introduce the function $$\label{eq:ut}
u_{n+1}^{(\Delta t,{{\rm hp}})}:={u^{\rm hp}}_{n+1}-(1-\Delta t){u^{\rm hp}}_n,$$ as well as $$\label{eq:ft}
f^{\Delta t}({u^{\rm hp}}_{n+1}):=\Delta tf({u^{\rm hp}}_n)+f'({u^{\rm hp}}_n)(u_{n+1}^{{\rm hp}}-{u^{\rm hp}}_n).$$ Then, rearranging terms, we can rewrite  in the following form: $$\label{start}
 \begin{aligned}
{\epsilon}\int_{\Omega}{ (u_{n+1}^{(\Delta t,{{\rm hp}})})'v'} {\,{\mathsf{d}}x}& = \int_{\Omega}{ f^{\Delta t} (u_{n+1}^{{\rm hp}}) v }{\,{\mathsf{d}}x}\qquad \forall v  \in V_{0}^h.
\end{aligned}$$

The aim of this section is to derive [*a posteriori*]{} residual-based bounds for . In order to measure the discrepancy between the finite element discretization  and the original problem , an obvious quantity to bound is the residual ${\mathsf{F}}_{{\epsilon}}({u^{\rm hp}}_{n+1})$ in $X'$. In order to proceed in this direction, we notice that the adaptively chosen damping parameter $\Delta t$ in the Newton method  will equal 1 sufficiently close to a root of ${\mathsf{F}}_{\epsilon}$. For this reason, as was proposed in [@AmreinWihler:15], we may focus on the ‘shifted’ residual ${\mathsf{F}}_{{\epsilon}}(u_{n+1}^{(\Delta t,{{\rm hp}})})$ in $X'$ instead; indeed, estimating this quantity turns out to be very natural in view of the linearized finite element discretization .

Robust Interpolation Estimates {#sc:int}
------------------------------

For the purpose of deriving an (upper) *a posteriori* residual estimate that is robust with respect to the singular perturbation parameter ${\epsilon}$ (for $0<{\epsilon}\ll 1$) as well as optimally scaled 