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 $$\mathbb {L}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , $$\mathbb {L}^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> , and $$\mathbb {H}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:math> norms. These theoretical results are further corroborated by several numerical experiments.