and the energy norm ${\left|\!\left|\!\left|u-u_{n+1}^{(\Delta t,h)}\right|\!\right|\!\right|}_{{\epsilon}}$ of the error are equivalent; see [@AmreinWihler:15 Theorem 4.5] for details.

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Following along the lines of [@AmreinWihler:15 §4.4.2] and [@MelenkWihler:15 Appendix], it is possible to prove *${\epsilon}$-robust local lower residual bounds* in terms of the error indicators $\eta_{K_j}$ and the data oscillation terms. This approach, however, results in local efficiency bounds that will be slightly suboptimal with respect to the local polynomial degrees due to the need of applying $p$-dependent norm equivalences (involving cut-off functions).

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An $hp$-Adaptive Newton-Galerkin Procedure {#sc:adaptivity}
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In this section, we will discuss how the *a posteriori* bound from Theorem \[thm:main\] can be exploited for the purpose of designing an $hp$-adaptive Newton-Galerkin algorithm for the numerical solution of . We will begin by revisiting an $hp$-adaptive testing strategy from [@FaWiWi14] (see also [@Wi11; @Wi11_2]).

$hp$-Adaptive Refinement Procedure {#sc:hpadapt}
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In order to $hp$-refine the $hp$-finite element space ${V^{\rm hp}({\mathcal{T}},{\bm p})}$, we shall apply an $hp$-adaptive algorithm which is based on the following two ingredients.

### (a) Element marking: {#a-element-marking .unnumbered}

The elementwise error indicators $\eta_{n,j}$ from Theorem \[thm:main\] are employed in order to mark elements for refinement. More precisely, we fix a parameter $\vartheta\in(0,1)$, and select elements to 