A decoupled and unconditionally convergent linear FEM integrator for the Landau–Lifshitz–Gilbert equation with magnetostriction

Abstract

To describe and simulate dynamic micromagnetic phenomena, we consider a coupled system of the nonlinear Landau-Lifshitz-Gilbert equation and the conservation of momentum equation. This coupling allows to include magnetostrictive effects into the simulations. Existence of weak solutions has recently been shown in [CARBOU ET AL., Math. Meth. Appl. Sci. (2011)]. In our contribution, we give an alternate proof which additionally provides an effective numerical integrator. The latter is based on linear finite elements in space and a linear-implicit Euler time-stepping. Despite the nonlinearity, only two linear systems have to be solved per timestep, and the integrator fully decouples both equations. Finally, we prove unconditional convergence-at least of a subsequence-towards, and hence existence of, a weak solution of the coupled system, as timestep size and spatial mesh-size tend to zero. We conclude the work with numerical experiments which study the discrete blow-up of the LLG equation as well as the influence of the magnetostrictive term on the discrete blow-up.