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

Quentin CarayolFrancis Collino — 2004

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave e i s ^ · v in terms of spherical harmonics { Y , m ( s ^ ) } | 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 ϵ as soon as L satisfies L + 1 2 v + C W 2 3 ( K ϵ - δ v γ ) v 1 3 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...

Error estimates in the fast multipole method for scattering problems. Part 2 : truncation of the Gegenbauer series

Quentin CarayolFrancis Collino — 2005

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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 . 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...

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

Quentin CarayolFrancis Collino — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

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

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

Quentin CarayolFrancis Collino — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave e i s ^ · v in terms of spherical harmonics { Y , m ( s ^ ) } | 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 as satisfies L + 1 2 v + C W 2 3 ( K ϵ - δ v γ ) v 1 3 where is the Lambert function and C , K , δ , γ are pure positive constants. Numerical experiments show that this asymptotic is optimal....

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