local interpolation operator $\tilde\pi_{K,m}:{\mathrm{H}}^{1+s}(K)\to \tilde{\mathcal{P}}_{m}(K)$, $s>1/2$ as follows: recalling the definition of $\tilde{\mathcal{P}}_{m}(K)$ given in

-   if $\partial K \cap \Gamma \not \in \mathcal F_h^\Gamma$ then $\tilde{\mathcal{P}}_{m}(K)= {\mathcal{P}}_{m}(K)$ and we take $\tilde\pi_{K,m} = \pi_{K,m}$, where $\pi_{K,m}$ is defined as in [@Monk Section 5.6];

-   if $\partial K \cap \Gamma=T  \in \mathcal F_h^\Gamma$ then $\tilde{\mathcal{P}}_{m}(K)={\mathcal{P}}_m(K) + {\mathcal{P}}_{m+1}^T(K)$ and $\tilde \pi_{K,m}$ is defined by changing the conditions defining $\pi_{K,m}$ on $T$ and on the edges composing $T$ into $$\label{freedomFb}
    \int_T \tilde\pi_{K,m} p q = \int_T p q \qquad \forall q \in {\mathcal{P}}_{m-2}(T)$$ and $$\label{freedomEb}
    \int_e \tilde\pi_{K,m} p q = \int_e p q \qquad \forall q \in {\mathcal{P}}_{m-1}(e), \quad \forall e\in {\mathcal{E}}(F)$$ respectively. The remaining degrees of freedom are the same as those defining $\pi_{K,m}$, see [@Monk Section 5.6].

We notice that $
\text{dim}({\mathcal{P}}_m(K) + {\mathcal{P}}_{m+1}^T(K))= \text{dim}({\mathcal{P}}_m(K)) + m+1
$ and the number of degrees of freedom defining $\tilde \pi_{K,m}$ is equal to the number of degrees of freedom of $\pi_{K,m}$ plus $\text{dim}({\mathcal{P}}_{m-2}(T)) - \text{dim}({\mathcal{P}}_{m-3}(T)) = m-1$ additional degrees of freedom on $T$ and one additional degree of freedom on each of the three edges of $T$, which gives a total of $\text{dim}({\mathcal{P}}_m(K)) + m+1$ degrees of freedom. Using this fact, it is 