On the Construction of Well-Conditioned Hierarchical Bases for $\mathcal{H}(div)$-Conforming $\mathbb{R}^n$ Simplicial Elements (Communications in Computational Physics, 2013)
Hierarchical bases of arbitrary order for  $\mathcal{H}(div)$-conforming triangular andtetrahedral elements are constructed with the goal of improving the conditioning of themass and stiffness matrices. For the basis with the triangular element, it is found numerically that the conditioning is acceptable up to the approximation of order four, andis better than a corresponding basis in the dissertation by Sabine Zaglmayr [High Order Finite Element Methods for Electromagnetic Field Computation, Johannes KeplerUniversität, Linz, 2006]. The sparsity of the mass matrices from the newly constructedbasis and from the one by Zaglmayr is similar for approximations up to order four.The stiffness matrix with the new basis is much sparser than that with the basis byZaglmayr for approximations up to order four. For the tetrahedral element, it is identified numerically that the conditioning is acceptable only up to the approximation oforder three. Compared with the newly constructed basis for the triangular element,the sparsity of the mass matrices from the basis for the tetrahedral element is relativelysparser.