  We present a new divergence-free and well-balanced hybrid FV/FE scheme for
the incompressible viscous and resistive MHD equations on unstructured
mixed-element meshes in 2 and 3 space dimensions. The equations are split into
subsystems. The pressure is defined on the vertices of the primary mesh, while
the velocity field and the normal components of the magnetic field are defined
on an edge-based/face-based dual mesh in two and three space dimensions,
respectively. This allows to account for the divergence-free conditions of the
velocity field and of the magnetic field in a rather natural manner. The
non-linear convective and the viscous terms are solved at the aid of an
explicit FV scheme, while the magnetic field is evolved in a divergence-free
manner via an explicit FV method based on a discrete form of the Stokes law in
the edges/faces of each primary element. To achieve higher order of accuracy, a
pw-linear polynomial is reconstructed for the magnetic field, which is
guaranteed to be divergence-free via a constrained L2 projection. The pressure
subsystem is solved implicitly at the aid of a classical continuous FE method
in the vertices of the primary mesh. In order to maintain non-trivial
stationary equilibrium solutions of the governing PDE system exactly, which are
assumed to be known a priori, each step of the new algorithm takes the known
equilibrium solution explicitly into account so that the method becomes exactly
well-balanced. This paper includes a very thorough study of the lid-driven MHD
cavity problem in the presence of different magnetic fields. We finally present
long-time simulations of Soloviev equilibrium solutions in several simplified
3D tokamak configurations even on very coarse unstructured meshes that, in
general, do not need to be aligned with the magnetic field lines.
