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. 2020 Mathematics Subject Classification. 65N30, 65N12, 65N15, 35Q79, 80A20, 76D05, 76R10.