The four-noded bi-linear rectangle element, which sometimes goes under the name Melosh element, is non-conforming unless the element sides are aligned. Out of curiosity I have implemented this element for two-dimensional elasticity problems. The figure shows a mesh with three such elements. The element sides are not all aligned, so the displacements are expected to be discontinuous.
The picture below illustrates the displacement-field for cantilever-type boundary conditions (left side is fixed, vertical forces on the right).
Although there is a gap between element 1 and 2 and an overlap between element 2 and 3, the overall field looks fine, and comparing with results for isoparametric quadrilaterals shows very similar maximum displacement for example.
This result, assuming there's no mistake in the implementation, suggests that it might be possible to give some form of convergence proof for the Melosh element also in case of non-aligned meshes (the unphysical gaps and overlaps seem to go zero as I refine the mesh). The question is thus whether such a proof can be found somewhere?
I should add that my question is motivated by pure curiosity; I'm not aware of a practical application for this element.