### Abstract

This paper is the three dimensional extension of the two dimensional work in Nakao et al. (Reliable Comput 9(5):359-372, 2003) and Watanabe et al. (J Math Fluid Mech 6:1-20, 2004) on a computer assisted proof of the existence of nontrivial steady state solutions for Rayleigh-Bénard convection based on the fixed point theorem using a Newton like operator. The differences are emerging of complicated types of bifurcation, direct attack on the problem without stream functions, and increased complexity of numerical computation. The last one makes it hard to proceed the verification of solutions corresponding to the points on bifurcation diagram for three dimensional case. Actually, this work should be the first result for the three dimensional Navier-Stokes problems which seems to be very difficult to solve by theoretical approaches.

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

Pages (from-to) | 389-406 |

Number of pages | 18 |

Journal | Numerische Mathematik |

Volume | 111 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 Jan |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Numerische Mathematik*,

*111*(3), 389-406. https://doi.org/10.1007/s00211-008-0191-5

**A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Bénard problems.** / Kim, Myoungnyoun; Nakao, Mitsuhiro T.; Watanabe, Yoshitaka; Nishida, Takaaki.

Research output: Contribution to journal › Article

*Numerische Mathematik*, vol. 111, no. 3, pp. 389-406. https://doi.org/10.1007/s00211-008-0191-5

}

TY - JOUR

T1 - A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Bénard problems

AU - Kim, Myoungnyoun

AU - Nakao, Mitsuhiro T.

AU - Watanabe, Yoshitaka

AU - Nishida, Takaaki

PY - 2009/1

Y1 - 2009/1

N2 - This paper is the three dimensional extension of the two dimensional work in Nakao et al. (Reliable Comput 9(5):359-372, 2003) and Watanabe et al. (J Math Fluid Mech 6:1-20, 2004) on a computer assisted proof of the existence of nontrivial steady state solutions for Rayleigh-Bénard convection based on the fixed point theorem using a Newton like operator. The differences are emerging of complicated types of bifurcation, direct attack on the problem without stream functions, and increased complexity of numerical computation. The last one makes it hard to proceed the verification of solutions corresponding to the points on bifurcation diagram for three dimensional case. Actually, this work should be the first result for the three dimensional Navier-Stokes problems which seems to be very difficult to solve by theoretical approaches.

AB - This paper is the three dimensional extension of the two dimensional work in Nakao et al. (Reliable Comput 9(5):359-372, 2003) and Watanabe et al. (J Math Fluid Mech 6:1-20, 2004) on a computer assisted proof of the existence of nontrivial steady state solutions for Rayleigh-Bénard convection based on the fixed point theorem using a Newton like operator. The differences are emerging of complicated types of bifurcation, direct attack on the problem without stream functions, and increased complexity of numerical computation. The last one makes it hard to proceed the verification of solutions corresponding to the points on bifurcation diagram for three dimensional case. Actually, this work should be the first result for the three dimensional Navier-Stokes problems which seems to be very difficult to solve by theoretical approaches.

UR - http://www.scopus.com/inward/record.url?scp=58149354411&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58149354411&partnerID=8YFLogxK

U2 - 10.1007/s00211-008-0191-5

DO - 10.1007/s00211-008-0191-5

M3 - Article

VL - 111

SP - 389

EP - 406

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 3

ER -