Fast BEM Solvers for 3D Poisson-Type Equations
Xuefei He; Kian-Meng Lim; and Siak-Piang Lim

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 35, No. 1, pp. 21-48, 2008
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Keywords Boundary element method (BEM), Poisson-type equation, non-linear equation, fast Fourier transform on multipoles (FFTM).
Abstract The boundary element method (BEM) is known to have the advantage of reducing the dimension of problem by discretizing only the boundary of the domain. But it becomes less attractive for solving Poisson-type equations, due to the need to evaluate the domain integral which is computationally expensive. In this paper, we present the extension of a recently developed fast algorithm for Laplace equation, based on fast Fourier transform on multipoles (FFTM), to solve large scale 3D Poisson-type equations. We combined the Laplace solver with two fast methods for handling the domain integral based on fast Fourier transform (FFT). The first method uses the FFT on multipoles to accelerate the domain integral, while the second method solves the domain integral as a particular solution using FFT. The particular solution method is found to be faster and more accurate, and it is extended to solve non-linear Poisson-type equations. The algorithm is shown to be efficient when it is used in the inner loop of the iterative solver for the non-linear equations.
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