### 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 |
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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 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Nakao, M. T., Yamamoto, N., & Nagatou, K. (1999). Numerical Verifications for Eigenvalues of Second-Order Elliptic Operators.

*Japan Journal of Industrial and Applied Mathematics*,*16*(3), 307-320.