Analysis of multilevel decomposition iterative methods for mixed finite element methods

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1. [1] I. BABUŠKA, 1973, The finite element method with Lagrangian multipliers, Numer, Math., 20, 179-192. Zbl0258.65108MR359352
2. [2] R. E. BANK and T. F. DUPONT, 1981, An optimal order process for solving elliptic finite element equations, Math. Comp., 36, 35-51. Zbl0466.65059MR595040
3. [3] R. E. BANK, T. F. DUPONT and H. YSERENTANT, 1988, The Hierarchical basis multigrid method, Numer. Math., 52, 427-458. Zbl0645.65074MR932709
4. [4] J. H. BRAMBLE, R. E. EWING, J. E. PASCIAK and A. H. SCHATZ, 1988, A preconditioning technique for the efficient solution of problems with local grid refinement, Compt. Meth. Appl. Mech. Eng., 67, 149-159. Zbl0619.76113
5. [5] J. H. BRAMBLE, J. E. PASCIAK, J. WANG and J. XU, 1991, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp., 57, 1-22. Zbl0754.65085MR1090464
6. [6] J. H. BRAMBLE, J. E. PASCIAK, J. WANG and J. XU, 1911, Convergence estimate for multigrid algorithms without regularity assumptions, Math. Comp., 57, 23-45. Zbl0727.65101MR1079008
7. [7] F. BREZZI, 1974, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers, R.A.I.R.O., Anal. Numér., 2, 129-151. Zbl0338.90047MR365287
8. [8] F. BREZZI, J. Jr. DOUGLAS, M. FORTIN and L. D. MARINI, 1987, Efficient rectangular mixed finite elements in two and three space variables, R.A.I.R.O., Anal. Numér., 21, 581-604. Zbl0689.65065MR921828
9. [9] F. BREZZI, J. Jr. DOUGLAS and L. D. MARINI, 1985, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 217-235. Zbl0599.65072MR799685
10. [10] J. Jr. DOUGLAS and J. E. ROBERTS, Global estimates for mixed finite element methods for second order elliptic equations, Math. Comp., 45, 39-52. Zbl0624.65109MR771029
11. [11] R. E. EWING and J. WANG, 1992, Analysis of the Schwarz algorithm for mixed finite element methods, R.A.I.R.O. M2AN, 26, 739-756. Zbl0765.65104MR1183415
12. [12] M. DRYJA and O. WIDLUND, 1987, An additive variant of the Schwarz alternating method for the case of many subregions, Technical Report, Courant Institute of Mathematical Sciences, 339. 
13. [13] M. DRYJA and O WIDLUND, 1989, Some domain decomposition algorithms for elliptic problems, Technical Report, Courant Institute of Mathematical Sciences, 438. Zbl0719.65084MR1038100
14. [14] R. FALK and J. OSBORN, 1980, Error estimates for mixed methods, R.A.I.R.O., Anal. Numér., 14, 249-277. Zbl0467.65062MR592753
15. [15] M. FORTIN, An analysis of the convergence of mixed finite element methods, R.A.I.R.O., Anal. Numér., 11, 341-354. Zbl0373.65055MR464543
16. [16] R. GLOWINSKI and M. F. WHEELER, 1988, Domain decomposition and mixed finite element methods for elliptic problems, Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Dijferential Equations (R. Glowinski, G. H. Golub, G. A. Meurant and J. Périaux, eds.). Zbl0661.65105MR972509
17. [17] W. HACKBUSCH, 1985, Multi-Grid Methods and Applications, Springer-Verlag, New York. Zbl0595.65106
18. [18] P. L. LIONS, 1988, On the Schwarz alternating method, Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations (R. Glowinski, G. H. Golub, G. A. Meurant and J. Périaux, eds.). Zbl0658.65090MR972509
19. [19] T. P. MATHEW, 1989, Domain Decomposition and Iterative Refinement Methods for Mixed Finite Element Discretizations of Elliptic Problems, Ph. D. Thesis, New York University. 
20. [20] P.-A. RAVIART and J.-M. THOMAS, 1977, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics (606), Springer-Verlag, Berlin and New York, 292-315. Zbl0362.65089MR483555
21. [21] H. A. SCHWARZ, 1869, Über einige Abbildungsaufgaben, Ges. Math. Abh., 11, 65-83. 
22. [22] J. WANG, 1992, Convergence analysis without regularity assumptions for multigrid algorithms based on SOR smoothing, SIAM J. Numer. Anal., 29, 987-1001. Zbl0753.65093MR1173181
23. [23] J. WANG, 1992, Convergence analysis of Schwarz algorithm and multilevel decomposition iterative methods I : selfadjoint and positive definite elliptic problems, in Iterative Methods in Linear Algebra, R. Beauwens and P. de Groen (eds.), North-Holland, Amsterdam. Zbl0785.65115MR1159720
24. [24] J. WANG, 1993, Convergence analysis of the Schwarz algorithm and multilevel decomposition iterative methods II : non-selfadjoint and indefinite elliptic problems, SIAM J. Numer. Anal., 30, 953-970. Zbl0777.65066MR1231322
25. [25] H. YSERENTANT, 1986, On the multi-level splitting of finite element spaces, Numer. Math., 49, 379-412. Zbl0608.65065MR853662

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