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Unconditionally optimal convergence analysis of second-order BDF scheme for Landau-Lifshitz Equation
Bolin Chen,Rong An
Keywords:Landau-Lifshitz equation, BDF2 scheme, Finite element method, Optimal error estimates.
      The Landau-Lifshitz equation is used to describe the evolution of spin fields in continuum ferromagnets and is a highly nonlinear parabolic problem with the constraint of unit length in the point-wise sense. This paper focuses on the unconditionally optimal error estimates of a linearized second-order BDF scheme for the numerical approximations of the solution to the Landau-Lifshitz equation. Since the point-wise constraint can be deduced from the partial differential equation, we do not take into account it in designing the numerical scheme. A rigorous error analysis is done and we derive the unconditionally optimal $\L^2$ error estimate by using the error splitting technique.Numerical result is shown to check the theoretical analysis.