A Mixed Finite Element Method with Exactly Divergence-free Velocities for Incompressible Magnetohydrodynamics

We introduce and analyze a mixed finite element method for the numerical discretization of a stationary incompressible magnetohydrodynamics problem, in two and three dimensions. The velocity field is discretized using divergence-conforming Brezzi-Douglas-Marini (BDM) elements and the magnetic field is approximated by curl-conforming Nédélec elements. The H¹-continuity of the velocity field is enforced by a DG approach. A central feature of the method is that it produces exactly divergence-free velocity approximations, and captures the strongest magnetic singularities. We prove that the energy norm error is convergent in the mesh size in general Lipschitz polyhedra under minimal regularity assumptions, and derive nearly optimal a-priori error estimates for the two-dimensional case. We present a comprehensive set of numerical experiments, which indicate optimal convergence of the proposed method for two-dimensional as well as three-dimensional problems.