Some Computer Assisted Proofs for Solutions of the Heat Convection Problems

This is a continuation of our previous results (Y. Watanabe, N. Yamamoto, T. Nakao, and T. Nishida, “A Numerical Verification of Nontrivial Solutions for the Heat Convection Problem,” to appear in the Journal of Mathematical Fluid Mechanics). In that work, the authors considered twodimensional Rayleigh-Bénard convection and proposed an approach to prove existence of steady-state solutions based on an infinite dimensional fixed-point theorem using a Newton-like operator with spectral approximation and constructive error estimates. We numerically verified several exact nontrivial solutions which correspond to solutions bifurcating from the trivial solution. This paper shows more detailed results of verification for given Prandtl and Rayleigh numbers. In particular, we found a new and interesting solution branch which was not obtained in the previous study, and it should enable us to present important information to clarify the global bifurcation structure. All numerical examples discussed are take into account of the effects of rounding errors in the floating point computations. 1. The Rayleigh-Bénard Problems We consider a plane horizontal layer (see Figure 1) of an incompressible viscous fluid heated from below. At the lower boundary z = 0 the layer of fluid is maintained at temperature T + δT , and the temperature of the upper boundary (z = h) is T . 360 MITSUHIRO T. NAKAO ET AL.