An unconditionally convergent BDF2-type finite element integrator for the inertial Landau-Lifshitz-Gilbert equation

We propose and analyze a fully discrete numerical scheme to approximate solutions to the inertial Landau-Lifshitz-Gilbert (iLLG) equation, a state-of-the-art model to describe the dynamics of the magnetization at subpicosecond time scales. The method employs first-order finite elements for the spatial discretization and the two-step backward differentiation formula (BDF2) for the time discretization, where the inertial term is treated by a convex combination of BDF2 and the backward Euler method. We prove well-posedness and unconditional stability of the scheme, and obtain explicit bounds for the error in the realization of the iLLG-inherent unit length constraint satisfied by the magnetization. Moreover, we derive the discrete energy law satisfied by the approximations, which resembles the inherent dissipative energy law of the model. Finally, with a constructive approach, we prove unconditional convergence of the approximations toward a weak solution of the problem. Numerical experiments assess the performance of the scheme and validate the theoretical findings.