Mixed finite element methods for the Landau--Lifshitz--Baryakhtar and the regularised Landau--Lifshitz--Bloch equations in micromagnetics

Abstract

The Landau--Lifshitz--Baryakhtar (LLBar) and the Landau--Lifshitz--Bloch (LLBloch) equations are nonlinear vector-valued PDEs which arise in the theory of micromagnetics to describe the dynamics of magnetic spin field in a ferromagnet at elevated temperatures. We consider the LLBar and the regularised LLBloch equations in a unified manner, thus allowing us to treat the numerical approximations for both problems at once. In this paper, we propose a semi-discrete mixed finite element scheme and two fully discrete mixed finite element schemes based on a semi-implicit Euler method and a semi-implicit Crank--Nicolson method to solve the problems. These numerical schemes provide accurate approximations to both the magnetisation vector and the effective magnetic field. Moreover, they are proven to be unconditionally energy-stable and preserve energy dissipativity of the system at the discrete level. Error analysis is performed which shows optimal rates of convergence in ${\mathbb{L}}^2$, ${\mathbb{L}}^{\infty }$, and ${\mathbb{H}}^1$ norms. These theoretical results are further corroborated by several numerical experiments.

References in corpus (2)

• Dynamic approach for micromagnetics close to the Curie temperature
• Global solutions of the Landau--Lifshitz--Baryakhtar equation

Cited by in corpus (5)

• The Landau--Lifshitz--Bloch equation with spin diffusion: Global strong solution and finite element approximation
• The stochastic Landau--Lifshitz--Baryakhtar equation: Global solution and invariant measure
• Error analysis of scalar auxiliary variable finite element methods for the Landau--Lifshitz--Bloch equation
• Numerical analysis of the Landau--Lifshitz--Bloch equation with spin-torques
• Global attractor and robust exponential attractors for some classes of fourth-order nonlinear evolution equations