### Abstract

Error estimates are introduced in a finite element approximation to thermal convection problems with the Joule heat. From the Joule heat, a nonlinear source term arises in the convection-diffusion part of problems. The derivation of the estimates is based on the fact that a finite element approximation to the electric potential appearing in such a nonlinear term is uniformly bounded in the cubic summable norm with derivatives up to the first order. The error estimates are optimal, and do not require any stability conditions. Numerical results show that the numerical convergence rates agree well with the theoretical ones, and that phenomena in a simplified electric glass furnace are influenced by the electric potential distributions associated with the location of electrodes.

Original language | English |
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Pages (from-to) | 193-210 |

Number of pages | 18 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 20 |

Issue number | 2 |

Publication status | Published - Jun 2003 |

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### All Science Journal Classification (ASJC) codes

- Engineering(all)
- Applied Mathematics

### Cite this

*Japan Journal of Industrial and Applied Mathematics*,

*20*(2), 193-210.

**A Finite Element Analysis of Thermal Convection Problems with the Joule Heat.** / Tagami, Daisuke; Itoh, Hajime.

Research output: Contribution to journal › Article

*Japan Journal of Industrial and Applied Mathematics*, vol. 20, no. 2, pp. 193-210.

}

TY - JOUR

T1 - A Finite Element Analysis of Thermal Convection Problems with the Joule Heat

AU - Tagami, Daisuke

AU - Itoh, Hajime

PY - 2003/6

Y1 - 2003/6

N2 - Error estimates are introduced in a finite element approximation to thermal convection problems with the Joule heat. From the Joule heat, a nonlinear source term arises in the convection-diffusion part of problems. The derivation of the estimates is based on the fact that a finite element approximation to the electric potential appearing in such a nonlinear term is uniformly bounded in the cubic summable norm with derivatives up to the first order. The error estimates are optimal, and do not require any stability conditions. Numerical results show that the numerical convergence rates agree well with the theoretical ones, and that phenomena in a simplified electric glass furnace are influenced by the electric potential distributions associated with the location of electrodes.

AB - Error estimates are introduced in a finite element approximation to thermal convection problems with the Joule heat. From the Joule heat, a nonlinear source term arises in the convection-diffusion part of problems. The derivation of the estimates is based on the fact that a finite element approximation to the electric potential appearing in such a nonlinear term is uniformly bounded in the cubic summable norm with derivatives up to the first order. The error estimates are optimal, and do not require any stability conditions. Numerical results show that the numerical convergence rates agree well with the theoretical ones, and that phenomena in a simplified electric glass furnace are influenced by the electric potential distributions associated with the location of electrodes.

UR - http://www.scopus.com/inward/record.url?scp=1642492006&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1642492006&partnerID=8YFLogxK

M3 - Article

VL - 20

SP - 193

EP - 210

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 2

ER -