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

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- by Peter Oswald PDF
- Math. Comp.
**68**(1999), 633-650 Request permission

## 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

- 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** - 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**, DOI 10.1090/S0025-5718-1991-1090464-8 - James H. Bramble, Joseph E. Pasciak, and Jinchao Xu,
*Parallel multilevel preconditioners*, Math. Comp.**55**(1990), no. 191, 1–22. MR**1023042**, DOI 10.1090/S0025-5718-1990-1023042-6 - James H. Bramble and Jinchao Xu,
*Some estimates for a weighted $L^2$ projection*, Math. Comp.**56**(1991), no. 194, 463–476. MR**1066830**, DOI 10.1090/S0025-5718-1991-1066830-3 - 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**, DOI 10.1090/conm/180/01955 - 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**, DOI 10.1007/s002110050172 - 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**, DOI 10.1007/978-3-322-91215-2 - P. Oswald,
*On estimates for hierarchic basis representations of finite element functions*, Forsch.-Ergebnisse FSU Jena, N/89/16, 1989. - P. Oswald,
*Stable subspaces splittings for Sobolev spaces and their applications*, Preprint Math/93/7, FSU Jena, September 1993. - 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**, DOI 10.1016/0377-0427(94)90330-1 - Jinchao Xu,
*Counterexamples concerning a weighted $L^2$ projection*, Math. Comp.**57**(1991), no. 196, 563–568. MR**1094965**, DOI 10.1090/S0025-5718-1991-1094965-8 - Jinchao Xu,
*Iterative methods by space decomposition and subspace correction*, SIAM Rev.**34**(1992), no. 4, 581–613. MR**1193013**, DOI 10.1137/1034116 - S. Minakshi Sundaram,
*On non-linear partial differential equations of the parabolic type*, Proc. Indian Acad. Sci., Sect. A.**9**(1939), 479–494. MR**0000088** - 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**, DOI 10.1017/S0962492900002385 - Xuejun Zhang,
*Multilevel Schwarz methods*, Numer. Math.**63**(1992), no. 4, 521–539. MR**1189535**, DOI 10.1007/BF01385873

## 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
- Received by editor(s): July 12, 1996
- Received by editor(s) in revised form: September 22, 1997
- © Copyright 1999 American Mathematical Society
- 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