Second-order N�d�lec tetrahedral element for computational electromagnetics

Abstract

The practical implementation of the second-order tetrahedral version of Nédélec's first family of curl-conforming elements [1] (Nédélec. Numerische Mathematik 1980; 35:315–341) is presented. Following the definition of the element given by Nédélec, the second-order vectorial basis functions of the element are deduced. The element thus obtained exhibits some important differences with respect to other second-order curl-conforming elements which have appeared in the literature. For example, the degrees of freedom associated to the faces of the tetrahedra are defined in terms of the surface integral of the tangential component of the field over the faces of the tetrahedra. The differences in the definition of the degrees of freedom lead to a different location of the nodes on the tetrahedron boundary, and also to a different set of basis functions. The basis functions thus obtained have all the same polynomial order than the order of the element, i.e., quadratic for the particular element presented here, and they lead to better conditioned FEM matrices than other second-order curl-conforming elements appeared in the literature. In order to analyse the features of the second-order tetrahedral element presented in this paper, it is used for the discretization of the double-curl vector formulation in terms of the electric or magnetic field, for the computation of the resonance modes of inhomogeneous and arbitrarily shaped three-dimensional cavities. Numerical results are given. A comparison of the rate of convergence achieved with the second-order element with respect to the first-order element is presented. Copyright © 2000 John Wiley & Sons, Ltd.