A decoupled, stable, and second-order integrator for the Landau--Lifshitz--Gilbert equation with magnetoelastic effects

Abstract

We consider the numerical approximation of a nonlinear system of partial differential equations modeling magnetostriction in the small-strain regime consisting of the Landau--Lifshitz--Gilbert equation for the magnetization and the conservation of linear momentum law for the displacement. We propose a fully discrete numerical scheme based on first-order finite elements for the spatial discretization. The time discretization employs a combination of the classical Newmark-$\beta$ scheme for the displacement and the midpoint scheme for the magnetization, applied in a decoupled fashion. The resulting method is fully linear and formally of second order in time. We derive the discrete energy law satisfied by the approximations and prove the stability of the scheme. Finally, we assess the performance of the proposed method in a collection of numerical experiments.