Convergent Geometric Integrator for the Landau-Lifshitz-Gilbert Equation in Micromagnetics

Abstract

We consider a finite element scheme of lowest order for the Landau-Lifshitz-Gilbert equation (LLG) which describes the dynamics of micromagnetism. In contrast to previous works, we examine LLG including the total magnetic field induced by several physical phenomena described in terms of exchange energy, anisotropy energy, magnetostatic energy, and Zeeman energy. Besides a strong nonlinearity and a non-convex side constraint, the non-local dependence of the demagnetization field from the magnetization represents a challenging task for the numerical integrator. Nevertheless, we prove unconditional convergence for the approximation of a weak solution.