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.
This preprint gave rise to the following definitive publication(s):
Jessika CAMAñO, Carlos GARCIA, Ricardo OYARZúA: Analysis of a new mixed-FEM for stationary incompressible magneto-hydrodynamics. Computers & Mathematics with Applications, vol. 127, pp. 65-79, (2022).