  We consider the approximation properties of finite element spaces on
quadrilateral meshes. The finite element spaces are constructed starting with a
given finite dimensional space of functions on a square reference element,
which is then transformed to a space of functions on each convex quadrilateral
element via a bilinear isomorphism of the square onto the element. It is known
that for affine isomorphisms, a necessary and sufficient condition for
approximation of order r+1 in L2 and order r in H1 is that the given space of
functions on the reference element contain all polynomial functions of total
degree at most r. In the case of bilinear isomorphisms, it is known that the
same estimates hold if the function space contains all polynomial functions of
separate degree r. We show, by means of a counterexample, that this latter
condition is also necessary. As applications we demonstrate degradation of the
convergence order on quadrilateral meshes as compared to rectangular meshes for
serendipity finite elements and for various mixed and nonconforming finite
elements.
