Error estimates for the three-field formulation with bubble stabilization

Brezzi, Franco; Marini, Donatella

doi:10.1090/S0025-5718-00-01250-3

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6842(online) ISSN 0025-5718(print)

Error estimates for the three-field formulation with bubble stabilization

Authors: Franco Brezzi and Donatella Marini
Journal: Math. Comp. 70 (2001), 911-934
MSC (2000): Primary 65N55, 65N30, 65N12, 65N15, 35J25
Posted: March 24, 2000
MathSciNet review: 1826573
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

In this paper we prove convergence and error estimates for the so-called 3-field formulation using piecewise linear finite elements stabilized with boundary bubbles. Optimal error bounds are proved in and in the broken norm for the internal variable , and in suitable weighted norms for the other variables $\lambda$ and $\psi$ .

1. Claudio Baiocchi, Franco Brezzi, and Leopoldo P. Franca, Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.), Comput. Methods Appl. Mech. Engrg. 105 (1993), no. 1, 125–141. MR 1222297 (94g:65058), http://dx.doi.org/10.1016/0045-7825(93)90119-I
2. C. Baiocchi, F. Brezzi, and L. D. Marini, Stabilization of Galerkin methods and applications to domain decomposition, (Paris, 1992) Lecture Notes in Comput. Sci., vol. 653, Springer, Berlin, 1992, pp. 345–355. MR 1228218 (94g:65119)
3. Petter E. Bjørstad and Olof B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal. 23 (1986), no. 6, 1097–1120. MR 865945 (88h:65188), http://dx.doi.org/10.1137/0723075
4. P.E. Bjørstad, M.S. Espedal and D.E. Keyes (eds.), Domain Decomposition Methods in Science and Engineering, Domain Decomposition Press, Bergen, 1998.
5. James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, The construction of preconditioners for elliptic problems by substructuring. IV, Math. Comp. 53 (1989), no. 187, 1–24. MR 970699 (89m:65098), http://dx.doi.org/10.1090/S0025-5718-1989-0970699-3
6. Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
7. F. Brezzi, L.P. Franca, L.D. Marini, and A. Russo, Stabilization techniques for domain decomposition methods with non-matching grids, in [4], pp. 1-11.
8. F. Brezzi and L. D. Marini, Macro hybrid elements and domain decomposition methods, Optimisation et contrôle (Sophia-Antipolis, 1992) Cépaduès, Toulouse, 1993, pp. 89–96. MR 1284966 (95c:65208)
9. F. Brezzi and L. D. Marini, A three-field domain decomposition method, Domain decomposition methods in science and engineering (Como, 1992), Contemp. Math., vol. 157, Amer. Math. Soc., Providence, RI, 1994, pp. 27–34. MR 1262602 (95a:65202), http://dx.doi.org/10.1090/conm/157/01402
10. A. Buffa, Error estimate for a stabilized domain decomposition method with nonmatching grid.
Numer. Math., to appear.
11. Tony F. Chan and Tarek P. Mathew, Domain decomposition algorithms, Acta numerica, 1994, Acta Numer., Cambridge Univ. Press, Cambridge, 1994, pp. 61–143. MR 1288096 (95f:65214)
12. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174 (58 #25001)
13. M. Dryja, A finite element-capacitance method for elliptic problems on regions partitioned into subregions, Numer. Math. 44 (1984), no. 2, 153–168. MR 753950 (86c:65131), http://dx.doi.org/10.1007/BF01410102
14. Tony F. Chan, Roland Glowinski, Jacques Périaux, and Olof B. Widlund (eds.), Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1064333 (91e:65010)
15. Gene H. Golub and David Mayers, The use of preconditioning over irregular regions, Computing methods in applied sciences and engineering, VI (Versailles, 1983), North-Holland, Amsterdam, 1984, pp. 3–14. MR 806766 (87h:65059)
16. Yu. A. Kuznetsov and M. F. Wheeler, Optimal order substructuring preconditioners for mixed finite element methods on nonmatching grids, East-West J. Numer. Math. 3 (1995), no. 2, 127–143. MR 1342888 (96c:65182)
17. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15 (German). Collection of articles dedicated to Lothar Collatz on his sixtieth birthday. MR 0341903 (49 #6649)
18. Barry F. Smith and Olof B. Widlund, A domain decomposition algorithm using a hierarchical basis, SIAM J. Sci. Statist. Comput. 11 (1990), no. 6, 1212–1220. MR 1068505 (91m:65125), http://dx.doi.org/10.1137/0911069
19. Charles H. Tong, Tony F. Chan, and C.-C. Jay Kuo, A domain decomposition preconditioner based on a change to a multilevel nodal basis, SIAM J. Sci. Statist. Comput. 12 (1991), no. 6, 1486–1495. MR 1129659 (92i:65070), http://dx.doi.org/10.1137/0912082
20. J. Xu, Theory of multilevel methods, PhD thesis, Cornell University, 1989.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|