We consider the coupled system of the Landau--Lifshitz--Gilbert equation and the conservation of linear momentum law to describe magnetic processes in ferromagnetic materials including magnetoelastic effects in the small-strain regime. For this nonlinear system of time-dependent partial differential equations, we present a decoupled integrator based on first-order finite elements in space and an implicit one-step method in time. We prove unconditional convergence of the sequence of discrete approximations towards a weak solution of the system as the mesh size and the time-step size go to zero. Compared to previous numerical works on this problem, for our method, we prove a discrete energy law that mimics that of the continuous problem and, passing to the limit, yields an energy inequality satisfied by weak solutions. Moreover, our method does not employ a nodal projection to impose the unit length constraint on the discrete magnetisation, so that the stability of the method does not require weakly acute meshes. Furthermore, our integrator and its analysis hold for a more general setting, including body forces and traction, as well as a more general representation of the magnetostrain. Numerical experiments underpin the theory and showcase the applicability of the scheme for the simulation of the dynamical processes involving magnetoelastic materials at submicrometer length scales.

翻译：我们考虑Landau-Lifshitz-Gilbert方程与线性动量守恒定律的耦合系统，以描述铁磁材料在小应变范围内包含磁弹性效应的磁化过程。针对这一含时非线性偏微分方程组，我们提出了一种基于空间一阶有限元和时间隐式单步法的解耦积分器。我们证明了当网格尺寸和时间步长趋于零时，离散近似序列无条件收敛于系统的弱解。与之前关于该问题的数值工作相比，我们的方法证明了一种离散能量律，该定律模仿了连续问题的能量律，并在取极限后得到弱解满足的能量不等式。此外，我们的方法未采用节点投影来施加离散磁化强度的单位长度约束，因此方法的稳定性不需要弱锐角网格。进一步地，我们的积分器及其分析适用于更一般的设定，包括体力和牵引力，以及更一般的磁应变表示。数值实验支撑了理论分析，并展示了该方案在亚微米尺度下涉及磁弹性材料的动态过程模拟中的适用性。