D. Praetorius,M. Ruggeri:

"Numerical integration of the Landau-Lifshitz-Gilbert equation";

Poster: Workshop on Recent advances in discontinuous Galerkin methods, Reading; 09-11-2014 - 09-12-2014.

The understanding of the magnetization dynamics, especially on a microscale, plays an important role in the design of many technological applications, e.g., magnetic sensors, recording heads, and magneto-resistive storage devices. In micromagnetism, it is well-accepted that the partial differential equation governing the dynamics of the magnetization is the Landau-Lifshitz-Gilbert equation (LLG). The reliable numerical integration of LLG faces several challenges due to the strong nonlinearity, possibly complicated and nonlocal field contributions, as well as an inherent non-convex side constraint which enforces length preservation. In general, when dealing with partial differential equations characterized by pointwise constraints, several numerical methods employ a nodal projection step. More precisely, at each time step the constraint is posed at the nodes of the triangulation via a projection of the computed solution onto the target manifold under consideration. To prove stability of such methods, some restrictive conditions on the step size or the underlying triangulation need to be imposed to allow the use of monotonicity arguments. We consider the tangent plane scheme. We show that the nodal projection step is actually not necessary for the finite element approximation to converge towards a weak solution of LLG. This leads to a violation of the constraint at the nodes of the triangulation, which is however controlled by the time step size, independently of the number of iterations. In particular, our analysis can therefore avoid a technical angle condition on the triangulation. Both versions of the integrator are computationally attractive and suitable for the development of decoupled algorithms for the numerical solution of coupled systems where LLG is coupled with another partial differential equation. As an example, we consider a multiscale model, where LLG is coupled with the magnetostatic Maxwell equations, and a spin-polarized transport model, where the coupling of LLG with a diffusion equation for the spin accumulation field is considered.

finite element method, Landau-Lifshitz-Gilbert equation, micromagnetism

http://publik.tuwien.ac.at/files/PubDat_230458.pdf

Created from the Publication Database of the Vienna University of Technology.