There has recently been interest in extending various finite element methods to more arbitrary partitions, particularly unstructured partitions of various polygons. Various methods aimed at this task have arisen, but of particular note, in a paper published in 2016, Floater and Lai produced a method for numerical solution of Poisson equations using polygonal splines, which are extensions of bivariate splines. This work first presents a method for numerical solution of partial differential equations which extends the method of Floater and Lai to solve very general second-order elliptic equations, but can also be used to approximate solutions of some mixed hyperbolic and parabolic equations. Next, this work will address a features common to many polygonal finite elements: a lack of global differentiability. This work provides a construction of $C^1$ local basis functions, particularly over quadrangulations, with some applications to function interpolation and smooth surface construction. The methods used to construct these functions, while computationally difficult, can be extended to higher regularity or to partitions of polygons with more vertices.