Given an initial density $\rho_0$ on $\Omega\subset \mathbb{R}^d$, it is well known that the Heat Equation
$$ \partial_t \rho(t)=\Delta\rho(t),~~~~~~~\rho(0)=\rho_0, ~~~~~(1)$$
dissipates the entropy $H$, defined (for a density $f$) as $H(f):=\int f\log f $, i.e $H(\rho(t))$ is decreasing as $t$ increases.
The question is : Suppose you solve $(1)$ via finite elements or finite difference, would this approximate solution also dissipate entropy (at every discrete time point)?