## 4 Numerical Examples and Validation of the Software
### Examples in 2d
#### 4.2.2 Examples on a Non-Convex Domain
\[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]3 Species distributions \(c_{A}\) (**a**,**d**,**g**), \(c_{B}\) (**b**,**e**,**h**) and \(c_{C}\) (**c**,**f**,**i**) in the 1 st, 94 th and 188 th ideation of the augmented Lagrangian algorithm with adaptive mesh refinement, w.r.t. the parameters that are given in \[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]4  
\[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]5 Concentration profiles \(c_{A}\), (**a**), \(c_{B}\), (**b**), \(c_{C}\) (**c**) as approximative result to (3) with parameters in \[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]6  
are parallel in both cases, uniform and adaptive refinement. Furthermore, one obtains that all error-terms converge to zero. Furthermore, it can be seen that the given error estimators that were induced by adaptive refinement seem to converge with a higher rate, since the corresponding graphs in the \(\log\log\) plot shown in Figure 21 are steeper than the graphs associated to uniform refinement. Similar to the previous example, one sees, in Figure 22, a uniform triangulation of the L-shaped domain in Figure 22 on the left side and right side one generated by the adaptive algorithm. The triangulation that is generated by the AFEM-loop is dominantly refined in the neighbourhood of the non-convex vertex. This clearly coincides with the theory about singular solutions to elliptic PDEs, cf. [33] and their treatment, cf. [27; 30]. When applying the augmented Lagrangian algorithm to problem (4) and applying the adaptive scheme to treat the iterate problem (5), one needs 35,057 iterations to guarantee non-negative species distributions, but it is also clear that a close fit to non-negative species distributions are already obtained after 17,528 i  
\[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]7 Development of the error estimators for the uniform and adaptive mesh refinement, for the approximation of (3) with parameters that are given in Table 2 on the L-shaped domain. \[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]8 Triangulations generated by uniform (**left**) and adaptive (**right**) refinement for the approximate solution of (3) with the parameters that are given in the \[f_{A}(x):= f_{B}(x):=f_{C}(x):=-2. \tag{32}\]9  
As last point of the discussion of the numerical experiments in 2d, the evaluation of the CPU running time has to be discussed. Table 5 presents the results discussed in the following. Table 1.0 Species distributions \(c_{A}\) (**a**,**d**,**g**), \(c_{B}\) (**b**,**e**,**h**) and \(c_{C}\) (**c**,**f**,**i**) in the 1 st, 141 st, and 283 th iteration of the augmented Lagrangian algorithm with adaptive mesh refinement, w.r.t. the parameters given in Table 1.1  
First, note that the number of iterations as well as the computation time reduce by using adaptive schemes. The reduction of computation time is for the the convex square domain is somehow remarkable, since the gain of efficiency, when using adaptive schemes, for convex domains is, in comparison to uniform refinement, rather low. The observed reduction of CPU time can be explained by the reduction of iterations in the augmented Lagrangian scheme, cf. Algorithm 1. However, the gain of efficiency, when using adaptive schemes to non-convex domains is remarkable, since a reduction of over 50,000 iterations of the augmented Lagrangian algorithm to under 300 is an extreme gain of efficiency. **Remark 7**. : _Note that the high number of needed iterations of the augmented Lagrangian algorithm for the \(L\)-shaped domain results of the singularity at the non-convex vertex of the geometry, which is a common result for non-convex domains. The adaptive scheme refines dominantly at the non-convex vertex of the geometry, which enhances the approximation in a way such that the augmented Lagrangian algorithm becomes enhanced, as can be seen in Table 1.2_  
When summarizing the results from this Section 4.2, it can been seen that the augmented Lagrangian scheme is more cost intensive than its correspondence in 1d.