## 4 Numerical Examples and Validation of the Software
### Theoretical Model  
In this section, an abstract model for the static metal deposition in galvanic cells will be discussed. As a model that is falling into the discussed model types, the static metal deposition with speciation in a single diffusion boundary layer will be studied, c.f. [1; 2; 3].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter Set** & **Dimension** & **CPU Time** & **CPU Time** & **CPU Time** \\  & & **AALA** & **CALA** & **PDAS** \\ \hline cf. Table 2 & 1d & 9.9426e + 03 s & 8.8871e + 03 s & 4.1808e + 03 s \\ cf. Table 2 & 2d & 742.0187 s & 4.4895e + 04 s & 1.1331e + 04 s \\ cf. Table 3 & 1d & 4.7214e + 04 s & 2.0356e + 05 s & - \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of the augmented Lagrangian scheme discussed in this paper with adaptive mesh refinement (AALA), the classical augmented Lagrangian algorithm (CALA), and the Primal-Dual Active-Set Strategy (PDAS), for different space dimensions and parameter sets.  
A few remarks on the physical setting will be made before the model will be discussed in detail. In an electrolyte, a metal \(M\), in this case Cu, and a ligand \(L\), in this case \(\beta\)-ala, is assumed, with the reaction  
\[\text{Cu}+(\beta-\text{ala})\rightleftharpoons\text{Cu}(\beta-\text{ala}). \tag{37}\]  
Furthermore, in the bath, there is an anode and a cathode, where, at the cathode, the metal is deposited. In the global setting, there is a transport of Cu-ions from the anode, with anodic current density \(j_{A}\), to the cathode, with cathodic current density \(j_{\text{C}}\). In this model, one assumes that \(j_{\text{C}}=j_{A}\).  
For the model, assume that there exists a laminar boundary layer at the cathode, which includes the diffusion boundary layer with thickness of \(\delta_{x}=10^{-4}\) m. Furthermore, assume that, outside the diffusion boundary layer, the concentration \(c_{\text{Cr}^{2+}}\) of \(Cu^{2+}\), the concentration \(c_{\text{Cu}^{2+}}\) of \(c_{\beta-\text{ala}}\) and \(c_{\text{Cu}-\beta-\text{ala}}\) are constant. Furthermore, the concentration the bath is set by \(c_{S}^{b}=0.5\ \frac{\text{mol}}{\ell}\), for each \(S\in\{\text{Cu}^{2+},\beta-\text{ala},\text{Cu}(\beta-\text{ala})\}\).  
Furthermore, let the only deposited species at the cathode, at position \(x=0\), be the Cu-ions and the species distribution in the diffusion boundary layer geometrically only be dependant on the distance to the cathode and the concentrations in the diffusion-boundary layer can be described by the diffusion-reaction problem (4), with diffusion coefficients and reaction rate constants, as given in Table 7, and let no further reactive force be given.  
When translating the assumptions above into boundary-data one obtains for all \(S\in\{\text{Cu},\beta-\text{ala},\text{Cu}-(\beta-\text{ala})\}\) the identity \(c_{S}(\delta_{x})=c_{S}^{b}\). At the cathode for the species \(S\in\{S\in\{\beta-\text{ala},\text{Cu}-(\beta-\text{ala})\}\), the identities \(\frac{\text{d}}{\text{d}x}c_{S}=0\) will be used and \(\frac{\text{d}}{\text{d}x}c_{\text{Cu}}=\frac{j}{D_{\text{Cu}^{2}}\ell}\), where \(F\) is Faraday's constant and \(z=2\) is the valency of the copper-ions Cu\({}^{2+}\). The absence of reactive forces is modeled by defining the functions \(f_{A},f_{B},f_{\text{C}}\in L^{2}(\Omega)\) through \(f_{A}=f_{\text{B}}=f_{\text{C}}=0\).  
For the diffusion coefficients cf. to supplementary material of [4] and for the kinetic parameters, cf. [3].