A hybrid boundary integral-PDE approach for the approximation of the demagnetization potential in micromagnetics

• Published in: Advances in computational mathematics Vol. 51; no. 3
• Main Authors: Arjmand, Doghonay, Calzada, Víctor Martínez
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.06.2025
• Subjects: Algorithms; Approximation; Boundary conditions; Computing costs; Convergence; Demagnetization; Demagnetization potential; Elliptic PDEs in infinite domains; Green's functions; Long range interactions; Magnetic domains; Magnetization; Mathematical analysis; Mathematics; Micromagnetism; Partial differential equations; Singularity (mathematics)
• ISSN: 1019-7168; 1572-9044; 1572-9044
• DOI: 10.1007/s10444-025-10233-z

The demagnetization field in micromagnetism is given as the gradient of a potential that solves a partial differential equation (PDE) posed in Rd. In its most general form, this PDE is supplied with continuity condition on the boundary of the magnetic domain, and the equation includes a discontinuity in the gradient of the potential over the boundary. Typical numerical algorithms to solve this problem rely on the representation of the potential via the Green’s function, where a volume and a boundary integral terms need to be accurately approximated. From a computational point of view, the volume integral dominates the computational cost and can be difficult to approximate due to the singularities of the Green’s function. In this article, we propose a hybrid model, where the overall potential can be approximated by solving two uncoupled PDEs posed in bounded domains, whereby the boundary conditions of one of the PDEs are obtained by a low cost boundary integral. Moreover, we provide a convergence analysis of the method under two separate theoretical settings: periodic magnetization and high-frequency magnetization. Numerical examples are given to verify the convergence rates.