A finite element approximation of a current-induced magnetization dynamics model

Micromagnetics is a continuum theory describing magnetization patterns inside ferromagnetic media. The dynamics of a ferromagnetic material are governed by the Landau-Lifshitz equation. This equation is highly nonlinear and  has a non-convex constraint. In this work, a finite element approximation of a current-induced magnetization dynamics model is proposed. The model consists of a modified Landau-Lifshitz-Gilbert (LLG) equation incorporating spin transfer torque. The scheme preserves a non-convex constraint, requires only a linear solver at each time step and is easily applicable to the limiting cases.  As the time and space steps tend to zero, a proof of convergence of the numerical solution to a (weak) solution of the modified LLG equation is given. Numerical results are presented to show the effect of the injected current on magnetization switching.