Analysis of the Schwarz algorithm for mixed finite elements methods

R. E. Ewing; J. Wang

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (1992)

  • Volume: 26, Issue: 6, page 739-756
  • ISSN: 0764-583X

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Ewing, R. E., and Wang, J.. "Analysis of the Schwarz algorithm for mixed finite elements methods." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 26.6 (1992): 739-756. <http://eudml.org/doc/193683>.

@article{Ewing1992,
author = {Ewing, R. E., Wang, J.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Schwarz alternating algorithm; domain decomposition; second-order elliptic differential equations; Neumann boundary conditions; Galerkin method; mixed finite element method; convergence},
language = {eng},
number = {6},
pages = {739-756},
publisher = {Dunod},
title = {Analysis of the Schwarz algorithm for mixed finite elements methods},
url = {http://eudml.org/doc/193683},
volume = {26},
year = {1992},
}

TY - JOUR
AU - Ewing, R. E.
AU - Wang, J.
TI - Analysis of the Schwarz algorithm for mixed finite elements methods
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 1992
PB - Dunod
VL - 26
IS - 6
SP - 739
EP - 756
LA - eng
KW - Schwarz alternating algorithm; domain decomposition; second-order elliptic differential equations; Neumann boundary conditions; Galerkin method; mixed finite element method; convergence
UR - http://eudml.org/doc/193683
ER -

References

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  1. [1] I. BABUŠKA, The finite element method with Lagrangian multipliers, Numer. Math., 20 (1973), 179-192. Zbl0258.65108MR359352
  2. [2] I. BABUŠKA, On the Schwarz algorithm in the theory of differential equations of mathematical physics, Tchecosl. Math. J., 8 (1958), 328-342 (in Russian). Zbl0083.11301
  3. [3] J. H. BRAMBLE, R. E. EWING, J. E. PASCIAK and A. H. SCHATZ, A preconditioning technique for the efficient solution of problems with local grid refinement, Compt. Meth. Appl. Mech. Eng., 67 (1988), 149-159. Zbl0619.76113
  4. [4] J. H. BRAMBLE, J. E. PASCIAK, J. WANG and J. XU, Convergence estimates for product iterative methods with applications to domain decomposition and multigrid, Math. Comp. (to appear). Zbl0754.65085MR1090464
  5. [5] J. H. BRAMBLE, J. E. PASCIAK, J. WANG and J. XU, Convergence estimate for multigrid algorithms without regularity assumptions, Math. Comp. (to appear). Zbl0727.65101MR1079008
  6. [6] F. BREZZI, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers, R.A.I.R.O., Modél. Math. Anal. Numér., 2 (1974), 129-151. Zbl0338.90047MR365287
  7. [7] F. BREZZI, J. Jr. DOUGLAS, R. DURÁN and L. D. MARINI, Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51 (1987), 237-250. Zbl0631.65107MR890035
  8. [8] F. BREZZI, J. Jr. DOUGLAS, R. FORTIN and L. D. MARINI, Efficient rectangular mixed finite elements in two and three space variables, R.A.I.R.O., Modél. Math. Anal. Numér., 21 (1987), 581-604. Zbl0689.65065MR921828
  9. [9] F. BREZZI, J. Jr. DOUGLAS and L. D. MARINI, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 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 (1985), 39-52. Zbl0624.65109MR771029
  11. [11] J. Jr. DOUGLAS and J. WANG, Superconvergence of mixed finite element methods on rectangular domains, Calcolo, 26 (1989), 121-134. Zbl0714.65084MR1083049
  12. [12] J. Jr. DOUGLAS and J. WANG, A new family of mixed finite element spaces over rectangles, submitted. Zbl0806.65109MR1288240
  13. [13] R. E. EWING and J. WANG, Analysis of mixed finite element methods on locally-refined grids, submitted. Zbl0772.65071
  14. [14] R. E. EWING and J. WANG, Analysis of multilevel decomposition iterative methods for mixed finite element methods, submitted to R.A.I.R.O., Modél. Math. Anal. Numér. Zbl0823.65035
  15. [15] P. G. CIARLET, « The Finite Element Method for Elliptic Problems », North-Holland, New York, 1978. Zbl0383.65058MR520174
  16. [16] M. DRYJA and O. WIDLUND, An additive variant of the Schwarz alternating method for the case of many subregions, Technical Report, Courant Institute of Mathematical Sciences, 339 (1987). 
  17. [17] M. DRYJA and O. WIDLUND, Some domain decomposition algorithms for elliptic problems, Technical Report, Courant Institute of Mathematical Sciences, 438 (1989). Zbl0719.65084MR1038100
  18. [18] R. FALK and J. OSBORN, Error estimates for mixed methods, R.A.I.R.O.,Modél. Math. Anal. Numér., 14 (1980), 249-277. Zbl0467.65062MR592753
  19. [19] M. FORTIN, An analysis of the convergence of mixed finite element methods, R.A.I.R.O., Modél. Math. Anal. Numér, 11 (1977), 341-354. Zbl0373.65055MR464543
  20. [20] R. GLOWINSKI and M. F. WHEELER, Domain decomposition and mixed finite element methods for elliptic problems, In the 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.), 1988. Zbl0661.65105MR972509
  21. [21] P. L. LIONS, On the Schwarz alternating method, In the 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.), 1988. Zbl0658.65090MR972510
  22. [22] T. P. MATHEW, « Domain Decomposition and Iterative Refinement Methods for Mixed Finite Element Discretizations of Elliptic Problems», Ph. D. Thesis, New York University, 1989. 
  23. [23] A. M. MATSOKIN and S. V. NEPOMNYASCHIKH, A Schwarz alternating method in a subspace, Soviet Math., 29(10) (1985), 78-84. Zbl0611.35017
  24. [24] P.-A. RAVIART and J.-M. THOMAS, A mixed finite element method for 2nd order elliptic problems, In Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. (606), Springer-Verlag, Berlin and New York, 1977, 292-315. Zbl0362.65089MR483555
  25. [25] H. A. SCHWARZ, Über einige Abbildungsaufgaben, Ges. Math. Abh., 11(1869), 65-83. 
  26. [26] J. WANG, Convergence analysis without regularity assumptions for multigrid algorithme based on SOR smoothing, SIAM J. Numer. Anal, (to appear). Zbl0753.65093MR1173181
  27. [27] J. WANG, Convergence analysis of Schwarz algorithm and multilevel decomposition iterative methods I : self adjoint and positive definite elliptic problems, SIAM J. Numer. Anal, (submitted) and in the « Proceeding of International Conference on Iterative Methods in Linear Algebra », Belgium, 1991. Zbl0785.65115MR1159720
  28. [28] J. WANG, Convergence analysis of Schwarz algorithm and multilevel decomposition iterative methods II : non-self adjoint and indefinite elliptic problems, SIAM J. Numer. Anal, (submitted). Zbl0777.65066
  29. [29] J. WANG, Asymptotic expansions and L∞-error estimates for mixed finite element methods for second order elliptic problems, Numer. Math., 55 (1989), 401-430. Zbl0676.65109MR997230

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