This script compares the rate of convergence for a quantity of interest with both uniform
and residual-based adaptive refinements using a known problem called backward facing step.
$$
\begin{equation*}
\begin{aligned}
&\text{Given constants}, \mu , \text{and}, \gamma, &&\\
&\text{find}, \mathbf{u}: \Omega \in \mathbb{R}^2, \text{such that:} && \\
& - {\rm div}(\boldsymbol{\sigma}) = 0 && \text{in}, \Omega\\
&\phantom{-div(u)}\mathbf{u} = \mathbf{0} && \text{on}, \Gamma_{D} \\
&\phantom{-div(u} u_{x} = u_{\rm max}(x_1 - h) \cfrac{H - (x_1 - h)}{\left(\frac{H}{2}\right)^2} && \text{on}, \Gamma_{in} \\
&\phantom{-div} \boldsymbol{\sigma} \cdot \mathbf{n} = \mathbf{0} && \text{on}, \Gamma_{out} \\
\end{aligned}
\end{equation*}
$$
with
$$
\begin{align*}
\boldsymbol{\sigma} = \mu (2 \boldsymbol{\varepsilon} + \gamma {\rm tr}(\boldsymbol{\varepsilon}) \mathbf{I}) \quad \quad
\boldsymbol{\varepsilon} = \frac{1}{2} \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^{\rm T} \right)
\end{align*}
$$
$$
\begin{equation*}
\begin{aligned}
&\text{Find}, \mathbf{u} \in \mathcal{S} \text{such that:} &&\\
& \mathcal{B}(\mathbf{u},\mathbf{v}) = \mathcal{F} (\mathbf{v}) && \forall \mathbf{v} \in \mathcal{V}
\end{aligned}
\end{equation*}
$$
with
$$
\begin{align*}
\mathcal{B}(\mathbf{u},\mathbf{v}) &= \int_{\Omega} \left[ 2 \mu \boldsymbol{\varepsilon} \left( \frac{1}{2} \left( \nabla \mathbf{v} + (\nabla \mathbf{v})^{\rm T} \right) \right) + \gamma \mu , {\rm div}(\mathbf{u}) , {\rm div}(\mathbf{v}) \right], {\rm dV}\\
\mathcal{F}(\mathbf{v}) &= \int_{\Gamma_{out}} \mathbf{v} \cdot (\boldsymbol{\sigma} \cdot \mathbf{n}) , {\rm dS} = 0
\end{align*}
$$
$$
\begin{equation*}
\mathcal{R}^h(\mathbf{v}) := B(\mathbf{u}^h, \mathbf{v}) - \mathcal{F}(\mathbf{v}) \phantom{ {\sup_{\mathbf{v} \in \mathcal{V}\setminus 0}} \frac{\mathcal{R}^H(\mathbf{v})}{| \mathbf{v} |_{\mathcal{V}}}}
\end{equation*}
$$