Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series

Abstract

Abstract.We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave e iŝ· v in terms of spherical harmonics {Y ,m (ŝ)} |m|≤ ≤∞ . We consider the truncated series where the summation is performed over the ( , m)'s satisfying |m| ≤ ≤ L. We prove that if v = | v| is large enough, the truncated series gives rise to an error lower than as soon aswhere 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.Mathematics Subject Classification. 33C10, 33C55, 41A80.

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

“…This formula gives an upper bound of the true L for large values of v. However, according to the numerical results we presented in [6], it seems to be optimal.…”

“…We recall here some results we obtained in [6] for the Jacobi-Anger series, or more precisely for a bounding series.…”

“…This is the reason why we chose to divide this study into three different parts. In the first one [6], we studied the truncation error in the series of formula (1) when u is large, which amounts to analyze the Jacobi-anger series. This first paper contains some techniques and results which will be useful in the present article, devoted to the truncation error for finite u.…”

“…The reasoning that was done for the Jacobi-Anger series in [6] is difficult to repeat here, since we do not have an appropriate formula at our disposal, like Christoffel-Darboux formula, to apply Abel's transformation. However, we can give some elements which show, in an intuitive way, a similarity between both series.…”

Comparison of two wave element methods for the Helmholtz problem

“…8 the accuracy of the UWVF started to deteriorate before the largest C values). This formula for the basis dimension stems from the analysis of truncation of the Jacobi-Anger series in the context of the fast multipole method [4]. It can be expected that a similar analysis is needed for studying the approximating properties of plane waves.…”