Heterogeneous multiscale methods for the Landau-Lifshitz equation

Abstract

In this paper, we present a finite difference heterogeneous multiscale method for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient. The approach combines a higher order discretization and artificial damping in the so-called micro problem to obtain an efficient implementation. The influence of different parameters on the resulting approximation error is discussed. Numerical examples for both periodic as well as more general coefficients are given to demonstrate the functionality of the approach.

A finite element based Heterogeneous Multiscale Method for the Landau-Lifshitz equation

“…A detailed discussion of a micro problem similar to the one discussed here is given in [19]. As suggested there, also in this paper we use artificial damping in the micro problem and choose α ≈ 1, which leads to an improved constant in the error estimate in Theorem 3.1 and thus convergence of the approximation errors for shorter final times η.…”

“…Further examples of how the upscaling is influenced by the choice of η, µ and µ are given in [19] for a similar but slightly different micro problem. We here conclude that choosing η, µ and µ large enough results in upscaling errors determined only by H. Given a smaller value of H, lower errors can be obtained by selecting the parameters η and as a consequence µ larger.…”

“…Several ways to set up HMM for a periodic version of (1.1) are discussed in [20]. A finite difference based implementation of one of these approaches, the so-called field model, is studied in [19]. In this article, we focus on a variation of the so-called flux model and investigate how to combine a finite element macro scheme with a finite difference discretized micro model.…”

“…This makes it possible to use the approach for more general geometries and gives a high flexibility. Additionally, in contrast to [19], the effective field H ε considered in this paper is more general and contains not only the exchange term but also applied field and demagnetization.…”