Summary
The treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence. For the linear equations which arise from the discretization of the Poisson equation, a convergence factor 0,5 is established independent of the shape of the domain and of the regularity of the solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Babuška, I., Rheinboldt, W.: Mathematical problems of computational decisions in the finite element method. Technical report TR-426, University of Maryland, 1975
Bank, R.E., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput.36, 35–51 (1981) and: Analysis of a two-level scheme for solving finite element equations. Numer. Math. (1981, in press)
Collatz, L.: Numerische Behandlung von Differentialgleichungen. Berlin-Göttingen-Heidelberg: Springer 1951
Gunn, J.E.: The solution of difference equations by semi-explicit iterative techniques. SIAM J. Numer. Anal.2, 24–45 (1965)
Hackbusch, W.: On the multigrid method applied to difference equations. Computing20, 291–306 (1978)
Meis, T., Marcowitz, U.: Numerische Behandlung partieller Differentialgleichungen. Berlin-Heidelberg-New York: Springer 1978
Nicolaides, R.A.: On thel 2-convergence of an algorithm for solving finite element equations. Math. Comput.31, 892–906 (1977)
Nicolaides, R.A.: On some theoretical and practical aspects of multigrid methods. Math. Comput.33, 933–952 (1979)
Varga, R.S.: Matrix Iterative Analysis. Englewood Cliffs: Prentice-Hall 1962
Wesseling, P.: A convergence proof for a multiple grid method. In: Numerical analysis. Proceedings Dundee 1979 (G.A. Watson, ed.), pp. 164–183. Lecture Notes in Mathematics, Vol. 733. Berlin-Heidelberg-New York: Springer 1980
Author information
Authors and Affiliations
Additional information
Dedicated to Professor Dr.Dr.h.c. Lothar Collatz on the occasion of his 70 th birthday
Rights and permissions
About this article
Cite this article
Braess, D. The contraction number of a multigrid method for solving the Poisson equation. Numer. Math. 37, 387–404 (1981). https://doi.org/10.1007/BF01400317
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01400317