no. 3
A multilevel preconditioner for the mortar method for nonconforming P 1 finite element
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 429-444.

A multilevel preconditioner based on the abstract framework of the auxiliary space method, is developed for the mortar method for the nonconforming P 1 finite element or the lowest order Crouzeix-Raviart finite element on nonmatching grids. It is shown that the proposed preconditioner is quasi-optimal in the sense that the condition number of the preconditioned system is independent of the mesh size, and depends only quadratically on the number of refinement levels. Some numerical results confirming the theory are also provided.

DOI : 10.1051/m2an/2009003
Classification : 65F10, 65N30, 65N55
Mots clés : Crouzeix-Raviart FE, mortar method, multilevel preconditioner, auxiliary space method 
@article{M2AN_2009__43_3_429_0,
     author = {Rahman, Talal and Xu, Xuejun},
     title = {A multilevel preconditioner for the mortar method for nonconforming $P_1$ finite element},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {429--444},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {3},
     year = {2009},
     doi = {10.1051/m2an/2009003},
     mrnumber = {2527400},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2009003/}
}
TY  - JOUR
AU  - Rahman, Talal
AU  - Xu, Xuejun
TI  - A multilevel preconditioner for the mortar method for nonconforming $P_1$ finite element
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2009
SP  - 429
EP  - 444
VL  - 43
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2009003/
DO  - 10.1051/m2an/2009003
LA  - en
ID  - M2AN_2009__43_3_429_0
ER  - 
%0 Journal Article
%A Rahman, Talal
%A Xu, Xuejun
%T A multilevel preconditioner for the mortar method for nonconforming $P_1$ finite element
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2009
%P 429-444
%V 43
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2009003/
%R 10.1051/m2an/2009003
%G en
%F M2AN_2009__43_3_429_0
Rahman, Talal; Xu, Xuejun. A multilevel preconditioner for the mortar method for nonconforming $P_1$ finite element. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 429-444. doi : 10.1051/m2an/2009003. http://www.numdam.org/articles/10.1051/m2an/2009003/

[1] Y. Achdou and Yu.A. Kuznetsov, Subtructuring preconditioners for finite element methods on nonmatching grids. J. Numer. Math. 3 (1995) 1-28. | MR | Zbl

[2] Y. Achdou, Yu.A. Kuznetsov and O. Pironneau, Substructuring preconditioner for the Q 1 mortar element method. Numer. Math. 71 (1995) 419-449. | MR | Zbl

[3] Y. Achdou, Y. Maday and O.B. Widlund, Iterative substructing preconditioners for mortar element methods in two dimensions. SIAM J. Numer. Anal. 36 (1999) 551-580. | MR | Zbl

[4] F. Ben Belgacem, The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-198. | MR | Zbl

[5] F. Ben Belgacem and Y. Maday, The mortar element method for three dimensional finite elements. RAIRO Modél. Math. Anal. Numér. 31 (1997) 289-302. | EuDML | Numdam | MR | Zbl

[6] 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, Vol. XI, Collège de France Seminar, H. Brezis and J.L. Lions Eds., Pitman Research Notes in Mathematics Series 299, Longman Scientific & Technical, Harlow (1994) 13-51. | MR | Zbl

[7] D. Braess and W. Dahmen, Stability estimates of the mortar finite element method for 3-dimensional problems. J. Numer. Math. 6 (1998) 249-264. | MR | Zbl

[8] D. Braess, W. Dahmen and C. Wieners, A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal. 37 (2000) 48-69. | MR | Zbl

[9] D. Braess, P. Deuflhard and K. Lipnikov, A subspace cascadic multigrid method for the mortar elements. Computing 69 (2002) 202-225. | MR | Zbl

[10] S. Brenner, Preconditioning complicated finite elements by simple finite elements. SIAM J. Sci. Comput. 17 (1996) 1269-1274. | MR | Zbl

[11] P.G. Ciarlet, The Finite Element Method for Elliptic Problem. North-Holland, Amsterdam (1978). | MR | Zbl

