We consider the numerical approximation of a continuum model of antiferromagnetic and ferrimagnetic materials. The state of the material is described in terms of two unit-length vector fields, which can be interpreted as the magnetizations averaging the spins of two sublattices. For the static setting, which requires the solution of a constrained energy minimization problem, we introduce a discretization based on first-order finite elements and prove its $\Gamma$-convergence. Then, we propose and analyze two iterative algorithms for the computation of low-energy stationary points. The algorithms are obtained from (semi-)implicit time discretizations of gradient flows of the energy. Finally, we extend the algorithms to the dynamic setting, which consists of a nonlinear system of two Landau-Lifshitz-Gilbert equations solved by the two fields, and we prove unconditional stability and convergence of the finite element approximations toward a weak solution of the problem. Numerical experiments assess the performance of the algorithms and demonstrate their applicability for the simulation of physical processes involving antiferromagnetic and ferrimagnetic materials.

翻译：本文考虑反铁磁和亚铁磁材料连续模型数值逼近问题。材料状态由两个单位长度向量场描述，可解释为两个子晶格自旋平均后的磁化强度。针对需解约束能量最小化问题的静态情形，我们引入基于一阶有限元的离散化方法并证明其$\Gamma$-收敛性。随后提出并分析两种用于计算低能稳态点的迭代算法，该算法源自能量梯度流的（半）隐式时间离散化。最后将算法拓展至动态情形——即由两个场求解的非线性Landau-Lifshitz-Gilbert方程组，并证明有限元逼近无条件稳定且收敛于问题的弱解。数值实验评估了算法性能，验证其在涉及反铁磁和亚铁磁材料物理过程模拟中的适用性。