Iterative solution and preconditioning for the tangent plane scheme in computational micromagnetics

TL;DR

This paper presents an efficient iterative solution and preconditioning strategy for the tangent plane scheme used in simulating ferromagnetic dynamics via the Landau-Lifshitz-Gilbert equation, improving computational performance.

Contribution

It introduces a novel preconditioning approach based on Householder reflections for constrained linear systems in micromagnetics simulations, with proven linear convergence.

Findings

Preconditioners are effective and nearly independent of time-step size.

GMRES algorithm converges linearly with the proposed preconditioning.

Numerical experiments confirm theoretical convergence and efficiency.

Abstract

The tangent plane scheme is a time-marching scheme for the numerical solution of the nonlinear parabolic Landau-Lifshitz-Gilbert equation (LLG), which describes the time evolution of ferromagnetic configurations. Exploiting the geometric structure of LLG, the tangent plane scheme requires only the solution of one linear variational form per time-step, which is posed in the discrete tangent space determined by the nodal values of the current magnetization. We develop an effective solution strategy for the arising constrained linear systems, which is based on appropriate Householder reflections. We derive possible preconditioners, which are (essentially) independent of the time-step, and prove that the preconditioned GMRES algorithm leads to linear convergence. Numerical experiments underpin the theoretical findings.