  In this paper we propose and analyze a mixed DG method and an HDG method for
the stationary Magnetohydrodynamics (MHD) equations with two types of boundary
(or constraint) conditions. The mixed DG method is based a recent work proposed
by Houston et. al. for the linearized MHD. With two novel discrete Sobolev
embedding type estimates for the discontinuous polynomials, we provide a priori
error estimates for the method on the nonlinear MHD equations. In the smooth
case, we have optimal convergence rate for the velocity, magnetic field and
pressure in the energy norm, the Lagrange multiplier only has suboptimal
convergence order. With the minimal regularity assumption on the exact
solution, the approximation is optimal for all unknowns. To the best of our
knowledge, this is the first a priori error estimates of DG methods for
nonlinear MHD equations. In addition, we also propose and analyze the first
divergence-free HDG method for the problem with several unique features
comparing with the mixed DG method.
