  In this paper we study finite element method for three-dimensional
incompressible resistive magnetohydrodynamic equations, in which the velocity,
the current density, and the magnetic induction are divergence-free. It is
desirable that the discrete solutions should also satisfy divergence-free
conditions exactly especially for the momentum equations. Inspired by
constrained transport method,we devise a new stable mixed finite element method
that can achieve the goal. We also prove the well-posedness of the discrete
solutions. To solve the resulting linear algebraic equations, we propose a
GMRES solver with an augmented Lagrangian block preconditioner. By numerical
experiments, we verify the theoretical results and demonstrate the
quasi-optimality of the discrete solver with respect to the number of degrees
of freedom
