### 抄録

In order to verify the solutions of nonlinear boundary value problems by Nakao's computer-assisted numerical method, it is required to find a constant, as sharp as possible, in the a priori error estimates for the finite element approximation of some simple linear problems. For singularly perturbed problems, however, generally it is known that the perturbation term produces a bad effect on the a priori error estimates, i.e., leads to a large constant, if we use the usual approximation methods. In this paper, we propose some verification algorithms for solutions of singularly perturbed problems with nonlinearity by using the constant obtained in the a priori error estimates based on the exponential fitting method with Green's function. Some numerical examples which confirm us the effectiveness of our method are presented.

元の言語 | 英語 |
---|---|

ページ（範囲） | 111-131 |

ページ数 | 21 |

ジャーナル | Japan Journal of Industrial and Applied Mathematics |

巻 | 22 |

発行部数 | 1 |

DOI | |

出版物ステータス | 出版済み - 2 2005 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Engineering(all)
- Applied Mathematics

### これを引用

*Japan Journal of Industrial and Applied Mathematics*,

*22*(1), 111-131. https://doi.org/10.1007/BF03167479

**A numerical verification method for solutions of singularly perturbed problems with nonlinearity.** / Hashimoto, Kouji; Abe, Ryohei; Nakao, Mitsuhiro T.; Watanabe, Yoshitaka.

研究成果: ジャーナルへの寄稿 › 記事

*Japan Journal of Industrial and Applied Mathematics*, 巻. 22, 番号 1, pp. 111-131. https://doi.org/10.1007/BF03167479

}

TY - JOUR

T1 - A numerical verification method for solutions of singularly perturbed problems with nonlinearity

AU - Hashimoto, Kouji

AU - Abe, Ryohei

AU - Nakao, Mitsuhiro T.

AU - Watanabe, Yoshitaka

PY - 2005/2

Y1 - 2005/2

N2 - In order to verify the solutions of nonlinear boundary value problems by Nakao's computer-assisted numerical method, it is required to find a constant, as sharp as possible, in the a priori error estimates for the finite element approximation of some simple linear problems. For singularly perturbed problems, however, generally it is known that the perturbation term produces a bad effect on the a priori error estimates, i.e., leads to a large constant, if we use the usual approximation methods. In this paper, we propose some verification algorithms for solutions of singularly perturbed problems with nonlinearity by using the constant obtained in the a priori error estimates based on the exponential fitting method with Green's function. Some numerical examples which confirm us the effectiveness of our method are presented.

AB - In order to verify the solutions of nonlinear boundary value problems by Nakao's computer-assisted numerical method, it is required to find a constant, as sharp as possible, in the a priori error estimates for the finite element approximation of some simple linear problems. For singularly perturbed problems, however, generally it is known that the perturbation term produces a bad effect on the a priori error estimates, i.e., leads to a large constant, if we use the usual approximation methods. In this paper, we propose some verification algorithms for solutions of singularly perturbed problems with nonlinearity by using the constant obtained in the a priori error estimates based on the exponential fitting method with Green's function. Some numerical examples which confirm us the effectiveness of our method are presented.

UR - http://www.scopus.com/inward/record.url?scp=15944399076&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15944399076&partnerID=8YFLogxK

U2 - 10.1007/BF03167479

DO - 10.1007/BF03167479

M3 - Article

AN - SCOPUS:15944399076

VL - 22

SP - 111

EP - 131

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 1

ER -