### Abstract

This paper is a continuation of the preceding study ([2]) in which we described an automatic proof by computer, utilizing Schauder's fixed point theorem, of the existence of weak solutions for Dirichlet problems of second order. We newly formulate a verification method using the Newton-like method and Sadovskii's fixed point theorem for the codensing map. This approach enables us to remove the magnitude limit of the spectral radius of operator appeared in the previous work. We show some numerical examples which confirm us that the method is really applicable to problems having large spectral radius.

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

Pages (from-to) | 477-488 |

Number of pages | 12 |

Journal | Japan Journal of Applied Mathematics |

Volume | 7 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1990 Oct |

Externally published | Yes |

### Fingerprint

### Keywords

- boundary value problems
- error estimates
- finite element method
- fixed point theorem

### ASJC Scopus subject areas

- Engineering(all)
- Applied Mathematics

### Cite this

*Japan Journal of Applied Mathematics*,

*7*(3), 477-488. https://doi.org/10.1007/BF03167855

**A numerical approach to the proof of existence of solutions for elliptic problems II.** / Nakao, Mitsuhiro T.

Research output: Contribution to journal › Article

*Japan Journal of Applied Mathematics*, vol. 7, no. 3, pp. 477-488. https://doi.org/10.1007/BF03167855

}

TY - JOUR

T1 - A numerical approach to the proof of existence of solutions for elliptic problems II

AU - Nakao, Mitsuhiro T.

PY - 1990/10

Y1 - 1990/10

N2 - This paper is a continuation of the preceding study ([2]) in which we described an automatic proof by computer, utilizing Schauder's fixed point theorem, of the existence of weak solutions for Dirichlet problems of second order. We newly formulate a verification method using the Newton-like method and Sadovskii's fixed point theorem for the codensing map. This approach enables us to remove the magnitude limit of the spectral radius of operator appeared in the previous work. We show some numerical examples which confirm us that the method is really applicable to problems having large spectral radius.

AB - This paper is a continuation of the preceding study ([2]) in which we described an automatic proof by computer, utilizing Schauder's fixed point theorem, of the existence of weak solutions for Dirichlet problems of second order. We newly formulate a verification method using the Newton-like method and Sadovskii's fixed point theorem for the codensing map. This approach enables us to remove the magnitude limit of the spectral radius of operator appeared in the previous work. We show some numerical examples which confirm us that the method is really applicable to problems having large spectral radius.

KW - boundary value problems

KW - error estimates

KW - finite element method

KW - fixed point theorem

UR - http://www.scopus.com/inward/record.url?scp=0000871632&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000871632&partnerID=8YFLogxK

U2 - 10.1007/BF03167855

DO - 10.1007/BF03167855

M3 - Article

AN - SCOPUS:0000871632

VL - 7

SP - 477

EP - 488

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 3

ER -