Optimal Error Analysis of Euler and BDF2 Semi-Renormalized FEMs for the Landau-Lifshitz-Slonczewski Equation (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 ofpreserving the non-convex constraint is the renormalized method. In this paper, weconsider linearized backward Euler and BDF2 semi-renormalized finite element methods, where the renormalized numerical solution is not used in the discretizations ofthe time derivative and only used in linearized explicit parts. By using the piecewiselinear finite element to make the spatial discretization, optimal $L^2$ error estimates arederived, i.e., $\mathcal{O}(τ+h^2)$ for the Euler scheme and $\mathcal{O}(τ^2+h^2)$ for the BDF2 scheme under some CFL type conditions. Finally, numerical results are shown to support thetheoretical convergence rates.