Fully discrete finite element schemes for the nonstationary $3$D magneto-micropolar equations

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.