Stable Lifting of Polynomial Traces on Triangles

Abstract

We construct a right inverse of the trace operator $u \mapsto (u|_{\partial T}, \partial_n u|_{\partial T})$ on the reference triangle $T$ that maps suitable piecewise polynomial data on $\partial T$ into polynomials of the same degree and is bounded in all $W^{s, q}(T)$ norms with $1 < q <\infty$ and $s \geq 2$. The analysis relies on new stability estimates for three classes of single edge operators. We then generalize the construction for $m$th-order normal derivatives, $m \in \mathbb{N}_0$.