Approximation of the inductionless MHD problem using a stabilized finite element method

In this work, a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method is presented. The MHD problem couples the Navier-Stokes and a Darcy-type problem for the electric potential via Lorentz’s force in the momentum equation of the Navier-Stokes equations and the currents generated by the moving fluid in Ohm’s law. The key feature of the FE formulation resides in the design of the stabilization terms, which serve several purposes. First, the formulation is suitable for convection dominated flows. Second, there is no need to use interpolation spaces constrained to an inf-sup condition in both problems and therefore, equal-order interpolation spaces can be used for all the unknowns. Finally, this formulation leads to a coupled linear system; this monolithic approach is effective, since the coupling can be dealt by effective preconditioning and iterative solvers.