Abstract
The numerical solution of the Poisson equation is treated by a multigrid method for a uniform grid. The convergence rate can be estimated even for the iteration with a V-cycle independently of the shape of the domain as long as it is convex and polygonal. The smoothing effect of the Gauß-Seidel relaxation is described by a discrete seminorm which is weaker than the energy norm.
This is a preview of subscription content, log in via an institution to check access.
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
R.E. Bank and T. Dupont, An optimal order process for solving finite element equations. Math.Comp. 36 (1981), 35–51.
R.E.Bank and T.Dupont, Analysis of a two-level scheme for solving finite element equations. Report CNA-159, Austin 1980.
D. Braess, The contraction number of a multigrid method for solving the Poisson equation. Numer. Math. 37 (1981), 387–404.
D.Braess, The convergence rate of a multigrid method with Gauß-Seidel relaxation for the Poisson equation. (submitted)
A. Brandt, Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31 (1977) 333–390.
W. Hackbusch, On the convergence of multi-grid iterations. Beiträge zur Numer. Math. 9 (1981), 213–239.
J.F.Maitre and F.Musy, The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems. (These proceedings).
Th.Meis and H.-W.Branca, Schnelle Lösung von Randwertaufgaben. ZAMM 62.
R.A. Nicolaides, On the l2-convergence of an algorithm for solving finite element equations. Math. Comp. 31 (1977), 892–906.
R.A. Nicolaides, On some theoretical and practical aspects of multigrid methods. Math. Comp. 33 (1979), 933–952.
M.Ries, U.Trottenberg and G.Winter, A note on MGR methods, Preprint, Bonn 1981.
U.Trottenberg, private communication.
R.Verfürth, The contraction number of a multigrid method with mesh ratio 2 for solving Poisson’s equation. (in preparation).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1982 Springer-Verlag
About this paper
Cite this paper
Braess, D. (1982). The convergence rate of a multigrid method with Gauss-Seidel relaxation for the poisson equation. In: Hackbusch, W., Trottenberg, U. (eds) Multigrid Methods. Lecture Notes in Mathematics, vol 960. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069934
Download citation
DOI: https://doi.org/10.1007/BFb0069934
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11955-5
Online ISBN: 978-3-540-39544-7
eBook Packages: Springer Book Archive