  We consider the numerical approximation of the inertial
Landau-Lifshitz-Gilbert (iLLG) equation, which describes the dynamics of the
magnetization in ferromagnetic materials at subpicosecond time scales. We
propose and analyze two fully discrete numerical schemes based on two different
approaches: The first method is based on a reformulation of the problem as a
linear constrained variational formulation for the time derivative of the
magnetization. The second method exploits a reformulation of the problem as a
first order system in time for the magnetization and the angular momentum. Both
schemes are implicit, based on first-order finite elements, and the constructed
numerical approximations satisfy the inherent unit-length constraint of iLLG at
the vertices of the underlying mesh. For both schemes, we establish a discrete
energy law and prove convergence of the approximations towards a weak solution
of the problem. Numerical experiments validate the theoretical results and show
the applicability of the methods for the simulation of ultrafast magnetic
processes.
