zbMATH — the first resource for mathematics

Improved convergence bounds for two-level methods with an aggressive coarsening and massive polynomial smoothing. (English) Zbl 1398.65051
Summary: An improved convergence bound for the polynomially accelerated two-level method of J. Brousek et al. [ETNA, Electron. Trans. Numer. Anal. 44, 401–442 (2015; Zbl 1327.65058), Section 5] is proven. This method is a reinterpretation of the smoothed aggregation method with an aggressive coarsening and massive polynomial smoothing of the second author et al. [SIAM J. Sci. Comput. 21, No. 3, 900–923 (1999; Zbl 0952.65099)], and its convergence rate estimate is improved here quantitatively. Next, since the symmetrization of the method requires two solutions of the coarse problem, a modification of the method is proposed that does not have this disadvantage, and a qualitatively better convergence result for the modification is established. In particular, it is shown that a bound of the convergence rate of the method with a multiply (\(k\)-times) smoothed prolongator is asymptotically inversely proportional to \(d^{2k}\), where \(d\) is the degree of the smoothing polynomial. In earlier works, this acceleration effect is only quadratic. Finally, for another modified multiply smoothed method, it is proved that this convergence improvement is not limited only to an asymptotic regime but holds true everywhere.

65F10 Iterative numerical methods for linear systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
Full Text: DOI Link
[1] J. H. BRAMBLE, J. E. PASCIAK, J. P. WANG,ANDJ. XU, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp., 57 (1991), pp. 23-45. · Zbl 0727.65101
[2] A. BRANDT, Algebraic multigrid theory: the symmetric case, Appl. Math. Comput., 19 (1986), pp. 23-56. · Zbl 0616.65037
[3] M. BREZINA, P. VAN ˇEK,ANDP. S. VASSILEVSKI, An improved convergence analysis of smoothed aggregation algebraic multigrid, Numer. Linear Algebra Appl., 19 (2012), pp. 441-469. · Zbl 1274.65315
[4] J. BROUSEK, P. FRANKOVÁ, M. HANUŠ, H. KOPINCOVÁ, R. KUŽEL, R. TEZAUR, P. VAN ˇEK,AND Z. VASTL, An overview of multilevel methods with aggressive coarsening and massive polynomial smoothing, Electron. Trans. Numer. Anal., 44 (2015), pp. 401-442. http://etna.ricam.oeaw.ac.at/vol.44.2015/pp401-442.dir/pp401-442.pdf
[5] P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. · Zbl 0383.65058
[6] P. VAN ˇEK, Acceleration of convergence of a two-level algorithm by smoothing transfer operators, Appl. Math., 37 (1992), pp. 265-274.
[7] , Fast multigrid solver, Appl. Math., 40 (1995), pp. 1-20. · Zbl 0824.65016
[8] P. VAN ˇEK ANDM. BREZINA, Nearly optimal convergence result for multigrid with aggressive coarsening and polynomial smoothing, Appl. Math., 58 (2013), pp. 369-388. · Zbl 1289.65064
[9] P. VAN ˇEK, M. BREZINA,ANDJ. MANDEL, Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math., 88 (2001), pp. 559-579. · Zbl 0992.65139
[10] P. VAN ˇEK, M. BREZINA,ANDR. TEZAUR, Two-grid method for linear elasticity on unstructured meshes, SIAM J. Sci. Comput., 21 (1999), pp. 900-923. · Zbl 0952.65099
[11] P. VAN ˇEK, J. MANDEL,ANDM. BREZINA, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56 (1996), pp. 179-196.
[12] J. XU ANDL. ZIKATANOV, The method of alternating projections and the method of subspace corrections in Hilbert space, J. Amer. Math. Soc., 15 (2002), pp. 573-597. · Zbl 0999.47015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.