  We devise variants of classical nonconforming methods for symmetric elliptic
problems. These variants differ from the original ones only by transforming
discrete test functions into conforming functions before applying the load
functional. We derive and discuss conditions on these transformations implying
that the ensuing method is quasi-optimal and that its quasi-optimality constant
coincides with its stability constant. As applications, we consider the
approximation of the Poisson problem with Crouzeix-Raviart elements and higher
order counterparts and the approximation of the biharmonic problem with Morley
elements. In each case, we construct a computationally feasible transformation
and obtain a quasi-optimal method with respect to the piecewise energy norm on
a shape regular mesh.
