Abstract
In previous works (Nakao et al., Reliab. Comput., 9(5):359–372, 2003; Watanabe et al., J. Math. Fluid Mech., 6(1):1–20, 2004), the authors considered the numerical verification method of solutions for two-dimensional heat convection problems known as Rayleigh-Bénard problem. In the present paper, to make the arguments self-contained, we first summarize these results including the basic formulation of the problem with numerical examples. Next, we will give a method to verify the bifurcation point itself, which should be an important information to clarify the global bifurcation structure, and show a numerical example. Finally, an extension to the three dimensional case will be described.
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References
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, London (1961)
Curry, J.H.: Bounded solutions of finite dimensional approximations to the Boussinesq equations. SIAM J. Math. Anal. 10, 71–79 (1979)
Getling, A.V.: Rayleigh-Bénard Convection: Structures and Dynamics. Advanced Series in Nonlinear Dynamics, vol. 11. World Scientific, Singapore (1998)
Kawanago, T.: A symmetry-breaking bifurcation theorem and some related theorems applicable to maps having unbounded derivatives. Jpn. J. Ind. Appl. Math. 21, 57–74 (2004)
Kim, M.-N., Nakao, M.T., Watanabe, Y., Nishida, T.: A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Bénard problems. Numer. Math. 111, 389–406 (2009)
Kearfott, R.B., Kreinovich, V. (eds.): Applications of Interval Computations. Kluwer Academic, Dordrecht (1996)
Nagatou, K., Yamamoto, N., Nakao, M.T.: An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optim. 20, 543–565 (1999)
Nakao, M.T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim. 22(3&4), 321–356 (2001)
Nakao, M.T., Watanabe, Y., Yamamoto, N., Nishida, T.: Some computer assisted proofs for solutions of the heat convection problems. Reliab. Comput. 9(5), 359–372 (2003)
Nakao, M.T., Hashimoto, K., Watanabe, Y.: A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems. Computing 75, 1–14 (2005)
Nishida, T., Ikeda, T., Yoshihara, H.: Pattern formation of heat convection problems. In: Miyoshi, T., et al. (eds.) Proc. Internat. Symp. Math. Modeling and Numer. Simul. in Cont. Mech. (2000). Lecture Notes in Comput. Sci. Engin., vol. 19, pp. 209–218. Springer, Berlin (2002)
Watanabe, Y., Yamamoto, N., Nakao, M.T., Nishida, T.: A numerical verification of nontrivial solutions for the heat convection problem. J. Math. Fluid Mech. 6(1), 1–20 (2004)
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, 2004–2013 (1998)
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An erratum to this article can be found at http://dx.doi.org/10.1007/s10915-010-9355-4
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Nakao, M.T., Watanabe, Y., Yamamoto, N. et al. Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems. J Sci Comput 43, 388–401 (2010). https://doi.org/10.1007/s10915-009-9303-3
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DOI: https://doi.org/10.1007/s10915-009-9303-3