On the Validation of Curl-Conforming Higher-Order Basis Functions using the Method of Manufactured Solutions

The authors have developed in [1–3] a family of mixed-order curl-conforming basis functions for the three main shapes used in the Finite Element Method (FEM): tetrahedra, triangular prisms, and hexahedra. We have followed a systematic approach that can be summarized in the following steps: first, we define a reference hexahedron where we obtain the basis functions as the dual basis regarding a set of unisolvent functionals (the so-called degrees of freedom in [4, 5]) acting on the Nédélec space, [4, 5], that need to be discretized. Then, we compute the basis functions in the real element using covariants and contravariants Piola mappings to ensure the curl-conformity property between elements. to obtain the functions as the dual basis with respect to properly discretized Nédélec degrees of freedom, following [4, 5], and yielding simple closed-form expressions for the case of second-order basis functions and reference elements.