Bernstein-Bezier Bases for Tetrahedral Finite Elements

TL;DR

This paper introduces a new set of Bernstein-Bezier basis functions for tetrahedral finite elements that conform to various function spaces and respect differential operators, enabling efficient computation and accurate approximation.

Contribution

It develops a novel Bernstein-Bezier basis for tetrahedral finite elements that conform to H(curl), H(div), and L2 spaces, with efficient computation procedures.

Findings

Basis functions respect differential operators.

Efficient computation of mass and stiffness matrices.

Numerical results demonstrate approximation accuracy.

Abstract

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.