Smooth Bezier Surfaces over Arbitrary Quadrilateral Meshes

We solve the following problem: given a polynomial of order $n$ and the corresponding $B\acute{e}zier$ 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.