The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation
Author:
Dietrich Braess
Journal:
Math. Comp. 42 (1984), 505-519
MSC:
Primary 65N20
DOI:
https://doi.org/10.1090/S0025-5718-1984-0736449-6
MathSciNet review:
736449
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Abstract: The convergence rate of a multigrid method for the numerical solution of the Poisson equation on a uniform grid is estimated. The results are independent of the shape of the domain as long as it is convex and polygonal. On the other hand, pollution effects become apparent when the domain contains reentrant corners. To estimate the smoothing of the Gauss-Seidel relaxation, the smoothness is measured by comparing the energy norm with a (weaker) discrete seminorm.
- [1] 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, https://doi.org/10.1090/S0025-5718-1981-0595040-2
- [2] R. E. Bank & T. Dupont, Analysis of a Two-Level Scheme for Solving Finite Element Equations, Report CNA-159, Center for Numerical Analysis, Austin, 1980.
- [3] Dietrich Braess, The contraction number of a multigrid method for solving the Poisson equation, Numer. Math. 37 (1981), no. 3, 387–404. MR 627112, https://doi.org/10.1007/BF01400317
- [4] Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), no. 138, 333–390. MR 431719, https://doi.org/10.1090/S0025-5718-1977-0431719-X
- [5] A. Brandt, private communication.
- [6] Achi Brandt and Nathan Dinar, Multigrid solutions to elliptic flow problems, Numerical methods for partial differential equations (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 42, Academic Press, New York-London, 1979, pp. 53–147. MR 558216
- [7] H. Foerster, K. Stüben & U. Trottenberg, "Non-standard multigrid techniques using checkered relaxation and intermediate grids," in Elliptic Problem Solvers (M. Schultz, ed.), Academic Press, New York, 1981.
- [8] W. Hackbusch, "On the convergence of multi-grid iterations," Beiträge Numer. Math., v. 9, 1981, pp. 213-239.
- [9] Wolfgang Hackbusch, Survey of convergence proofs for multigrid iterations, Special topics of applied mathematics (Proc. Sem., Ges. Math. Datenverarb., Bonn, 1979) North-Holland, Amsterdam-New York, 1980, pp. 151–164. MR 585154
- [10] S. F. McCormick and J. W. Ruge, Multigrid methods for variational problems, SIAM J. Numer. Anal. 19 (1982), no. 5, 924–929. MR 672568, https://doi.org/10.1137/0719067
- [11] T. Meis and H.-W. Branca, Schnelle Lösung von Randwertaufgaben, Z. Angew. Math. Mech. 62 (1982), no. 5, T263–T270. MR 677307, https://doi.org/10.1002/zamm.19820620504
- [12] W. L. Miranker and V. Ya. Pan, Methods of aggregation, Linear Algebra Appl. 29 (1980), 231–257. MR 562764, https://doi.org/10.1016/0024-3795(80)90245-1
- [13] R. A. Nicolaides, On the 𝑙² convergence of an algorithm for solving finite element equations, Math. Comp. 31 (1977), no. 140, 892–906. MR 488722, https://doi.org/10.1090/S0025-5718-1977-0488722-3
- [14] R. A. Nicolaides, On some theoretical and practical aspects of multigrid methods, Math. Comp. 33 (1979), no. 147, 933–952. MR 528048, https://doi.org/10.1090/S0025-5718-1979-0528048-4
- [15] Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, https://doi.org/10.1090/S0025-5718-1974-0373325-9
- [16] Manfred Ries, Ulrich Trottenberg, and Gerd Winter, A note on MGR methods, Linear Algebra Appl. 49 (1983), 1–26. MR 688373, https://doi.org/10.1016/0024-3795(83)90091-5
- [17] Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
- [18] R. Verfürth, "The contraction number of a multigrid method with mesh ratio 2 for solving Poisson's equation," Linear Algebra Appl. (To appear.)
- [19] 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, https://doi.org/10.1137/0720066
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Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1984-0736449-6
Article copyright:
© Copyright 1984
American Mathematical Society