In this paper we extend recent results obtained for the Navier–Stokes equations to propose and analyze a new fully mixed virtual element method (mixed-VEM) for the stationary two-dimensional Boussinesq equations appearing in non-isothermal flow phenomena. The model consists of a Navier–Stokes type system, modeling the velocity and the pressure of the fluid, coupled to an advection-diffusion equation for the temperature. The variational formulation is based on the introduction of the additional unknowns given by a modified pseudostress tensor, which depends on the pressure, and the diffusive and convective terms of the fluid, and the pseudoheat vector, which involves the temperature, its gradient, and the velocity. As a consequence of the former, the pressure is eliminated from the system, but computed afterwards via a post-processing formula. In turn, for the Galerkin approximation we follow the approach employed in a previous work introducing for the first time an L spaces-based mixed-VEM for the Navier–Stokes equations, and couple it with a similar mixed-VEM for the convection–diffusion equation modeling the temperature. The solvability analysis of the resulting coupled discrete scheme is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach-Nečas-Babuška theorem, both in subspaces of Banach spaces. The first Strang lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of the pseudostress tensor, the pseudoheat vector, and the post-processed pressure. Finally, several numerical results, illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory, are reported.

中文翻译：

---

基于Banach空间的平稳二维Boussinesq方程的全混合虚元法

在本文中，我们扩展了纳维-斯托克斯方程的最新结果，提出并分析了一种新的完全混合虚拟单元方法（混合 VEM），用于解决非等温流动现象中出现的稳态二维 Boussinesq 方程。该模型由纳维-斯托克斯型系统组成，对流体的速度和压力进行建模，并与温度的平流扩散方程相结合。变分公式基于引入由修正的赝应力张量给出的附加未知数，该张量取决于压力、流体的扩散项和对流项以及赝热矢量，其中涉及温度、其梯度和速度。由于前者，压力从系统中消除，但随后通过后处理公式计算。反过来，对于伽辽金近似，我们遵循先前工作中采用的方法，首次引入用于纳维-斯托克斯方程的基于 L 空间的混合 VEM，并将其与用于对流扩散的类似混合 VEM 耦合方程模拟温度。通过使用适当的定点参数以及 Babuška-Brezzi 理论和 Banach-Nečas-Babuška 定理的离散版本（均在 Banach 空间的子空间中），对所得耦合离散格式进行可解性分析。第一个斯特朗引理用于导出虚拟单元解以及赝应力张量、赝热矢量和后处理压力的完全可计算近似的先验误差估计。最后，报告了几个数值结果，说明了混合 VEM 方案的性能并确认了理论预测的收敛速度。