Abstract
We consider the numerical approximation of the Landau–Lifshitz–Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii–Moriya interaction, which is the most important ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. We propose and analyze three tangent plane integrators, for which we prove (unconditional) convergence of the finite element solutions towards a weak solution of the problem. The analysis is constructive and also establishes existence of weak solutions. Numerical experiments demonstrate the applicability of the methods for the simulation of practically relevant problem sizes.
Highlights
1.1 State of the art Magnetic skyrmions are topologically protected vortex-like magnetization configurations [30, 49, 61], which have been theoretically predicted [19,20,21, 55] and Extended author information available on the last page of the article.experimentally observed [47, 54] in several magnetic systems
The Dzyaloshinskii–Moriya interaction (DMI) is modeled by an energy contribution, which is linear in the first spatial derivatives of the magnetization and is added to the micromagnetic energy for chiral ferromagnets
As for the mathematical literature, the existence of isolated skyrmions emerging as energy minimizers of two-dimensional micromagnetic models and their dynamic stability have been investigated in [25, 45], whereas chiral domain walls in ultrathin ferromagnetic films have been studied in [48]
Summary
Extended author information available on the last page of the article. experimentally observed [47, 54] in several magnetic systems. The most important ingredient for the enucleation and the stabilization of magnetic skyrmions is the socalled Dzyaloshinskii–Moriya interaction (DMI) (see [26, 46]) It is a short-range effect, sometimes referred to as antisymmetric exchange, which exerts a torque on the magnetization inducing neighboring spins to be perpendicular to each other. The integrator proposed in [5], which considers the case in which the energy only comprises the exchange contribution, requires only the solution of one linear system per time-step, is formally of first order in time, and is unconditionally convergent towards a weak solution of the problem, i.e., the numerical analysis of the scheme does not require to impose any restrictive CFL-type coupling condition on the timestep size and the spatial mesh size. Adapting ideas from [5, 15], the recent work [39] proposes a similar predictor-corrector scheme based on a linear mass-lumped variational formulation of LLG
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