# A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations

## Abstract

We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman–Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman–Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in $$\mathrm {L}^4$$. As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable $$\mathrm {H}(\mathrm {div})$$-type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram and Banach–Nečas–Babuška theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart–Thomas spaces of order $$k\ge 0$$ for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree $$\le k + 1$$ are employed for the velocity, and piecewise polynomials of degree $$\le k$$ for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.

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Correspondence to Sergio Caucao.

This work was partially supported by CONICYT-Chile through the Project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal, the Project PAI77190084 of the PAI Program: Convocatoria Nacional Subvención a la Instalación en la Academia, and Fondecyt Project 11121347; by Centro de Investigación en Ingeniería Matemática ($$\hbox {CI}^2$$MA), Universidad de Concepción; and by Universidad del Bío-Bío through DIUBB Project 151408 GI/VC.