## ABSTRACT

In the previous chapter, we have considered polynomials that are linear in the tetrahedron, and used them to form the matrix that linearizes a nonlinear operator deﬁned on RN or CN (the set of real-valued or complex-valued grid functions). In this chapter, we extend this framework to the more diﬃcult case of functions that, in each tetrahedron in the mesh, can be viewed as a polynomial of degree ﬁve. Furthermore, the functions are rather smooth: they are continuous in the entire mesh and have continuous gradients across edges in the entire mesh. They are therefore suitable to help extend a given grid function deﬁned on the individual nodes into a smooth function deﬁned in the entire mesh. This smooth extension is called the spline of the original grid function [24] [32].