Convergence estimates for multigrid algorithms without regularity assumptions
HTML articles powered by AMS MathViewer
- by James H. Bramble, Joseph E. Pasciak, Jun Ping Wang and Jinchao Xu PDF
- Math. Comp. 57 (1991), 23-45 Request permission
Abstract:
A new technique for proving rate of convergence estimates of multigrid algorithms for symmetric positive definite problems will be given in this paper. The standard multigrid theory requires a "regularity and approximation" assumption. In contrast, the new theory requires only an easily verified approximation assumption. This leads to convergence results for multigrid refinement applications, problems with irregular coefficients, and problems whose coefficients have large jumps. In addition, the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications.References
-
O. Axelsson and P. S. Vassilevski, Algebraic multilevel preconditioning methods, II, (preprint).
- D. Bai and A. Brandt, Local mesh refinement multilevel techniques, SIAM J. Sci. Statist. Comput. 8 (1987), no. 2, 109–134. MR 879406, DOI 10.1137/0908025
- Randolph E. Bank and Craig C. Douglas, Sharp estimates for multigrid rates of convergence with general smoothing and acceleration, SIAM J. Numer. Anal. 22 (1985), no. 4, 617–633. MR 795944, DOI 10.1137/0722038
- Randolph E. Bank and Todd Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), no. 153, 35–51. MR 595040, DOI 10.1090/S0025-5718-1981-0595040-2
- D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the $V$-cycle, SIAM J. Numer. Anal. 20 (1983), no. 5, 967–975. MR 714691, DOI 10.1137/0720066
- James H. Bramble and Joseph E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), no. 180, 311–329. MR 906174, DOI 10.1090/S0025-5718-1987-0906174-X
- 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, The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems, Math. Comp. 51 (1988), no. 184, 389–414. MR 930228, DOI 10.1090/S0025-5718-1988-0930228-6
- 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, Joseph E. Pasciak, and Jinchao Xu, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, Math. Comp. 56 (1991), no. 193, 1–34. MR 1052086, DOI 10.1090/S0025-5718-1991-1052086-4
- 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
- Achi Brandt, Algebraic multigrid theory: the symmetric case, Appl. Math. Comput. 19 (1986), no. 1-4, 23–56. Second Copper Mountain conference on multigrid methods (Copper Mountain, Colo., 1985). MR 849831, DOI 10.1016/0096-3003(86)90095-0
- Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), no. 138, 333–390. MR 431719, DOI 10.1090/S0025-5718-1977-0431719-X W. Hackbusch, Multi-grid methods and applications, Springer-Verlag, New York, 1985. R. B. Kellogg, Interpolation between subspaces of a Hilbert space, Tech. Note BN-719, Univ. of Maryland, Inst. Fluid Dynamics and Appl. Math., 1971.
- Michal Kočvara and Jan Mandel, A multigrid method for three-dimensional elasticity and algebraic convergence estimates, Appl. Math. Comput. 23 (1987), no. 2, 121–135. MR 896973, DOI 10.1016/0096-3003(87)90034-8
- J.-F. Maitre and F. Musy, Algebraic formalisation of the multigrid method in the symmetric and positive definite case—a convergence estimation for the $V$-cycle, Multigrid methods for integral and differential equations (Bristol, 1983) Inst. Math. Appl. Conf. Ser. New Ser., vol. 3, Oxford Univ. Press, New York, 1985, pp. 213–223. MR 849375
- Jan Mandel, Étude algébrique d’une méthode multigrille pour quelques problèmes de frontière libre, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 18, 469–472 (French, with English summary). MR 750748
- Jan Mandel, Multigrid convergence for nonsymmetric, indefinite variational problems and one smoothing step, Appl. Math. Comput. 19 (1986), no. 1-4, 201–216. Second Copper Mountain conference on multigrid methods (Copper Mountain, Colo., 1985). MR 849837, DOI 10.1016/0096-3003(86)90104-9
- J. Mandel, S. McCormick, and R. Bank, Variational multigrid theory, Multigrid methods, Frontiers Appl. Math., vol. 3, SIAM, Philadelphia, PA, 1987, pp. 131–177. MR 972757 P. Vassilevski, Iterative methods for solving finite element equations based on multilevel splitting of the matrix, Bulgarian Academy of Sciences, Sofia, Bulgaria, (preprint).
- Jinchao Xu, Convergence estimates for some multigrid algorithms, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 174–187. MR 1064343 J. Xu, Theory of multilevel methods, Ph.D. Thesis, Cornell University and Penn State University, Dept. Math. Rep AM-48, 1989.
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Math. Comp. 57 (1991), 23-45
- MSC: Primary 65J10; Secondary 65N55
- DOI: https://doi.org/10.1090/S0025-5718-1991-1079008-4
- MathSciNet review: 1079008