(untitled) (Advances in Applied Mathematics and Mechanics, 2025)
The Landau-Lifshitz-Slonczewski equation is used to describe the magnetization dynamics under the influence of a spin-polarized current in terms of a spin-velocity field in ferromagnetic materials. This equation is a strongly nonlinear evolution equation with a non-convex constraint of unit sphere. A natural method of preserving the non-convex constraint is the renormalized method. In this paper, we consider linearized backward Euler and BDF2 semi-renormalized finite element methods, where the renormalized numerical solution is not used in the discretizations of the time derivative and only used in linearized explicit parts. By using the piecewise linear finite element to make the spatial discretization, optimal L^2 error estimates are derived, i.e., O(τ+h^2) for the Euler scheme and O(τ^2+h^2) for the BDF2 scheme under some CFL type conditions. Finally, numerical results are shown to support the theoretical convergence rates.