  This article on nonconforming schemes for $m$ harmonic problems
simultaneously treats the Crouzeix-Raviart ($m=1$) and the Morley finite
elements ($m=2$) for the original and for modified right-hand side $F$ in the
dual space $V^*:=H^{-m}(\Omega)$ to the energy space $V:=H^{m}_0(\Omega)$. The
smoother $J:V_{\rm nc} \to V$ in this paper is a companion operator, that is a
linear and bounded right-inverse to the nonconforming interpolation operator
$I_{\rm nc} :V\to V_{\rm nc}$, and modifies the discrete right-hand side
$F_h:=F\circ J \in V_{\rm nc}^*$. The best-approximation property of the
modified scheme from Veeser et al. (2018) is recovered and complemented with an
analysis of the convergence rates in weaker Sobolev norms. Examples with
oscillating data show that the original method may fail to enjoy the
best-approximation property but can also be better than the modified scheme.
The a~posteriori analysis of this paper concerns data oscillations of various
types in a class of right-hand sides $F\in V^*$. The reliable error estimates
involve explicit constants and can be recommended for explicit error control of
the piecewise energy norm. The efficiency follows solely up to data
oscillations and examples illustrate this can be problematic.
