## 6 Discussion and Conclusions  
Although, in this article, the rigorous mathematical proofs are not done, in Section 4 various numerical examples are discussed, which show convergence. Hence, the convergence of all included algorithms can be stated as a founded conjecture. Although some mathematical considerations are made in this paper, some rigorous mathematical work still has to be done. In detail, the following formal proofs have to be made in some future work:  
1. There exist, at least locally, a unique solution to (3) and (4). 2. The iterated minimization problems (5) are, at least locally, uniquely solvable and the sequence of solutions \(\big{(}(u_{S}^{(n)},p_{S}^{(n)})_{S\in(A,B,C)}\big{)}\in X\), generated by the augmented Lagrangian method, converges to the solution \((u_{S},P_{S})_{S\in(A,B,C)}\in X\) of (4). 3. For a triangulation \(\mathcal{T}\), with low enough diameter \(0<h\) and a stable starting point of the quasi-Newton scheme described by solutions of (13), every iterate problem of (13) is uniquely solvable. 4. The quasi-Newton method given by the iterative solution \(x_{h}^{(n)}\) of (13) converges in \(X_{h}\). i.e., there exists a \(x_{h}\in X_{h}\), such that \(x_{h}=\lim\limits_{n\to\infty}x_{h}^{(n)}\). 5. For \(h\downarrow 0\) the limits described in (iv) converge against the solution of (5). 6. The adaptive scheme that is described in Algorithm 3 converges. Furthermore, the numerical experiments show that the application of the practical error \(\sigma\), as defined in Definition and Remark 1, can be used in two space dimensions. In addition, the behavior seems to indicate the needed norm equivalences in two space dimensions. Furthermore, in one space dimension, it can be seen that the practical error estimator is not sufficient for the required norm equivalence, but the numerical examples nevertheless show good results, which allow the use as above. Although the numerical schemes are still cost intensive, they converge and, hence, give physical behavior for three compound systems. In addition, as to be seen in Section 5, in this article, a numerical scheme was introduced that is able to treat the high rate constants given in real-world problems, cf. supplement to [4]. Hence, the numerical scheme can be used in most relevant settings when considering three-component systems. Furthermore, note that the algorithm that is described in this section seems, due to the discussions for the numerical examples, to be especially efficient for nonlinear and non-convex geometries. Especially, the efficiency for those two cases makes the algorithm interesting for further applications. The model that is presented in the Section 5 gives a first glance at the process of metal deposition, but one has to expect that the model is improvable. Firstly, the assumption of the existence of a uniform laminar boundary layer on any complex geometry is questionable since fluid dynamics tend to be strongly geometry dependant, cf. [56]. Furthermore in galvanic processes H\({}_{2}\) development at the cathode can be expected, which questions the existence of a laminar boundary layer in many cases. Although the concentration gradients in a galvanic bath can be assumed to be low in some cases, the species-transport in the bath, as described in [1; 2; 3], cannot necessarily be neglected. Furthermore, the processes in the bath can be described by a multi physics coupling of fluid dynamics, diffusion, reactions, and electro-magnetics. Furthermore, as seen in [3], the reaction kinetics that were discussed in this article are not complete. Hence, a further discussion of this type of model needs to be done. As is well known, most of the real world processes do not reduce to static problems, but, nevertheless, important limiting factors can be obtained from static laws. Furthermore, from a technical numerical perspective, time-dependent laws can be approximately treated as systems of static laws, c.f. [57] or [58]. Hence, this article gives a good first foundation for further discussions and investigations.