### Abstract

We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space. By using the finite element approximation and constructive error estimates, we calculate the eigenvalue bound with smallest absolute value to evaluate the norm of the inverse of the linearized operator. Utilizing this bound we derive a verification condition of the Newton-Kantorovich type. Numerical examples are presented.

Original language | English |
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Pages (from-to) | 543-565 |

Number of pages | 23 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 20 |

Issue number | 5-6 |

Publication status | Published - 1999 Jul |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Analysis
- Signal Processing
- Computer Science Applications
- Control and Optimization

### Cite this

*Numerical Functional Analysis and Optimization*,

*20*(5-6), 543-565.

**Approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness.** / Nagatou, K.; Yamamoto, N.; Nakao, M. T.

Research output: Contribution to journal › Article

*Numerical Functional Analysis and Optimization*, vol. 20, no. 5-6, pp. 543-565.

}

TY - JOUR

T1 - Approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness

AU - Nagatou, K.

AU - Yamamoto, N.

AU - Nakao, M. T.

PY - 1999/7

Y1 - 1999/7

N2 - We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space. By using the finite element approximation and constructive error estimates, we calculate the eigenvalue bound with smallest absolute value to evaluate the norm of the inverse of the linearized operator. Utilizing this bound we derive a verification condition of the Newton-Kantorovich type. Numerical examples are presented.

AB - We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space. By using the finite element approximation and constructive error estimates, we calculate the eigenvalue bound with smallest absolute value to evaluate the norm of the inverse of the linearized operator. Utilizing this bound we derive a verification condition of the Newton-Kantorovich type. Numerical examples are presented.

UR - http://www.scopus.com/inward/record.url?scp=0033154252&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033154252&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033154252

VL - 20

SP - 543

EP - 565

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 5-6

ER -