[12] M. Dryja, An iterative substructuring method for elliptic mortar finite element problems with discontinous coefficients, in Domain Decomposition Methods 10, J. Mandel, C. Farhat and X.C. Cai Eds., Contemp. Math. 218 (1998) 94-103. | MR | Zbl

[13] M. Dryja, A. Gantner, O. Widlund and B. Wohlmuth, Multilevel additive Schwarz preconditioner for nonconforming mortar finite element methods. J. Numer. Math. 12 (2004) 23-38. | MR | Zbl

[14] J. Gopalakrishnan and J.P. Pasciak, Multigrid for the mortar finite element method. SIAM J. Numer. Anal. 37 (2000) 1029-1052. | MR | Zbl

[15] R.H.W. Hoppe and B. Wohlmuth, Adaptive multilevel iterative techniques for nonconforming finite element discretizations. J. Numer. Math. 3 (1995) 179-198. | MR | Zbl

[16] C. Kim, R. Lazarov, J. Pasciak and P. Vassilevski, Multiplier spaces for the mortar finite element method in three dimensions. SIAM J. Numer. Anal. 39 (2001) 519-538. | MR | Zbl

[17] L. Marcinkowski, The mortar element method with locally nonconforming elements. BIT Numer. Math. 39 (1999) 716-739. | MR | Zbl

[18] L. Marcinkowski, Additive Schwarz method for mortar discretization of elliptic problems with P 1 nonconforming finite element. BIT Numer. Math. 45 (2005) 375-394. | MR | Zbl

[19] L. Marcinkowski and T. Rahman, Neumann - Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems. BIT Numer. Math. 48 (2008) 607-626. | MR | Zbl

[20] S.V. Nepomnyaschikh, Fictitious components and subdomain alternating methods. Sov. J. Numer. Anal. Math. Modelling 5 (1990) 53-68. | MR | Zbl

[21] P. Oswald, Preconditioners for nonconforming elements. Math. Comp. 65 (1996) 923-941. | MR | Zbl

[22] T. Rahman, X. Xu and R. Hoppe, Additive Schwarz method for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients. Numer. Math. 101 (2005) 551-572. | MR | Zbl

[23] T. Rahman, P.E. Bjørstad and X. Xu, Crouzeix-Raviart FE on nonmatching grids with an approximate mortar condition. SIAM J. Numer. Anal. 46 (2008) 496-516. | MR | Zbl

[24] M. Sarkis, Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Nonconforming Elements. Tech. Report 671, Ph.D. Thesis, Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, USA (1994).

[25] M. Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using nonconforming elements. Numer. Math. 77 (1997) 383-406. | MR | Zbl

[26] Z.C. Shi and X. Xu, Multigrid for the Wilson mortar element method. Comput. Methods Appl. Math. 1 (2001) 99-112. | EuDML | MR | Zbl

[27] P. Vassilevski and J. Wang, An application of the abstract multilevel theory to nonconforming finite element methods. SIAM J. Numer. Anal. 32 (1995) 235-248. | MR | Zbl

[28] B. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38 (2000) 989-1012. | MR | Zbl

[29] B. Wohlmuth, A multigrid method for saddlepoint problems arising from mortar finite element discretizations. Electron. Trans. Numer. Anal. 11 (2000) 43-54. | EuDML | MR | Zbl

[30] J. Xu, Theory of Multilevel Methods. Ph.D. Thesis, Cornell University, USA (1989).

[31] J. Xu, Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992) 581-613. | MR | Zbl

[32] J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grid. Computing 56 (1996) 215-235. | MR | Zbl

[33] X. Xu and J. Chen, Multigrid for the mortar element method for P 1 nonconforming element. Numer. Math. 88 (2001) 381-398. | MR | Zbl

[34] H. Yserentant, Old and new convergence proofs for multigrid methods. Acta Numer. (1993) 285-326. | MR | Zbl

[35] S. Zhang and Z. Zhang, Treatments of discontinuity and bubble functions in the multigrid method. Math. Comp. 66 (1997) 1055-1072. | MR | Zbl

Cité par Sources :