Mixed virtual element approximation for the five-field formulation of the steady Boussinesq problem with temperature-dependent parameters

Abstract: In this work, we develop recent research on the fully mixed virtual element method (mixed-VEM) based on the Banach space for the stationary Boussinesq equation to suggest and analyze a new mixed-VEM for the stationary two-dimensional Boussinesq equation with temperature-dependent parameters in terms of the pseudostress, vorticity, velocity, pseudoheat vector and temperature fields. The well-posedness of the continuous formulation is analyzed utilizing a fixed-point strategy, a smallness assumption on the data, and some additional regularities on the solution. The discretization for the mentioned variables is based on the coupling $\mathbb{H}(\mathbf{div}{6/5})$ -- and $\mathbf{H}(\mathrm{div}{6/5})$ -- conforming virtual element techniques. The proposed scheme is rewritten as an equivalent fixed point operator equation, so that its existence and stability estimates have been proven. In addition, an a priori convergence analysis is established by utilizing the C\'ea 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.