Error estimates in the fast multipole method for scattering problems. Part 1 : truncation of the Jacobi-Anger series
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 2, pp. 371-394.

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave eis^·v→ in terms of spherical harmonics {Yℓ,m(s^)}|m|≤ℓ≤∞. We consider the truncated series where the summation is performed over the (ℓ,m)’s satisfying |m|≤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+12≃v+CW23(Kϵ-δvγ)v13 where W is the Lambert function and C,K,δ,γ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates for the error in the fast multipole method for scattering computation.

DOI : 10.1051/m2an:2004017
Classification : 33C10, 33C55, 41A80
Mots-clés : Jacobi-Anger, fast multipole method, truncation error 
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     author = {Carayol, Quentin and Collino, Francis},
     title = {Error estimates in the fast multipole method for scattering problems. {Part} 1 : truncation of the {Jacobi-Anger} series},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {371--394},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {2},
     year = {2004},
     doi = {10.1051/m2an:2004017},
     mrnumber = {2069152},
     zbl = {1077.41027},
     language = {en},
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Carayol, Quentin; Collino, Francis. Error estimates in the fast multipole method for scattering problems. Part 1 : truncation of the Jacobi-Anger series. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 2, pp. 371-394. doi : 10.1051/m2an:2004017. https://www.numdam.org/articles/10.1051/m2an:2004017/

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