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

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.

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