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：本文研究描述铁磁材料中磁化过程（含小应变范围内磁弹性效应）的Landau-Lifshitz-Gilbert方程与线动量守恒定律耦合系统。针对该含时非线性偏微分方程组，我们提出一种基于空间一阶有限元与时间隐式单步法的解耦积分算法。证明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至系统弱解。与已有数值工作相比，我们的方法满足离散能量律以模拟连续问题特性，并通过极限过程获得弱解满足的能量不等式。此外，该方法无需采用节点投影约束离散磁化强度单位长度，因此稳定性不依赖于弱锐角网格条件。更广泛而言，所提积分器及其分析框架适用于包含体力、面力以及更一般磁应变表示的情形。数值实验验证了理论结果，展示了该方法在亚微米尺度下模拟含磁弹性材料动力学过程的适用性。