Finite volume element method for Landau-Lifshitz equation
For researchers in micromagnetics, this method provides an efficient numerical solver for the Landau-Lifshitz equation, reducing computational cost while maintaining accuracy.
The paper develops a finite volume element method with Gauss-Seidel projection for the Landau-Lifshitz equation, achieving computational complexity comparable to solving the scalar heat equation implicitly. Numerical experiments validate the efficiency gain and enable simulation of 2D magnetic textures.
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.