| Title:
|
A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization (English) |
| Author:
|
Wang, Chunmei |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
59 |
| Issue:
|
6 |
| Year:
|
2014 |
| Pages:
|
653-672 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by $(1+\log (H/h))^2$, where $H$ and $h$ are mesh sizes. Finally, numerical tests are presented to verify the theoretical results. (English) |
| Keyword:
|
FETI-DP |
| Keyword:
|
Crouzeix-Raviart element |
| Keyword:
|
nonstandard mortar condition |
| Keyword:
|
preconditioner |
| MSC:
|
65N22 |
| MSC:
|
65N30 |
| MSC:
|
65N38 |
| MSC:
|
65N55 |
| idZBL:
|
Zbl 06391455 |
| idMR:
|
MR3277732 |
| DOI:
|
10.1007/s10492-014-0078-y |
| . |
| Date available:
|
2014-11-10T09:15:48Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143993 |
| . |
| Reference:
|
[1] Bernardi, C., Maday, Y., Patera, A. T.: A new nonconforming approach to domain decomposition: The mortar element method.H. Brezis et al. Nonlinear Partial Differential Equations and their Applications Collège de France Seminar, Vol. XI, Paris, France, 1989-1991 Logman Scientific & Technical. Pitman Res. Notes Math. Ser. 299 (1994), 13-51. Zbl 0797.65094, MR 1268898 |
| Reference:
|
[2] Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods (3rd ed.).Texts in Applied Mathematics 15 Springer, New York (2008). Zbl 1135.65042, MR 2373954 |
| Reference:
|
[3] Dryja, M., Widlund, O. B.: 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. National Autonomous University of Mexico (UNAM), México (2003), 27-38 (electronic). MR 2093732 |
| Reference:
|
[4] Farhat, C., Lesoinne, M., Pierson, K.: A scalable dual-primal domain decomposition method.Numer. Linear Algebra Appl. 7 (2000), 687-714. Zbl 1051.65119, MR 1802366, 10.1002/1099-1506(200010/12)7:7/8<687::AID-NLA219>3.0.CO;2-S |
| Reference:
|
[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 (2001), 1523-1544. MR 1813746, 10.1002/nme.76 |
| Reference:
|
[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 (2005), 2159-2175. Zbl 1080.65117, MR 2139242, 10.1137/S0036142903423381 |
| Reference:
|
[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 (2002), 159-179 (electronic). Zbl 1032.65031, MR 1921914, 10.1137/S0036142901388081 |
| Reference:
|
[8] Mandel, J., Tezaur, R.: On the convergence of a dual-primal substructuring method.Numer. Math. 88 (2001), 543-558. Zbl 1003.65126, MR 1835470, 10.1007/s211-001-8014-1 |
| Reference:
|
[9] Mandel, J., Tezaur, R., Farhat, C.: A scalable substructuring method by Lagrange multipliers for plate bending problems.SIAM J. Numer. Anal. 36 (1999), 1370-1391 (electronic). Zbl 0956.74059, MR 1706770, 10.1137/S0036142997289896 |
| Reference:
|
[10] Marcinkowski, L.: The mortar element method with locally nonconforming elements.BIT 39 (1999), 716-739. Zbl 0944.65115, MR 1735101, 10.1023/A:1022343324625 |
| Reference:
|
[11] Marcinkowski, L.: A mortar element method for some discretizations of a plate problem.Numer. Math. 93 (2002), 361-386. Zbl 1036.74046, MR 1941401, 10.1007/s002110100389 |
| Reference:
|
[12] Marcinkowski, L.: Additive Schwarz method for mortar discretization of elliptic problems with $P_1$ nonconforming finite elements.BIT 45 (2005), 375-394. Zbl 1080.65118, MR 2176199, 10.1007/s10543-005-7123-x |
| Reference:
|
[13] Marcinkowski, L.: A mortar finite element method for fourth order problems in two dimensions with Lagrange multipliers.SIAM J. Numer. Anal. 42 (2005), 1998-2019 (electronic). Zbl 1076.74055, MR 2139234, 10.1137/S0036142902387574 |
| Reference:
|
[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 electronic only (2011). Zbl 1205.65318, MR 2871856 |
| Reference:
|
[15] Marcinkowski, L., Rahman, T.: Neumann-Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems.BIT 48 (2008), 607-626. Zbl 1180.65164, MR 2447988, 10.1007/s10543-008-0167-y |
| Reference:
|
[16] Marcinkowski, L., Rahman, T.: A FETI-DP method for Crouzeix-Raviart finite element discretizations.Comput. Methods Appl. Math. 12 (2012), 73-91. Zbl 1284.65182, MR 3041002, 10.2478/cmam-2012-0005 |
| Reference:
|
[17] Pierson, K. H.: 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). |
| Reference:
|
[18] Rahman, T., Bjørstad, P., Xu, X.: The Crouzeix-Raviart FE on nonmatching grids with an approximate mortar condition.SIAM J. Numer. Anal. 46 (2008), 496-516. Zbl 1160.65344, MR 2377273, 10.1137/060663593 |
| Reference:
|
[19] Sarkis, M.: Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements.Numer. Math. 77 (1997), 383-406. Zbl 0884.65119, MR 1469678, 10.1007/s002110050292 |
| Reference:
|
[20] Tezaur, R.: Analysis of Lagrange multiplier based domain decomposition.PhD thesis, University of Colorado at Denver, Denver (1998). MR 2697697 |
| Reference:
|
[21] Toselli, A., Widlund, O.: Domain Decomposition Methods-Algorithms and Theory.Springer Series in Computational Mathematics 34 Springer, Berlin (2005). Zbl 1069.65138, MR 2104179 |
| Reference:
|
[22] Wohlmuth, B. I.: Discretization Methods and Iterative Solvers Based on Domain Decomposition.Lecture Notes in Computational Science and Engineering 17 Springer, Berlin (2001). Zbl 0966.65097, MR 1820470 |
| . |
