Monday, June 24th at 4:20 pm. John Von Neumann Seminar Room, Beauchef 851, 7th floor.

Speaker: Jessika Camaño Valenzuela from the Universidad Católica de la Santísima Concepcion.

Title: «Analysis of a mixed FEM 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 convenient 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 formulation, 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 Banach 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 $k\geq 0$ for the pseudostress tensor, discontinuous piecewise polynomial elements of degree $k$ for the velocity and the translation of the velocity gra\-dient, 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 {\it a priori} error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.