## Preprint 2023-08

## Lady Angelo, Jessika Camaño, Sergio Caucao:

### A five-field mixed formulation for stationary magnetohydrodynamic flows in porous media

### Abstract:

We introduce and analyze a new mixed variational formulation for a stationary magnetohydrodynamic flows in porous media problem, whose governing equations are given by the steady Brinkman– Forchheimer equations coupled with the Maxwell equations. Besides the velocity, magnetic field and a Lagrange multiplier asssociated to the divergence-free condition of the magnetic field, a con- venient translation of the velocity gradient and the pseudostress tensor are introduced as further unknowns. As a consequence, we obtain a five-field Banach spaces-based mixed variational formu- lation, where the aforementioned variables are the main unknowns of the system. The resulting mixed scheme is then written equivalently as a fixed-point equation, so that the well-known Ba- nach theorem, combined with classical results on nonlinear monotone operators and a sufficiently small data assumption, are applied to prove the unique solvability of the continuous and discrete systems. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. The finite element discretization involves Raviart–Thomas elements of order k ≥ 0 for the pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity and the translation of the velocity gradient, N ́ed ́elec elements of degree k for the magnetic field and Lagrange elements of degree k + 1 for the associated Lagrange multiplier. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.

This preprint gave rise to the following definitive publication(s):

**Lady ANGELO, Jessika CAMAñO, Sergio CAUCAO: ***A five-field mixed formulation for stationary magnetohydrodynamic flows in porous media*. Computer Methods in Applied Mechanics and Engineering, vol. 414, Art. Num. 116158, (2023).