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

Carayol, Quentin  ; Collino, Francis

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 38 (2004) no. 2, pp. 371-394

Résumé

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.

Détail

DOI : 10.1051/m2an:2004017

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