identity $$\label{eq:Fres}
\left \langle {\mathsf{F}}_{{\epsilon}}(u_{n+1}^{(\Delta t,{{\rm hp}})}),v \right \rangle
=\sum_{j=1}^{N-1}a_j+\sum_{j=1}^N(b_j+c_j),$$ where $$\begin{aligned}
{2}
a_j&:= {\epsilon}{\left\llbracket(u_{n+1}^{(\Delta t,{{\rm hp}})})'\right\rrbracket}(x_j)(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)(x_j),&\qquad&1\le j\le N-1,\nonumber\\
b_j&:=\int_{K_j}{\left\{f^{\Delta t}({u^{\rm hp}}_{n+1})+{\epsilon}(u_{n+1}^{(\Delta t,{{\rm hp}})})''\right\}({\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v-v)}{\,{\mathsf{d}}x},&&1\le j\le N,\label{bounds}\\
c_j&:=\int_{K_j}{\left\{f^{\Delta t}({u^{\rm hp}}_{n+1})-f(u_{n+1}^{(\Delta t,{{\rm hp}})})\right\}v}{\,{\mathsf{d}}x},&&1\le j\le N;\nonumber\end{aligned}$$ see [@AmreinWihler:15 Proposition 4.3].

We estimate the terms $a_j$, $b_j$, and $c_j$ individually. First let $1\le j\le N-1$. Then, $a_j$ from  can be estimated using Corollary \[cr:int\] as follows: $$\begin{aligned}
|a_j|&\le {\epsilon}\left|{\left\llbracket(u_{n+1}^{(\Delta t,{{\rm hp}})})'\right\rrbracket}(x_j)\right||(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)(x_j)|\\
&\preccurlyeq\gamma_j^{\nicefrac12}{\epsilon}\left|{\left\llbracket(u_{n+1}^{(\Delta t,{{\rm hp}})})'\right\rrbracket}(x_j)\right|\left({\left|\!\left|\!\left|v\right|\!\right|\!\right|}^2_{{\epsilon},\widetilde K_j}+{\left|\!\left|\!\left|v\right|\!\right|\!\right|}^2_{{\epsilon},\widetilde{K}_{j+1}}\right)^{\nicefrac12}.\end{aligned}$$ Applying the Cauchy-Schwarz inequality leads to $$\begin{aligned}
\left|\sum_{j=1}^{N-1}a_j\right|
&\preccurlyeq\left(\sum_{j=1}^{N-1}{{\epsilon}^2\gamma_j\left|{\left\llbracket(u_{n+1}^{(\Delta t,{{\rm hp}})})'\right\rrbracket}(x_j)\right|^2}\right)^{\nicefrac12}
\left(\sum_{j=1}^{N-1}\left({\left|\!\left|\!\left|v\right|\!\right|\!\right|}^2_{{\epsilon},\widetilde K_j}+{\left|\!\left|\!\left|v\right|\!\right|\!\right|}^2_{{\epsilon},\widetilde{K}_{j+1}}\right)\right)^{\nicefrac12}\\
&\preccurlyeq\left(\sum_{j=1}^{N-1}{{\epsilon}^2\gamma_j\left|{\left\llbracket(u_{n+1}^{(\Delta t,{{\rm hp}})})'\right\rrbracket}(x_j)\right|^2}\right)^{\nicefrac12}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon}}.\end{aligned}$$ Furthermore, again using Corollary \[cr:int\], we see that $$\begin{aligned}
\left|\sum_{j=1}^Nb_j\right|
&\preccurlyeq\sum_{j=0}^N\alpha_j^{\nicefrac12}{\left\|f^{\Delta t}({u^{\rm hp}}_{n+1})+{\epsilon}(u_{n+1}^{(\Delta t,{{\rm hp}})})''\right\|}_{0,K_j}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_j}\\
&\preccurlyeq\left(\sum_{j=1}^N\alpha_j{\left\|f^{\Delta t}({u^{\rm hp}}_{n+1})+{\epsilon}(u_{n+1}^{(\Delta t,{{\rm hp}})})''\right\|}_{0,K_j}^2\right)^{\nicefrac12}\left(\sum_{j=1}^N{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde{K}_j}^2\right)^{\nicefrac12}\\
&\preccurlyeq\left(\sum_{j=1}^N\alpha_j{\left\|f^{\Delta t}({u^{\rm hp}}_{n+1})+{\epsilon}(u_{n+1}^{(\Delta t,{{\rm hp}})})''\right\|}_{0,K_j}^2\right)^{\nicefrac12}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon}}.\end{aligned}$$ Similarly, there holds $$\begin{aligned}
\left|\sum_{j=1}^Nc_j\right|&
\preccurlyeq\sum_{j=1}^N {\left\|f^{\Delta t}({u^{\rm hp}}_{n+1})-f(u_{n+1}^{(\Delta t,{{\rm hp}})})\right\|}_{0,K_j}{\left\|v\right\|}_{0,K_j}\\
&\preccurlyeq\left(\sum_{j=1}^N{\left\|f^{\Delta t}({u^{\rm hp}}_{n+1})-f(u_{n+1}^{(\Delta t,{{\rm hp}})})\right\|}_{0,K_j}^2\right)^{\nicefrac12}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon}}.\end{aligned}$$ Now, applying the Cauchy-Schwarz inequality to  we see that $$\begin{aligned}
{\left|\left \langle {\mathsf{F}}_{{\epsilon}}(u_{n+1}^{(\Delta t,{{\rm hp}})}),v \right \rangle\right|}
&\preccurlyeq\left(\delta_{n,\Omega}^2+\sum_{j=1}^N{\eta_{n,j}^{2}}\right)^{\nicefrac{1}{2}}{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon}}.
\end{aligned}$$ Dividing by ${\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon}}$, and taking the supremum for all $v\in H^1_0(\Omega)$, completes the proof.

Suppose that there exist constants $\lambda>-C_P^{-2}{\epsilon}$ and $L>0$, with $C_P$ being the Poincaré constant from , such that $$\begin{aligned}
(f(x)-f(y))(x-y)\le-\lambda(x-y)^2,\qquad
|f(x)-f(y)|\le L|x-y|,\end{aligned}$$ for all $x,y\in\mathbb{R}$. Then, the residual norm $ {\left|\!\left|\!\left|{\mathsf{F}}_{\epsilon}(u_{n+1}^{(\Delta t,h)})\right|\!\right|\!\right|}_{X',{\epsilon}}$ 