Preprint 2024-11
Lady Angelo, Jessika Camaño, Sergio Caucao:
A skew-symmetric-based mixed FEM for stationary MHD flows in highly porous media
Abstract:
We propose and analyze a new mixed variational formulation for the coupling of the convective Brinkman--Forchheimer and Maxwell equations for stationary magnetohydrodynamic flows in highly porous media. Besides the velocity, magnetic field, and a Lagrange multiplier associated with the divergence-free condition of the magnetic field, our approach introduces a convenient translation of the velocity gradient and the pseudostress tensor as additional unknowns. Consequently, we obtain a five-field mixed variational formulation within a Banach space framework, where the aforementioned variables are the main unknowns of the system, exploiting the skew-symmetric property of one of the involved operators. The resulting mixed scheme is then equivalently written as a fixed-point equation, allowing the application of the well-known Banach theorem, combined with classical results on nonlinear monotone operators and a sufficiently small data assumption, 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 the 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 velocity gradient translation, Nédélec elements of degree k for the magnetic field, and continuous piecewise polynomial elements of degree k + 1 for the Lagrange multiplier. We establish stability, convergence, and optimal a priori error estimates for the corresponding Galerkin scheme. Theoretical results are illustrated by numerical tests.
