms, using overlapping subregions, for mortar finite element methods, Siam J. Numer. Anal., 44 (2006), pp. 1514–1534. · Zbl 1155.65090
[28] D. Lazarov and S. Margenov, On a Two-Level Parallel MIC(0) Preconditioning of Crouzeix–Raviart Non-conforming FEM Systems, Lect. Notes Comput. Sci., vol. 2542, Springer, London, 2002. · Zbl 1032.65045
[29] L. Marcinkowski, The mortar element method with locally nonconforming elements, BIT, 39 (1999), pp. 716–739. · Zbl 0944.65115
[30] L. Marcinkowski, Additive Schwarz method for mortar discretization of elliptic problems with P1 nonconforming finite element, BIT, 45 (2005), pp. 375–394. · Zbl 1080.65118
[31] P. Oswald, On a hierarchical basis multilevel method with nonconforming P1 elements, Numer. Math., 62 (1992), pp. 189–212. · Zbl 0767.65078
[32] T. Rahman, P. E. Bjørstad, and X. Xu, The Crouzeix–Raviart FE on nonmatching grids with an approximate mortar condition, Siam J. Numer. Anal., 46 (2008), pp. 496–516. · Zbl 1160.65344
[33] T. Rahman, X. Xu, and R. Hoppe, Additive Schwarz methods for the Crouzeix–Raviart mortar finite element for elliptic problems with discontinuous coefficients, Numer. Math., 101 (2003), pp. 551–572. · Zbl 1087.65117
[34] M. Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements, Numer. Math., 77 (1997), pp. 383–406. · Zbl 0884.65119
[35] B. F. Smith, P. E. Bjørstad, and W. D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, 1996. · Zbl 0857.65126
[36] D. Stefanica and A. Klawonn, The FETI method for mortar finite elements, in 11th International Conference on Domain Decomposition Methods (London, 1998), DDM.org, Augsburg, 1999, pp. 121–129.
[37] A. Toselli and O. Widlund, Domain Decomposition Methods – Algorithms and Theory, Springer Ser. Comput. Math., vol. 34, Springer, Berlin, 2005. · Zbl 1069.65138
[38] P. S. Vassilievsky and J. Wang, An application of the abstract multilevel theory to nonconforming finite element methods, SIAM J. Numer. Anal., 32 (1995), pp. 235–248. · Zbl 0828.65125
[39] C. Wieners and B. Wohlmuth, The coupling of mixed and conforming finite element discretizations, in Proceedings of the 10th International Conference on Domain Decomposition (Boulder, Colorado, June 1997), J. Mandel, C. Farhat, and X. C. Cai, eds., Contemp. Math., vol. 218, Am. Math. Soc., Providence, RI, 1998, pp. 453–459. · Zbl 0910.65091
[40] B. I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM J. Numer. Anal., 38 (2000), pp. 989–1012. · Zbl 0974.65105
[41] X. Xu and J. Chen, Multigrid for the mortar element method for P1 nonconforming element, Numer. Math., 88 (2001), pp. 381–398. · Zbl 0989.65141
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.