The purpose of this article is to study the convergence of a low order finite element approximation for a natural convection problem. We prove that the discretization based on P1 polynomials for every variable (velocity, pressure and temperature) is well-posed if used with a penalty term in the divergence equation, to compensate the loss of an inf-sup condition. With mild assumptions on the pressure regularity, we recover convergence for the Navier-Stokes-Boussinesq system, provided the penalty term is chosen in accordance with the mesh size. We express conditions to obtain optimal order of convergence. We illustrate theoretical convergence results with extensive examples. The computational cost that can be saved by this approach is also assessed.

翻译：本条的目的是研究自然对流问题低定序有限元素近似值的趋同性。我们证明,每个变量(速度、压力和温度)基于P1多元值的离散性,如果在差异方程式中以惩罚术语使用的话,就完全可以用来弥补每个变量(速度、压力和温度)的分解性,以补偿一个内溢条件的损失。如果对压力的规律性有轻微的假设,我们就可以恢复纳维埃-斯托克斯-波西奈斯克系统的趋同性,只要根据网目大小选择惩罚术语。我们提出达到最佳趋同顺序的条件。我们用大量的例子来说明理论上的趋同性结果。我们还可以评估通过这种方法可以节省的计算成本。