### Abstract

A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Bénard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set. including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.

Original language | English |
---|---|

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 |

Externally published | Yes |

### Fingerprint

### Keywords

- Computer assisted proof
- Fixed-point theorem
- Rayleigh-Bénard convection

### ASJC Scopus subject areas

- Mathematical Physics
- Condensed Matter Physics
- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Mathematical Fluid Mechanics*,

*6*(1), 1-20. https://doi.org/10.1007/s00021-003-0077-3

**A numerical verification of nontrivial solutions for the heat convection problem.** / Watanabe, Yoshitaka; Yamamoto, Nobito; Nakao, Mitsuhiro T.; Nishida, Takaaki.

Research output: Contribution to journal › Article

*Journal of Mathematical Fluid Mechanics*, vol. 6, no. 1, pp. 1-20. https://doi.org/10.1007/s00021-003-0077-3

}

TY - JOUR

T1 - A numerical verification of nontrivial solutions for the heat convection problem

AU - Watanabe, Yoshitaka

AU - Yamamoto, Nobito

AU - Nakao, Mitsuhiro T.

AU - Nishida, Takaaki

PY - 2004

Y1 - 2004

N2 - A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Bénard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set. including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.

AB - A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Bénard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set. including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.

KW - Computer assisted proof

KW - Fixed-point theorem

KW - Rayleigh-Bénard convection

UR - http://www.scopus.com/inward/record.url?scp=10044237618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10044237618&partnerID=8YFLogxK

U2 - 10.1007/s00021-003-0077-3

DO - 10.1007/s00021-003-0077-3

M3 - Article

VL - 6

SP - 1

EP - 20

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

SN - 1422-6928

IS - 1

ER -