Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Substructuring preconditioners for the three fields domain decomposition method


Author: Silvia Bertoluzza
Journal: Math. Comp. 73 (2004), 659-689
MSC (2000): Primary 65N55, 65N22
DOI: https://doi.org/10.1090/S0025-5718-03-01550-3
Published electronically: October 17, 2003
MathSciNet review: 2031400
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of preconditioners based on substructuring, for the discrete Steklov-Poincaré operator arising in the three fields formulation of domain decomposition in two dimensions. Under extremely general assumptions on the discretization spaces involved, an upper bound is provided on the condition number of the preconditioned system, which is shown to grow at most as $\log(H/h)^2$ ($H$ and $h$ denoting, respectively, the diameter and the discretization mesh-size of the subdomains). Extensive numerical tests--performed on both a plain and a stabilized version of the method--confirm the optimality of such bound.


References [Enhancements On Off] (What's this?)

  • 1. Y. Achdou, Y. Maday, and O. Widlund.
    Substructuring preconditioners for the mortar method in dimension two.
    SIAM J. Numer. Anal., 36:551-580, 1999. MR 99m:65233
  • 2. C. Baiocchi, F. Brezzi, and D. Marini.
    Stabilization of Galerkin methods and application to domain decomposition.
    In Future Tendencies in Computer Science, Control and Applied Mathematics. 1992. MR 94g:65119
  • 3. C. Bernardi, Y. Maday, and A.T. Patera.
    A new nonconforming approach to domain decomposition: The mortar element method.
    In H. Brezis & J.-L. Lions, editors, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, volume XI of Notes Math. Ser. 299, pages 13-51. 1994. MR 95a:65201
  • 4. S. Bertoluzza.
    Analysis of a stabilized three fields domain decomposition method.
    Technical Report 1175, I.A.N.-C.N.R., 2000.
  • 5. S. Bertoluzza.
    Wavelet stabilization of the Lagrange multiplier method.
    Numer. Math., 86:1-28, 2000. MR 2001h:65142
  • 6. S. Bertoluzza, C. Canuto, and A. Tabacco.
    Stable discretization of convection-diffusion problems via computable negative order inner products.
    SINUM, 38:1034-1055, 2000. MR 2001m:65163
  • 7. S. Bertoluzza and A. Kunoth.
    Wavelet stabilization and preconditioning for domain decomposition.
    I.M.A. Jour. Numer. Anal., 20:533-559, 2000. MR 2001h:65156
  • 8. S. Bertoluzza and G. Manzini.
    Wavelet stabilization of the three fields domain decomposition method: Implementation and numerical tests.
    In preparation.
  • 9. J. H. Bramble, J. E. Pasciak, and A. H. Schatz.
    The construction of preconditioners for elliptic problems by substructuring.
    Math. of Comp., 47(175):103-134, 1986. MR 87m:65174
  • 10. J.H. Bramble, J.E. Pasciak, and P.S. Vassilevski.
    Computational scales of Sobolev norms with application to preconditioning.
    Math. Comp., 69:463-480, 2000. MR 2000k:65088
  • 11. F. Brezzi and M. Fortin.
    Mixed and Hybrid Finite Element Methods.
    Springer, 1991. MR 92d:65187
  • 12. F. Brezzi, L. Franca, D. Marini, and A. Russo.
    Stabilization techniques for domain decomposition methods with nonmatching grids.
    In Proc. IX Domain Decomposition Methods Conference.
  • 13. F. Brezzi and D. Marini.
    Error estimates for the three-field formulation with bubble stabilization.
    Math. of Comp.,
    70:911-934, 2001. MR 2002b:65159
  • 14. F. Brezzi and D. Marini.
    A three-field domain decomposition method.
    In A. Quarteroni, J. Periaux, Y.A. Kuznetsov, and O.B. Widlund, editors, Domain Decomposition Methods in Science and Engineering, volume 157 of American Mathematical Society, Contemporary Mathematics, pages 27-34, 1994. MR 95a:65202
  • 15. A. Cohen.
    Numerical analysis of wavelet methods.
    In P.G. Ciarlet and J.L. Lions, editors, Handbook in Numerical Analysis, volume VII. Elsevier Science Publishers, North Holland, 2000. MR 2002c:65252
  • 16. W. Dahmen and A. Kunoth.
    Multilevel preconditioning.
    Numer. Math., 63:315-344, 1992. MR 93j:65065
  • 17. J.L. Lions and E. Magenes.
    Non-homogeneous Boundary Value Problems and Applications.
    Springer, 1972. MR 50:2670, MR 50:2671
  • 18. P. Le Tallec and T. Sassi.
    Domain decomposition with nonmatching grids: Augmented Lagrangian approach.
    Math. Comp., 64:1367-1396, 1995. MR 95m:65212

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N55, 65N22

Retrieve articles in all journals with MSC (2000): 65N55, 65N22


Additional Information

Silvia Bertoluzza
Affiliation: Istituto di Matematica Applicata e Tecnologie Informatiche del Consiglio Nazionale delle Ricerche, v. Ferrata 1, 27100 Pavia, Italy
Email: silvia.bertoluzza@imati.cnr.it

DOI: https://doi.org/10.1090/S0025-5718-03-01550-3
Keywords: Three fields formulation, domain decomposition, stabilized methods, preconditioning
Received by editor(s): November 6, 2000
Received by editor(s) in revised form: February 22, 2002
Published electronically: October 17, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society