Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

On the robustness of the BPX-preconditioner with respect to jumps in the coefficients


Author: Peter Oswald
Journal: Math. Comp. 68 (1999), 633-650
MSC (1991): Primary 65N22, 65N55, 65F10
DOI: https://doi.org/10.1090/S0025-5718-99-01041-8
MathSciNet review: 1620239
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We determine the worst case behavior of the standard BPX-preconditioner for elliptic problems with arbitrary coefficient jumps along the boundaries of the coarsest partition. The counterexamples are also useful for other problems.


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

  • 1. J. H. Bramble, Multigrid methods, Pitman Research Notes in Mathematical Sciences 294, Longman Sci.&Techn., Harlow, Essex, 1993. MR 95b:65002
  • 2. J. H. Bramble, J. E. Pasciak, J. Wang, J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp. 57 (1991), 1-22. MR 92d:65094
  • 3. J. H. Bramble, J. E. Pasciak, J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990) 1-22. MR 90k:65170
  • 4. J. H. Bramble, J. Xu, Some estimates for a weighted $L^2$ projection, Math. Comp. 56 (1991), 463-476. MR 91k:65140
  • 5. M. Dryja, Multilevel methods for elliptic problems with discontinuous coefficients in three dimensions, Proceedings Seventh International Conference on Domain Decomposition, Contemporary Mathematics, vol. 180 (D. E. Keyes, J. Xu, eds.), AMS, Providence, 1994, pp. 43-47. MR 95j:65137
  • 6. M. Dryja, M. V. Sarkis, O. B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math. 72 (1996), 313-348. MR 96h:65134
  • 7. P. Oswald, Multilevel finite element approximation : theory and application, Teubner Skripten zur Numerik, Teubner, Stuttgart, 1994. MR 95k:65110
  • 8. P. Oswald, On estimates for hierarchic basis representations of finite element functions, Forsch.-Ergebnisse FSU Jena, N/89/16, 1989.
  • 9. P. Oswald, Stable subspaces splittings for Sobolev spaces and their applications, Preprint Math/93/7, FSU Jena, September 1993.
  • 10. J. Wang, New convergence estimates for multilevel algorithms for finite-element approximations, J. Comput. Appl. Math. 50 (1994), 593-604. MR 95d:65105
  • 11. J. Xu, Counterexamples concerning a weighted $L_2$-projection into finite element spaces, Math. Comp. 57 (1991), 563-568. MR 92b:65090
  • 12. J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), 581-613. MR 93k:65029
  • 13. H. Yserentant, On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), 379-412. MR 88d:65068
  • 14. H. Yserentant, Old and new convergence proofs for multigrid methods, Acta Numerica 1993, Cambr. Univ. Press, Cambridge, 1993, pp. 285-326. MR 94i:65128
  • 15. X. Zhang, Multilevel additive Schwarz methods, Numer. Math. 63 (1992), 521-539. MR 93h:65047

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65N22, 65N55, 65F10

Retrieve articles in all journals with MSC (1991): 65N22, 65N55, 65F10


Additional Information

Peter Oswald
Affiliation: Bell Laboratories, Lucent Technologies, 600 Mountain Ave., Rm. 2C-403, Murray Hill, NJ 07974-0636
Email: poswald@research.bell-labs.com

DOI: https://doi.org/10.1090/S0025-5718-99-01041-8
Keywords: Finite element multilevel preconditioners, robustness, elliptic problems with variable coefficients
Received by editor(s): July 12, 1996
Received by editor(s) in revised form: September 22, 1997
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society