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.

In this paper we present and analyze a new mixed finite element method for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations with varying porosity. Our approach is based on the introduction of the pseudostress and the gradient of the porosity times the velocity, as further unknowns. As a consequence, we obtain a mixed variational formulation within a Banach spaces framework, with the velocity and the aforementioned tensors as the only unknowns. The pressure, the velocity gradient, the vorticity, and the shear stress can be computed afterwards via postprocessing formulae. A fixed-point strategy, along with monotone operators theory and the classical Banach theorem, are employed to prove the well-posedness of the continuous and discrete systems. Specific finite element subspaces satisfying the required discrete stability condition are defined, and optimal a priori error estimates are derived. Finally, several numerical examples illustrating the performance and flexibility of the method and confirming the theoretical rates of convergence, are reported.

We propose and analyze a new mixed finite element method for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations. In addition to the original fluid variables, the pseudostress is introduced as an auxiliary unknown, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the aforementioned tensor and the velocity. As a consequence, we obtain a mixed variational formulation consisting of a nonlinear perturbation of, in turn, a perturbed saddle point problem in a Banach spaces framework. In this way, and differently from the techniques previously developed for this model, no augmentation procedure needs to be incorporated into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that recently established solvability results for perturbed saddle-point problems in Banach spaces, along with the well-known Banach–Nečas–Babuška and Banach theorems, are applied to prove the well-posedness of the continuous and discrete systems. The finite element discretization involves Raviart–Thomas elements of order for the pseudostress tensor and discontinuous piecewise polynomial elements of degree for the velocity. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method. In particular, the case of flow through a 2D porous media with fracture networks is considered.