An efficient and unconditionally stable finite element scheme for an active magnetic fluid model

In this article, based on the laws of mass and momentum conservation and the coupled effects of electromagnetism and hydrodynamics, we derive an active magnetic fluid model to qualitatively describe the collective and organized motion of magnetotactic bacteria at high concentrations. To solve the developed model, we construct a decoupled, linear and unconditionally stable finite element scheme by employing an auxiliary variable, the implicit-explicit approach and the pressure projection method. The numerical scheme has three distinct properties: (i) the velocity field can be approximated by lower order finite element; (ii) the velocity and magnetic fields are decoupled by adding a stabilization term; (iii) the velocity and pressure are split using the pressure projection method. We also prove the unique solvability and unconditional stability of the proposed scheme. Numerical examples are presented to validate the theoretical results. Comparisons between simulation and laboratory results demonstrate that the active magnetic fluid model can accurately capture the complex dynamics of active magnetic fluid motion.