  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.
