## 4 Numerical Examples and Validation of the Software
### Numerical Examples in 1d
The augmented Lagrangian scheme converges to set a non-negative species distributions for this example, as shown in the subfigures (a), (d), (g). Furthermore, as predicted in Remark 1, the localized evaluation of the least-squares functional, which is given by the estimator \(\eta\), does not converge to zero in all examples. In this case, for the uniform refinement, the practical error estimator converges \(\lim\limits_{h\downarrow 0}\sigma=0\). Additionally, this example also shows the convergence of the multipliers \(\mu_{A}\), \(\mu_{B}\), and \(\mu_{C}\) as well, which also seems to converge. The parameter \(\alpha\) grows exponentially over the iterations of the augmented Lagrangian algorithm, as shown in the Figure 4.  
In Figure 5, the development of the augmented Lagrangian algorithm is shown, applying the adaptive algorithm in each iteration of (5). The augmented Lagrangian algorithm takes, like for uniform mesh refinement, 85 iterations to guarantee non-negative species distributions, as can be seen in this case. Just observing the discrete solutions \(c_{A,\mu}\), cf. Figure 5a, \(c_{B,h}\), (d), and \(c_{C,h}\) (g), and the Lagrangian multipliers \(\mu_{A}\) (b), \(\mu_{B}\) (e), and \(\mu_{C}\) (h),  
Figure 1: Concentration profiles with negative concentrations \(c_{A}\) (a), \(c_{B}\) (b) and \(c_{C}\) (c), obtained by approximation of (3), for the parameters that are given in Table 1.  
Figure 2: Development of the error estimators for the uniform and adaptive mesh refinement, for the approximation of (3) with the parameters given in Table 1.  
one obtains the same result as for the uniform refinement. The limit of the solutions seems to be identical, but the interesting aspect is the estimators, which are given in the subfigures (c), (f), and (i). The practical errors \(\sigma\) seem to converge to a value \(b\
eq 0\), as well as the estimators \(\eta\) do, as expected to a value \(a\
eq 0\). Figure 3: Concentrationprofiles (**a**,**d**,**g**), evaluation of the Lagrangian multipliers (**b**,**e**,**h**) and estimators (**c**,**f**,**i**) during the augmented Lagrangian scheme in the 1-st, 43-rd, and 85-th itration w.r.t. parameters that are given in Table 1 and uniform mesh refinement. Applying the augmented Lagrangian Algorithm 1 onto (4) with the parameters that are given in Table 2 and applying uniform refinement on the mesh, \(\mathcal{T}\) one obtains, after 89 iterations of the augmented Lagrangian scheme, non-negative species distributions, as seen in Figure 7.  
Figure 5: Concentration profiles (**a**,**d**,**g**), evaluation of the Lagrangian multipliers (**b**,**e**,**h**) and estimators (**c**,**f**,**i**) during the augmented Lagrangian scheme in the 1-st, 43-rd and 85-th itartion w.r.t. parameters that are given in Table 1 and adaptive mesh refinement. Figure 6: Development of the error estimators for the uniform and adaptive mesh refinement, for the approximation of (3) with the parameters given in Figure 4.0  
The estimator \(\eta\) converges not to 0 in each iteration of the augmented Lagrangian method, as expected, but the practical estimator does converge to 0 in each iteration, as can be seen in Figure 4.1  
In Figure 8, the application of the augmented Lagrangian algorithm is shown for multiple iterations with the application of adaptive mesh refinement, as described above. The algorithm needed, as for treating the iterate problems with uniform refinement, 89 iterations. Figure 4.2 Concentration profiles (**a**,**d**,**g**), evaluation of the Lagrangian multipliers (**b**,**e**,**h**) and estimators (**c**,**f**,**i**) during the augmented Lagrangian scheme in the 1-st, 44-th, and 89-th itartion w.r.t. parameters that are given in Table 2 and uniform mesh refinement. As it can be seen, the algorithm outputs non-negative species distributions. In principle, equivalent results in the output can be seen in Figure 4.3 However, the principal behavior of the practical error estimator differs while using adaptive schemes from the same methodology but using uniform mesh refinement.