Error estimates in the fast multipole method for scattering problems. Part 2 : truncation of the Gegenbauer series

Carayol, Quentin; Collino, Francis

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 1, pp. 183-221.

We perform a complete study of the truncation error of the Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, $\frac{{\mathrm{e}}^{i|\overrightarrow{u}-\overrightarrow{v}|}}{4\pi i|\overrightarrow{u}-\overrightarrow{v}|}$, which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices $\ell \le L$. 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 $L+\frac{1}{2}\simeq v+C{W}^{\frac{2}{3}}\left(K\left(\alpha \right){ϵ}^{-\delta }{v}^{\gamma }\right)v^{\frac{1}{3}}$ where $W$ is the Lambert function, $K\left(\alpha \right)$ depends only on $\alpha =\frac{|\overrightarrow{u}|}{|\overrightarrow{v}|}$ and $C,\delta ,\gamma$ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates of the error in the fast multipole method for scattering computation.

[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] J.A. Barcelo, A. Ruiz and L. Vega, Weighted estimates for the Helmholtz equation and some applications. J. Funct. Anal. 150 (1997) 356-382. | Zbl

[4] H. Bateman, Higher transcendental Functions. McGraw-Hill (1953). | MR

[5] Q. Carayol, Développement et analyse d'une méthode multipôle multiniveau pour l'électromagnétisme. Ph.D. thesis, Université Paris VI Pierre et Marie Curie, rue Jussieu 75005 Paris (2002).

[6] Q. Carayol and F. Collino, Error estimates in the fast multipole method for scattering problems. part 1: Truncation of the jacobi-anger series. ESAIM: M2AN 38 (2004) 371-394. | Numdam | Zbl

[7] T.M. Cherry, Uniform asymptotic formulae for functions with transition points. Trans. AMS 68 (1950) 224-257. | Zbl

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

[9] 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.

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

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

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

[13] E. Darve and P. Havé, Efficient fast multipole method for low frequency scattering. J. Comput. Physics 197 (2004) 341-363. | Zbl

[14] B. Dembart and E. Yip, Accuracy of fast multipole methods for maxwell's equations. IEEE Comput. Sci. Engrg. 5 (1998) 48-56.

[15] 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

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

[17] 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

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

[19] 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

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

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

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