Bernstein-Bezier Bases for Tetrahedral Finite Elements

We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2 -conforming finite elements of arbitrary order on a tetrahedron. The basis functions are expressed in terms of Bernstein polynomials and augment the natural H1 -conforming Bernstein basis. The basis functions respect the differential operators, namely, the gradients of the high-order H1 -conforming Bernstein-Bezier basis functions form part of the H(curl)-conforming basis, and the curl of the high-order, non-gradients H(curl)-conforming basis functions form part of the H(div)-conforming basis, and the divergence of the high-order, non-curl H(div)-conforming basis functions form part of the L2-conforming basis. Procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for (piece-wise) constant coefficients on affine tetrahedra. Numerical results are presented to illustrate the use of the basis to approximate representative problems.