Hierarchical vector bases for 3D problems

Hierarchical curl- and divergence-conforming vector bases for two- and three-dimensional cells were derived and discussed in a four paper series published by Graglia, Peterson et al. in the IEEE Trans. Antennas Propagation ([1] vol. 59, pp. 950–959, 2011; [2] vol. 59, pp. 2766–2773, 2011; [3] vol. 60, pp. 3314–3324, 2012; [4] vol. 60, pp. 5215–5227, 2012). Relatively few results for 3D problems are reported in these papers due to an emphasis on (a) the procedures used to derive the hierarchical vector bases and (b) the linear independence of the basis functions. Conversely, results for two-dimensional problems obtained using hierarchical vector bases functions on triangular and quadrilateral cells are reported in two short papers by Peterson and Graglia ([5] “Scale factors and matrix conditioning associated with triangular-cell hierarchical vector basis functions,” IEEE AWPL, vol. 9, pp. 40–43, 2010. [6] “Evaluation of hierarchical vector basis functions for quadrilateral cells” IEEE Trans. Mag., vol. 47, no. 5, pp. 1190–1193, 2011).