### Abstract

This paper is an extension of the preceding study (Nakao, this journal, 1991) in which we described a numerical verification method of the solution for one-space dimensional parabolic problems, to the several-space dimensional case. Here, numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer. We reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed-point problem of a compact operator on certain function spaces. As in the preceding study based upon a simple C^{0} finite-element approximation and its constructive a priori error estimates, a numerical verification procedure is presented with some numerical examples.

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

Pages (from-to) | 401-410 |

Number of pages | 10 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 50 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 1994 May 20 |

Externally published | Yes |

### Fingerprint

### Keywords

- Error estimates
- Finite-element method
- Fixed-point theorem
- Parabolic problem

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

### Cite this

*Journal of Computational and Applied Mathematics*,

*50*(1-3), 401-410. https://doi.org/10.1016/0377-0427(94)90316-6

**On computational proofs of the existence of solutions to nonlinear parabolic problems.** / Nakao, Mitsuhiro T.; Watanabe, Yoshitaka.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 50, no. 1-3, pp. 401-410. https://doi.org/10.1016/0377-0427(94)90316-6

}

TY - JOUR

T1 - On computational proofs of the existence of solutions to nonlinear parabolic problems

AU - Nakao, Mitsuhiro T.

AU - Watanabe, Yoshitaka

PY - 1994/5/20

Y1 - 1994/5/20

N2 - This paper is an extension of the preceding study (Nakao, this journal, 1991) in which we described a numerical verification method of the solution for one-space dimensional parabolic problems, to the several-space dimensional case. Here, numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer. We reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed-point problem of a compact operator on certain function spaces. As in the preceding study based upon a simple C0 finite-element approximation and its constructive a priori error estimates, a numerical verification procedure is presented with some numerical examples.

AB - This paper is an extension of the preceding study (Nakao, this journal, 1991) in which we described a numerical verification method of the solution for one-space dimensional parabolic problems, to the several-space dimensional case. Here, numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer. We reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed-point problem of a compact operator on certain function spaces. As in the preceding study based upon a simple C0 finite-element approximation and its constructive a priori error estimates, a numerical verification procedure is presented with some numerical examples.

KW - Error estimates

KW - Finite-element method

KW - Fixed-point theorem

KW - Parabolic problem

UR - http://www.scopus.com/inward/record.url?scp=0028423596&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028423596&partnerID=8YFLogxK

U2 - 10.1016/0377-0427(94)90316-6

DO - 10.1016/0377-0427(94)90316-6

M3 - Article

AN - SCOPUS:0028423596

VL - 50

SP - 401

EP - 410

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-3

ER -