  The temperature in natural convection problems is, under mild data
assumptions, uniformly bounded in time. This property has not yet been proven
for the standard finite element method (FEM) approximation of natural
convection problems with nonhomogeneous partitioned Dirichlet boundary
conditions, e.g., the differentially heated vertical wall and
Rayleigh-B\'{e}nard problems. For these problems, only stability in time,
allowing for possible exponential growth of $\| T^{n}_{h} \| $, has been proven
using Gronwall's inequality. Herein, we prove that the temperature
approximation can grow at most linearly in time provided that the first mesh
line in the finite element mesh is within $\mathcal{O} (Ra^{-1})$ of the
nonhomogeneous Dirichlet boundary.
