Polynomial extension operators. Part III

In this concluding part of a series of papers on tetrahedral polynomial extension operators, the existence of a polynomial extension operator in the Sobolev space $\boldsymbol{H}({div})$ is proven constructively. Specifically, on any tetrahedron $K$, given a function $w$ on the boundary $\partial K$ that is a polynomial on each face, the extension operator applied to $w$ gives a vector function whose components are polynomials of at most the same degree in the tetrahedron. The vector function is an extension in the sense that the trace of its normal component on the boundary $\partial K$ coincides with $w$. Furthermore, the extension operator is continuous from $H^{-1/2}(\partial K)$ into $\boldsymbol{H}({div},K)$. The main application of this result and the results of this series of papers is the existence of commuting projectors with good $hp$-approximation properties.