AI 中文总结
该研究针对LLG方程提出线性集中质量有限元方法，结合混合有限元-有限差分框架，实现无条件能量耗散与逐节点长度保持，经数值实验验证其精度与鲁棒性。
AI 中文摘要
针对拟均匀三角形网格上高度非线性的Landau-Lifshitz-Gilbert（LLG）方程，我们开发了一种线性、无条件能量耗散的集中质量有限元方法。该方法基于投影策略构建，用于满足非凸逐点约束|𝐦|=1，而标准有限元离散化同时实现该约束与无条件能量稳定性仍具挑战性。核心创新是统一的混合有限元-有限差分框架，这构成了所提方法设计与分析的基础。在格式构造中，我们利用集中质量有限元方法的弱形式与节点结构，同时结合合适的插值算子和受有限差分离散化启发的逐节点长度保持机制。该组合产生了一种线性格式，其可保持逐节点单位长度约束并满足离散能量耗散定律。相同的混合框架在误差分析中也发挥核心作用，其中有限元的弱形式与拟均匀网格结构结合基于插值的逐节点有限差分方法，以控制强非线性阻尼项并建立最优阶误差估计。更重要的是，所提方法提供了一个统一框架，系统整合了有限元方法的几何灵活性与有限差分方法的逐节点约束保持特性，从而为设计和分析约束耗散系统的保结构离散化提供了通用策略。包括经典 blow-up 模拟在内的数值实验，证实了该方法的预测精度、能量耗散特性及鲁棒性。
英文摘要
We develop a linear, unconditionally energy-dissipative, mass-lumped finite element method for the highly nonlinear Landau--Lifshitz--Gilbert (LLG) equation on quasi-uniform triangular meshes. The method is built on a projection strategy for enforcing the nonconvex pointwise constraint $|\mathbf{m}| = 1$, whose simultaneous preservation with unconditional energy stability remains challenging for standard finite element discretizations. The key innovation is a unified hybrid finite element-finite difference framework that underlies both the design and the analysis of the proposed method. In the scheme construction, we exploit the weak formulation and nodal structure of mass-lumped finite element method, while incorporating suitable interpolation operators and a node-wise length-preserving mechanism inspired by finite difference discretizations. This combination yields a linear scheme that preserves the node-wise unit-length constraint and satisfies a discrete energy dissipation law. The same hybrid framework also plays a central role in the error analysis, where the weak formulation and quasi-uniform mesh structure of finite elements are combined with interpolation-based and nodewise finite difference method to control the strongly nonlinear damping term and to establish an optimal-order error estimate. More importantly, the proposed method provides a unified framework that systematically integrates the geometric flexibility of finite element method with the pointwise constraint-preserving property of finite difference method, and thus offers a general strategy for designing and analyzing structure-preserving discretizations of constrained dissipative systems. Numerical experiments, including a classical blow-up simulation, confirm the predicted accuracy, energy dissipation, and robustness of the method.
Comments24 pages, 16 figures
