  We solve the following problem: given a polynomial of order $n$ and the
corresponding $B\'ezier$ tensor product patches over an unstructured regular
quadrilateral mesh of any valence, find a solution to the $G^{1}1$ or $C^{1}1$
approximation (resp. interpolation) problem ! Constraints defining regularity
conditions across patches have to be satisfied. The resulting number of free
degrees of freedom must be such that for instance the interpolation problem has
a solution. This is similar to studying the minimal determining set (MDS) for a
$C^{1}$ continuity construction. The givenunstructured quadrilateral mesh can
include a cubic boundary curve. The final surface approximation or PDE solution
is obtained by energy methods. We completely solve the problem and show that
there is always a solution for $n\ge 5$ and under some mesh restrictions for
$n=4$. From a practical point of view, the present paper provides a way to
build first order smooth interpolation/approximation and solutions to partial
differential equations for arbitrary structures of quadrilateral meshes.
