## 3 Numerical Scheme
### Remarks on Existence and Convergence Theory to the Numerical Scheme
#### 3.1.1 Notes on the Existence Theory  
In this section, some theoretical notes on the existence theory will be given. While the existence theory in one space dimension, thanks to the ODE theory and, in particular, thanks to the theorem of Picard-Lindelolf, cf. [35], and for linear PDEs is well known, cf. [33], the existence theory for nonlinear PDEs is, in most of the cases, not known, cf. [34]. The following discussion only tangents the cases known in literature. **Remark 3** (The 1d-case). : _In this remark, a the 1d case will be discussed._  
1. _The system of second order ODEs, as given by (_2_a)-(_2_e_), can equivalently reformulated into a System of first order ODEs, which are uniquely solvable due to the theorem of Picard-Lindelolf, cf. [35]._
2. _The assertion in (i) gives directly the solvability of (_3_)-(_5_), due to the fact that in the derivation of the least squares functional exactly the needed first order reformulation was used, which was needed for the assertion._  
After the remark above on ODE theory, a further remark on the linear PDE theory will by given:  
**Remark 4** (The linear 2d-case). : _In this remark, the discussion of linear variants of (_2_a)-(_2_e_) will be treated. i.e., during this remark \(k_{1}=0\) is assumed. In this case the continuous space \(X\) can be extended to \(X=\left(H^{1}(\Omega)\times H(\text{div},\Omega)\right)^{3}\), which makes the theory reasonable easier since in this case \(X\) is an Hilbert space._  
1. _First, similar as in_ _[_52_]__, the case of minimal assumptions to the problem formulation will be discussed. Essentially one assumes_ \(0<D_{A},D_{B},D_{C}\) _and the injectivity of the linear operator_ \(\mathcal{L}:H^{1}_{D}(\Omega)^{3}\to\left(H^{1}_{D}(\Omega)^{3}\right)^{*}=H^{ -1}_{D}(\Omega)^{3}\)_, used in the operator Equation (_2_a)-(_2_e_). For this, one assumes that_ \(k_{2},\gamma_{A},\gamma_{B}\) _and_ \(\gamma_{C}\) _are assumed to be chosen in a way that_ \(0\) _is no eigenvalue of_ \(\mathcal{L}\)_._  
_Indeed, the injectivity, i.e., there is no \(0\)-Eigenvalue of the corresponding operator, assumption together with the assumption of the existence of solutions for given \(f_{A},f_{B},f_{C}\in L^{2}(\Omega)\), gives the unique solvability due to [PERSON]'s alternative, cf. [53]._  
_The results above directly apply to the PDE equation that is given by_  
\[0 =D_{A}\text{div}\ p_{A}+k_{2}c_{C}+f_{A}, \text{in}\ \Omega, \tag{19a}\] \[0 =p_{A}-\
abla c_{A}, \text{in}\ \Omega,\] (19b) \[0 =D_{B}\text{div}\ p_{B}+k_{2}c_{C}+f_{B}, \text{in}\ \Omega,\] (19c) \[0 =p_{B}-\
abla c_{B}, \text{in}\ \Omega,\] (19d) \[0 =D_{C}\text{div}\ p_{C}-k_{2}c_{C}+f_{C}, \text{in}\ \Omega,\] (19e) \[0 =p_{C}-\
abla c_{C}, \text{in}\ \Omega,\] (19f) \[g_{A,D} =c_{A}|_{\Gamma_{D}},\ g_{B,D}=c_{B}|_{\Gamma_{D}},\ \text{g}_{C,D}=c_{C}|_{\Gamma_{D}},\] (19g) \[g_{A,N} =\
abla c_{A}|_{\Gamma_{N}}\cdot\
u_{N},\ g_{B,N}=\
abla c_{B}|_{\Gamma_{N}}\cdot\
u_{N},\ g_{C,N}=\
abla c_{C}|_{\Gamma_{N}}\cdot \
u_{N}. \tag{19h}\]and one obtains unique solvability of (19a)-(19b). The corresponding Operator is denoted by \(\mathcal{O}:\big{(}H^{1}(\Omega)\times H(\text{div};\Omega)\big{)}^{3}\to \big{[}\big{(}H^{1}(\Omega)\times H(\text{div};\Omega)\big{)}^{3}\big{]}^{*}\), the RHS is given by a functional \(g\in\big{[}\big{(}H^{1}(\Omega)\times H(\text{div};\Omega)\big{)}^{3}\big{]}^{*}\). This solvability of (19a)-(19b) directly implies the unique solvability of (3). * Note that in (i) only necessary conditions for the unique solvability are discussed. Some sufficient conditions can be obtained by the decoupling of the PDE. Where in fact the sufficient conditions for the unique solvability can be obtained by sufficient conditions to scalar Diffusion reaction equations, cf. [33; 54].