Lp-Based analysis of a mixed FEM with exactly divergence-free magnetic-field for stationary MHD

In this work, we develop an Lp-based analysis of a mixed finite element method for the stationary incompressible magnetohydrodynamics (MHD) problem, providing an exactly divergence-free approximation of the magnetic field together with a direct approximation of the electric field. The method is based on introducing the electric field as an additional unknown, leading to a mixed formulation in which the primary magnetic variables are the electric and magnetic fields, together with a Lagrange multiplier enforcing the divergence-free constraint on the magnetic field, while the hydrodynamic variables are the velocity and pressure. The associated Galerkin scheme is constructed by employing lowest-order Nédélec and Raviart–Thomas elements for the electric and magnetic fields, respectively, discontinuous piecewise constants for the Lagrange multiplier, and any inf–sup stable finite element pair for the velocity–pressure approximation, such as the Mini element. The well-posedness of the continuous and discrete problems is established within this Lp framework, by means of the Banach–Nečas–Babuška theorem and the Banach fixed-point theorem, under a sufficiently small-data assumption and, for the discrete case, quasi-uniformity of the mesh. Finally, we derive a Céa-type estimate and obtain the corresponding theoretical convergence rates under a minimal regularity assumption.