a (possibly varying) step size $\Delta t_n$, is well-defined as long as the iterations are being performed.

\[al:full\] Given a (coarse) starting mesh $\mathcal{T}$ in $\Omega$, with an associated (low-order) polynomial degree distribution $\bm p$, and an initial guess $ u_{0}^{{{\rm hp}}} \in  {V^{\rm hp}({\mathcal{T}},{\bm p})}$. Set $n\gets 0$.

Determine the Newton step size parameter $\Delta t_n$ based on ${u^{\rm hp}}_n$ by the adaptive procedure from Algorithm \[al:zs\]. Compute the FEM solution $u_{n+1}^{{{\rm hp}}}$ from  with step size $\Delta t_n$ on the space ${V^{\rm hp}({\mathcal{T}},{\bm p})}$. Furthermore, obtain $ u_{n+1}^{(\Delta t_n,{{\rm hp}})} $ in , and evaluate the corresponding error indicators $ \{\eta_{n,j}\}_j $, and $\delta_{n,\Omega} $ from  and , respectively.

Numerical Experiments
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Let us provide some comments on how Algorithm \[al:full\] is implemented in the ensuing examples.

-   The Newton transform ${\mathsf{N}_{{\mathsf{F}}}}({u^{\rm hp}}_n)$ required for the computation of the step size parameter $\Delta t_n$ in step ([1:]{}) is approximated using the $hp$-FEM on the current mesh.

-   If the criterion  in step ([3:]{}) is satisfied, the mesh is refined, and then step ([2:]{}) is repeated based on the previously computed solution $u_{n+1}^{{{\rm hp}}}$ as interpolated on the refined mesh.

-   The linear systems resulting from the finite element 