### Abstract

We propose some numerical methods for the automatic proof of existence of weak solutions for parabolic initial boundary value problems with one space dimension. It also means that one can obtain a posteriori error bounds for the approximate solutions of the problems. Based upon Schauder's fixed-point theorem, a verification condition is formulated and, by the use of finite-element approximation and its error estimates for a simple parabolic problem, we present a numerical verification algorithm of exact solutions in a computer. Some numerical examples which are verified by the method are illustrated.

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

Pages (from-to) | 323-334 |

Number of pages | 12 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 38 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 1991 Dec 23 |

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

**Solving nonlinear parabolic problems with result verification. Part I : one-space dimensional case.** / Nakao, Mitsuhiro T.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 38, no. 1-3, pp. 323-334. https://doi.org/10.1016/0377-0427(91)90179-N

}

TY - JOUR

T1 - Solving nonlinear parabolic problems with result verification. Part I

T2 - one-space dimensional case

AU - Nakao, Mitsuhiro T.

PY - 1991/12/23

Y1 - 1991/12/23

N2 - We propose some numerical methods for the automatic proof of existence of weak solutions for parabolic initial boundary value problems with one space dimension. It also means that one can obtain a posteriori error bounds for the approximate solutions of the problems. Based upon Schauder's fixed-point theorem, a verification condition is formulated and, by the use of finite-element approximation and its error estimates for a simple parabolic problem, we present a numerical verification algorithm of exact solutions in a computer. Some numerical examples which are verified by the method are illustrated.

AB - We propose some numerical methods for the automatic proof of existence of weak solutions for parabolic initial boundary value problems with one space dimension. It also means that one can obtain a posteriori error bounds for the approximate solutions of the problems. Based upon Schauder's fixed-point theorem, a verification condition is formulated and, by the use of finite-element approximation and its error estimates for a simple parabolic problem, we present a numerical verification algorithm of exact solutions in a computer. Some numerical examples which are verified by the method are illustrated.

KW - error estimates

KW - finite-element method

KW - fixed-point theorem

KW - Parabolic problem

UR - http://www.scopus.com/inward/record.url?scp=0003924558&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0003924558&partnerID=8YFLogxK

U2 - 10.1016/0377-0427(91)90179-N

DO - 10.1016/0377-0427(91)90179-N

M3 - Article

AN - SCOPUS:0003924558

VL - 38

SP - 323

EP - 334

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-3

ER -