The Landau--Lifshitz--Bloch equation (LLBE) describes the evolution of magnetic spin field in a ferromagnet at high temperatures. We consider a viscous (pseudo-parabolic) regularisation of the LLBE for temperatures higher than the Curie temperature, which we call the $\epsilon$-LLBE. Variants of the $\epsilon$-LLBE are applicable to model pattern formation, phase transition, and heat conduction for non-simple materials, among other things. In this paper, we show well-posedness of the $\epsilon$-LLBE and the convergence of the solution $\boldsymbol{u}^\epsilon$ of the regularised equation to the solution $\boldsymbol{u}$ of the LLBE as $\epsilon\to 0^+$. As a by-product of our analysis, we show the existence and uniqueness of regular solution to the LLBE for temperatures higher than the Curie temperature. Furthermore, we propose a linear fully discrete conforming finite element scheme to approximate the solution of the $\epsilon$-LLBE. Error analysis is performed to show unconditional stability and optimal uniform-in-time convergence rate for the schemes. Several numerical simulations corroborate our theoretical results.

翻译：Landau--Lifshitz--Bloch 方程（LLBE）描述了高温铁磁体中磁自旋场的演化过程。针对高于居里温度的情形，我们考虑该方程的一种黏性（拟抛物型）正则化形式，称为 $\epsilon$-LLBE。$\epsilon$-LLBE 的变体可应用于非简单材料中的图案形成、相变及热传导等过程的建模。本文证明了 $\epsilon$-LLBE 的适定性，并证得正则化方程的解 $\boldsymbol{u}^\epsilon$ 在 $\epsilon\to 0^+$ 时收敛至 LLBE 的解 $\boldsymbol{u}$。作为分析的副产物，我们证明了高于居里温度时 LLBE 正则解的存在唯一性。此外，我们提出了一种线性全离散协调有限元格式来逼近 $\epsilon$-LLBE 的解。通过误差分析，证明了该格式的无条件稳定性及关于时间的最优一致收敛阶。数值模拟结果验证了理论分析的正确性。