This paper discusses the theory and numerical method of two-scale analysis for the multiscale Landau-Lifshitz-Gilbert equation in composite ferromagnetic materials. The novelty of this work can be summarized in three aspects: Firstly, the more realistic and complex model is considered, including the effects of the exchange field, anisotropy field, stray field, and external magnetic field. The explicit convergence orders in the $H^1$ norm between the classical solution and the two-scale solution are obtained. Secondly, we propose a robust numerical framework, which is employed in several comprehensive experiments to validate the convergence results for the Periodic and Neumann problems. Thirdly, we design an improved implicit numerical scheme to reduce the required number of iterations and relaxes the constraints on the time step size, which can significantly improve computational efficiency. Specifically, the projection and the expansion methods are given to overcome the inherent non-consistency in the initial data between the multiscale problem and homogenized problem.

翻译：本文探讨复合铁磁材料中多尺度Landau-Lifshitz-Gilbert方程的两尺度分析理论及数值方法。本研究的新颖性可概括为三个方面：其一，考虑了更真实、更复杂的模型，包括交换场、各向异性场、杂散场和外加磁场的影响，获得了经典解与两尺度解在$H^1$范数下的显式收敛阶；其二，提出了一个稳健的数值框架，并应用于多个综合实验，以验证周期问题和Neumann问题的收敛结果；其三，设计了一种改进的隐式数值格式，减少了所需迭代次数，并放宽了对时间步长的限制，从而显著提高了计算效率。具体而言，给出了投影方法和扩展方法，以克服多尺度问题与均匀化问题在初始数据上的固有不一致性。