Balac, Stéphane; Chupin, Laurent; Martin, Sébastien Computation of the magnetic potential induced by a collection of spherical particles using series expansions. (English) Zbl 1446.65196 ESAIM, Math. Model. Numer. Anal. 54, No. 4, 1073-1109 (2020). Summary: In Magnetic Resonance Imaging there are several situations where, for simulation purposes, one wants to compute the magnetic field induced by a cluster of small metallic particles. Given the difficulty of the problem from a numerical point of view, the simplifying assumption that the field due to each particle interacts only with the main magnetic field but does not interact with the fields due to the other particles is usually made. In this paper we investigate from a mathematical point of view the relevancy of this assumption and provide error estimates for the scalar magnetic potential in terms of the key parameter that is the minimal distance between the particles. A special attention is paid to obtain explicit and relevant constants in the estimates. When the “non-interacting assumption” is deficient, we propose to compute a better approximation of the magnetic potential by taking into account pairwise magnetic field interactions between particles that enters in a general framework for computing the scalar magnetic potential as a series expansion. MSC: 65N99 Numerical methods for partial differential equations, boundary value problems 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 41-04 Software, source code, etc. for problems pertaining to approximations and expansions 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) 78A30 Electro- and magnetostatics 92C55 Biomedical imaging and signal processing 35Q92 PDEs in connection with biology, chemistry and other natural sciences Keywords:series expansion; error estimates; spherical surface harmonics; Green’s representation formula; magnetic potential; magnetic resonance imaging Software:SHTns; FreeFem++; DLMF PDFBibTeX XMLCite \textit{S. Balac} et al., ESAIM, Math. Model. Numer. 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