The Landau--Lifshitz--Bloch equation with spin diffusion: Global strong solution and finite element approximation

TL;DR

This paper proves the existence of a unique global strong solution for the spin-diffusion Landau--Lifshitz--Bloch system under small initial data and introduces a decoupled finite element scheme with proven convergence.

Contribution

It establishes well-posedness of the SDLLB system and develops a computationally efficient, convergent finite element method for solving it.

Findings

Existence of a unique global strong solution for small initial data.

A decoupled linearized finite element scheme with optimal convergence rate.

Numerical experiments confirm theoretical convergence results.

Abstract

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 $\Omega\subset \mathbb{R}^d$, where $d\leq 3$, 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.