A guided tour of interpolatory vector basis functions

Fully interpolatory higher-order vector basis functions of the Nedelec variety were defined by Graglia et al. (see IEEE Trans. Antennas Propagat., vol.45, 1997) in a unified manner for the most common element shapes. These functions include the curl-conforming type typically used to solve the vector Helmholtz (curl-curl) equation and the divergence-conforming functions usually employed to deal with the electric-field integral equation. Both classes of vector basis function are conveniently expressed in terms of Silvester-Lagrange polynomials similar in form to those used to define scalar interpolation functions in multiple dimensions. The construction process offers a procedure for generating vector basis functions that is as systematic as the almost universal description of scalar interpolation functions, the widespread adoption of which is due in large part to the contributions and publications of Silvester (1990). The authors suggest how the Silvester-Lagrange polynomials are used to construct higher-order elements for triangles. They review the construction of these functions, and use pictures to illustrate the functions, their interpolation points within common element shapes, and the indexing scheme identifying a specific basis function.