Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-30T22:11:58.116Z Has data issue: false hasContentIssue false
  • English
  • Français

A multilevel preconditioner for the mortar method for nonconforming P 1 finite element

Published online by Cambridge University Press:  07 February 2009

Talal Rahman  and
Xuejun Xu
Talal Rahman
Affiliation:
Department of Mathematics, University of Bergen, c/o Center for Integrated Petroleum Research, Allegt. 41, 5007 Bergen, Norway. [email protected] Present address: Faculty of Engineering, Bergen University College, 5020 Bergen, Norway.
Xuejun Xu
Affiliation:
LSEC, Institute of Computational Mathematics, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, P.R. China. [email protected]
Get access

Abstract

A multilevel preconditioner based on the abstract framework of theauxiliary space method, is developed for the mortar method for thenonconforming P 1 finite element or the lowest orderCrouzeix-Raviart finite element on nonmatching grids. It is shownthat the proposed preconditioner is quasi-optimal in the sense thatthe condition number of the preconditioned system is independent ofthe mesh size, and depends only quadratically on the number ofrefinement levels. Some numerical results confirming the theory arealso provided.

Keywords

Crouzeix-Raviart FEmortar methodmultilevel preconditionerauxiliary space method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 3 , May 2009 , pp. 429 - 444
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achdou, Y. and Kuznetsov, Yu.A., Subtructuring preconditioners for finite element methods on nonmatching grids. J. Numer. Math. 3 (1995) 1–28.
Achdou, Y., Kuznetsov, Yu.A. and Pironneau, O., Substructuring preconditioner for the Q 1 mortar element method. Numer. Math. 71 (1995) 419–449. CrossRef
Achdou, Y., Maday, Y. and Widlund, O.B., Iterative substructing preconditioners for mortar element methods in two dimensions. SIAM J. Numer. Anal. 36 (1999) 551–580. CrossRef
Ben Belgacem, F., The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173–198. CrossRef
F. Ben Belgacem and Y. Maday, The mortar element method for three dimensional finite elements. RAIRO Modél. Math. Anal. Numér. 31 (1997) 289–302.
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.
Braess, D. and Dahmen, W., Stability estimates of the mortar finite element method for 3-dimensional problems. J. Numer. Math. 6 (1998) 249–264.
Braess, D., Dahmen, W. and Wieners, C., A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal. 37 (2000) 48–69. CrossRef
Braess, D., Deuflhard, P. and Lipnikov, K., A subspace cascadic multigrid method for the mortar elements. Computing 69 (2002) 202–225. CrossRef
Brenner, S., Preconditioning complicated finite elements by simple finite elements. SIAM J. Sci. Comput. 17 (1996) 1269–1274. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problem. North-Holland, Amsterdam (1978).
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.
Dryja, M., Gantner, A., Widlund, O. and Wohlmuth, B., Multilevel additive Schwarz preconditioner for nonconforming mortar finite element methods. J. Numer. Math. 12 (2004) 23–38. CrossRef
Gopalakrishnan, J. and Pasciak, J.P., Multigrid for the mortar finite element method. SIAM J. Numer. Anal. 37 (2000) 1029–1052. CrossRef
Hoppe, R.H.W. and Wohlmuth, B., Adaptive multilevel iterative techniques for nonconforming finite element discretizations. J. Numer. Math. 3 (1995) 179–198.
Kim, C., Lazarov, R., Pasciak, J. and Vassilevski, P., Multiplier spaces for the mortar finite element method in three dimensions. SIAM J. Numer. Anal. 39 (2001) 519–538. CrossRef
Marcinkowski, L., The mortar element method with locally nonconforming elements. BIT Numer. Math. 39 (1999) 716–739. CrossRef
Marcinkowski, L., Additive Schwarz method for mortar discretization of elliptic problems with P 1 nonconforming finite element. BIT Numer. Math. 45 (2005) 375–394. CrossRef
Marcinkowski, L. and Rahman, T., Neumann – Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems. BIT Numer. Math. 48 (2008) 607–626. CrossRef
Nepomnyaschikh, S.V., Fictitious components and subdomain alternating methods. Sov. J. Numer. Anal. Math. Modelling 5 (1990) 53–68. CrossRef
Oswald, P., Preconditioners for nonconforming elements. Math. Comp. 65 (1996) 923–941. CrossRef
Rahman, T., Xu, X. and Hoppe, R., Additive Schwarz method for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients. Numer. Math. 101 (2005) 551–572. CrossRef
Rahman, T., Bjørstad, P.E. and Crouzeix-Raviart FE, X. Xu on nonmatching grids with an approximate mortar condition. SIAM J. Numer. Anal. 46 (2008) 496–516. CrossRef
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).
Sarkis, M., Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using nonconforming elements. Numer. Math. 77 (1997) 383–406. CrossRef
Shi, Z.C. and Multigrid, X. Xu for the Wilson mortar element method. Comput. Methods Appl. Math. 1 (2001) 99–112. CrossRef
Vassilevski, P. and Wang, J., An application of the abstract multilevel theory to nonconforming finite element methods. SIAM J. Numer. Anal. 32 (1995) 235–248. CrossRef
Wohlmuth, B., A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38 (2000) 989–1012. CrossRef
Wohlmuth, B., A multigrid method for saddlepoint problems arising from mortar finite element discretizations. Electron. Trans. Numer. Anal. 11 (2000) 43–54.
J. Xu, Theory of Multilevel Methods. Ph.D. Thesis, Cornell University, USA (1989).
Iterative, J. Xu methods by space decomposition and subspace correction. SIAM Rev. 34 (1992) 581–613.
The, J. Xu auxiliary space method and optimal multigrid preconditioning techniques for unstructured grid. Computing 56 (1996) 215–235.
Xu, X. and Chen, J., Multigrid for the mortar element method for P 1 nonconforming element. Numer. Math. 88 (2001) 381–398. CrossRef
H. Yserentant, Old and new convergence proofs for multigrid methods. Acta Numer. (1993) 285–326.
Zhang, S. and Zhang, Z., Treatments of discontinuity and bubble functions in the multigrid method. Math. Comp. 66 (1997) 1055–1072. CrossRef