Second-Order Nédélec Curl-Conforming Hexahedral Element for Computational Electromagnetics

We follow a systematic approach to obtain mixed-order curl-conforming basis functions for the hexahedron that are compatible with basis functions for tetrahedra and triangular prisms previously published. The approach is mathematically sound since we obtain the functions as the dual basis with respect to properly discretized Nédélec degrees of freedom. Well-conditioned bases without the need for added orthogonalization procedures are obtained. We provide simple closed-form expressions for second-order basis functions in a reference hexahedron in terms of integer coefficients and monomials. The expressions are ready to use as long as the appropriate geometric mappings are made. We apply the Method of Manufactured Solutions (MMS) to a finite-element double curl vector wave formulation for verification purposes; specifically, we conduct a study of the non-symmetrical structure of the corresponding tensor product finite-element space. We also solve generalized eigenvalue problems for well-known cavities. We provide the open-source code for generating the coefficients, evaluating the basis functions, and computing the finite-element matrices involved in some of the numerical solutions shown in the article.