a linear second-order diffusion-reaction problem, which is coercive on $X$. In consequence, for any given $ u_n \in X $, it has a unique solution $u_{n+1}\in X$; see [@AmreinDiss15] for details.

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Linearized Finite Element Approach
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In order to provide a numerical approximation of , we will discretize the weak formulation  by means of an $hp$-finite element method, which, in combination with the Newton iteration, constitutes a Newton-Galerkin approximation scheme. Furthermore, in view of deriving an $hp$-adaptive algorithm, we will develop an ${\epsilon}$-robust $hp$-version [*a posteriori*]{} residual analysis for the linear finite element formulation.

### $hp$-Spaces

We introduce a partition ${\mathcal{T}}=\{K_j\}_{j=1}^N$ of $N\ge 1$ (open) elements $K_j=(x_{j-1},x_j)$, $j=1,2,\ldots,N$, on $\Omega=(a,b)$, with $a=x_0<x_1<x_2<\ldots<x_{N-1}<x_N=b$. The length of an element $K_j$ is denoted by $h_j=x_j-x_{j-1}$, $j=1,2,\ldots,N$. For each element $K_j \in {\mathcal{T}}$, it will be convenient to introduce the patch $\widetilde K_j = \bigcup \{K'\in{\mathcal{T}}\,|\, 
\overline{K'}\cap \overline{K_j} \ne \emptyset\}$ as the union of $K_j$ and of the elements adjacent to it. In addition, to each element $K_j$ we associate a polynomial degree $p_j\ge 1$, $j=1,2,\ldots,N$. These numbers are stored in a polynomial degree vector ${\bm p}=(p_1,p_2,\ldots,p_N)$. Then, we define an $hp$-finite element space by $${V^{\rm hp}({\mathcal{T}},{\bm p})}=\left\{v\in H^1_0(\Omega):\,
v|_{K_j}\in\mathbb{P}_{p_j}(K_j),\,j=1,2,\ldots,N\right\},$$ where, for $p\ge 