On the computation of Riccati-Bessel functions

Peter Maličký; Marianna Maličká

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

Quentin Carayol, Francis Collino (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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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, $\frac{e^{i | \vec{u} - \vec{v} |}}{4 π i | \vec{u} - \vec{v} |}$ , 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 $ℓ \leq L$ . We prove that if $v = | \vec{v} |$ is large enough, the truncated series gives rise to an error lower than $ϵ$ as soon as $L$ satisfies $L + \frac{1}{2} ≃ v + C W^{\frac{2}{3}} (K (α) ϵ^{- δ} v^{γ}) v^{\frac{1}{3}}$ where $W$ is the Lambert function, $K (α)$ depends only on...

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