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Smoothed prolongation multigrid with rapid coarsening and massive smoothing. (English) Zbl 1249.65272

Assuming no regularity, it is impossible to improve the convergence of the multigrid method by adding more smoothing steps, i.e., rapid coarsening cannot be compensated by massive smoothing. The paper is concerned with a version of the algebraic multigrid method which consists in the rapid coarsening of the second grid as compared with the first, finest grid. As a consequence, the computation is very quickly moved to coarse grids and is very efficient.
On the other hand, the price we have to pay for this advantage is a very massive smoothing of the prolongation operator. This idea was first published by the author (and his coauthors) for the two-grid method. In the present paper the author first formulates the algorithm of the multigrid method and then presents the convergence analysis. He employs simpler and more general tools.
The method is analyzed for the V-cycle of the multigrid method and generalized. The principal statement of the paper is proven. The extension to W-cycle is apparent. A particular prolongation smoother is proposed. In conclusion, the author shows an application of the procedure presented to the regular multigrid method for a problem with an \(H_0^1\) equivalent norm.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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References:

[1] J.H. Bramble, J.E. Pasciak, J. Wang, J. Xu: Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comput. 57 (1991), 23–45. · Zbl 0727.65101 · doi:10.1090/S0025-5718-1991-1079008-4
[2] M. Brezina, C. Heberton, J. Mandel, P. Vaněk: An iterative method with convergence rate chosen a priori UCD/CCM. Report No. 140 (1999).
[3] P. Vaněk, M. Brezina, R. Tezaur: Two-grid method for linear elasticity on unstructured meshes. SIAM J. Sci Comput. 21 (1999), 900–923. · Zbl 0952.65099 · doi:10.1137/S1064827596297112
[4] P. Vaněk, J. Mandel, M. Brezina: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56 (1996), 179–196. · Zbl 0851.65087 · doi:10.1007/BF02238511
[5] P. Vaněk: Acceleration of convergence of a two-level algorithm by smoothing transfer operators. Appl. Math. 37 (1992), 265–274. · Zbl 0773.65021
[6] P. Vaněk, M. Brezina, J. Mandel: Convergence of algebraic multigrid based on smoothed aggregations. Numer. Math. 88 (2001), 559–579. · Zbl 0992.65139 · doi:10.1007/s211-001-8015-y
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