Commun. Comput. Phys., 6 (2009), pp. 625-638.

Fast Spectral Collocation Method for Surface Integral Equations of Potential Problems in a Spheroid

Zhenli Xu 1, Wei Cai 1*

1 Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA.

Received 15 September 2008; Accepted (in revised version) 13 November 2008
Available online 13 February 2009


This paper proposes a new technique to speed up the computation of the matrix of spectral collocation discretizations of surface single and double layer operators over a spheroid. The layer densities are approximated by a spectral expansion of spherical harmonics and the spectral collocation method is then used to solve surface integral equations of potential problems in a spheroid. With the proposed technique, the computation cost of collocation matrix entries is reduced from $O(M^2N^4)$ to $O(MN^4)$, where $N^2$ is the number of spherical harmonics (i.e., size of the matrix) and M is the number of one-dimensional integration quadrature points. Numerical results demonstrate the spectral accuracy of the method.

AMS subject classifications: 31B10, 33C90, 65R20, 76M22

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Key words: Boundary integral equations, three-dimensional potential problems, collocation spectral methods, spherical harmonics, Fourier series, hypergeometric functions.

*Corresponding author.
Email: (Z. Xu), (W. Cai)

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