Summary
The Fully Adaptive Multigrid Method (FAMe) is a concept for the effective solution of elliptic problems including robust and efficient iterative solution, error estimation, and self-adaptive refinement. In this paper we introduce a variant of the FAMe similar in structure to a multigrid V-cycle and a multiplicative multilevel Schwarz method. This variant permits a convergence analysis showing that the FAMe provides optimal convergence rates when the classical methods do. The FAMe, however, will be more efficient in a local refinement context by exploiting the locality of the computations and will be more robust, because it naturally provides diagnostic information that can serve as rigorous error bounds.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Bai and A. Brandt. Local mesh refinement multilevel techniques. SIAM J. Sci. Stat. Comput., 8 (2): 109–134, March 1987.
R. Bank. PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. Frontiers in Applied Mathematics. SIAM, Philadelphia, 1990.
D. Braess and W. Hackbusch. A new convergence proof for the multigrid method including the V-cycle. SIAM J. Numer. Anal., 20: 967–975, 1983.
J. Bramble, J. Pasciak, and J. Xu. Parallel multilevel preconditioners. Math. Comp., 31: 333–390, 1990.
N. Decker, J. Mandel, and S. Parter. On the role of regularity in multigrid methods. In S. McCormick, editor, Multigrid Methods: Theory, Applications, Supercomputing: Proceedings of the Third Copper Mountain Conference on Multigrid Methods, April 5–10, 1987, New York, 1988. Marcel Dekker.
P. Leinen. Ein schneller adaptiver Lóser für elliptische Randwertprobleme auf Seriell-und Parallelrechnern. Dissertation, Universität Dortmund, 1990.
J.-F. Maitre and F. Musy. Multigrid methods: Convergence theory in a variational framework. SIAM J. Numer. Anal., 21: 657–671, 1984.
J. Mandel, S. McCormick, and J. Ruge. An algebraic theory for multigrid methods for variational problems. SIAM J. Numer. Anal., 25 (1): 91–110, February 1988.
S. McCormick. Multilevel Adaptive Methods for Partial Differential Equations, volume 6 of Frontiers in Applied Mathematics. SIAM, Philadelphia, 1989.
P. Oswald. On discrete norm estimates related to multilevel preconditioners in the finite element method. In Proceedings of the International Conference on Constructive Theory of Functions, Varna, 1991,pages 203–214, Sofia, 1992. Bulg. Acad. Sci.
M. Rivara. Algorithms for refining triangular grids suitable for adaptive and multigrid techniques. International Journal for Numerical Methods in Engineering, 20: 745–756, 1984.
U. Rüde. Adaptive higher order multigrid methods. In W. Hackbusch and U. Trotten-berg, editors, Proceedings of the Third European Conference on Multigrid Methods, October 1–4, 1990, pages 339–351, Basel, 1991. Birkhäuser. International Series of Numerical Mathematics, Vol. 98.
U. Rüde. On the multilevel adaptive iterative method. In T. Manteuffel, editor, Preliminary proceedings of the 2nd Copper Mountain Conference on Iterative Methods, April 9–14, 1992. University of Colorado at Denver, 1992. Accepted for publication in SIAM J. Sci. Stat. Comput. and also available as TU-Bericht I - 9216.
U. Rüde. Fully adaptive multigrid methods. SIAM J. Numer. Analysis,30(1):230248, February 1993.
U. Rüde. Mathematical and computational techniques for multilevel adaptive methods, volume 13 of Frontiers in Applied Mathematics. SIAM, Philadelphia, 1993.
O. B. Widlund. Optimal iterative refinement methods. In T. F. Chan, R. Glowinsky, J. Périaux, and O. B. Widlund, editors, Domain Decomposition Methods, Philadelphia, 1989. SIAM.
H. Yserentant. Two preconditioners based on the multi-level splitting of finite element spaces. Numer. Math., 58: 163–184, 1990.
X. Zhang. Multilevel additive Schwarz methods. Tech. Report 582, New York University, Courant Institute, Department of Computer Science, 1991.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer Fachmedien Wiesbaden
About this chapter
Cite this chapter
Rüde, U. (1994). On the V-Cycle of the Fully Adaptive Multigrid Method. In: Hackbusch, W., Wittum, G. (eds) Adaptive Methods — Algorithms, Theory and Applications. Notes on Numerical Fluid Mechanics (NNFM). Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14246-1_17
Download citation
DOI: https://doi.org/10.1007/978-3-663-14246-1_17
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-07646-7
Online ISBN: 978-3-663-14246-1
eBook Packages: Springer Book Archive