the second estimate, we apply, for $1\le j\le N-1$, a multiplicative trace inequality; see [@MelenkWihler:15 Lemma A.1]): $$\begin{aligned}
\left|(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)(x_j)\right|^2
&\le h_j^{-1}{\left\|v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v\right\|}^2_{L^2(K_j)}\\
&\quad+2{\left\|v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v\right\|}_{L^2(K_j)}{\left\|(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)'\right\|}_{L^2(K_j)}.\end{aligned}$$ Then, invoking the above bounds as well as the estimates from Proposition \[pr:intp\], we get $$\begin{aligned}
\big|(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)(x_j)\big|^2
&\le C_I^2\left(\alpha_jh_j^{-1}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_j}^2
+2\sqrt{\alpha_j}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_j}{\left\|v'\right\|}_{L^2(\widetilde K_j)}\right)\\
&\le C_I^2\left(\alpha_jh_j^{-1}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_j}^2
+2\sqrt{{\epsilon}^{-1}\alpha_j}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_j}^2\right)\\
&\le C_I^2\beta_j{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_j}^2,\end{aligned}$$ with $\beta_j$ from . Since $x_j$ is also a boundary point of $K_{j+1}$, we similarly obtain that $$\big|(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)(x_j)\big|^2
\le C_I^2\beta_{j+1}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_{j+1}}^2.$$ Therefore, $$\begin{aligned}
\big|(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)(x_j)\big|^2
&=\frac{\beta_{j+1}}{\beta_{j}+\beta_{j+1}}\big|(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)(x_j)\big|^2\\
&\quad +\frac{\beta_{j}}{\beta_{j}+\beta_{j+1}}\big|(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)(x_j)\big|^2\\ &\le
C_I^2\gamma_j\left({\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_{j}}^2+{\left|\!\left|\!\left|v\right|\!\right|\!\right|}^2_{{\epsilon},\widetilde K_{j+1}}\right),\end{aligned}$$ with $\gamma_j$ from . Thus, we have shown the second estimate.

Robust *A Posteriori* Residual Estimate
---------------------------------------

Based on the previous interpolation analysis in Section \[sc:int\], we are now in a position to prove the following main result.

\[thm:main\] \[thm:1\] For $u_{n+1}^{(\Delta t,{{\rm hp}})}$ from  there holds the upper *a posteriori* bound $$\label{eq:upperbound}
{\left|\!\left|\!\left|{\mathsf{F}}_{\epsilon}(u_{n+1}^{(\Delta t,{{\rm hp}})})\right|\!\right|\!\right|}_{X',{\epsilon}}^2
\preccurlyeq \delta_{n,\Omega}^2+\sum_{j=1}^N{\eta_{n,j}^{2}},$$ with $$\label{Femerror}
\begin{split}
\eta_{n,j}^2:&= \alpha_{j} {\left\|f^{\Delta t}({u^{\rm hp}}_{n+1})+{\epsilon}(u_{n+1}^{(\Delta t,{{\rm hp}})})''\right\|}_{0,K_j}^2\\
&\quad+\frac{1}{2}{\epsilon}^2\gamma_j\left|{\left\llbracket(u_{n+1}^{(\Delta t,{{\rm hp}})})'\right\rrbracket}(x_j)\right|^2
+\frac{1}{2}{\epsilon}^2\gamma_{j-1}\left|{\left\llbracket(u_{n+1}^{(\Delta t,{{\rm hp}})})'\right\rrbracket}(x_{j-1})\right|^2,
\end{split}$$ for $1\le j\le N$, and $$\label{Newtonerror}
\delta_{n,\Omega}:={\left\|f^{\Delta t}(u_{n+1}^{{{\rm hp}}})-f(u_{n+1}^{(\Delta t,{{\rm hp}})})\right\|}_{0,\Omega}.$$ Here, $\alpha_j$ and $\gamma_j$ are defined in  and , respectively, and we 