Numerical analysis of the Landau--Lifshitz--Bloch equation with spin-torques

The current-induced magnetisation dynamics in a ferromagnet at elevated temperatures can be described by the Landau--Lifshitz--Bloch (LLB) equation with spin-torque terms. First, we establish the existence and uniqueness of the global strong solution to the model in spatial dimensions $d=1,2,3$, with an additional smallness assumption on the initial data if $d=3$. We then propose a fully discrete linearly implicit finite element scheme for the problem and prove that it approximates the solution with an optimal order of convergence, provided the exact solution possesses adequate regularity. Furthermore, we propose an unconditionally energy-stable finite element method to approximate the LLB equation without spin current. This scheme also converges to the exact solution at an optimal order, and is shown to be energy-dissipative at the discrete level. Finally, some numerical simulations to complement the theoretical analysis are included.