## 4 Numerical Examples and Validation of the Software
### Numerical Examples in 1d
The estimator \(\eta\), cf. remark and Definition 1, converges, but not to 0, as expected, while the convergence of the practical error \(\sigma\) seem to also be given, as can be seen in Figure 8c,f,i. Figure 4.4 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 adaptive mesh refinement. The developments of the parameter \(\alpha\) during the augmented Lagrangian algorithm are for the adaptive and uniform refinement identical for linear reaction terms and they are very similar to the development for the pure diffusive case given in Figure 4.5  
The third example that is discussed in this section is given by the parameters in Figure 4.6 The discrete solutions to (3) w.r.t. the parameters above are close to identical to the discrete solution given in Figure 1, as discussed previously. In fact no differences are visible. Hence, the corresponding figure will not be displayed in this paper. The evaluation of the error estimators, which are defined in remark and Definition 1, cf. Figure 9, are yielding that the numerical scheme converges for both the uniform and adaptive refinement scheme. Furthermore, it is shown in Figure 9 that the localized evaluation of the non-linear least-squares functional, as beforehand \(\eta\), seems to be equivalent to the exact error for uniform and adaptive refinement. Furthermore, up to a value of approximately \(10^{4}\) DoF, the corresponding practical estimators \(\sigma\) are decreasing for uniform and adaptive refinement. At this stage, the convergence of the numerical schemes is given from a practical viewpoint. Applying the augmented Lagrangian algorithm, cf. Algorithm 1, to (4) with the parameters that are given in the Table 3 and applying uniform refinement, one obtains non-negative species distributions after 89 iterations of the augmented Lagrangian scheme, c.f. Figure 4.7 Furthermore, it is observable that the practical error estimator \(\eta\), cf. definition and Remark 1, converges, but not to 0 in each setting. Furthermore, it is observable that the practical \(\sigma\), cf. definition and Remark 1, converges to zero for each iteration. Applying the augmented Lagrangian algorithm to (4) with the parameters that are given in Table 3 and applying adaptive mesh refinement to the iterative defined problem (5), one obtains the results given in Figure 4.8 The iterates converge to a species distributions with non-negative species concentrations, as seen in the corresponding figure. After 89 iterations, non-negative species distributions are obtained. Furthermore, it is visible that the estimator \(\eta\), cf. remark and Definition 1, that is based on the piecewise evaluation of the corresponding least-squares functional converges, but not to 0. In contrast to the setting shown in Figure 11, the practical error \(\sigma\), cf. remark and Definition 1, one obtains convergence, but not to 0, as beforehand. Figure 4.9 Development of the error estimators for the uniform and adaptive mesh refinement, for the approximation of (3) with parameters given in Figure 1:0  
Figure 1:1 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 iteration w.r.t. the parameters given in Table 3 and uniform mesh refinement. The developments of the parameter \(\alpha\) during the augmented Lagrangian algorithm for the adaptive and uniform refinement are identical for the full reaction terms and they are very similar to the development for the pure diffusive case, as given in Figure 4, also in this case, a figure is not given in this paper. As a remaining part of the discussions to the 1d examples, a comparison of the CPU running times of the different examples will be discussed.