Finite volume element method for Landau-Lifshitz equation

TL;DR

This paper introduces a finite volume element method combined with a Gauss-Seidel projection approach for simulating the Landau-Lifshitz equation, improving computational efficiency and accuracy in micromagnetics.

Contribution

The paper develops a novel FVEM with GSPM for Landau-Lifshitz equations, providing error analysis, energy law validation, and demonstrating efficiency and capability in simulating magnetic textures.

Findings

The method achieves accurate approximation errors in space.

Energy law is preserved in the numerical scheme.

Significant acceleration in simulations compared to traditional methods.

Abstract

The Landau-Lifshitz equation describes the dynamics of magnetization in ferromagnetic materials. Due to the essential nonlinearity and nonconvex constraint, it is typically solved numerically. In this paper, we developed a finite volume element method (FVEM) with the Gauss-Seidel projection method (GSPM) for the micromagnetics simulations. We provide the approximation error in space and depict the energy law when the FVEM is adopted. Owing to the GSPM for time-marching, the discrete system is decoupled component by component, making the computational complexity comparable to that of solving the scalar heat equation implicitly. This significantly accelerates real simulations. We present several numerical experiments to validate the theoretical analysis and the efficiency gain. Additionally, we study the blow-up solution and efficiently simulate the 2D magnetic textures using the proposed method.