Abstract
We consider a numerical method to get a guaranteed bound of the optimal constant in the error estimates of a finite element method with linear triangular elements in the plane. The problem is reduced to a kind of smallest eigenvalue problem for an elliptic operator in a certain function space on the reference triangle. In order to solve the problem, we formulate a numerical verification procedure based on finite element approximations and constuctive error estimates. Consequently, we obtain a sufficiently sharp bound of the desired constant by a computer assisted proof. In this paper, we provide the basic idea and outline the concept of verification procedures as well as show the final numerical result. The detailed description of procedures for actual computations will be presented in the forthcoming paper [11].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Arbenz, P.: Computable finite element error bounds for Poisson’s equation. IMA J. Numer. Anal. 2, 475–479 (1982).
Grisvard, P.: Elliptic problems in nonsmooth domain. Boston: Pitman, 1985.
Kearfott, R. B., Kreinovich, V. (eds.): Applications of interval computations. Dordrecht: Kluwer, 1996.
Ladyzhenskaya, O. A.: The mathematical theory of viscous incompressible flow, 2nd edn. New York: Gordon and Breach, 1969.
Lehmann, R.: Computable error bounds in the finite-element method. IMA J. Numer. Anal. 6, 265–271 (1986).
Nakao, M. T.: A numerical approach to the proof of existence of solutions for elliptic problems. Jpn. J. Appl. Math. 5, 313–332 (1988).
Nakao, M. T.: Solving nonlinear elliptic problems with result verification using an H-1 residual iteration. Computing [Suppl.] 9, 161–173 (1993).
Nakao, M. T., Yamamoto, N.: Numerical verification of solutions for nonlinear elliptic problems using L.L∞ residual method. J. Math. Anal. Appl. 217, 246–262 (1998).
Nakao, M. T., Yamamoto, N., Kimura, S.: On best constant in the optimal error estimates for the H 00 -projection into piecewise polynomial spaces. J. Approx. Theory 93, 491–500 (1998).
Nakao, M. T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Functional Anal. Optim. (to appear).
Nakao, M. T., Yamamoto, N.: A guaranteed bound of the optimal constant in the error estimates for linear triangular element: Part II: Details, submitted.
Natterer, F.: Berechenbare Fehler schranken für die Methode der Finite Elemente. International Series of Numerical Mathematics, Vol. 28, 109–121 Basel: Birkhäuser, 1975.
Schultz, M.H.: Spline anlysis. London: Prentice-Hall, 1973.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Wien
About this paper
Cite this paper
Nakao, M.T., Yamamoto, N. (2001). A Guaranteed Bound of the Optimal Constant in the Error Estimates for Linear Triangular Element. In: Alefeld, G., Chen, X. (eds) Topics in Numerical Analysis. Computing Supplementa, vol 15. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6217-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-7091-6217-0_13
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-83673-6
Online ISBN: 978-3-7091-6217-0
eBook Packages: Springer Book Archive