| Abstract: |
| In this talk, we introduce and analyze a discontinuous Galerkin method for the stationary incompressible magnetohydrodynamic system subject to Navier-type boundary conditions for both the velocity and the magnetic field.
We establish a new discrete Sobolev inequality in the $L^p$ setting, which plays a central role in the analysis, in particular for proving the well-posedness and convergence of the scheme. The existence and uniqueness of a discrete solution are obtained by means of the Banach fixed point theorem under a smallness assumption on the data.
Furthermore, we derive a priori error estimates in a natural energy norm for both the velocity and the magnetic field.
To the best of our knowledge, this is the first work providing a complete analysis of a discontinuous Galerkin method for the nonlinear coupled magnetohydrodynamic system with Navier-type boundary conditions imposed simultaneously on the velocity and the magnetic field. |
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