Trivariate Cr Polynomial Macroelements

Trivariate C^r macroelements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1,2, these spaces reduce to well-known macroelement spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining sets. We also show that the spaces approximate smooth functions to optimal order.