Interpolation-operator orthogonal, hierarchical H¹-conforming basis functions for tetrahedral finite elements

A new $H^{1}$-conforming finite-element basis is proposed. The basis functions are hierarchical in terms of the polynomial order and feature symmetries in terms of barycentric coordinates. The corresponding element matrices exhibit high levels of sparsity and moderate condition numbers in the higher-order case. Moreover, the basis functions are pairwise orthogonal with respect to the finite-element interpolation operator, which is given in explicit form. Thus, the basis possesses not only hierarchical but also interpolatory properties. The resulting interpolation scheme is efficient because it requires the evaluation of only a single scalar product per basis function. An example is presented, where the proposed interpolation scheme is used to construct inhomogeneous Dirichlet boundary conditions. The numerical results confirm that interpolation yields the same asyptotic rate of convergence as $L^{2}$ approximation, which is much more expensive.