Stable Lifting of Polynomial Traces on Triangles

We construct a right inverse of the trace operator $u\mapsto (u{\mathrm{\mid }}_{\mathrm{\partial }T},{\mathrm{\partial }}_{n}u{\mathrm{\mid }}_{\mathrm{\partial }T})$ on the reference triangle $T$ that maps suitable piecewise polynomial data on $\mathrm{\partial }T$ into polynomials of the same degree and is bounded in all $W^{s,q}(T)$ norms with $1<q<\mathrm{\infty }$ and $s\ge 2$. 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, $m\in {\mathbb{N}}_0$.