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

TL;DR

This paper investigates the errors introduced by different upscaling strategies in Heterogeneous Multiscale Methods applied to the Landau-Lifshitz equation with rapidly oscillating coefficients, providing error estimates based on averaging techniques.

Contribution

The paper introduces and analyzes multiple upscaling approaches for the Landau-Lifshitz equation with oscillatory coefficients, deriving error bounds related to averaging parameters.

Findings

Averaging errors depend on oscillation scale and averaging domain size.

Error estimates are provided for periodic micro problems.

The micro problem is consistently modeled by the Landau-Lifshitz equation with oscillatory coefficients.

Abstract

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.