Finite element formalism for micromagnetism

The aim of this work is to present the details of the finite element approach that was developed for solving the Landau‐Lifschitz‐Gilbert (LLG) equations in order to be able to treat problems involving complex geometries.

There are several possibilities to solve the complex LLG equations numerically. The method is based on a Galerkin‐type finite element approach. The authors start with the dynamic LLG equations, the associated boundary condition and the constraint on the magnetization norm. They derive the weak form required by the finite element method. This weak form is afterwards integrated on the domain of calculus.

The authors compared the results obtained with our finite element approach with the ones obtained by a finite difference method. The results being in very good agreement, it can be stated that the approach is well adapted for 2D micromagnetic systems.

The future work implies the generalization of the method to 3D systems. To optimize the approach spatial transformations for the treatment of the magnetostatic problem will be implemented.

The paper presents a special way of solving the LLG equations. The time integration a backward Euler method has been used, the time derivative being calculated as a function of the solutions at times n and n+1. The presence of the constraint on the magnetization norm induced a special two‐step procedure for the calculation of the magnetization at instant n+1.