A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system

Abstract

A new fully-mixed formulation is advanced for the stationary Oberbeck-Boussinesq problem when viscosity depends on both temperature and concentration of a solute. Following recent ideas in the context of mixed methods for Boussinesq and Navier-Stokes systems, the velocity gradient and the Bernoulli stress tensor are taken as additional field variables in the momentum and mass equilibrium equations. Similarly, the gradients of temperature and concentration together with a Bernoulli vector are considered as unknowns in the heat and mass transfer equations. Consequently, a dual-mixed approach with Dirichlet data is defined in each sub-system, and the well-known Banach and Brouwer theorems are combined with Babuška-Brezzi's theory in each independent set of equations, yielding the solvability of the continuous and discrete schemes. We show that our development also applies to the case where the equations of thermal energy and solute transport are coupled through cross-diffusion. Appropriate finite element subspaces are specified, and optimal a priori error estimates are derived. Furthermore, a reliable and efficient residual-based a posteriori error estimator is proposed. Several numerical examples illustrate the performance of the fully-mixed scheme and of the adaptive refinement algorithm driven by the error estimator.

A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations

“…Then, similarly to [17] and [19], we combine a fixed-point argument, classical results on nonlinear monotone operators, Babuška-Brezzi's theory in Banach spaces, sufficiently small data assumptions, and the well known Banach fixed-point theorem, to establish existence and uniqueness of solution of both the continuous and discrete formulations. In this regard, and since the formulation for the double-diffusion equations is similar to the ones employed in [17,18], our present analysis certainly makes use of the corresponding results available there. In addition, applying an ad-hoc Strang-type lemma in Banach spaces, we are able to derive the corresponding a priori error estimates.…”

“…Next, in order to derive a new fully-mixed formulation for (2.1)-(2.5), and unlike [13], we do not employ any augmentation procedure and simply proceed as in [17] (see also [18]). More precisely, we now introduce as further unknowns the velocity gradient t, the pseudostress tensor σ, the temperature/concentration gradient t j , and suitable auxiliary variables ρ j depending on t j , u, and φ j , all of which are defined, respectively, by…”

“…In this section we follow [17] and [18] to derive a fully-mixed formulation in a Banach spaces framework for the coupled system given by (2.8). To this end, we first multiply the third equation of (2.8) by a test function v associated with the unknown u, which formally yields…”

“…In order to overcome this, in recent years there has arisen an increasing development on Banach spaces-based mixed finite element methods to solve a wide family of single and coupled nonlinear problems in continuum mechanics (see, e.g. [5], [6], [10], [11], [14], [15], [17], and [18]). This kind of procedures shows two advantages at least: no augmentation is required, and the spaces to which the unknowns belong are the natural ones arising from the application of the Cauchy-Schwarz and Hölder inequalities to the terms resulting from the testing and integration by parts of the equations of the model.…”

“…To this end, we proceed as in [17] and introduce the velocity gradient and pseudostress tensors as auxiliary unknowns, and subsequently eliminate the pressure unknown using the incompressibility condition. In turn, we follow [17,18] and adopt a dual-mixed formulation for the double-difussion equations making use of the temperature/concentration gradients and Bernoulli-type vectors as further unknowns. Then, similarly to [17] and [19], we combine a fixed-point argument, classical results on nonlinear monotone operators, Babuška-Brezzi's theory in Banach spaces, sufficiently small data assumptions, and the well known Banach fixed-point theorem, to establish existence and uniqueness of solution of both the continuous and discrete formulations.…”