Interpolatory Curl-Conforming Vector Bases for Pyramid Cells

We have recently shown that hierarchical higher-order complete curl-conforming and divergence-conforming bases for pyramids can be obtained by multiplying the lowest-order basis functions by hierarchical scalar multipliers defined by Jacobi polynomials. This paper extends this technique and builds curl-conforming interpolatory bases for pyramids by replacing the hierarchical polynomials with appropriate combinations of interpolatory polynomials of Silvester. Our curl-conforming bases for the pyramid are tangentially continuous with those of adjacent differently shaped cells of the same order and type (i.e., hierarchical or interpolatory) available for years in the literature. This allows numerical electromagnetic solvers using zero-order vector basis functions to be transformed into higher order solvers that work with hybrid meshes simply by adding a few routines to compute the multiplicative polynomials and their first derivatives. Hierarchical bases, including ours of previous papers, are in general more convenient than interpolatory ones for using p-adaptive techniques, while the interpolatory bases such as those shown here are more easily implemented because the recurrence relations of Silvester polynomials are much simpler than those associated with hierarchical multipliers. Numerical results that verify the correctness of our new bases are also reported.