Numerical Homogenization of Landau-Lifshitz Equation with Rough Coefficients

TL;DR

This paper presents a novel numerical homogenization method for the nonlinear Landau-Lifshitz equation with complex, multiscale coefficients, enabling efficient and accurate simulations of magnetic systems with microstructural heterogeneity.

Contribution

It introduces localized basis functions within the GRPS framework for coarse approximation of the Landau-Lifshitz equation with rough coefficients, addressing computational challenges.

Findings

Achieves significant computational savings

Maintains accuracy in multiscale magnetic simulations

Demonstrates robustness across various microstructures

Abstract

In this work, we develop a numerical homogenization approach for the fully nonlinear Landau-Lifshitz equation with rough coefficients, including non-periodicity and nonseparable scales. Direct numerical resolution of such multiscale problems on fine meshes incurs prohibitive computational costs. To address this challenge, we propose an efficient coarse scale approximation through localized basis functions derived from energy minimization within the Generalized Rough Polyharmonic Splines (GRPS) framework. These basis functions preserve critical multiscale features while operating on a computationally tractable coarse mesh. The nonlinear, vectorial, and non-symmetric nature of the Landau-Lifshitz equation necessitates careful design of variational formulations for basis construction. We introduce several such formulations, each tailored to specific structural aspects of the problem. Through systematic numerical experiments, we demonstrate that our approach achieves significant computational savings without compromising accuracy, offering a robust framework for simulating multiscale magnetic systems with complex microstructures.