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