### Abstract

In this paper, we consider a numerical technique to verify the exact eigenvalues and eigenfunctions of second-order elliptic operators in some neighborhood of their approximations. This technique is based on Nakao's method [9] using the Newton-like operator and the error estimates for the C^{0} finite element solution. We construct, in computer, a set containing solutions which satisfies the hypothesis of Schauder's fixed point theorem for compact map on a certain Sobolev space. Moreover, we propose a method to verify the eigenvalue which has the smallest absolute value. A numerical example is presented.

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

Pages (from-to) | 307-320 |

Number of pages | 14 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 16 |

Issue number | 3 |

Publication status | Published - 1999 Oct |

Externally published | Yes |

### Fingerprint

### Keywords

- Eigenvalue problem
- Elliptic operators
- Error estimates
- Finite element solution

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Japan Journal of Industrial and Applied Mathematics*,

*16*(3), 307-320.

**Numerical Verifications for Eigenvalues of Second-Order Elliptic Operators.** / Nakao, M. T.; Yamamoto, N.; Nagatou, K.

Research output: Contribution to journal › Article

*Japan Journal of Industrial and Applied Mathematics*, vol. 16, no. 3, pp. 307-320.

}

TY - JOUR

T1 - Numerical Verifications for Eigenvalues of Second-Order Elliptic Operators

AU - Nakao, M. T.

AU - Yamamoto, N.

AU - Nagatou, K.

PY - 1999/10

Y1 - 1999/10

N2 - In this paper, we consider a numerical technique to verify the exact eigenvalues and eigenfunctions of second-order elliptic operators in some neighborhood of their approximations. This technique is based on Nakao's method [9] using the Newton-like operator and the error estimates for the C0 finite element solution. We construct, in computer, a set containing solutions which satisfies the hypothesis of Schauder's fixed point theorem for compact map on a certain Sobolev space. Moreover, we propose a method to verify the eigenvalue which has the smallest absolute value. A numerical example is presented.

AB - In this paper, we consider a numerical technique to verify the exact eigenvalues and eigenfunctions of second-order elliptic operators in some neighborhood of their approximations. This technique is based on Nakao's method [9] using the Newton-like operator and the error estimates for the C0 finite element solution. We construct, in computer, a set containing solutions which satisfies the hypothesis of Schauder's fixed point theorem for compact map on a certain Sobolev space. Moreover, we propose a method to verify the eigenvalue which has the smallest absolute value. A numerical example is presented.

KW - Eigenvalue problem

KW - Elliptic operators

KW - Error estimates

KW - Finite element solution

UR - http://www.scopus.com/inward/record.url?scp=0000218518&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000218518&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0000218518

VL - 16

SP - 307

EP - 320

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 3

ER -