Mitsuhiro T. Nakao, Yoshitaka Watanabe, Nobito Yamamoto, Takaaki Nishida:
Some Computer Assisted Proofs for Solutions of the Heat Convection Problems,
Reliable Computing, Vol.9, No.5 (2003) pp.359-372.
(Special Issue Proceedings of the Validated Computing 2002 conference, May 23-25, 2002, Toronto, Canada, Guest Editor: R. Baker Kearfott)
This is a continuation of our previous results ([RB] to appear in Journal of Mathematical Fluid Mechanics). In [RB], the authors considered the two-dimensional Rayleigh-B\'enard convection and proposed an approach to prove the exsistence of the steady-state solutions based on the infinite dimensional fixed-point theorem using Newton-like operator with the spectral approximation and the constructive error estimates. We numerically verified several exact non-trivial solutions which correspond to the bifurcated solutions from the trivial solution. This paper shows more detailed results of verification for the given Prandtl and Rayleigh numbers. Particularly, we found a new and interesting solution branch which was not obtained in the previous study, and it should enable us to present an important information to clarify the global bifurcation structure. All numerical results discussed are taken into account of the effects of rounding errors in the floating point computations.
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