The Landau--Lifshitz--Baryakhtar (LLBar) and the Landau--Lifshitz--Bloch (LLBloch) equations are nonlinear vector-valued PDEs which arise in the theory of micromagnetics to describe the dynamics of magnetic spin field in a ferromagnet at elevated temperatures. We consider the LLBar and the regularised LLBloch equations in a unified manner, thus allowing us to treat the numerical approximations for both problems at once. In this paper, we propose a semi-discrete mixed finite element scheme and two fully discrete mixed finite element schemes based on a semi-implicit Euler method and a semi-implicit Crank--Nicolson method to solve the problems. These numerical schemes provide accurate approximations to both the magnetisation vector and the effective magnetic field. Moreover, they are proven to be unconditionally energy-stable and preserve energy dissipativity of the system at the discrete level. Error analysis is performed which shows optimal rates of convergence in $\mathbb{L}^2$, $\mathbb{L}^\infty$, and $\mathbb{H}^1$ norms. These theoretical results are further corroborated by several numerical experiments.

翻译：Landau--Lifshitz--Baryakhtar（LLBar）方程与Landau--Lifshitz--Bloch（LLBloch）方程是非线性向量值偏微分方程，源于微磁学理论，用于描述铁磁体在高温下磁化矢量场的动力学行为。本文以统一框架处理LLBar方程与正则化LLBloch方程，从而实现对两类问题数值逼近的同步处理。我们提出了半离散混合有限元格式，以及基于半隐式Euler方法与半隐式Crank--Nicolson方法的两种全离散混合有限元格式来求解这些问题。这些数值格式能够对磁化矢量与有效磁场同时提供精确逼近，并被证明具有无条件能量稳定性，在离散层面保持了系统的能量耗散特性。误差分析表明，该方法在$\mathbb{L}^2$、$\mathbb{L}^\infty$和$\mathbb{H}^1$范数下均达到最优收敛阶。若干数值实验进一步验证了这些理论结果。