标量辅助变量有限元方法在Landau-Lifshitz-Bloch方程中的误差分析
Error analysis of scalar auxiliary variable finite element methods for the Landau--Lifshitz--Bloch equation
AI总结：
本文提出并分析了两种基于标量辅助变量方法的完全离散有限元方案，用于在高温 regime 中求解LLB方程，实现了二阶时间精度和无条件能量稳定性。
摘要：
Landau-Lifshitz-Bloch（LLB）方程是描述铁磁材料在高温下磁化动力学的已建立的微磁学模型。本文提出并分析了两种基于标量辅助变量（SAV）方法的完全离散、符合有限元方案，用于在居里点以上的高温 regime 中求解LLB方程。第一种方案采用半隐式欧拉时间离散化，而第二种则基于线性外推BDF2方法。两种方案均为线性、对能量范数无条件稳定的，并满足一个涉及基于SAV的能量功能的离散能量定律，该能量功能近似于真实的微磁学能量。在适当的正则性假设下，我们建立了无条件能量稳定性，并推导了在$\mathbb{L}^2, \mathbb{H}^1$和$\mathbb{L}^\infty$范数下的最优阶误差估计。据我们所知，这是首次对完全离散SAV方法在准线性向量值问题上的严谨误差分析，也是首次在高温 regime 中针对LLB方程的线性、能量稳定的方案，实现了二阶时间精度。
英文摘要：
The Landau--Lifshitz--Bloch (LLB) equation is a well-established micromagnetic model for describing magnetisation dynamics in ferromagnets at elevated temperatures. In this paper, we propose and analyse two fully discrete, conforming finite element schemes based on the scalar auxiliary variable (SAV) approach for solving the LLB equation in the high-temperature regime above the Curie point. The first scheme employs a semi-implicit Euler time discretisation, while the second is based on a linearly extrapolated BDF2 method. Both schemes are linear, unconditionally stable with respect to the energy norm, and satisfy a discrete energy law involving the SAV-based energy functional that approximates the true micromagnetic energy. Under suitable regularity assumptions, we establish unconditional energy stability and derive optimal-order error estimates in $\mathbb{L}^2, \mathbb{H}^1$, and $\mathbb{L}^\infty$ norms. To the best of our knowledge, this is the first rigorous error analysis of a fully discrete SAV-based method for a quasilinear vector-valued problem, as well as the first linear, energy-stable scheme for the LLB equation in the high-temperature regime that achieves second-order temporal accuracy. Numerical experiments validate the theoretical convergence rates and compare the efficiency of the proposed schemes with existing methods; additional tests with adaptive time stepping illustrate the behaviour of the SAV-based modified energy in relation to the physical energy.
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