Mixed finite element approximation of incompressible MHD problems based on weighted regularization Original Research Article

We introduce and analyze a new mixed finite element method for the numerical approximation of stationary incompressible magneto-hydrodynamics (MHD) problems in polygonal and polyhedral domains. The method is based on standard inf–sup stable elements for the discretization of the hydrodynamic unknowns and on nodal elements for the discretization of the magnetic variables. In order to achieve convergence in non-convex domains, the magnetic bilinear form is suitably modified using the weighted regularization technique recently developed in [Numer. Math. 93 (2002) 239]. We first discuss the well-posedness of this approach and establish a novel existence and uniqueness result for non-linear MHD problems with small data. We then derive quasi-optimal error bounds for the proposed finite element method and show the convergence of the approximate solutions in non-convex domains. The theoretical results are confirmed in a series of numerical experiments for a linear two-dimensional Oseen-type MHD problem, demonstrating that weighted regularization is indispensable for the resolution of the strongest magnetic singularities.