We present a novel exactly divergence-free and well-balanced hybrid finite volume / finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes. The algorithm is based on the splitting the equations into several subsystems so that each of them can be discretized with a particular scheme to preserve some fundamental structural features of the MHD system at the discrete level.
The use of face-based staggered grids allows to account for the divergence-free conditions of the velocity and magnetic fields in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme. Then, the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the edges. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L^2 projection. Finally, a classical continuous finite element approach is employed to compute the pressure. Besides, we account for the known equilibrium solution at each step of the new algorithm so that the method becomes exactly well-balanced.
Validation of the methodology includes a MHD lid-driven cavity benchmark and long-time simulations of Soloviev equilibrium solutions in simplified 3D tokamak configurations which show the capability of the method to maintain stationary equilibria exactly over very long integration times in general grids.