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. In this paper, we focus on the regime above the Curie temperature. We first establish the existence and uniqueness of a global strong solution to the model in spatial dimensions $d=1,2,3$, under an additional smallness assumption on the initial data if $d=3$. Relevant smoothing and decay estimates are also derived. We then propose a fully discrete, linearly implicit finite element scheme for the problem and prove that it achieves optimal-order convergence, assuming adequate regularity of the exact solution. In addition, we introduce an unconditionally energy-stable finite element method for the case of negligible non-adiabatic torque. This scheme is also shown to converge optimally and, in the absence of current, preserves energy dissipation at the discrete level. Finally, we present numerical simulations that support the theoretical analysis and demonstrate the performance of the proposed methods.