Newton framework from the previous Section \[sc:anewton\], and apply an $hp$-discretization methodology for the resulting sequence of (locally) linearized problems.

Newton Linearization
--------------------

In order to apply an adaptive Newton method as introduced in Section \[sc:anewton\] to the nonlinear problem , note that the Fréchet-derivative of $ {\mathsf{F}}_{{\epsilon}} $ from  at $u\in X$ is given by $$\left \langle {\mathsf{F}}_{{\epsilon}}'(u)w,v\right \rangle = \int_{\Omega}\left\{{\epsilon}w'v'-f'(u)wv\right\}{\,{\mathsf{d}}x},\qquad v,w\in X=H^1_0(\Omega),$$ where we write $f'\equiv\partial_uf$. We note that, if $f'(u)\in L^{1}(\Omega)$, then ${\mathsf{F}}_{{\epsilon}}'(u)$ is a well-defined linear and bounded mapping from $X$ to $X'$; see [@AmreinWihler:15 Lemma A.1].

Then, given an initial guess $u_0\in X$ for , the Newton method  is to find $u_{n+1}\in X$ from $u_n\in X$, $n\ge 0$, such that $${\mathsf{F}}_{\epsilon}'(u_n)(u_{n+1}-u_n)=-\Delta t_n{\mathsf{F}}_{\epsilon}(u_n),$$ in $X'$. Equivalently, $$\label{eq:weak}
a_{\epsilon}(u_n; u_{n+1},v)=a_{\epsilon}(u_n; u_{n},v)-\Delta t_n\ell_{\epsilon}(u_n;v)\qquad\forall v\in X,$$ where, for [*fixed*]{} $u\in X$, $$\begin{aligned}
a_{{\epsilon}}(u;w,v)&:=\int_{\Omega}\left\{{\epsilon}w'v'-f'(u)wv\right\}{\,{\mathsf{d}}x},\\
l_{{\epsilon}}(u;v)&:=\int_{\Omega}\left\{{\epsilon}u'v'-f(u)v\right\}{\,{\mathsf{d}}x}\end{aligned}$$ are bilinear and linear forms on $X\times X$ and $X$, respectively.

Incidentally, if there are constants $\underline{\lambda},\overline{\lambda}\ge 0$ with ${\epsilon}C_P^{-2}>\overline{\lambda}$ such that $ -\underline\lambda \leq f'(u) \leq \overline{\lambda}$ holds for all $u\in\mathbb{R} $, where $C_P:=\nicefrac{(b-a)}{\pi}>0$ is the (best) constant in the Poincaré inequality on $\Omega=(a,b)$, i.e.,$$\label{eq:P}
\|w\|_{0}\le C_P\|w'\|_{0}\qquad\forall w \in X,$$ see [@HLP52 Theorem 257], then, it is elementary to show that the formulation  is 