Energy-stable mixed finite element methods for a ferrofluid flow model

In this paper, we develop a class of mixed finite element methods for the ferrofluid flow model proposed by Shliomis (1972). We show that the energy stability of the weak solutions to the model is preserved exactly for both the semi-discrete and fully discrete finite element solutions. Furthermore, we prove the existence and uniqueness of the discrete solutions and derive optimal error estimates for both the semi-discrete and fully discrete schemes. Numerical experiments confirm the theoretical results.