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

A. J. Meir, Paul G. Schmidt
1999 SIAM Journal on Numerical Analysis
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
more » ... oblem 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. Introduction. Magnetohydrodynamics (MHD) is the theory of the macroscopic interaction of electrically conducting fluids and electromagnetic fields. Applications arise in astronomy and geophysics as well as in connection with numerous engineering problems, such as liquid-metal cooling of nuclear reactors, electromagnetic casting of metals, MHD power generation, and MHD ion propulsion. We refer to  or  for general information and to  for more specific references. Assuming the fluid to be incompressible, viscous, and finitely conducting, MHD flow is governed by the Navier-Stokes and pre-Maxwell equations, coupled via Ohm's law and the Lorentz force (see, for example, [20, Chapter 2]). While the fluid may be confined to a bounded region of space, it typically interacts with the outside world (in particular, with current-carrying external conductors) through the universal electromagnetic field. This interaction entails formidable difficulties in the mathematical analysis and numerical solution of realistic MHD flow problems. In particular, while the Navier-Stokes equations are posed in the body of conducting fluid, Maxwell's equations need to be solved in all of space, and interior and exterior fields must be suitably matched at the interfaces separating media with different electromagnetic properties. Only in special circumstances, most notably if the fluid is confined by perfectly conducting walls, can attention be restricted to the fluid region itself. A fair amount of mathematical work has been devoted to this case, that is, the case of MHD flow with "ideal" boundaries