Acceleration of convergence for the higher-order Whitney element FEM analysis of antennas and electromagnetic scattering

Several higher order generalizations of edge-based Whitney elements can be found in the literature. Their performance with respect to the convergence of the iterative solver exhibits significant deviations. However, we have shown that the majority of such elements actually belong to the same class and are related to each other by an affine transformation. Hence, an optimal, with respect to convergence element can be derived, if the transformation is defined such that the eigenvalues of the element matrix are as clustered as possible. We provide a general explicit formula for the 3-D transformation of degrees of freedom (DoF) and basis functions (BF) and investigate its effects on the convergence rate. The results, based on a 2-D scattering problem and a 3-D patch antenna analysis, demonstrate a remarkable improvement of convergence, especially in the case of severely ill-conditioned matrices that arise due to the presence of different materials in the same problem. We also propose the use of different transformations in different regions and, despite the lack of a clear optimization criterion, an additional improvement is observed, again, in the case of the most ill-conditioned matrices.