An interpolatory spectral element method using curl-conforming vector basis functions on tetrahedra

The Nedelec basis functions are commonly used in the finite element solution of electromagnetic field problems. Higher-order Nedelec-type of basis functions are constructed to be either hierarchical or interpolatory. Due to a lack of an explicit expression for interpolation functions through an arbitrary set of nodes on a tetrahedron, equispaced nodes, for which explicit expressions do exist, are commonly used in the finite element formulation. The poor interpolation properties of those functions make the finite element matrices poorly conditioned for higher orders. Here we use the Vandermonde matrix to express the interpolatory vector basis on an arbitrary set of nodes in terms of a hierarchical basis utilizing orthonormal polynomials on a tetrahedron. In an effort to increase efficiency, integration and differentiation operations are developed using matrix-matrix multiplications. Numerical results are given which verify the efficacy of the approach.