The Landau–Lifshitz–Bloch Equation With Spin Diffusion: Global Strong Solution and Finite Element Approximation

The spin‐diffusion Landau–Lifshitz–Bloch (SDLLB) system is a nonlinearly coupled system of quasilinear vector‐valued PDEs which models the interaction between spin‐polarised currents and magnetisation at high temperatures. The aim of this paper is twofold. Firstly, assuming the initial data is sufficiently small, we show the existence of a unique global strong solution to the SDLLB equation in a bounded domain , where , thus ensuring well‐posedness of the model. Secondly, we propose a decoupled linearised fully‐discrete finite element scheme to solve the problem. Despite the strong nonlinearity of the system, the proposed scheme only requires the solution of two completely decoupled linear systems per time‐step. Assuming adequate regularity of the exact solution and a certain time‐step constraint, we rigorously show that the numerical scheme converges at an optimal rate. Several numerical experiments corroborate our theoretical results.