Dra. Camaño-Valenzuela, Jessika

2026, Angelo, Lady, Dra. Camaño-Valenzuela, Jessika, 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.

2016, Dra. Camaño-Valenzuela, Jessika, Gatica, Gabriel, Oyarzúa, Ricardo, Tierra, Giordano

A new mixed variational formulation for the Navier--Stokes equations with constant density and variable viscosity depending nonlinearly on the gradient of velocity, is proposed and analyzed here. Our approach employs a technique previously applied to the stationary Boussinesq problem and to the Navier--Stokes equations with constant viscosity, which consists firstly of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, and the pressure. Next, by using an equivalent statement suggested by the incompressibility condition, the pressure is eliminated, and in order to handle the nonlinear viscosity, the gradient of velocity is incorporated as an auxiliary unknown. Furthermore, since the convective term forces the velocity to live in a smaller space than usual, we overcome this difficulty by augmenting the variational formulation with suitable Galerkin-type terms arising from the constitutive and equilibrium equations, the aforementioned relation defining the additional unknown, and the Dirichlet boundary condition. The resulting augmented scheme is then written equivalently as a fixed point equation, and hence the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. No discrete inf-sup conditions are required for the well-posedness of the Galerkin scheme, and hence arbitrary finite element subspaces of the respective continuous spaces can be utilized. In particular, given an integer K >_0, piecewise polynomials of degree _< K for the gradient of velocity, Raviart--Thomas spaces of order K for the pseudostress, and continuous piecewise polynomials of degree _< K + 1 for the velocity, constitute feasible choices. Finally, optimal a priori error estimates are derived, and several numerical results illustrating the good performance of the augmented mixed finite element method and confirming the theoretical rates of convergence are reported.