Abstract
A numerical verification method of the solution for the stationary Navier-Stokes equations is described. This method is based on the infinite dimensional fixed point theorem using the Newton-like operator. We present a verification algorithm which generates automatically on a computer a set including the exact solution. Some numerical examples are also discussed.
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
Ciarlet, P. G.: The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
Girault, V. and Raviart, P. A.: Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin, Heidelberg, 1986.
Kearfott, R. B. and Kreinovich, V.: Applications of Interval Computations, Kluwer Academic Publishers, Netherlands, 1996.
Knüppel, O.: PROFIL/BIAS-A Fast Interval Library, Computing 53 (1994), pp. 277–287.
Kulisch, U. et. al.: C-XSC, A C++ Class Library for Extended Scientific Computing, Springer-Verlag, New York, 1993.
Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Flow, Gordon Breach, New York, 1969.
Nakao, M. T.: A Numerical Verification Method for the Existence of Weak Solutions for Nonlinear Boundary Value Problems, J. Math. Anal. Appl. 164 (1992), pp. 489–507.
Nakao M. T.: Solving Nonlinear Elliptic Problems with Result Verification Using an H−1 Type Residual Iteration, Computing, Suppl. 9 (1993), pp. 161–173.
Nakao, M. T., Yamamoto, N., and Kimura, S.: On Best Constant in the Optimal Error Estimates for the Ho-projection into Piecewise Polynomial Spaces, J. Approximation Theory 93 (1998), pp. 491–500.
Nakao, M. T., Yamamoto, N., and Watanabe, Y.: A Posteriori and Constructive A Priori Error Bounds for Finite Element Solutions of the Stokes Equations, J. Comput. Appl. Math. 91 (1998), pp. 137–158.
Nakao, M. T., Yamamoto, N., and Watanabe, Y.: Constructive L2 Error Estimates for Finite Element Solutions of the Stokes Equations, Reliable Computing 4 (2) (1998), pp. 115–124.
Nakao, M. T., Yamamoto, N., and Watanabe, Y.: Guaranteed Error Bounds for the Finite Element Solutions of the Stokes Problem, in: Alefeld, G., Frommer, A., and Lang, B. (eds): Scientific Computing and Validated Numerics, Proceedings of International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics SCAN-95, Mathematical Research 90 (1996), pp. 258–264.
Rump, S. M.: On the Solution of Interval Linear Systems, Computing 47 (1992), pp. 337–353.
Talenti, G.: Best Constant in Sobolev Inequality, Ann. Math. Pure Appl. 110 (1976), pp 353–372.
Watanabe, Y., and Nakao, M. T.: Numerical Verifications of Solutions for Nonlinear Elliptic Equations, Japan J. Indust. Appl. Math. 10 (1993) pp. 165–178.
Yamamoto, N.: A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach’s Fixed-point Theorem, SIAM J. Numer. Anal. 35 (1998) pp. 2004–2013.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Watanabe, Y., Yamamoto, N., Nakao, M.T. (1999). A Numerical Verification Method of Solutions for the Navier-Stokes Equations. In: Csendes, T. (eds) Developments in Reliable Computing. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1247-7_27
Download citation
DOI: https://doi.org/10.1007/978-94-017-1247-7_27
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5350-3
Online ISBN: 978-94-017-1247-7
eBook Packages: Springer Book Archive