  We consider the approximation of weakly T-coercive operators. The main
property to ensure the convergence thereof is the regularity of the
approximation (in the vocabulary of discrete approximation schemes). In a
previous work the existence of discrete operators $T_n$ which converge to $T$
in a discrete norm was shown to be sufficient to obtain regularity. Although
this framework proved usefull for many applications for some instances the
former assumption is too strong. Thus in the present article we report a weaker
criterium for which the discrete operators $T_n$ only have to converge
point-wise, but in addition a weak T-coercivity condition has to be satisfied
on the discrete level. We apply the new framework to prove the convergence of
certain $H^1$-conforming finite element discretizations of the damped
time-harmonic Galbrun's equation, which is used to model the oscillations of
stars. A main ingredient in the latter analysis is the uniformly stable
invertibility of the divergence operator on certain spaces, which is related to
the topic of divergence free elements for the Stokes equation.
