Jessika Camaño, Carlos Garcia, Ricardo Oyarzúa:
Analysis of a new mixed-FEM for stationary incompressible magneto-hydrodynamics
In this paper we propose and analyze a new mixed finite element method for a stationary magnetohydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax-Milgram lemma, the Banach-Necas-Babuska and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis for the discrete scheme can be carried out by means of a quasi-uniformity assumption of the mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimally convergent. Finally, some numerical results illustrating the good performance of the method are provided.