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.

翻译：本文基于Banach空间上的全混合虚拟元方法（mixed-VEM），针对具有温度依赖参数的稳态二维Boussinesq方程，以伪应力、涡量、速度、伪热矢量和温度场为变量，提出并分析了一种新的混合虚拟元方法。利用不动点策略、数据的小量假设以及解的正则性条件，分析了连续问题的适定性。对于上述变量的离散化，采用了$\mathbb{H}(\mathbf{div}_{6/5})$——和$\mathbf{H}(\mathrm{div}_{6/5})$——共形虚拟元技术。所提出的方案被改写为一个等价的不动点算子方程，从而证明了其存在性和稳定性估计。此外，通过Céa估计和所有变量在其自然范数下的适当数据假设，建立了先验收敛性分析，展示了最优收敛速率。最后，通过若干数值算例验证了所提方法的有效性。