Upscaling errors in Heterogeneous Multiscale Methods for the Landau-Lifshitz equation

In this paper, we consider several possible ways to set up Heterogeneous Multiscale Methods for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient, which can be seen as a means to modeling rapidly varying ferromagnetic materials. We then prove estimates for the errors introduced when approximating the relevant quantity in each of the models given a periodic problem, using averaging in time and space of the solution to a corresponding micro problem. In our setup, the Landau-Lifshitz equation with highly oscillatory coefficient is chosen as the micro problem for all models. We then show that the averaging errors only depend on $\varepsilon$, the size of the microscopic oscillations, as well as the size of the averaging domain in time and space and the choice of averaging kernels.