Commun. Comput. Phys., 9 (2011), pp. 649-667.


An FFT Based Fast Poisson Solver on Spherical Shells

Yin-Liang Huang 1*, Jian-Guo Liu 2, Wei-Cheng Wang 3

1 Department of Applied Mathematics, National University of Tainan, Tainan 70005, Taiwan.
2 Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA.
3 Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan.

Received 6 May 2009; Accepted (in revised version) 8 June 2009
Available online 17 September 2010
doi:10.4208/cicp.060509.080609s

Abstract

We present a fast Poisson solver on spherical shells. With a special change of variable, the radial part of the Laplacian transforms to a constant coefficient differential operator. As a result, the Fast Fourier Transform can be applied to solve the Poisson equation with $O(N^3\log N)$ operations. Numerical examples have confirmed the accuracy and robustness of the new scheme.

AMS subject classifications: 35Q86, 65N06, 65N15, 65N22, 65T50

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Key words: Poisson equation, spherical coordinate, FFT, spectral-finite difference method, fast diagonalization, high order accuracy, error estimate, trapezoidal rule, Euler-Maclaurin formula, Bernoulli numbers.

*Corresponding author.
Email: liang@mail.nutn.edu.tw (Y.-L. Huang), jian-guo.liu@duke.edu (J.-G. Liu), wangwc@math.nthu.edu.tw (W.-C. Wang)
 

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