  We consider the numerical approximation of a continuum model of
antiferromagnetic and ferrimagnetic materials. The state of the material is
described in terms of two unit-length vector fields, which can be interpreted
as the magnetizations averaging the spins of two sublattices. For the static
setting, which requires the solution of a constrained energy minimization
problem, we introduce a discretization based on first-order finite elements and
prove its $\Gamma$-convergence. Then, we propose and analyze two iterative
algorithms for the computation of low-energy stationary points. The algorithms
are obtained from (semi-)implicit time discretizations of gradient flows of the
energy. Finally, we extend the algorithms to the dynamic setting, which
consists of a nonlinear system of two Landau-Lifshitz-Gilbert equations solved
by the two fields, and we prove unconditional stability and convergence of the
finite element approximations toward a weak solution of the problem. Numerical
experiments assess the performance of the algorithms and demonstrate their
applicability for the simulation of physical processes involving
antiferromagnetic and ferrimagnetic materials.
