may further improve the performance of the Newton-Galerkin approach. In the $hp$-literature, several adaptive strategies and algorithms have been proposed; see, e.g., the overview article [@MiMc11] and the references therein. In particular, we point to the smoothness estimation techniques proposed in [@eibner-melenk06; @HoustonSeniorSuliENUMATH; @HoustonSuliHPADAPT; @Ma94], and the related approach [@FaWiWi14; @Wi11_2; @Wi11] involving continuous Sobolev embeddings. The latter works will be applied in the current paper in order to drive the $hp$-adaptive finite element refinements; in combination with the adaptive Newton linearizations this will lead to a fully $hp$-adaptive Newton-Galerkin procedure.

*Problem formulation and notation:* Henceforth, we suppose that a (not necessarily unique) solution $u\in X:=H^1_0(\Omega)$ of  exists; here, we denote by $H^1_0(\Omega)$ the standard Sobolev space of functions in $H^1(\Omega)=W^{1,2}(\Omega)$ with zero trace on $\partial\Omega:=\{a,b\}$. Furthermore, signifying by $X'=H^{-1}(\Omega)$ the dual space of $X$, and upon defining the map ${\mathsf{F}}_{{\epsilon}}: X\rightarrow X'$ through $$\label{eq:Fweak}
\left \langle {\mathsf{F}}_{{\epsilon}}(u),v\right \rangle :=  \int_{\Omega}\left\{{\epsilon}u'v'-f(u)v\right\}{\,{\mathsf{d}}x}\qquad \forall v\in X,$$ where $\left\langle\cdot,\cdot\right\rangle$ is the dual product in $X'\times X$, the above problem  can be written as a nonlinear operator equation in $X'$: $$\label{eq:F0}
u\in X:\qquad {\mathsf{F}}_{\epsilon}(u)=0.$$ In addition, for open subsets $D\subseteq\Omega$, we introduce the norm $${\left|\!\left|\!\left|u\right|\!\right|\!\right|}_{{\epsilon},D}:=\Bigl({\epsilon}{\left\|u'\right\|}_{0,D}^2 +{\left\|u\right\|}_{0,D}^2 \Bigr)^{\nicefrac{1}{2}},$$ where $\|\cdot\|_{0,D}$ denotes the $L^2$-norm on $D$. 