Well-Posedness and Finite Element Approximation for the Landau-Lifshitz-Gilbert Equation with Spin-Torques

TL;DR

This paper establishes the well-posedness of a modified Landau-Lifshitz-Gilbert equation incorporating spin-torques and develops a convergent finite element method for its numerical approximation, supported by simulations.

Contribution

It proves existence and uniqueness of solutions for the non-homogeneous LLG equation and introduces a convergent finite element scheme for its numerical solution.

Findings

Existence and uniqueness of high regularity local solutions

Development of a convergent finite element method

Numerical simulations demonstrating the approach

Abstract

Spin currents act on ferromagnets by exerting a torque on the magnetisation. This torque is modelled by appending additional terms to the Landau-Lifshitz-Gilbert equation motivating the study of the non-homogeneous Landau-Lifshitz-Gilbert equation. We first prove the existence and uniqueness of high regularity local solutions to this equation using the Faedo-Galerkin method. Then we construct a numerical method for the problem and prove that it converges to a global weak solution of the PDE. Numerical simulations of the problem are also included.