Discontinuous Galerkin method for the incompressible Magnetohydrodynamic system with Navier-type boundary condition

In this work, we introduce and analyze a discontinuous Galerkin method (DG) for the stationary Magnetohydrodynamic system (MHD) with Navier-type boundary condition for both the velocity and the magnetic field. We prove a new discrete Sobolev inequality for the L p-norm which is the key ingredient in the study of the well-posedness and the convergence of the DG scheme. The existence of the discrete solution is proved by using Brouwer's theorem under assumption of sufficiently small data. We provide a priori error estimates in terms of a natural energy norm for the velocity and the magnetic fields. The novelty of this work is that, to the author's knowledge, this is the first time that a DG method, applied to the nonlinear coupled MHD system, with Navier-type boundary conditions for both the velocity and the magnetic field, is proposed and completely analyzed