Summary
A simple and direct application of multigrid techniques for solving a standard finite difference approximation for the first boundary value problem of the biharmonic equation is described. Numerical results show that the convergence factors of the iterative multigrid method are h-independent and correspond to the good interior smoothing factors of the relaxation used. Moreover the convergence factors of our method are comparable to those of corresponding mu1tigrid iterations for the Poisson equation while its computational effort per iteration is higher by a factor of 2.3–3 (depending on the chosen variant). Furthermore numerical results for the full-multigrid method (FMG) are presented. As in the case of Poisson’s equation, FMG turns out to be an efficient approximate direct solver for the biharmonic equation with complexity O(N2) when solving the problem on a NxN grid, yielding solutions up to discretization errors.
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
Bjørstad, P.: Numerical solution of the biharmonic equation. Ph.D. Thesis, Dept. of Computer Science, Stanford University, 1981.
Bramble, J.H.: A second order finite difference analog of the first biharmonic boundary value problem. Numerische Mathematik, 9, 1966.
Brandt, A.: Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD-Studie No. 85, Gesellschaft für Mathematik und Datenverarbeitung, St. Augustin, 1984.
Buzbee, B.L., Dorr, F.W.: The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions. SIAM 3. Numer. Anal., Vol. 11, 1974.
Holstein, H.: A multigrid treatment of Stokesian boundary conditions. Lecture, held at the Oberwolfach-conference on Multigrid Methods (9.12.–15.12.1984).
Kuttler, J.R.: A finite difference approximation for the eigenvalues of the clamped plate. Numerische Mathematik, 17, 1971.
Linden, J.: Dissertation, Universität Bonn, to appear.
Papamanolis, G.: Multigrid methods in fluid dynamics. Master Thesis, Dept. of Computer Science, University of Wales, Aberystwyth, 1984.
Stüben, K., Trottenberg, U.: Multigrid methods: fundamental algorithms, model problem analysis and applications. Proceedings of the conference on Multigrid Methods, Köln 1981 (W. Hackbusch, U. Trottenberg, eds.). Lecture Notes in Mathematics, 960, Springer Verlag, Berlin, 1982.
Vajtersic, M.: A fast algorithm for solving the first biharmonic boundary value problem. Computing 23, 1979.
Zlamal, M.: Discretization and error estimates for elliptic boundary value problems of the fourth order. SIAM J. Numer. Anal., Vol. 4, 1967.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer Fachmedien Wiesbaden
About this chapter
Cite this chapter
Linden, J. (1985). A Multigrid Method for Solving the Biharmonic Equation on Rectangular Domains. In: Braess, D., Hackbusch, W., Trottenberg, U. (eds) Advances in Multi-Grid Methods. Notes on Numerical Fluid Mechanics, vol 11. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14245-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-663-14245-4_7
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-08085-3
Online ISBN: 978-3-663-14245-4
eBook Packages: Springer Book Archive