A multigrid method based on the unconstrained product space for mortar finite element discretizations

SIAM Journal on Numerical Analysis

Volumn 39, Issue 1, 2002, Pages 192-213

  Wohlmuth, Barbara I ^a   Krause, Rolf H ^b  

^a UNIVERSITY OF AUGSBURG   (Germany)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Dual space; Lagrange multiplier; Level dependent bilinear forms; Mortar finite elements; Multigrid methods; Nonmatching triangulations

Indexed keywords

EID: 0041761670     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142999360676     Document Type: Article

References (32)

1. Y. ACHDOU AND Y. KUZNETSOV, Substructuring preconditioners for finite element methods on nonmatching grids, East-West J. Numer. Math., 3 (1995), pp. 1-28.
2. 1 mortar element method, Numer. Math., 71 (1995), pp. 419-449.
3. Y. ACHDOU, Y. MADAY, AND O. WIDLUND, Iterative substructuring preconditioners for mortar element methods in two dimensions, SIAM J. Numer. Anal., 36 (1999), pp. 551-580.
4. P. BASTIAN, K. BIRKEN, K. JOHANNSEN, S. LANG, N. NEUSS, H. RENTZ-REICHERT, AND C. WIENERS, UG - a flexible software toolbox for solving partial differential equations, Computing and Visualization in Science, 1 (1997), pp. 27-40.
5. R. BECKER AND P. HANSBO, A Finite Element Method for Domain Decompositions with Non-matching Grids, Preprint 3613, INRIA, Sophia Antipolis, France, 1999.
6. F. BEN BELGACEM, The mortar finite element method with Lagrange multipliers, Numer. Math, 84 (1999), pp. 173-197.
7. F. BEN BELGACEM AND Y. MADAY, The mortar element method for three dimensional finite elements, RAIRO Modél. Math. Anal. Numér., 31 (1997), pp. 289-302.
8. C. BERNARDI AND Y. MADAY, Raffinement de maillage en elements finis par la methode des joints, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), pp. 373-377.
9. C. BERNARDI, Y. MADAY, AND A. PATERA, Domain decomposition by the mortar element method, in Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, H. Kaper et al., eds., Reidel, Dordrecht, 1993, pp. 269-286.
10. C. BERNARDI, Y. MADAY, AND T. PATERA, A new nonconforming approach to domain decomposition: The mortar element method, in Nonlinear Partial Differential Equations and Their Applications, Longman, Harlow, UK, 1994, pp. 13-51.
11. D. BRAESS AND W. DAHMEN, Stability estimates of the mortar finite element method for 3-dimensional problems, East-West J. Numer. Math., 6 (1998), pp. 249-264.
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.
13. D. BRAESS, P. DEUFLHARD, AND K. LIPNIKOV, A Cascadic Conjugate Gradient Method for Domain Decomposition with Non-Matching Grids, Preprint SC99-07, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, 1999.
14. D. BRAESS, M. DRYJA, AND W. HACKBUSCH, Multigrid method for nonconforming fediscretisations with application to nonmatching grids, Computing, 63 (1999), pp. 1-25.
15. D. BRAESS AND R. SARAZIN, An efficient smoother for the Stokes problem, Appl. Numer. Math., 23 (1997), pp. 3-19.
16. S. BRENNER AND R. SCOTT, Mathematical Theory of Finite Element Methods, Texts Appl. Math. 15, Springer-Verlag, New York, 1994.
17. M. CASARIN AND O. WIDLUND, A hierarchical preconditioner for the mortar finite element method, Electron. Trans. Numer. Anal., 4 (1996), pp. 75-88.
18. M. DRYJA, An additive Schwarz method for elliptic mortar finite element problems in three dimensions, in Proceedings of the 9th International Conference on Domain Decomposition, P. Bjørstad, M. Espedal, and D. Keyes, eds., Domain Decomposition Press, Bergen, 1998, pp. 88-96.
19. B. ENGELMANN, R. HOPPE, Y. ILIASH, Y. KUZNETSOV, Y. VASSILEVSKI, AND B. WOHLMUTH, Adaptive macro-hybrid finite element methods, in Proceedings of the 2nd European Conference on Numerical Methods, H. Bock, F. Brezzi, R. Glowinski, G. Kanschat, Y. Kuznetsov, J. Périaux, and R. Rannacher, eds., World Scientific, Singapore, 1998, pp. 294-302.
20. J. GOPALAKRISHNAN AND J. E. PASCIAK, Multigrid for the mortar finite element method, SIAM J. Numer. Anal., 37 (2000), pp. 1029-1052.
21. W. HACKBUSCH, Multigrid Methods and Applications, Springer-Verlag, Berlin, New York, 1985.
22. R. HOPPE, Y. ILIASH, Y. KUZNETSOV, Y. VASSILEVSKI, AND B. WOHLMUTH, Analysis and parallel implementation of adaptive mortar finite element methods. East-West J. Numer. Math., 6 (1998), pp. 223-248.
23. Y. KUZNETSOV, Efficient iterative solvers for elliptic finite element problems on nonmatching grids, Russian J. Numer. Anal. Math. Modelling, 10 (1995), pp. 187-211.
24. Y. KUZNETSOV, Overlapping domain decomposition with non-matching grids, in Proceedings of the 9th International Conference on Domain Decomposition, P. Bjørstad, M. Espedal, and D. Keyes, eds., Domain Decomposition Press, Bergen, 1998, pp. 606-617.
25. 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), pp. 9-15.
26. R. STENBERG, Mortaring by a method of J.A. Nitsche, in Computational Mechanics: New Trends and Applications, E. O. S. Idelsohn and E. Dvorkin, eds., CIMNE, Barcelona, 1998.
27. C. WIENERS AND B. WOHLMUTH, The coupling of mixed and conforming finite element discretizations, in Proceedings of the 10th International Conference on Domain Decomposition, Contemp. Math., J. Mandel, C. Farhat, and X.-C. Cai, eds., AMS, Providence, RI, 1998, pp. 546-553.
28. C. WIENERS AND B. WOHLMUTH, A general framework for multigrid methods for mortar finite elements, in Numerical Techniques for Composite Materials, Notes Numer. Fluid Mech., Vieweg, Braunschweig, to appear.
29. B. WOHLMUTH, A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM J. Numer. Anal., 38 (2000), pp. 989-1012.
30. B. WOHLMUTH, A multigrid method for saddlepoint problems arising from mortar finite element discretizations, Electron. Trans. Numer. Anal., 11 (2000), pp. 43-54.
31. B. WOHLMUTH, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Lect. Notes Comput. Sci. Eng. 17, Springer-Verlag, Heidelberg, 2001.
32. W. ZULEHNER, A Class of Smoothers for Saddle Point Problems, Tech. rep., Fakultät Mathematik, Universität Linz, Linz, Austria, 1998.