# Eigenvalues of the Laplacian through boundary integral equations

Research output: Journal Publications and Reviews (RGC: 21, 22, 62) › 21_Publication in refereed journal › peer-review

## Author(s)

## Detail(s)

Original language | English |
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Pages (from-to) | 597-609 |

Journal / Publication | SIAM Journal on Matrix Analysis and Applications |

Volume | 12 |

Issue number | 3 |

Publication status | Published - Jul 1991 |

Externally published | Yes |

## Link(s)

DOI | ## DOI |
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Permanent Link | https://scholars.cityu.edu.hk/en/publications/publication(f18ee2a0-114f-43a3-9db9-15a5fb0641c5).html |

## Abstract

A numerical method for the eigenvalue problem of the Laplacian in two-dimensional domains is developed in this paper. This method requires $O( N )$ operations for calculating one eigenvalue in each iteration step, where N is the number of boundary points in the discretization. It is based on the boundary integral formulation which reduces the computation of the eigenvalues to the zeros of the function $\mu _1 ( \lambda )$ defined as the smallest eigenvalue of a related matrix. Iteration methods such as the Lanczos method are used to compute $\mu_1 ( \lambda )$, which requires the multiplication of an $N \times N$ matrix with a vector. The multipole expansion techniques developed for the potential problems by Rokhlin [J. Comput. Phys., 60 (1985), pp. 187–207] are applied and extended here, and the number of operations is reduced to $O( N )$ for this multiplication. The zeros of $\mu _1 ( \lambda )$ are found by the method of quadratic interpolation. A method for finding the kth eigenvalue with the value of k prespecified is also presented. It is based on continuously tracing the eigenvalue while the domain is deforming to (or from) the unit disk. Only five values of $\mu _1 $ are required in this tracing process.

## Citation Format(s)

**Eigenvalues of the Laplacian through boundary integral equations.**/ LU, Ya Yan; Yau, Shing-Tung.

In: SIAM Journal on Matrix Analysis and Applications, Vol. 12, No. 3, 07.1991, p. 597-609.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62) › 21_Publication in refereed journal › peer-review