In this paper we propose and analyze, utilizing mainly tools and abstract results from Banach spaces rather than from Hilbert ones, a new fully-mixed finite element method for the stationary Boussinesq problem with temperature-dependent viscosity. More precisely, following an idea that has already been applied to the Navier–Stokes equations and to the fluid part only of our model of interest, we first incorporate the velocity gradient and the associated Bernoulli stress tensor as auxiliary unknowns. Additionally, and differently from earlier works in which either the primal or the classical dual-mixed method is employed for the heat equation, we consider here an analogue of the approach for the fluid, which consists of introducing as further variables the gradient of temperature and a vector version of the Bernoulli tensor. The resulting mixed variational formulation, which involves the aforementioned four unknowns together with the original variables given by the velocity and temperature of the fluid, is then reformulated as a fixed point equation. Next, we utilize the well-known Banach and Brouwer theorems, combined with the application of the Babuška-Brezzi theory to each independent equation, to prove, under suitable small data assumptions, the existence of a unique solution to the continuous scheme, and the existence of solution to the associated Galerkin system for a feasible choice of the corresponding finite element subspaces. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the performance of the fully-mixed scheme and confirming the theoretical rates of convergence.
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.
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