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 Free Access
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.
- 1. James H. Bramble, Multigrid methods, Pitman Research Notes in Mathematics Series, vol. 294, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1247694
- 2. James H. Bramble, Joseph E. Pasciak, Jun Ping Wang, and Jinchao Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp. 57 (1991), no. 195, 1–21. MR 1090464, https://doi.org/10.1090/S0025-5718-1991-1090464-8
- 3. James H. Bramble, Joseph E. Pasciak, and Jinchao Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), no. 191, 1–22. MR 1023042, https://doi.org/10.1090/S0025-5718-1990-1023042-6
- 4. James H. Bramble and Jinchao Xu, Some estimates for a weighted 𝐿² projection, Math. Comp. 56 (1991), no. 194, 463–476. MR 1066830, https://doi.org/10.1090/S0025-5718-1991-1066830-3
- 5. Maksymilian Dryja, Multilevel methods for elliptic problems with discontinuous coefficients in three dimensions, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993) Contemp. Math., vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 43–47. MR 1312376, https://doi.org/10.1090/conm/180/01955
- 6. Maksymilian Dryja, Marcus V. Sarkis, and Olof B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math. 72 (1996), no. 3, 313–348. MR 1367653, https://doi.org/10.1007/s002110050172
- 7. Peter Oswald, Multilevel finite element approximation, Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics], B. G. Teubner, Stuttgart, 1994. Theory and applications. MR 1312165
- 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. Jun Ping Wang, New convergence estimates for multilevel algorithms for finite-element approximations, Proceedings of the Fifth International Congress on Computational and Applied Mathematics (Leuven, 1992), 1994, pp. 593–604. MR 1284291, https://doi.org/10.1016/0377-0427(94)90330-1
- 11. Jinchao Xu, Counterexamples concerning a weighted 𝐿² projection, Math. Comp. 57 (1991), no. 196, 563–568. MR 1094965, https://doi.org/10.1090/S0025-5718-1991-1094965-8
- 12. Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, https://doi.org/10.1137/1034116
- 13. H. Yserentant, On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), 379-412. MR 88d:65068
- 14. Harry Yserentant, Old and new convergence proofs for multigrid methods, Acta numerica, 1993, Acta Numer., Cambridge Univ. Press, Cambridge, 1993, pp. 285–326. MR 1224685, https://doi.org/10.1017/S0962492900002385
- 15. Xuejun Zhang, Multilevel Schwarz methods, Numer. Math. 63 (1992), no. 4, 521–539. MR 1189535, https://doi.org/10.1007/BF01385873
Retrieve articles in Mathematics of Computation 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