In this paper we consider three kinds of fully discrete time-stepping schemes for the nonstationary $3$D magneto-micropolar equations that describes the microstructure of rigid microelements in electrically conducting fluid flow under some magnetic field. The first scheme is comprised of the Euler semi-implicit discretization in time and conforming finite element/stabilized finite element in space. The second one is based on Crank-Nicolson discretization in time and extrapolated treatment of the nonlinear terms such that skew-symmetry properties are retained. We prove that the proposed schemes are unconditionally energy stable. Some optimal error estimates for the velocity field, the magnetic field, the micro-rotation field and the fluid pressure are obtained. Furthermore, we establish some fully discrete first-order decoupled time-stepping algorithms. Numerical tests are provided to check the theoretical rates and unconditionally energy stable.

翻译：本文考虑三类针对非定常三维磁微极方程的全离散时间推进格式，该方程描述了在磁场作用下导电流体中刚性微元结构的微观特性。第一类格式采用时间上的欧拉半隐式离散与空间上的协调有限元/稳定化有限元方法。第二类格式基于时间上的Crank-Nicolson离散与非线性项的延拓处理，从而保留斜对称性质。我们证明了所提格式是无条件能量稳定的。获得了速度场、磁场、微旋转场及流体压力的最优误差估计。此外，我们建立了一些全离散的一阶解耦时间推进算法。数值实验验证了理论收敛阶及无条件能量稳定性。