# 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

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topCarayol, 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

top- M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. Dover, New-York (1964). Zbl0171.38503
- 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. Zbl0973.65092
- J.A. Barcelo, A. Ruiz and L. Vega, Weighted estimates for the Helmholtz equation and some applications. J. Funct. Anal.150 (1997) 356–382. Zbl0890.35028
- H. Bateman, Higher transcendental Functions. McGraw-Hill (1953).
- 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).
- Q. Carayol and F. Collino, Error estimates in the fast multipole method for scattering problems. part 1: Truncation of the jacobi-anger series. ESAIM: M2AN38 (2004) 371–394. Zbl1077.41027
- T.M. Cherry, Uniform asymptotic formulae for functions with transition points. Trans. AMS68 (1950) 224–257. Zbl0036.06102
- W.C. Chew, J.M. Jin, E. Michielssen and J.M. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Artech House (2001).
- R. Coifman, V. Rokhlin and S. Greengard, The Fast Multipole Method for the wave equation: A pedestrian prescription. IEEE Antennas and Propagation Magazine35 (1993) 7–12.
- D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Springer-Verlag 93 (1992). Zbl0760.35053
- E. Darve, The fast multipole method. I. Error analysis and asymptotic complexity. SIAM J. Numer. Anal.38 (2000) 98–128 (electronic). Zbl0974.65033
- E. Darve, The fast multipole method: Numerical implementation. J. Comput. Physics160 (2000) 196–240. Zbl0974.78012
- E. Darve and P. Havé, Efficient fast multipole method for low frequency scattering. J. Comput. Physics197 (2004) 341–363. Zbl1073.65133
- B. Dembart and E. Yip, Accuracy of fast multipole methods for maxwell's equations. IEEE Comput. Sci. Engrg.5 (1998) 48–56.
- 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. Zbl0852.31006
- I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series, and products, 5th edition. Academic Press (1994). Zbl0918.65002
- 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). Zbl0924.65116
- L. Lorch, Alternative proof of a sharpened form of Bernstein's inequality for Legendre polynomials. Applicable Anal.14 (1982/83) 237–240. Zbl0505.33007
- 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.
- J.C. Nédélec, Acoustic and Electromagnetic Equation. Integral Representation for Harmonic Problems. Springer-Verlag 144 (2001). Zbl0981.35002
- J. Rahola, Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems. BIT36 (1996) 333–358. Zbl0854.65122
- G.N. Watson, A treatise on the theory of Bessel functions. Cambridge University Press (1966). Zbl0174.36202

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