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

Abstract

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 finite-element 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.