## 3 Numerical Scheme
### Remarks on Existence and Convergence Theory to the Numerical Scheme
#### 3.1.2 Notes on the Convergence Theory  
In this section, a few notes on the convergence theory will be made. Although a proof of convergence of the solutions to the discretisation of (5), will be given in this section, the convergence analysis will be incomplete. For the rest of the necessary convergence theorems, some notes on what to be shown will be given. First, some remarks on the quasi-Newtonian scheme, as given by the iterative solution of (13) and the Homotopy loop will be made. The remark below gives a sketch of the proof of convergence of the discussed quasi-Newtonian scheme. A mathematical rigorous proof has to be given in a future paper. **Remark 5**. : _(i) First note that the least squares functional \(LS\big{(}(v_{S},q_{S})_{S\in\{A,B,C\}}\big{)}\) is Frechet differentiable and explicitly \(A\big{(}(v_{S},q_{S})_{S\in\{A,B,C\}}\big{)}=v_{A}v_{B}\) is Frechet differentiable. Recall that the Notion of Frechet derivatives yields that for given \(x\in X\) and for all \(0<\varepsilon\), there exists a \(0<\delta\), such that_  
\[\big{\|}A(x+h)-A(x)-A^{\prime}(x)d\big{\|}_{Y}\leq\varepsilon\|d\|_{X},\qquad \forall\|d\|_{X}<\delta.\]  
_This yields that there exists a constant \(0<c\), such that for a solution \(x_{h}^{(n)}\in X\) of (13) the following estimate holds true, if the initial value \(x_{0}\in X_{h}\) is sufficiently close to the exact solution to (8):_  
\[ \begin{array}{l}\Big{|}LS_{A^{(3)}}\left(x_{h}^{(n)},\frac{\lambda^{(k)}}{ \alpha^{k}}\right)-LS_{A^{(3)}}\left(x_{h}^{(n)},\frac{\lambda^{(k)}}{\alpha^ {k}};\mathcal{T},n\right)\Big{|}\\ \leq\varepsilon\|x_{h}^{(n)}-x_{h}^{(n-1)}\|_{X}\\ \quad\quad+c\sum\limits_{S\in\{A,B,C\}}\Big{[}\big{\|}(1-\Pi_{0})P_{S,h}^{(n) }\big{\|}_{L^{2}(\Omega\times\mathbb{R}^{d})}+\big{\|}(1-\Pi_{0})c_{S,h}^{(n) }\big{\|}_{L^{2}(\Omega)}\Big{]}\end{array}  \tag{21}\]  
_As discussed in Remark 2, a direct application of a classical quasi-Newtonian scheme to (8) yields a convergent sequence \((y^{n})\subset X_{h}\), if the initial value \(y_{0}\) is sufficiently close to the solution \(x_{h}^{*}\in X_{h}\) to (8) and it follows with inequality (21), the convergence of \(x_{h}^{(n)}\). A similar interpretation, as in Remark 2 of the sequences \(y_{h^{*}}^{k}x_{h}^{(k)}\subset X_{h}\) and the inequality (21), indicate that there exists a constant \(c_{1}>0\), such that for initial values \(y^{(0)}\) sufficiently close to the solution \(x_{h}^{*}\) of (8) and the initial value \(x_{h}^{(0)}\) are sufficiently close to the limit of the quasi-Newtonian scheme, the following inequality holds true:_  
\[\|x_{h}^{(n)}-y_{h}^{(n)}\|_{X}\] \[\quad\leq c_{1}\sum_{S\in\{A,B,C\}}\Bigl{[}\left\|(1-\Pi_{0})p_{S,h }^{(n)}\right\|_{L^{2}(\Omega;\mathbb{R}^{d})}+\left\|(1-\Pi_{0})c_{S,h}^{(n)} \right\|_{L^{2}(\Omega)}\Bigr{]}.\]  
_It directly follows_  
\[\lim_{h\downarrow 0}\|x_{h}^{(n)}-y_{h}^{(n)}\|_{X_{h}}=0. \tag{22}\]  
_The discussion (i) indicates that the quasi-Newtonian scheme described by the iterative solution by (13) has the same issue with stability as classical quasi-Newtonian schemes, cf._ _[_17_]__. To stabilize the quasi-Newtonian scheme, it was coupled to a homotopy method, cf._ _[_32_]__. As already discussed, the idea of the method is to solve for a continuous function_ \(H:X_{h}\times[0,1]\to X_{h}\)_, with_ \(H(x_{h},0)=id_{X_{h}}\) _and_ \(H(x_{h},1)=LS_{x_{h}}\Bigl{(}x_{h},\frac{u^{(k)}}{x_{k}};\mathcal{T},n\Bigr{)}\) _and a decomposition_ \(0=t_{0}<\ldots<t_{j}<\ldots<t_{M}\) _to solve the minimization problem that is given by:_ _Find_ \(x_{h}^{(j)}\in X_{h}\) _such that_  
\[x_{h}^{(j)}=\operatorname*{argmin}_{x_{h}\in X_{h}}H(x_{h},t_{j}). \tag{23}\]  
_As a solution step, the quasi-Newtonian scheme (13) can be used. For \(j=0\), an initial value \(x_{0}\) for the algorithm is given through \(x_{0}\) and for \(j>0\) the \(x_{h}^{(j-1)}\) is given as an initial value for the quasi-Newtonian scheme._  
In the following lemma, a theoretical statement will be proven, which will be essential for the proof of a simple convergence result for solutions of (8).