be refined according to the *Dörfler marking* criterion: $$\label{eq:dorf}\tag{D}
\vartheta\sum_{j=1}^N\eta_{n,j}^2\le \sum_{j'=1}^M\eta_{n,j'}^2.$$ Here, the indices $j'$ are chosen such that the error indicators $\eta_{j'}$ from  are sorted in descending order, and $M$ is minimal.

### (b) $hp$-refinement criterion: {#b-hp-refinement-criterion .unnumbered}

The decision of whether a marked element in step (a) is refined with respect to $h$ (element bisection) or $p$ (increasing the local polynomial order by 1) is based on a smoothness testing approach. Specifically, if the (numerical) solution is considered smooth on a marked element $K_j$, then the polynomial degree is increased by 1 on that particular element (no element bisection), otherwise the element is bisected (retaining the current polynomial degree $p_j$ on both subelements). In order to evaluate the smoothness of the solution ${u^{(\Delta t,\rm hp)}_{n+1}}$ on a marked element $K_j$, we employ an elementwise smoothness indicator as introduced in [@FaWiWi14 Eq. (3)], $$\label{eq:hp}\tag{F}
\mathcal{F}^{p_j}_j\left[{u^{(\Delta t,\rm hp)}_{n+1}}\right]:=
\frac{{\left\|\frac{{\mathsf{d}}^{p_j-1}}{{\mathsf{d}}x^{p_j-1}}{u^{(\Delta t,\rm hp)}_{n+1}}\right\|}_{L^\infty(K_j)}}{h_j^{-\nicefrac12}{\left\|\frac{{\mathsf{d}}^{p_j-1} {u^{(\Delta t,\rm hp)}_{n+1}}}{{\mathsf{d}}x^{p_j-1}}\right\|}_{L^2(K_j)}+\frac{1}{\sqrt2}h_j^{\nicefrac12}{\left\|\frac{{\mathsf{d}}^{p_j} {u^{(\Delta t,\rm hp)}_{n+1}}}{{\mathsf{d}}x^{p_j}}\right\|}_{L^2(K_j)}},$$ if $\frac{{\mathsf{d}}^{p_j-1}}{{\mathsf{d}}x^{p-1}}{u^{(\Delta t,\rm hp)}_{n+1}}|_{K_j}\not\equiv 0$, and $\mathcal{F}^{p_j}_j[{u^{(\Delta t,\rm hp)}_{n+1}}]=1$ otherwise. Here, the basic idea is to consider the continuous Sobolev embedding $H^1(K_j)\hookrightarrow L^\infty(K_j)$, which implies that $$\sup_{v\in H^1(K_j)}\frac{{\left\|v\right\|}_{L^\infty(K_j)}}{h_j^{-\nicefrac12}{\left\|v\right\|}_{L^2(K_j)}+\frac{1}{\sqrt2}h_j^{\nicefrac12}{\left\|v'\right\|}_{L^2(K_j)}}\le 1;$$ see [@FaWiWi14 Proposition 1]. In particular, it follows that $\mathcal{F}_j^{p_j}[{u^{(\Delta t,\rm hp)}_{n+1}}]\le 1$. For 