The paper presents results on piecewise polynomial approximations of tensor product type in Sobolev-Slobodecki spaces by various interpolation and projection techniques, on error estimates for quadrature rules and projection operators based on hierarchical bases, and on inverse inequalities.

Hierarchical bases of arbitrary order for (div)-conforming triangular and tetrahedral elements are constructed with the goal of improving the conditioning of the mass 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, and is better than a corresponding basis in the dissertation by Sabine Zaglmayr [High Order Finite Element Methods for Electromagnetic Field Computation, Johannes Kepler Universität, Linz, 2006]. The sparsity of the mass matrices from the newly constructed basis 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 by Zaglmayr for approximations up to order four. For the tetrahedral element, it is identified numerically that the conditioning is acceptable only up to the approximation of order 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 relatively sparser.

We construct a well-conditioned hierarchical basis for triangular H(curl)-conforming elements with selected orthogonality. The basis functions are grouped into edge and interior functions, and the later is further grouped into normal and bubble functions. In our construction, the trace of the edge shape functions are orthonormal on the associated edge. The interior normal functions, which are perpendicular to an edge, and the bubble functions are both orthonormal among themselves over the reference element. The construction is made possible with classic orthogonal polynomials, viz., Legendre and Jacobi polynomials. For both the mass matrix and the quasi-stiffness matrix, better conditioning of the new basis is shown by a comparison with the basis previously proposed by Ainsworth and Coyle [Comput. Methods. Appl. Mech. Engrg., 190 (2001), 6709-6733].