1$, we denote by $\mathbb{P}_p$ the space of all polynomials of degree at most $p$. We say that the pair $({\mathcal{T}},{\bm p})$ of a partition ${\mathcal{T}}$ and of a degree vector ${\bm p}$ is $\mu$-shape regular, for some constant $\mu>0$ independent of $j$, if $$\label{eq:gamma-shape-regular}
\mu^{-1} h_{j+1} \leq h_{j} \leq \mu h_{j+1}, 
\qquad  
\mu^{-1} p_{j+1} \leq p_{j} \leq \mu p_{j+1}, 
\qquad j=1,\ldots,N-1,$$ i.e., if both the element sizes and polynomial degrees of neighboring elements are comparable.

### Linear $hp$-FEM

We define the numerical approximation of  in an iterative way as follows: For $n=0$, we let ${V^{\rm hp}({\mathcal{T}},{\bm p})}$ be an initial $hp$-discretization space, and ${u^{\rm hp}}_{0} \in {V^{\rm hp}({\mathcal{T}},{\bm p})}$ a suitable initial guess for the Newton iteration. Furthermore, for $n\ge 0$, and given ${u^{\rm hp}}_n \in {V^{\rm hp}({\mathcal{T}},{\bm p})}$, we find a finite element approximation ${u^{\rm hp}}_{n+1}\in {V^{\rm hp}({\mathcal{T}},{\bm p})}$ of  as the solution of the weak formulation $$\label{eq:fem}
a_{\epsilon}({u^{\rm hp}}_n; {u^{\rm hp}}_{n+1},v)=a_{\epsilon}({u^{\rm hp}}_n; {u^{\rm hp}}_n,v)-\Delta t\ell_{\epsilon}({u^{\rm hp}}_n;v)\qquad\forall v\in {V^{\rm hp}({\mathcal{T}},{\bm p})}.$$ Here, $\Delta t$ takes the role of a (possibly varying) step size parameter in the adaptive Newton scheme as described earlier in Section \[sc:newton\].

[*A Posteriori*]{} Residual Analysis
====================================

Recalling , for given $\Delta t>0$, let us 