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

Oswald, Peter

doi:10.1090/S0025-5718-99-01041-8

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