Summary
The multigrid full approximation scheme (FAS MG) is a well-known solver for nonlinear boundary value problems. In this paper we restrict ourselves to a class of second order elliptic mildly nonlinear problems and we give local conditions, e.g. a local Lipschitz condition on the derivative of the continuous operator, under which the FAS MG with suitably chosen parameters locally converges. We prove quantitative convergence statements and deduce explicit bounds for important quantities such as the radius of a ball of guaranteed convergence, the number of smoothings needed, the number of coarse grid corrections needed and the number of FAS MG iterations needed in a nested iteration. These bounds show well-known features of the FAS MG scheme.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bank, R.E., Douglas, C.C.: Sharp estimates for multigrid rates of convergence with general smoothing and acceleration. SIAM J. Numer. Anal.22, 617–633 (1985)
Brandt, A.: Guide in Multigrid Development. In: Multigrid Methods (W. Hackbusch, U. Trottenberg, eds.) pp. 220–312. Proceedings, Köln-Porz, 1981. Lect. Notes Math., Vol. 960. Berlin, Heidelberg, New York: Springer 1982
Brezis, H.: Operateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert. Amsterdam, London. North-Holland 1973; New York: American Elsevier 1973
Brezis, H., Crandall, M.G., Pazy, A. Perturbations of Nonlinear Maximal Monotone Sets in Banach space. Commun. Pure Appl. Math.23, 123–144, 1970
Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam, New York, Oxford: North-Holland 1978
Hackbusch, W.: Multigrid Methods and Applications. Berlin, Heidelberg, New York, Tokyo: Springer 1985
Hackbusch, W., Trottenberg, U. (eds.). Multigrid Methods. Proceedings, Köln-Porz, 1981. Lect. Notes Math., Vol. 960. Berlin, Heidelberg, New York: Springer 1982
Necas, J.: Les methodes directes en theorie des equations elliptiques. Paris, Masson etC ie 1967
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. New York, London: Academic Press 1970
Bank, R.E., Rose, D.J.: Analysis of a Multilevel Iterative Method for Nonlinear Finite Element Equations. Math. Comput.39, 453–465 (1982)
Maitre, J.F., Musy, F.: Multigrid Methods: Convergence Theory in a Variational Framework. SIAM J. Numer. Anal.21, 657–671 (1984)
Deimling, K.: Nonlinear Functional Analysis. Berlin, Heidelberg, New York Springer 1985
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Reusken, A. Convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems. Numer. Math. 52, 251–277 (1987). https://doi.org/10.1007/BF01398879
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01398879