  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.
