Abstract In this paper, we establish the unconditionally optimal error estimates of linearized second-order backward difference formula (BDF2) Galerkin finite element methods (FEMs) for the Landau-Lifshitz equation describing the magnetic behavior in ferromagnetic materials. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, we analyze the temporal error by introducing a time-discrete system and derive some regularity of the solution of the time-discrete system. Second, by the above achievements, we obtain the τ-independent spatial error and the boundedness of the numerical solution in L ∞ -norm. Then, the optimal L 2 and H 1 error estimates for r-th order FEMs ( r = 1 , 2 ) are derived without any restriction on the time step size. Numerical results in both two and three dimensional spaces are presented to confirm the theoretical predictions and demonstrate the efficiency of the methods.

中文翻译：

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Landau-Lifshitz 方程线性化二阶 BDF Galerkin FEM 的无条件最优误差估计

摘要 在本文中，我们建立了描述铁磁材料磁行为的 Landau-Lifshitz 方程的线性二阶后向差分公式 (BDF2) Galerkin 有限元方法 (FEM) 的无条件最优误差估计。通过使用时空误差分裂技术，我们将精确解和数值解之间的误差分为两部分，称为时间误差和空间误差。首先，我们通过引入时间离散系统来分析时间误差，并推导出时间离散系统的解的一些规律性。其次，通过上述成果，我们得到了与τ无关的空间误差和L∞范数下数值解的有界性。然后，对于第 r 阶 FEM 的最优 L 2 和 H 1 误差估计（ r = 1 ，2 ) 的导出不受时间步长的任何限制。给出了二维和三维空间中的数值结果，以确认理论预测并证明方法的效率。