  Accurately predicting the demographics of dark matter (DM) substructure is of
paramount importance for many fields of astrophysics, including gravitational
lensing, galaxy evolution, halo occupation modeling, and constraining the
nature of dark matter. Because of its strongly non-linear nature, DM
substructure is typically modeled using N-body simulations, which reveal that
large fractions of DM subhaloes undergo complete disruption. In this paper we
use both analytical estimates and idealized numerical simulations to
investigate whether this disruption is mainly physical, due to tidal heating
and stripping, or numerical (i.e., artificial). We show that, contrary to naive
expectation, subhaloes that experience a tidal shock $\Delta E$ that exceeds
the subhalo's binding energy, $|E_{\rm b}|$, do not undergo disruption, even
when $\Delta E/|E_{\rm b}|$ is as large as 100. Along the same line, and
contrary to existing claims in the literature, instantaneously stripping matter
from the outskirts of a DM subhalo also does not result in its complete
disruption, even when the instantaneous remnant has positive binding energy. In
addition, we show that tidal heating due to high-speed (impulsive) encounters
with other subhaloes (`harassment'), is negligible compared to the tidal
effects due to the host halo. Hence, we conclude that, in the absence of
baryonic processes, the complete, physical disruption of CDM substructure is
extremely rare, and that most disruption in numerical simulations therefore
must be artificial. We discuss various processes that have been associated with
numerical overmerging, and conclude that inadequate force-softening is the most
likely culprit.
