The paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau–Lifshitz equation with the point-wise constraint $|{\textbf{m}}|=1$. The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been used in many areas in the past several decades. However, error analysis for the commonly used method has not been done since the classical energy approach cannot be applied directly. In this paper we present an optimal $\textbf{L}^2$ error analysis of the backward Euler sphere projection method by using quadratic or higher order finite elements under a time step condition $\tau =O(\epsilon _0 h)$ with some small $\epsilon _0>0$. The analysis is based on more precise estimates of the extra error caused by the sphere projection in both $\textbf{L}^2$ and $\textbf{H}^1$ norms, and the classical estimate of dual norm. Numerical experiment is provided to confirm our theoretical analysis.

中文翻译：

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Landau-Lifshitz 方程的后向 Euler 投影 FEM 分析

本文重点分析了铁磁材料磁化动力学的欧拉投影伽辽金有限元法 (FEM)，由具有逐点约束 $|{\textbf{m}}|= 的 Landau-Lifshitz 方程描述1美元。该方法基于简单的球面投影，将数值解在每个时间步长投影到一个单位球面上，该方法在过去几十年中已在许多领域得到应用。然而，由于不能直接应用经典能量方法，因此尚未对常用方法进行误差分析。在本文中，我们提出了在时间步长条件 $\tau =O(\epsilon _0 h)$ 下使用二次或更高阶有限元的反向欧拉球投影方法的最优 $\textbf{L}^2$ 误差分析。有一些小的 $\epsilon _0>0$。该分析基于对 $\textbf{L}^2$ 和 $\textbf{H}^1$ 范数中球体投影引起的额外误差的更精确估计，以及对偶范数的经典估计。提供了数值实验来证实我们的理论分析。