## 4 Numerical Examples and Validation of the Software
### Numerical Examples in 1d  
For the numerical examples presented in this section, let \(\Omega:=]0,2[\). For the examples that are discussed in this section, let an exact solutions \(c_{A},c_{B},c_{C}\) of the unrestrained problem (3) be given through  
\[c_{A}:=c_{B}:=c_{C}:=x(x-2)+\frac{1}{2} \tag{27}\]  
and let \(\Gamma_{D}=\partial\Omega\) and \(g_{A,D}=g_{B,D}=g_{C,D}\equiv 1\). Afterwards, the functions \(f_{A},f_{B},f_{C}\in L^{2}(\Omega)\) are given through  
\[f_{A} =-2D_{A}-k_{2}\left(x(x-2)+\frac{1}{2}\right)+k_{1}\left(x(x-2)+ \frac{1}{2}\right)^{2}, \tag{28a}\] \[f_{B} =-2D_{B}-k_{2}\left(x(x-2)+\frac{1}{2}\right)+k_{1}\left(x(x-2)+ \frac{1}{2}\right)^{2},\] (28b) \[f_{C} =-2D_{C}+k_{2}\left(x(x-2)+\frac{1}{2}\right)-k_{1}\left(x(x-2)+ \frac{1}{2}\right)^{2}. \tag{28c}\]  
As breaking conditions for the simulations, in this subsection, the fulfillment of the following conditions were set:  
1. The breaking condition from \(\mathbf{S3}\) in Algorithm 1 is fulfilled with a tolerance \(\varepsilon:=10^{-4}\). 2. The number of nodes in the current mesh increases \(10,000\) nodes. 3. More than 100,000 iterations of the augmented Lagrangian algorithm were performed. Before the numerical examples are discussed, note that the exact error is given through:  
\[\|x-x_{h}\|_{X}^{2}=\sum_{S\in\{A,B,C\}}\left[\left\|\
abla c_{S}-\
abla u_{S} \right\|_{L^{2}(\Omega)}^{2}+\left\|p_{S,h}-\
abla u_{S}\right\|_{L^{2}(\Omega )}^{2}+\left\|\frac{\partial}{\partial x}p_{S,h}-\frac{\partial^{2}}{\partial x ^{2}}u_{S}\right\|_{L^{2}(\Omega)}^{2}\right]. \tag{29}\]  
Simulating the unrestrained problem (3), for the the given RHS, as defined above, for the parameters that are defined in the Tables 1-3, with the reduced numerical scheme, see Remark 1, one obtains approximations of the defined exact solution (27). For a number of \(10,000\) DoFs, one obtains the results that are shown in Figure 1, which were simulated for the parameters defined in Table 1. Because no visible difference to the other experiments can be seen, Figure 1 will be the only given pictures of approximate solutions to (3) in this section. Furthermore, note that, for the simulations, the following setup was chosen:  
For all simulations of the 'classical' concentration-profiles, with no restrictions on the positivity of the concentrations, the reduced method, as described in Remark 1, was used. Furthermore, in all cases of simulations of (3), the estimtor \(\eta\) was used. For the adaptive refinement of (5), the practical error estimator \(\sigma\) was used for all approximations of solutions of (5). Furthermore, for all adaptive refinement Dorfler marking, cf. Remark 1, was used as marking strategy with common bulk parameter \(\theta=0.5\). The first example that will be discussed in the following in more detail is given by the parameters in Table 1. As mentioned before, Figure 1 provides the simulations with a minimum number of \(10^{4}\) nodes in the AFEM loop as breaking-condition. By the evaluation of the error estimators for this example, cf. Figure 2, one can observe convergence of the estimators, except for the practical estimator \(\sigma\) for the uniform \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Species** & \(D\) **in**\(\frac{\mathbf{m}^{2}}{\mathbf{s}}\) & \(k_{1}\) **in**\(\frac{\mathbf{1}}{\mathbf{s}}\) & \(k_{2}\) **in**\(\frac{\mathbf{m}^{3}}{\mathbf{m}\mathbf{s}}\) & \(\gamma_{A}\) \\ \hline \(A\) & 1/2 & - & - & 0 \\ \(B\) & 1/3 & - & - & 0 \\ \(C\) & 1/4 & 1 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Third example of physical parameters of the diffusion–reaction model for the complexation of \(K\). refinement. As expected, the rest of the estimators are parallel, from which the equivalence of estimators can be formulated as a conjecture. In Figure 3, some iterations of the augmented Lagrangian scheme with uniform refinement are shown. The augmented Lagrangian scheme needed 85 steps to obtain non-negative species distributions.