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.

翻译：微磁学中的退磁场由势函数的梯度给出，该势函数求解定义在R^d空间中的偏微分方程。其最一般形式中，该偏微分方程需满足磁畴边界上的连续性条件，且方程中包含势函数梯度在边界上的不连续性。求解该问题的典型数值算法通过格林函数表示势函数，需精确近似体积积分项与边界积分项。从计算角度而言，体积积分主导计算成本，且因格林函数的奇异性难以近似。本文提出一种混合模型，通过求解两个解耦的、定义在有界区域上的偏微分方程来逼近总势函数，其中一个偏微分方程的边界条件通过低计算成本的边界积分获得。此外，我们分别在周期性磁化与高频磁化两种理论框架下，提供了该方法的收敛性分析。数值算例验证了收敛阶。