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

Quentin Carayol; Francis Collino

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 1, page 183-221
  • ISSN: 0764-583X

Abstract

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

How to cite

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Carayol, Quentin, and Collino, Francis. "Error estimates in the Fast Multipole Method for scattering problems Part 2: Truncation of the Gegenbauer series." ESAIM: Mathematical Modelling and Numerical Analysis 39.1 (2010): 183-221. <http://eudml.org/doc/194256>.

@article{Carayol2010,
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, $\frac\{ \{\rm e\}^\{i |\vec\{u\}-\vec\{v\}|\}\}\{4 \pi i |\vec\{u\}-\vec\{v\}|\}$, 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 $\ell \le L$. 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 $L+\frac\{1\}\{2\} \simeq v + C W^\{\frac\{2\}\{3\}\}(K(\alpha) \epsilon^\{-\delta\} v^\gamma )\, v^\{\frac\{1\}\{3\}\}$ where W is the Lambert function, $K(\alpha)$ depends only on $\alpha=\frac\{|\vec\{u\}|\}\{|\vec\{v\}|\}$ 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. },
author = {Carayol, Quentin, Collino, Francis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Gegenbauer; fast multipole method; truncation error.; truncation error},
language = {eng},
month = {3},
number = {1},
pages = {183-221},
publisher = {EDP Sciences},
title = {Error estimates in the Fast Multipole Method for scattering problems Part 2: Truncation of the Gegenbauer series},
url = {http://eudml.org/doc/194256},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Carayol, Quentin
AU - Collino, Francis
TI - Error estimates in the Fast Multipole Method for scattering problems Part 2: Truncation of the Gegenbauer series
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 1
SP - 183
EP - 221
AB - 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, $\frac{ {\rm e}^{i |\vec{u}-\vec{v}|}}{4 \pi i |\vec{u}-\vec{v}|}$, 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 $\ell \le L$. 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 $L+\frac{1}{2} \simeq v + C W^{\frac{2}{3}}(K(\alpha) \epsilon^{-\delta} v^\gamma )\, v^{\frac{1}{3}}$ where W is the Lambert function, $K(\alpha)$ depends only on $\alpha=\frac{|\vec{u}|}{|\vec{v}|}$ 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.
LA - eng
KW - Gegenbauer; fast multipole method; truncation error.; truncation error
UR - http://eudml.org/doc/194256
ER -

References

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