| Issue |
ESAIM: M2AN
Volume 39, Number 1, January-February 2005
|
|
|---|---|---|
| Page(s) | 183 - 221 | |
| DOI | https://doi.org/10.1051/m2an:2005008 | |
| Published online | 15 March 2005 | |
Error estimates in the Fast Multipole Method for scattering problems Part 2: Truncation of the Gegenbauer series
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:
26
October
2004
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,
,
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 
.
We prove that if 
is large enough,
the truncated series gives rise to an error lower than ϵ
as soon as L satisfies
where W is the Lambert function,
depends only on 
and
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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