Mixed Virtual Element Approximation for the Five-Field Formulation of the Steady Boussinesq Problem with Temperature-Dependent Parameters

In this work, we extend recent research on the fully mixed virtual element method based on Banach spaces for the stationary Boussinesq equation to suggest and analyze a new mixed-virtual element technique for the stationary generalized Boussinesq equations, where viscosity and thermal conductivity are temperature-dependent. Besides the original thermo-fluid variables, the pseudostress and vorticity tensors of the fluid and the pseudoheat vector of the thermal system are introduced as auxiliary unknowns, and the incompressibility condition is then used to remove the pressure, which is later computed using a postprocessing formula. The resulting highly coupled formulation is then written equivalently as uncoupled problems thanks to a fixed-point strategy, so that the classical Banach Theorem, combined with the corresponding Babuška–Brezzi theory, the Banach–Nečas–Babuška Theorem, smallness-of-data assumptions, and a slight higher-regularity assumption for the exact solution, are employed to establish the unique solvability of the continuous formulation. The virtual element discretization of the uncoupled problems is based on the \( \mathbb {H}({\textbf{div}}_{6/5}) \) $\mathbb{H}({\mathbf{div}}_{6/5})$ - and \( \textbf{H}({\textrm{div}}_{6/5}) \) $\mathbf{H}({\text{div}}_{6/5})$ -conforming virtual element techniques. The discrete analysis is conducted in a similar manner by establishing the discrete inf-sup condition and using the discrete versions of the aforementioned theorems to demonstrate the existence of the discrete solution and derive its stability estimates. In addition, a priori error estimates are established by utilizing the Céa estimate and a suitable assumption on data for all variables in their natural norms showing an optimal rate of convergence. Finally, several numerical examples are presented to illustrate the performance of the proposed method.