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 Abstract 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, 76M22Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Boundary integral equations, three-dimensional potential problems, collocation spectral methods, spherical harmonics, Fourier series, hypergeometric functions. *Corresponding author. Email: zxu7@uncc.edu (Z. Xu), wcai@uncc.edu (W. Cai) |