A finite element scheme for the Landau–Lifshitz–Bloch equation

The phenomenological Landau–Lifshitz equation (LL) suggested by Landau and Lifshitz in 1935 to describe the precessional motion of spins in ferromagnetic materials has shown its limitations when the temperature is close to or above the Curie temperature. This model has been replaced by the Landau–Lifshitz–Bloch model (LLB), which proves its efficiency in modelling magnetic phenomena at all temperature ranges. In this work, we propose an implicit finite element scheme for the latter model. We show that the proposed scheme converges to a weak solution of the (LLB) equation. In practice, a nonlinear system must be solved at each step of time. So, we use a fixed point method to solve this system. Finally, some numerical experiments have been given to show the performance of our approach.