freedom; see Figure \[fig:slin0\] (right).

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![Example \[ex:slin0\]: Two positive solutions of  (left), and the corresponding performance data for the residuals (right).[]{data-label="fig:slin0"}](figures/Bratusolutions "fig:"){width="0.45\linewidth"} ![Example \[ex:slin0\]: Two positive solutions of  (left), and the corresponding performance data for the residuals (right).[]{data-label="fig:slin0"}](figures/Braturesiduals "fig:"){width="0.45\linewidth" height="0.375\linewidth"}

\[ex:slin1\] The focus of the next sequence of experiments is on the robustness of the *a posteriori* residual bound  with respect to the singular perturbation parameter ${\epsilon}$ as ${\epsilon}\to 0$. To this end, let us consider the *Ginzburg-Landau*-type equation, $$-{\epsilon}u''=u-u^3\ \text{on} \ (0,1),\qquad u(0)=u(1)=0.$$ Starting from the initial guess depicted in Figure \[fig:slin1\_1\] (left), we test the fully adaptive Newton-Galerkin Algorithm \[al:full\] for different choices of ${\epsilon}=\{10^{-i}\}_{i=2}^5$. As ${\epsilon}\to 0$ the resulting solutions feature ever stronger boundary layers, as well as an interior shock close to the center of the domain; see Figure \[fig:slin1\_1\] (right). The performance data in Figure \[fig:slin1\_2\] (left) shows that the residuals decay, firstly, fairly robust in ${\epsilon}$, and, secondly, exponentially fast with respect to the (square root) of the number of degrees of freedom (cf., e.g., [@Schwab98]). The final mesh for ${\epsilon}=10^{-5}$ is displayed in Figure \[fig:slin1\_2\] (right); whilst the singular effects have been properly resolved by suitable $hp$-refined elements, we notice that the 