Numerical Enclosures for Nontrivial Solutions of the Heat Convection Problems

Yoshitaka Watanabe, Nobito Yamamoto, Mitsuhiro T. Nakao and Takaaki Nishida
10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2002), September 24-27, 2002, Paris, France.

A computer assisted proof of nontrivial steady-state solutions for the two-dimensional Rayleigh-B\'enard convection is described.

The method is based on the infinite dimensional fixed-point theorem using Newton-like operator. We also proposes a numerical verification algorithm which generates automatically on a computer a set including the exact nontrivial solution. All discussed numerical examples are taken into account of the effects of rounding errors in the floating point computations.