ease of evaluation, note that, by taking the derivative of order $p_j-1$ in the definition , the smoothness indicator $\mathcal{F}^{p_j}_j[{u^{(\Delta t,\rm hp)}_{n+1}}]$ is evaluated for linear functions only; in this case, it can be shown that $$\frac12\approx\frac{\sqrt{3}}{\sqrt{6}+1}\le\mathcal{F}_j^{p_j}\left[{u^{(\Delta t,\rm hp)}_{n+1}}\right]\le 1;$$ cf. [@FaWiWi14 Section 2.2]. For a given smoothness testing parameter $\zeta\in(\nicefrac{\sqrt{3}}{(\sqrt{6}+1)},1)$, the numerical solution ${u^{(\Delta t,\rm hp)}_{n+1}}$ is classified smooth on $K_j$ if $$\label{eq:st}
\mathcal{F}^{p_j}_j\left[{u^{(\Delta t,\rm hp)}_{n+1}}\right]\ge\zeta,$$ and otherwise nonsmooth. Incidentally, representing the local solution ${u^{(\Delta t,\rm hp)}_{n+1}}|_{K_j}$ in terms of (local) Legendre polynomials (or more general Jacobi polynomials), any derivatives of ${u^{(\Delta t,\rm hp)}_{n+1}}$ can be evaluated exactly by means of appropriate recurrence relations. For further information, we refer to the paper [@FaWiWi14] (see also [@Wi11_2; @Wi11]), where the smoothness testing strategy described above has been introduced and discussed in detail.

Fully Adaptive Newton-Galerkin Method
-------------------------------------

We will now propose a procedure that provides an interplay of the adaptive Newton methods presented in Section \[sc:anewton\] and automatic $hp$-finite element mesh refinements based on the [*a posteriori*]{} error estimate from Theorem \[thm:1\] (as outlined in the previous Section \[sc:hpadapt\]). To this end, we make the assumption that the Newton sequence $\left\{u_{n+1}^{(\Delta t_n,{{\rm hp}})}\right\}_{n\ge 0} $ given by  and , with 