Why sub- and super-harmonic? My question might sound stupid, but here we go. Recall that a function $u$ is called subharmonic (superharmonic) if $\Delta u\ge 0 (\le 0)$. 1) Why is the former called subharmonic but not superharmonic, and vice versa for the latter? 2) Also, why the word sub- and super-? I'm curious about the origin of these names.

Sub-harmonic = under harmonic. If two functions $u, h$ have the same boundary values in a domain $\Omega$, and satisfy $\Delta u\ge 0$ and $\Delta h=0$, then $u\le h$ in $\Omega$: the graph of subharmonic function lies under the graph of harmonic function. (This is a consequence of the maximum principle applied to $u-h$.)

For superharmonic $u$ it's the other way around: $u\ge h$.

One can draw a comparison with a convex function $f:\mathbb{R}\to \mathbb{R}$. The graph of such a function on an interval $(a,b)$ lies below the secant line drawn through $(a, f(a))$ and $(b, f(b))$. The parallel subharmonic : harmonic :: convex : affine is often helpful.