Reformulated formulation and efficient fully discrete finite element method for a conductive ferrofluid model

In this paper, we consider numerical approximation of an electrically conductive ferrofluid model, which consists of Navier-Stokes equations, magnetization equation, and magnetic induction equation. To solve this highly coupled, nonlinear, and multiphysics system efficiently, we develop a decoupled, linear, second-order in time, and unconditionally energy stable finite element scheme. We incorporate several distinct numerical techniques, including reformulations of the equations and a scalar auxiliary variable to handle the coupled nonlinear terms,a symmetric implicit-explicit treatment for the symmetric positive definite nonlinearity, and stable finite element approximations. We also prove that the numerical scheme is provably uniquely solvable and unconditionally energy stable rigorously. A series of numerical examples are presented to illustrate the accuracy and performance of our scheme.