Error estimates in the fast multipole method for scattering problems. Part 1 : truncation of the Jacobi-Anger series

Carayol, Quentin; Collino, Francis

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 2, pp. 371-394.

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave ${\mathrm{e}}^{i\widehat{s}·\overrightarrow{v}}$ in terms of spherical harmonics ${\left\{Y_{\ell ,m}\left(\widehat{s}\right)\right\}}_{\left|m\right|\le \ell \le \infty }$. We consider the truncated series where the summation is performed over the $(\ell ,m)$’s satisfying $\left|m\right|\le \ell \le L$. We prove that if $v=|\overrightarrow{v}|$ is large enough, the truncated series gives rise to an error lower than $ϵ$ as soon as $L$ satisfies $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K{ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where $W$ is the Lambert function and $C,K,\delta ,\gamma$ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates for the error in the fast multipole method for scattering computation.

Classification : 33C10, 33C55, 41A80
Mots clés : Jacobi-Anger, fast multipole method, truncation error

[1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. Dover, New York (1964).

[2] S. Amini and A. Profit, Analysis of the truncation errors in the fast multipole method for scattering problems. J. Comput. Appl. Math. 115 (2000) 23-33. | Zbl

[3] Q. Carayol, Développement et analyse d'une méthode multipôle multiniveau pour l'électromagnétisme. Ph.D. Thesis, Université Paris VI Pierre et Marie Curie, Paris (2002).

[4] O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic pdes to the 2D Helmholtz problem. SIAM J. Numer. Anal. 35 (1998) 255-299. | Zbl

[5] W.C. Chew, J.M. Jin, E. Michielssen and J.M. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Artech House (2001).

[6] R. Coifman, V. Rokhlin and S. Greengard, The fast multipole method for the wave equation: A pedestrian prescription. IEEE Antennas and Propagation Magazine 35 (1993) 7-12.

[7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Springer-Verlag 93 (1992). | MR | Zbl

[8] R. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey and D.E. Knuth, On the Lambert $W$ function. Adv. Comput. Math. 5 (1996) 329-359. | Zbl

[9] E. Darve, The fast multipole method. I. Error analysis and asymptotic complexity. SIAM J. Numer. Anal. 38 (2000) 98-128 (electronic). | Zbl

[10] E. Darve, The fast multipole method: Numerical implementation. J. Comput. Phys. 160 (2000) 196-240. | Zbl

[11] E. Darve and P. Havé, Efficient fast multipole method for low frequency scattering. J. Comput. Phys. (to appear). | MR | Zbl

[12] M.A. Epton and B. Dembart, Multipole translation theory for the three-dimensional Laplace and Helmholtz equations. SIAM J. Sci. Comput. 16 (1995) 865-897. | Zbl

[13] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, 5th edn., Academic Press (1994). | MR | Zbl

[14] S. Koc, J. Song and W.C. Chew, Error analysis for the numerical evaluation of the diagonal forms of the scalar spherical addition theorem. SIAM J. Numer. Anal. 36 (1999) 906-921 (electronic). | Zbl

[15] L. Lorch, Alternative proof of a sharpened form of Bernstein's inequality for Legendre polynomials. Applicable Anal. 14 (1982/83) 237-240. | Zbl

[16] L. Lorch, Corrigendum: “Alternative proof of a sharpened form of Bernstein's inequality for Legendre polynomials” [Appl. Anal. 14 (1982/83) 237-240; MR 84k:26017]. Appl. Anal. 50 (1993) 47. | Zbl

[17] J.C. Nédélec, Acoustic and Electromagnetic Equation. Integral Representation for Harmonic Problems. Springer-Verlag 144 (2001). | MR | Zbl

[18] S. Ohnuki and W.C. Chew, Numerical accuracy of multipole expansion for 2-d mlfma. IEEE Trans. Antennas Propagat. 51 (2003) 1883-1890.

[19] J. Rahola, Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems. BIT 36 (1996) 333-358. | Zbl

[20] G.N. Watson, Bessel functions and Kapteyn series. Proc. London Math. Soc. (1916) 150-174. | JFM

[21] G.N. Watson, A treatise on the theory of Bessel functions. Cambridge University Press (1966). | JFM | MR | Zbl