### Abstract

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.

Original language | English |
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Pages (from-to) | 111-131 |

Number of pages | 21 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 22 |

Issue number | 1 |

Publication status | Published - 2005 Feb |

Externally published | Yes |

### Fingerprint

### Keywords

- A priori constant
- Finite element method
- Numerical verification
- Singularly perturbed problem

### ASJC Scopus subject areas

- Engineering(all)
- Applied Mathematics

### Cite this

*Japan Journal of Industrial and Applied Mathematics*,

*22*(1), 111-131.

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

Research output: Contribution to journal › Article

*Japan Journal of Industrial and Applied Mathematics*, vol. 22, no. 1, pp. 111-131.

}

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.

KW - A priori constant

KW - Finite element method

KW - Numerical verification

KW - Singularly perturbed problem

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

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

M3 - Article

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 -