## 3 Numerical Scheme
\begin{cases}\big{(}S^{1}(\mathcal{T})\times S^{1}(\mathcal{T})\big{)}^{ \times 3},&\text{for $d=1$},\\ \big{(}S^{1}(\mathcal{T})\times RT_{0}(\mathcal{T})\big{)}^{\times 3},&\text{for $d=2$}. \end{cases}  \tag{7}\]This leaves the following discrete problem to solve or to approximate:  
Find \(x_{h}\in X_{h}\) such that  
\[\big{(}\big{(}c_{S,LS,h},p_{S,LS,h}\big{)}_{S\in\{A,B,C\}}\big{)}= \operatorname*{argmin}_{(v_{S},q_{S})_{S\in\{A,B,C\}}\in X_{h}}LS_{\alpha_{j}} \big{(}(v_{S},q_{S})_{S\in\{A,B,C\}},\frac{\lambda^{(j)}}{\alpha_{j}}\big{)} \tag{8}\]  
As a fundamental step of the numerical scheme, a linearization of (8) is introduced. For this, a few calculations are necessary. First, remark that, in contrast to the non-linear functional \(LS\big{(}(v_{S,C},q_{S,RT})\big{)}\), the functional \(LS_{\alpha}\big{(}(v_{S,C},q_{S,RT}),\mu\big{)}\) is, for \(\alpha>0\), not differentiable. The non-differentiability lies in the evaluation of the function \(\big{(}\mu-v_{S,C}\big{)}_{+}\). Hence, for the linearization of \(\big{(}\mu-v_{S,C}\big{)}_{+}\), one has to use a different notion of differentiability than the common Frechet-differential. Therefore for the linearization of \(\big{(}\mu-v_{S,C}\big{)}_{+}\) in this article, the convex subdifferential \(\partial_{cvx}\) as defined, c.f. [44], was used. For a convex function \(f:Y\to\mathbb{R}\), for a Banach space \(Y\), its topological dual \(Y^{*}\), and the dual pairing \(\langle\bullet,\bullet\rangle_{Y^{*},Y}\), cf. [40], in a point \(y_{0}\in Y\) is given through  
\[\partial_{cvx}f(y_{0}):=\big{\{}g\in X^{*}\;\big{|}\;\big{\langle}g,x-x_{0} \big{\rangle}+f(y_{0})\leq f(y),\quad\forall y\in X\big{\}}.\]  
As it is commonly known, the subdifferential \(\partial_{cvx}\big{(}\big{[}\mu_{S}-c_{S}\big{]}(\bullet)\big{)}_{+}\) is for almost all \(x_{0}\in\Omega\) and all \(S\in\{A,B,C\}\), given through  
\[\partial_{cvx}\big{(}\big{[}\mu_{S}-c_{S}\big{]}(\bullet)\big{)}_{+}(x_{0})= \begin{cases}\{0\},&\text{for }\big{[}\mu_{S}-c_{S}\big{]}(x_{0})<0,\\ \{-1\},&\text{for }\big{[}\mu_{S}-c_{S}\big{]}(x_{0})>0,\\ \{-1,0\},&\text{for }\big{[}\mu_{S}-c_{S}\big{]}(x_{0})=0.\end{cases}  \tag{9}\]  
Furthermore, for \(c_{A,h},c_{B,h}\in S^{1}(\mathcal{T})\) and arbitrary fixed \(v_{A,h},v_{B,h}\), the following [PERSON] approximation is considered  
\[v_{A,h}v_{B,h}\approx c_{A,h}c_{B,h}+c_{A,h}\big{[}v_{B,h}-c_{B,h} \big{]}+c_{B,h}\big{[}v_{A,h}-c_{A,h}\big{]}\] \[= c_{A,h}v_{B,h}+v_{A,h}c_{B,h}-c_{A,h}c_{B,h}. \tag{10}\]  
An additional approximation of \(c_{S,h}\), for \(S\in\{A,B\}\), is made through  
\[c_{S,h}\approx\Pi_{0}c_{S,h}\]  
In the notation above, the map, \(\Pi_{0}:L^{2}(\Omega)\to P_{0}(\mathcal{T})\) denotes the orthogonal \(L^{2}\)-projection onto the space of piecewise constant functions, which is given through:  
\[P_{0}(\mathcal{T}):=\big{\{}v\in L^{2}(\Omega)\;\big{|}\;\forall T\in \mathcal{T}\exists c\in\mathbb{R}:\;v|_{T}\equiv c\big{\}}.\]  
i.e., for all \(f\in L^{2}(\Omega)\), the image of \(\Pi_{0}f\) is given as the solution of the following minimization problem, cf. [30]:  
\[\Pi_{0}f=\operatorname*{argmin}_{w_{0}\in P_{0}(\mathcal{T})}\big{\|}f-w_{0} \big{\|}_{L^{2}(\Omega)}^{2}. \tag{11}\]  
Furthermore, for \(f\in H^{1}(\Omega)\), due to the Poincare inequality, cf. [40] or [30], there exists a constant \(0<c\), such that  
\[\|f\|_{L^{2}(\Omega)}^{2}=\big{\|}\Pi_{0}f\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|} (1-\Pi_{0})f\big{\|}_{L^{2}(\Omega)}^{2}\leq\big{\|}\Pi_{0}f\big{\|}_{L^{2}( \Omega)}^{2}+ch^{2}, \tag{12}\]  
where the equality results from the pythagorean theorem, [40].