Analysis and Numerical Approximation of a Stationary Mhd Flow Problem with Nonideal Boundary∗

We are concerned with the steady flow of a conducting fluid, confined to a bounded region of space and driven by a combination of body forces, externally generated magnetic fields, and currents entering and leaving the fluid through electrodes attached to the surface. The flow is governed by the Navier–Stokes equations (in the fluid region) and Maxwell’s equations (in all of space), coupled via Ohm’s law and the Lorentz force. By means of the Biot–Savart law, we reduce the problem to a system of integro-differential equations in the fluid region, derive a mixed variational formulation, and prove its well-posedness under a small-data assumption. We then study the finiteelement approximation of solutions (in the case of unique solvability) and establish optimal-order error estimates. Finally, an implementation of the method is described and illustrated with the results of some numerical experiments.