Stable $C^1$-conforming finite element methods for the Landau--Lifshitz--Baryakhtar equation

The Landau--Lifshitz--Baryakhtar equation describes the evolution of magnetic spin field in magnetic materials at elevated temperature below the Curie temperature, when long-range interactions and longitudinal dynamics are taken into account. We propose two linear fully-discrete $C^1$-conforming methods to solve the problem, namely a semi-implicit Euler method and a semi-implicit BDF method, and show that these schemes are unconditionally stable. Error analysis is performed which shows optimal convergence rates in each case. Numerical results corroborate our theoretical results.