Dr. Caucao-Paillán, Sergio

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.

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 associated 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 for the pseudostress tensor, discontinuous piecewise polynomial elements of degree for the velocity and the translation of the velocity gradient, Nédélec elements of degree for the magnetic field and Lagrange elements of degree 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.

In this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and suitable Helmholtz decompositions, we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed finite element scheme on arbitrary (convex or non-convex) polygonal and polyhedral regions. On the other hand, inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove the efficiency of the proposed a posteriori error indicator. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.