Error bound of the multilevel fast multipole method for 3‐D scattering problems

The multilevel fast multipole method (MLFMM) is widely used to accelerate the solutions of acoustic and electromagnetic scattering problems. In the expansions and translation operators of the MLFMM for 3‐D scattering problems, some special functions are used, including spherical Bessel functions, sp...

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Bibliographic Details
Published inNumerical methods for partial differential equations Vol. 40; no. 6
Main Author Meng, Wenhui
Format Journal Article
LanguageEnglish
Published Hoboken, USA John Wiley & Sons, Inc 01.11.2024
Wiley Subscription Services, Inc
Subjects
3‐D scattering problems
Bessel functions
Elastostatics
Electromagnetic scattering
Error analysis
error bound
multilevel fast multipole method
Multipoles
Octrees
Operators (mathematics)
spherical Bessel functions
Spherical harmonics
Stokes flow
Truncation errors
Upper bounds
Wigner 3j$$ 3j $$ symbol
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Summary:The multilevel fast multipole method (MLFMM) is widely used to accelerate the solutions of acoustic and electromagnetic scattering problems. In the expansions and translation operators of the MLFMM for 3‐D scattering problems, some special functions are used, including spherical Bessel functions, spherical harmonics and Wigner 3j$$ 3j $$ symbol. This makes it difficult to analyze the truncation errors. In this paper, we first give sharp bounds for the truncation errors of the expansions used in the MLFMM, then derive the overall error formula of the MLFMM and estimate its upper bound, the result is finally applied to the cube octree structure. Some numerical examples are performed to validate the proposed results. The method in this paper can also be used to the MLFMM for other 3‐D problems, such as potential problems, elastostatic problems, Stokes flow problems and so on.
ISSN:0749-159X
1098-2426
DOI:10.1002/num.23148