Abstract
In the present paper we describe and classify these numerous variants such as BPX, hierachical basis, HBMG, local multi-grid etc. in the framework of multi-grid. We compare additive and multiplicative multi-grid and particularly investigate the different behaviour with respect to smoothing. Theoretical as well as numerical results are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Bank, R. E., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Preprint SC-87-2 (April 1987).
Bastian, P.: Locally refined solution of unsymmetric and nonlinear problems. In: Incomplete decompositions (Hackbusch, W., Wittum, G., eds.). Incomplete Decompositions NNFM, Vol. 41. Braunschweig: Vieweg 1993.
Bastian, P.: Adaptive parallele Mehrgitterverfahren. Teubner Skripten zur Numerik. Stuttgart: Teubner 1996.
Bastian, P., Wittum, G.: Adaptive multigrid methods: The UG concept. In: Adaptive methods — algorithms, theory and applications (Hackbusch, W., Wittum, G., eds.). NNFM, Vol. 46. Braunschweig: Vieweg 1994.
Bastian, P., Wittum, G.: On robust and adaptive multigrid methods. In: Multigrid methods IV (Hemker, P., Wesseling, P., eds.). Basel: Birkhäuser 1994.
Bey, J., Wittum, G.: On downwind numbered smoothing. ICA-Bericht N3/95. Appl. Numer. Math. (in press).
Bramble, J. H., Pasciak, J. E., Xu, J.: Parallel multilevel preconditioners. Math. Comput.55, 1–22 (1990).
Bramble, J. H., Pasciak, J. E., Wang, J., Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comp.57, 23–45 (1991).
Ecker, A., Zulehner, W.: On the smoothing property for the non-symmetric case. Numer. Lin. Alg. Appl.3, 161–172 (1996).
Greenbaum, A.: A multigrid method for multiprocessors. Appl. Math. Comp.19, 75–88 (1986).
Hackbusch, W.: Multi-grid methods and applications. Berlin, Heidelberg, New York, Tokyo: Springer 1985.
Hackbusch, W.: Elliptic differential equations. Berlin, Heidelberg, New York, Tokyo: Springer 1992.
Hackbusch, W.: Iterative solution of large sparse systems of equations. Berlin, Heidelberg, New York, Tokyo: Springer 1994.
Hackbusch, W.: A note on Reuskens lemma. Computing55, 181–189 (1995).
Neuß, N.: Homogenisierung und Mehrgitterverfahren. Dissertation, Heidelberg, 1995.
Reusken, A.: A new lemma in multigrid theory. Report RANA 91-07, Tu Eindhoven 1991.
Rivara, M.C.: Algorithms for refining triangular grids for adaptive and multigrid techniques. Int. J. Numer. Meth. Eng.20, 745–756 (1984).
Ruge, J., Stüben, K.: Algebraic multigrid.
Wesseling, P.: Theoretical and practical aspects of a multigrid method. SIAM J. Sci. Statist. Comp.3, 387–407 (1982).
Wittum, G.: On the robustness of ILU-smoothing. SIAM J. Sci. Stat. Comput.10, 699–717 (1989).
Wittum, G.: Linear iterations as smoothers in multi-grid methods. Impact Comput. Sci. Eng.1, 180–215 (1989).
Wittum, G.: Spektralverschobene Iterationen, ICA-Bericht N95/5, ICA, Universität Stuttgart, 1995. Numer. Math. (in press).
Xu, J.: Iterative methods by space decomposition and subspace correction: A unifying approach. SIAM Rev.34, 581–613 (1992).
Yserentant, H.: Über die Aufspaltung von Finite-Element-Räumen in Teilräume verschiedener Verfeinerungsstufen. Habilitationsschrift, RWTH Aachen, 1984.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bastian, P., Wittum, G. & Hackbusch, W. Additive and multiplicative multi-grid — A comparison. Computing 60, 345–364 (1998). https://doi.org/10.1007/BF02684380
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02684380