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

The demagnetization field in micromagnetism is given as the gradient of a potential which solves a partial differential equation (PDE) posed in R^d. 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 relies 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 is obtained by a low cost boundary integral. Moreover, we provide a convergence analysis of the method under two separate theoretical settings; periodic magnetisation, and high-frequency magnetisation. Numerical examples are given to verify the convergence rates.