Error estimates in the Fast Multipole Method for scattering problems Part 2: Truncation of the Gegenbauer series

Free access article

Issue		ESAIM: M2AN Volume 39, Number 1, January-February 2005


Page(s)		183 - 221
DOI		10.1051/m2an:2005008

M2AN, Vol. 39, N°1, pp. 183-221
DOI: 10.1051/m2an:2005008

Error estimates in the Fast Multipole Method for scattering problems Part 2: Truncation of the Gegenbauer series

Quentin Carayol¹ and Francis Collino²

¹ Dassault Aviation, 78 quai Marcel Dassault, Cedex 300, 92552 Saint-Cloud Cedex, France. quentin.carayol@dassault-aviation.fr
² CERFACS, 42 avenue G. Coriolis, 31057 Toulouse, France. Collino@cerfacs.fr

(Received: October 26, 2004. )

Abstract
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, $\frac{ {\rm e}^{i \vert\vec{u}-\vec{v}\vert}}{4 \pi i \vert\vec{u}-\vec{v}\vert}$ , 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 $v = \vert\vec{v}\vert$ is large enough, the truncated series gives rise to an error lower than $\epsilon$ as soon as L satisfies $L+\frac{1}{2} \simeq v + C W^{\frac{2}{3}}(K(\alpha) \epsilon^{-\delta} v^\gamma )\, v^{\frac{1}{3}}$ where W is the Lambert function, $K(\alpha)$ depends only on $\alpha=\frac{\vert\vec{u}\vert}{\vert\vec{v}\vert}$ 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.

Mathematics Subject Classification. 33C10, 33C55, 41A80.

Key words: Gegenbauer, fast multipole method, truncation error.

© EDP Sciences, SMAI 2005

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