A kernel-independent fast multipole algorithm Technical Report TR 2003-839 1

We present a new fast multipole method for particle simulations. The main feature of our algorithm is that is kernel independent, in the sense that no analytic expansions are used to represent the far field. Instead we use equivalent densities, which we compute by solving small Dirichlet-type boundary value problems. The translations from the sources to the induced potentials are accelerated by singular value decomposition in 2D and fast Fourier transforms in 3D. We have tested the new method on the single and double layer operators for the Laplacian, the modified Laplacian, the Stokes, the modified Stokes, the Navier, and the modified Navier operators in two and three dimensions. Our numerical results indicate that our method compares very well with the best known implementations of the analytic FMM method for both the Laplacian and modified Laplacian kernels. Its advantage is the (relative) simplicity of the implementation and its immediate extension to more general kernels.

[1]  S. Kapur,et al.  IES/sup 3/: efficient electrostatic and electromagnetic simulation , 1998 .

[2]  V. Rokhlin.  Rapid solution of integral equations of classical potential theory , 1985 .

[3]  L. Greengard,et al.  Regular Article: A Fast Adaptive Multipole Algorithm in Three Dimensions , 1999 .

[4]  L. Greengard,et al.  A new version of the fast multipole method for screened Coulomb interactions in three dimensions , 2002 .

[5]  L. Greengard,et al.  A new version of the Fast Multipole Method for the Laplace equation in three dimensions , 1997, Acta Numerica.

[6]  O. Bruno,et al.  A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications , 2001 .

[7]  Norman Yarvin,et al.  An Improved Fast Multipole Algorithm for Potential Fields on the Line , 1999 .

[8]  Norman Yarvin,et al.  Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators , 1998, SIAM J. Sci. Comput..

[9]  Junichiro Makino.  Yet Another Fast Multipole Method without Multipoles-Pseudoparticle Multipole Method , 1999 .

[10]  Jinsong Zhao,et al.  A fast method of moments solver for efficient parameter extraction of MCMs , 1997, DAC.

[11]  Piet Hut,et al.  A hierarchical O(N log N) force-calculation algorithm , 1986, Nature.

[12]  R. V. D. Geijn,et al.  A fast solution method for three‐dimensional many‐particle problems of linear elasticity , 1998 .

[13]  C. Leonard Berman.  Grid-Multipole Calculations , 1995, SIAM J. Sci. Comput..

[14]  Lexing Ying,et al.  The Embedded Boundary Integral Method (EBI) for the Incompressible Navier-Stokes equations , 2002 .

[15]  Kenichi Yoshida,et al.  Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D , 2001 .

[16]  Jacob K. White,et al.  A precorrected-FFT method for electrostatic analysis of complicated 3-D structures , 1997, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[17]  V. Popov,et al.  An O(N) Taylor series multipole boundary element method for three-dimensional elasticity problems , 2001 .

[18]  G. Rodin,et al.  Fast solution method for three-dimensional Stokesian many-particle problems , 2000 .

[19]  Zydrunas Gimbutas,et al.  A Generalized Fast Multipole Method for Nonoscillatory Kernels , 2003, SIAM J. Sci. Comput..

[20]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[21]  L. Greengard.  The Rapid Evaluation of Potential Fields in Particle Systems , 1988 .

[22]  R. Kress.  Linear Integral Equations , 1989 .

[23]  Christopher R. Anderson,et al.  An Implementation of the Fast Multipole Method without Multipoles , 1992, SIAM J. Sci. Comput..