In science and engineering, one is often required to provide the value f̃_i reached by the quantity f (s) at s_i out of the knowledge of n samples f_k =f (s_k) obtained for k = 1,2,... , n, with s_k≠s_i for all k. This is an interpolation problem if s_i is located within the same range defined by the n isolated points s_k and an extrapolation problem whenever s_i is outside of this range. The interpolation problem is a particular case of the more general curve fitting (or approximation) problem where the function f̃(s) re-constructed out of the discrete set of known data points goes exactly through these data points. The task of interpolating a function is closely related to the problem of representing an unknown function f(s) within a numerical solution procedure.

In this chapter, we consider the representation of vector functions (often referred to as “vector fields”) with low-order (constant and linear) polynomial basis functions on simple cells, such as triangles or quadrilateral cells in two dimensions or tetrahedrons and bricks in three dimensions. As will soon be apparent, there are multiple ways of defining vector basis functions, and therefore the approach requires some consideration. The proper representation of a function depends on what will be done with it-do we need to compute the curl of the function, for instance? If so, the representation might be different than if we need to compute the divergence of the function. We use the term curl conforming to denote the space of vector functions that maintain first-order tangential-vector continuity throughout the domain and can be differentiated via the curl operation, without producing unbounded or generalized functions (Dirac delta functions) in the process. The term divergence conforming is used to denote the complementary space of vector functions that maintain first-order normal-vector continuity throughout the domain and can therefore be differentiated via the divergence operation. (First order or C₀ continuity is continuity of the function itself, but not necessarily continuity of its first derivatives.) The simple low-order polynomial vector basis functions in widespread use are either curl conforming or divergence conforming; seldom we will use functions that maintain complete continuity and belong to both the curl-conforming and divergence-conforming spaces, although it is possible to define such functions.

Hierarchical scalar bases and hierarchical vector bases that span the Nédélec mixed-order spaces have been proposed for the most common two- and three-dimensional cells. These bases maintain appropriate continuity in a mesh containing multiple cell shapes. In particular, the vector bases maintain the tangential or normal continuity in the curl- or divergence-conforming case, respectively. The use of orthogonal scalar polynomials to systematically construct the basis functions is believed to offer a simpler approach for enhancing their linear independence as their polynomial order increases than the partial orthogonalization of the final vector functions. The process by which the basis functions were obtained is described in detail. Numerical results for the matrix condition numbers suggest that these hierarchical bases exhibit reasonable linear independence for large orders.

Singular scalar and vector basis functions have been described and used to analyze two-dimensional cavity resonators and waveguide structures. The singular functions are designed to provide non-analytic DoFs to efficiently model fields near sharp corners and edges. These functions are hierarchical, in the sense that a representation of order p contains the members of order p - 1. These functions are also additive, which provide maximum flexibility at the cost of somewhat higher matrix CNs. The philosophy behind these functions is discussed and appropriate combinations of singular and non-singular functions are proposed. The possibility of numerically constructing these singular basis functions should facilitate their use in problems containing field singularities arising from very general situations such as wedge tips made of penetrable material. Several types of singular vector bases have been proposed in the recent literature for application to 3D problems. While this topic is far from mature at the time of this writing, we briefly report the performance of the functions proposed in [30,45] for illustration.