## 4 Numerical Examples and Validation of the Software
### Examples in 2d
#### 4.2.2 Examples on a Non-Convex Domain  
In this section, two examples on non-convex domains will be discussed. In this section, let the domain be given \(\Omega=]0,1[^{2}\big{(}[0,1]\times[-1,0]\big{)}\). For the numerical experiments, the boundary conditions are defined analogously as in Section 4.2.1. The Dirichlet boundary is assumed to be given as \(\Gamma_{D}=\partial\Omega\), the Neumann-boundary given as \(\Gamma_{N}:=\varnothing\) and  
\[c_{A}|_{\Gamma_{D}}=c_{A}|_{\Gamma_{D}}=c_{A}|_{\Gamma_{D}}=\frac{1}{2}.\]  
Furthermore, let the RHSs \(f_{A},f_{B},f_{C}\in\mathrm{L}^{2}(\Omega)\) be given, as follows:  
\[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]  
The first example that is discussed in this subsection is given for the parameters defined in Table 1.  
When calculating an approximate solution to the minimization problem (3), one obtains discrete solutions in the form, as shown in Figure 16, for uniform mesh refinement. Because of the fact that the approximate solution w.r.t. adaptive mesh refinement only gives small differences to the displayed figure, it is not shown in this paper. Figure 15: Species distributions \(c_{A}\) (**a**,**d**,**g**), \(c_{B}\) (**b**,**e**,**h**) and \(c_{C}\) (**c**,**f**,**i**) in the 1 st, 80 th and 181 st ideation of the augmented Lagrangian algorithm with adaptive mesh refinement, w.r.t. the parameters that are given in Table 2.  
When evaluating the estimators \(\eta\) and \(\sigma\), as defined in definition and Remark 1, for the approximation of (3) for adaptive and uniform mesh refinement, one obtains that the estimators \(\eta\) and \(\sigma\) are restricted parallel to the refinement type, cf. Figure 17. Furthermore, one sees, as expected, that the estimators for adaptive refinement are steeper in the \(\log\log\) plot given in Figure 17, which indicates the expected increase of efficiency obtained by adaptive schemes. When comparing the triangulation generated by uniform refinement, cf. left image in Figure 18, and the triangulation, cf. right image in Figure 18, one observes that, while the uniform triangulation has a homogeneous distribution of triangles, in the adaptively generated mesh, the triangles accumulate in the non-convex vertex. This behavior is commonly known in the literature, cf. [27; 30]. Figure 16: Concentration profiles \(c_{A}\), (a), \(c_{B}\), (b), \(c_{C}\) (c) as approximate result to (3), with parameters, as given in Table 1.  
Figure 17: Developement of the error estimators for the uniform and adaptive mesh refinement, for the approximation of (3) with parameters that are given in Table 1 on the L-shaped domain. The evaluation of the augmented Lagrangian algorithm, cf. Figure 19, with the use of adaptive refinement with estimator \(\eta\) yields that 188 iterations of the augmented Lagrangian algorithm are needed to guarantee non-negative species distributions. The second example that is given in this section is, as before, given by the parameters that are defined in Table 2. When simulating the minimization problem (3), approximately one obtains an approximation of the form, as shown in Figure 20, which was obtained by simulation under uniform refinement. Because there are no visible differences between the picture generated under uniform refinement and generated under adaptive refinement, only the approximate solution under the usage of adaptive mesh-refinement will be given in this paper. \[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]0 Cont. \[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]1 Triangulations generated by uniform (**left**) and adaptive (**right**) refinement for the approximate solution of (3) with the parameters that are given in the \[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]2  
After the evaluation of the error estimators \(\eta\) and \(\sigma\), as defined in remark and Definition 1, one obtains that the practical error \(\sigma\) and the classical error estimator \(\eta\) are equal to the classical error estimator \(\eta\).