  A general theory for obtaining anisotropic interpolation error estimates for
macro-element interpolation is developed revealing general construction
principles. We apply this theory to interpolation operators on a macro type of
biquadratic $C^1$ finite elements on rectangle grids which can be viewed as a
rectangular version of the $C^1$ Powell-Sabin element. This theory also shows
how interpolation on the Bogner-Fox-Schmidt finite element space (or higher
order generalizations) can be analyzed in a unified framework. Moreover we
discuss a modification of Scott-Zhang type giving optimal error estimates under
the regularity required without imposing quasi uniformity on the family of
macro-element meshes used. We introduce and analyze an anisotropic
macro-element interpolation operator, which is the tensor product of
one-dimensional $C^1-P_2$ macro interpolation and $P_2$ Lagrange interpolation.
These results are used to approximate the solution of a singularly perturbed
reaction-diffusion problem on a Shishkin mesh that features highly anisotropic
elements. Hereby we obtain an approximation whose normal derivative is
continuous along certain edges of the mesh, enabling a more sophisticated
analysis of a continuous interior penalty method in another paper.
