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A Fast Collocation Method for Eigen-Problems of Weakly Singular Integral Operators

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Abstract

A multiscale collocation method is developed for solving the eigen-problem of weakly singular integral operators. We employ a matrix truncation strategy of Chen, Micchelli and Xu to compress the collocation matrix, which the compressed matrix has only \(\mathcal{O}(N\log N)\) nonzero entries, where N denotes the order of the matrix. This truncation leads to a fast collocation method for solving the eigen-problem. We prove that the fast collocation method has the optimal convergence order for approximation of the eigenvalues and eigenvectors. The power iteration method is used for solving the corresponding discrete eigen-problem. We present a numerical example to demonstrate how the methods can be used to compute a nonzero eigenvalue rapidly and efficiently.

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Authors and Affiliations

  1. Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzhou, 510275, P.R. China

    Zhongying Chen & Yongdong Zhang

  2. Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, West-Bengal, India

    Gnaneshwar Nelakanti

  3. Department of Mathematics, Syracuse University, Syracuse, NY, 13244-1150, USA

    Yuesheng Xu

Authors
  1. Zhongying Chen
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  2. Gnaneshwar Nelakanti
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  3. Yuesheng Xu
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  4. Yongdong Zhang
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Corresponding author

Correspondence to Yuesheng Xu.

Additional information

The research of Z. Chen and Y. Zhang was supported in part by the Natural Science Foundation of China under grants 10771224 and the Science and Technology Section of SINOPEC.

The research of G. Nelakanti was supported in part by the DST research project under serc fast tack proposals for young scientists: D.O.NO.SB/FTP/MS-12/2005, Dt, 10-11-2006, India, and the foundation of Doctoral Program of National Higher Education of China under grant 20030558008.

The research of Y. Xu was supported in part by the US National Science Foundation under grants DMS-0712827 and CCF-0833152, by the Natural Science Foundation of China under grants 10371122 and 10631080, by the Education Ministry of the People’s Republic of China under the Changjiang Scholar Chair Professorship Program through Sun Yat-Sen University, and by Institute of Mathematics, Chinese Academy of Sciences.

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Chen, Z., Nelakanti, G., Xu, Y. et al. A Fast Collocation Method for Eigen-Problems of Weakly Singular Integral Operators. J Sci Comput 41, 256 (2009). https://doi.org/10.1007/s10915-009-9295-z

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  • DOI: https://doi.org/10.1007/s10915-009-9295-z

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