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

Authors:
Zhongying Chen;Gnaneshwar Nelakanti;Yuesheng Xu;Yongdong Zhang
Affiliations:
Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzhou, P.R. China 510275;Department of Mathematics, Indian Institute of Technology, Kharagpur, India 721302;Department of Mathematics, Syracuse University, Syracuse, USA 13244-1150;Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzhou, P.R. China 510275
Venue:
Journal of Scientific Computing
Year:
2009

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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.